machine tools, nuclear power plants and metrology devices, such as shape meters. ...... analytical model generated by Friedel (2011) is a bi-supported beam in which ...... Trabalho de conclusão de curso (Engenharia Mecânica) â Escola de.
Luis Renato Chiarelli
Rotor-bearing systems: theory and numerical approach using MATLAB® and ANSYS®
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To cite this version: Chiarelli, L.R. Rotor-bearing systems: theory and numerical approach using MATLAB® and ANSYS®, OMNIScriptum, 2017.
Brazil 2017
ABOUT THE AUTHOR
PhD student at University of Campinas (UNICAMP) in the area of mechanical engineering, sub-area of solid mechanics and mechanical design. Master's degree at São Paulo University (USP) in mechanical design sub-area (2014), focusing on the design and static and dynamic modeling of a rotor assembly (shaft, bearings and turbine) using the finite elements and a aerostatic ceramic porous bearing using the finite difference method.
PREFACE
This work presents a review of rotordynamics and bearings (focusing in aerostatic bearings) based on Chiarelli (2014). During the chapters the analytical and numerical models are presented in order to teach to the reader how to approach problems involving static and dynamic characterization and prediction of rotor assemblies. The numerical models are implemented in common use software such as ANSYS® and MATLAB® and the programs are shown at the end of the work. This work is dedicated to Prof. Dr. Zilda de Castro Silveira who believed in and invested in my potential to start in academic research, to Prof. Dr. Rodrigo Nicoletti who provided the computational program for the dynamic analyzes of porous bearing, to all my family and friends.
LIST OF FIGURES
Figure 2.1: General characteristics of ultra-precision machines (Zong,2010). .................................. 3 Figure 2.2: Widely used bearings (Panzera, 2007) ............................................................................ 6 Figure 2.3: Radial ball bearing example (Nonato, 2009).................................................................... 8 Figure 2.4: Four-point contact ball bearing (a); single row ball bearing and angular contact (b) (SKF, 2013). ................................................................................................................................................. 9 Figure 2.5: Flow regime of operation of externally pressurized bearings (Purquério, 1990). ............ 10 Figure 2.6: Diagram of force (w) X gap (h) (a); feeding orifice (b) (Purquerio,1990). ........................ 11 Figure 2.7: Restrictors used in aerostatic bearings (Panzera, 2007). ................................................ 12 Figure 2.8: Porous sample in aqueous medium (New Way,2013). .................................................... 14 Figure 2.9: Image of the form of porous radial bearings (Miyatake, 2006) ........................................ 15 Figure 2.10: Experimental set-up used in Cranfield (Kwan, 1996). ................................................... 18 Figure 2.11: Static performance of axial porous bearings under various loads and different feed pressures (Kwan, 1996). .................................................................................................................... 18 Figure 2.12: Experimental scheme for obtaining characteristics of partially porous aerostatic bearings (Fujii, 2008). ........................................................................................................................................ 19 Figure 2.13: Dimensionless stiffness calculated by subtracting the minimum clearance from the applied load; thickness of the test piece (a) 3,5mm and (b) 4,5mm (Fujii, 2008). ............................. 20 Figure 2.14: Experimental apparatus; (a) with symmetrical loads and (b) cross loads (Yoshimoto, 2003) .................................................................................................................................................. 21 Figure 2.15: Relation between load and eccentricity ratio (Yoshimoto,2003). ................................... 21 Figure 2.16: Experimental apparatus for instability test of aerostatic radial bearings with metallic matrices (Miyatake, 2006). ................................................................................................................. 22 Figure 2.17: Method used to estimate the stiffness of the porous aerostatic bearing test bench (Friedel, 2011). ................................................................................................................................................. 23 Figure 3.1: Jeffcott rotor example - flexible shaft and rigid bearings (Yoon,2012). ........................... 26 Figure 3.2: Shaft element and their respective degrees of freedom. ................................................. 30 Figure 3.3: Example of shaft divided into three elements and supported on bearings at its ends. ... 38 Figure 3.4: Distribuition of the fluid speeds in plates (Powell, 1970). ................................................ 43 Figure 3.5: Schematic representation of the porous medium with cross-sectional area and constant thickness (Silveira, 2008). .................................................................................................................. 47 Figure 3.6: Aerostatic porous bearing diagram (Silveira, 2008). ........................................................ 47 Figure 3.7: Kinematics of fluid flow, velocity profiles (Nicoletti, 2008). .............................................. 48 Figure 4.1: Structure of the research line of porous ceramic aerostatic bearing. .............................. 51 Figure 4.2: (A) technical drawing of the bearing; (B) manufactured bearing. .................................... 52 Figure 4.3: Experimental set-up scheme (Friedel, 2011). .................................................................. 53 Figure 5.1: Proposed test models; the inner rings of the bearings are marked with the letter "M" and the turbine disc with the letter "T". ...................................................................................................... 55
Figure 5.2: Boundary conditions applied to the Axis subset of Models A and B using ANSYS® software. ............................................................................................................................................. 56 Figure 5.3: Boundary conditions applied to the Axis subset of Model A and B using Autodesk Simulator ® software. ......................................................................................................................................... 57 Figure 5.4: Modeling the turbine of Model C using ANSYS®. ........................................................... 57 Figure 5.5: Tetrahedral mesh (a) and hexahedral mesh (b) – ANSYS®. .......................................... 58 Figure 5.6: Tetrahedral mesh using Autodesk Simulator®. ............................................................... 58 Figure 5.7: Mesh composed of one-dimensional elements in ANSYS®. ........................................... 59 Figure 5.8: One-dimensional meshes generated in MATLAB® ......................................................... 59 Figure 5.9: Relation between the hexahedral mesh (A) and tetrahedral (B) X aspect ratio .............. 61 Figure 5.10: Relation between the hexahedral mesh (A) and tetrahedral (B) X skewness ............... 62 Figure 5.11: Relation between the hexahedral mesh (A) and tetrahedral (B) X Jacobian Ratio ....... 63 Figure 5.12: Strategy adopted for the evaluation of numerical models.............................................. 65 Figure 5.13: Deformations of Model A. .............................................................................................. 67 Figure 5.14: Deformations of Model B. .............................................................................................. 68 Figure 5.15: Deformations of Model C. .............................................................................................. 69 Figure 6.1: Models with hexahedral and tetrahedral meshes respectively. ....................................... 71 Figure 6.2: Model D and the generated mesh in MATLAB®. ............................................................. 72 Figure 6.3: Comparison between natural frequencies using different methods. ................................ 73 Figure 6.4: First natural frequency as a function of the slenderness ratio through different methods (A); Percentage difference between FEM and analytical method (B). ..................................................... 74 Figure 6.5: Models and meshes generated in MATLAB® environment. ............................................ 75 Figure 6.6: Strategy used in dynamic analysis of the rotating system. .............................................. 76 Figure 6.7: Comparison between the results of the Table 6.3. .......................................................... 77 Figure 6.8: Frequency responses obtained through FEM; Theory of Euler-Bernoulli (A); Theory of Timoshenko (B). ................................................................................................................................. 78 Figure 6.9: Comparison between the results of the Table 6.4. .......................................................... 79 Figure 6.10: Frequency responses obtained through FEM; Theory of Euler-Bernoulli (A); Theory of Timoshenko (B) .................................................................................................................................. 80 Figure 6.11: Comparison between the results of the Table 6.5. ........................................................ 81 Figure 6.12: Frequency responses obtained through FEM; Theory of Euler-Bernoulli (A); Theory of Timoshenko (B). ................................................................................................................................. 82 Figure 6.13: Mesh used in the finite difference method. .................................................................... 83 Figure 6.14: Aerostatic ceramic porous bearing scheme (Silveira et al., 2010)................................. 84 Figure 6.15: Basic algorithm used to obtain the dynamic coefficients of porous ceramic aerostatic bearing. ............................................................................................................................................... 84 Figure 6.16: Normal stiffness coefficients (a); Crossed stiffness coefficients (b); Normal damping coefficients (c); Crossed damping coefficients (d). ............................................................................ 86 Figure 6.17: Variation of the stiffness as a function of the axis rotation; Kyy*(A); Kzz*(B)................ 87 Figure 6.18: Variation of the stiffness as a function of the axis rotation; Kyz*(A); Kzy*(B)................ 88
Figure 6.19: Variation of the damping as a function of the axis rotation; Dyy(A); Dzz(B). ................ 89 Figure 6.20: Variation of the damping as a function of the axis rotation; Dyz(A); Dzy(B). ................ 90 Figure 6.21: Comparison between damped amplitudes (left) and undamped amplitudes (right) of Model A; angular contact rolling element bearings (a); ball bearing (b); rigid (c); Ps/Pa = 5 (-), Ps/Pa = 10(--), Ps/Pa = 15 (- -) e Ps/Pa = 20 (--)............................................................................................. 92 Figure 6.22: Comparison between damped amplitudes (left) and undamped amplitudes (right) of Model B; angular contact rolling element bearings (a); ball bearing (b); rigid (c); Ps/Pa = 5 (-), Ps/Pa = 10(--), Ps/Pa = 15 (- -) e Ps/Pa = 20 (--)............................................................................................. 93 Figure 6.23: Comparison between damped amplitudes (left) and undamped amplitudes (right) of Model C; angular contact rolling element bearings (a); ball bearing (b); rigid (c); Ps/Pa = 5 (-), Ps/Pa = 10(--), Ps/Pa = 15 (- -) e Ps/Pa = 20 (--)............................................................................................. 94 Figure 6.24: Campbell diagram of Model A. ...................................................................................... 95 Figure 6.25: Campbell diagram of Model B. ....................................................................................... 95 Figure 6.26: Campbell diagram of Model C. ..................................................................................... 96 Figure 6.27: FRFs undamped of the shaft supported by aerostatic ceramic porous bearings; Ps/Pa = 5 (-), Ps/Pa = 10(--), Ps/Pa = 15 (- -) e Ps/Pa = 20 (--). ........................................................................ 97 Figure 6.28: FRFs damped of the shaft supported by aerostatic ceramic porous bearings – Model A. ............................................................................................................................................................ 98 Figure 6.29: FRFs damped of the shaft supported by aerostatic ceramic porous bearings –Model B. ............................................................................................................................................................ 98 Figure 6.30: FRFs damped of the shaft supported by aerostatic ceramic porous bearings – Model C. ............................................................................................................................................................ 99
LIST OF TABLES
Table 2.1: Comparison of bearings (Panzera, 2007). ........................................................................ 7 Table 3.1: Example of assembly of part of the global translational mass matrix. .............................. 38 Table 3.2: Frequency response matrix check example {H}jk. ............................................................. 42 Table 3.3: Definition of the terms used in the Reynolds equation. ..................................................... 50 Table 5.1: Static stiffness obtained by different methods. ................................................................ 64 Table 5.2: Results and comparisons in the initial design with rigid rolling element bearings. ........... 66 Table 5.3: Results of Model A considering the stiffness of the bearings. .......................................... 67 Table 5.4: Stiffness of Model B with variations in rolling element bearings. ...................................... 68 Table 5.5: Stiffness of Model C with variations in rolling element bearings. ...................................... 69 Table 6.1: First natural frequencies of the detailed shaft using diferente meshes............................. 72 Table 6.2: Natural frequencies of Model D obtained through different theories. ............................... 72 Table 6.3: Natural frequencies of Model A obtained through different theories. ................................ 77 Table 6.4: Natural frequencies of Model B obtained through different theories. ................................ 79 Table 6.5: Natural frequencies of Model C obtained through different theories. ............................... 81 Table 6.6: Stiffness coefficients obtained at different supply pressures. ........................................... 85 Table 6.7: Damping coefficients obtained at different supply pressures............................................ 85 Table 6.8: Natural frequencies of the aerostatic bearing as a function of supply pressure. .............. 91 Table 6.9: First natural frequency of the system with all bearings. .................................................... 91 Table 6.10: Undamped natural frequencies of the shaft supported by aerostatic ceramic porous bearings. ............................................................................................................................................ 96
NOMENCLATURE ̅ = dimensionless gap ̅ = non-dimensional stiffness ̅ = dimension orthogonal to the gap and to the length of the aerostatic bearing [m] ̇ , Qm = mass flux [kg/s] ̅ = non-dimensional pressure ̅ ̅ = non-dimensional coordinates n
[Db ] = damping matrix of the rolling bearing [G] = gyroscopic matrix of the shaft [G0], [G1], [G2] = gyroscopic matices of the Timoshenko theory [Gd] = gyroscopic matrix of the disc [K] = stiffness matrix of the shaft [K0], [K1] = stiffness matices of the Timoshenko theory n
[Kb ] = stiffness matrix of the rolling bearing [M] = translational mass matrix of the shaft [M0], [M1], [M2] = translational mass matices of the Timoshenko theory [Md] = translational mass matrix of the disc [N] = rotational mass matrix of the shaft [N0], [N1], [N2] = rotational mass matices of the Timoshenko theory [Nd] = rotational mass matrix of the disc [q] = displacement array [φ] = shaft rotation shape functions matrix [ψ] = shaft shape funtions matrix {F} = force array {H} = frequency response function matrix 2
A = section area of the flux of the porous surface [m ] A1 = integration constant 2
Ae = shaft element area [m ] Ax , Ay = constants obtained through the initial condition of the disc B = integration constant c = assemble gap [m] Cg = global damping matrix cs = damping coefficient of the shaft [N/m] Dg = matrix build by Cg and Gg dw = derivative of the force function to obtain the simplified stiffness of the aerostatic [N] E = Young modulus [Pa] eu = distance of the unbalance [m] G = transversal elasticity modulus [Pa]
Gg = global gyroscopic matrix H = porous matrix thickness [m] h = bearing clearance [m] 4
I = moment of inertia [m ] 2
Id = diametral moment of inertia [kg.m ] [Ig]= identity matrix 2
Ip = moment of inertia polar [kg.m ] k = form factor K = simplified stiffness for aerostatic bearings [N/m] 2
K1 = coefficient of viscous permeability [m ] Kg = global stiffness matrix Ks = coefficient of stiffness to the lateral flexion in the axial center of the axis [N/m] L, l = length [m] Lt = perimeter of the bearing [m] m = mass [kg] Mg = global mass matrix P,p = pressure of the gas [Pa] Ps = supply pressure [Pa] R = gas constant [J/kg.K] r = radius [m] r0 = external radius [m] Rg = state model matrix ri = internal radius [m] S = complex constant [rad/s] s = dimension [m] Sg = state model matrix T = absolute gas temperature [K] t = time [s] U = linear velocity of the rotor surface [m/s] u,v,w = velocities of the gas [m/s] uC = distance from the geometric center of the disk in polar coordinates [m] uxc , uyc = distances from the geometric center of the disc [m] uxG , uyG = distances from the unbalanced mass point of the disc [m] V = rotor translational velocity in the gap [m/s] Vinj = air injection velocity [m/s] x,y,z = Cartesian axes [m] X,Y,Z = Cartesian axes [m] Y1 , Y2 = degrees of freedom of movement on the Y direction Z1 , Z2 = degrees of freedom of movement on the Z direction α1 , α2 = degrees of freedom of rotation on the Z direction
β1 , β2 = degrees of freedom of rotation on the Y direction Γ = non-dimensional parameter of the porous matrix 2
η = dynamic gas viscosity [N.s/m ] o
θ = angle of the distance of the geometric center of the disk in polar coordinates [ ] λ = eigenvalues Λ = non-dimensional parameter of the velocity μ = mass of the element per unit length [kg/m] 3
ρ = density [kg/m ] Φ = coefficient used in the Timoshenko theory due to cross shear Ψ = non-dimensional of excitation ψ1 , ψ2 , ψ3 , ψ4 = shape functions of the shaft element ω = angular velocity of the disc shaft with unbalanced mass [rad/s] Ω = angular velocity of the shaft element [rad/s] ωcr = critical angular speed of the shaft [rad/s] ωn = undamped natural frequency of the shaft [rad/s] = non-dimensional time
SUMMARY ABOUT THE AUTHOR
PREFACE
LIST OF FIGURES
LIST OF TABLES
NOMENCLATURE
1. INTRODUCTION
1
1.1. Objectives ............................................................................................................................... 2
2. LITERATURE REVIEW
3
2.1. Considerations of ultra-precision machines ............................................................................ 3 2.2. Definition and types of bearings.............................................................................................. 4 2.2.1. Rolling element bearings ............................................................................................... 8 2.2.2. Aerostatic bearings........................................................................................................ 9 2.2.3. Constructive types of bearings and flow restrictors ...................................................... 11 2.2.4. Aeorstatic bearings of porous materials ........................................................................ 14 2.2.4.1. Development and structure of aerostatic bearings .......................................... 14
3. THEORETICAL CONCEPTS
25
3.1. Considerations about rotor dynamics ..................................................................................... 25 3.1.1. Whirl, synchronous and asynchronous motion ............................................................. 27 3.1.2. Basic modeling of rotor dynamics ................................................................................. 27 3.1.3. Undamped free vibrations ............................................................................................. 28 3.1.4. The finite element method ............................................................................................. 29 3.1.4.1. Euler-Bernoulli theory ....................................................................................... 29 3.1.4.2. Timoshenko theory ........................................................................................... 32 3.1.4.3. Rolling bearings matrices ................................................................................. 36 3.1.4.4. Disc matrices .................................................................................................... 37 3.1.4.5. Assembling the matrices (global matrices) ...................................................... 37 3.1.4.6. Eigenvalues and eigenvectors ......................................................................... 39 3.1.4.7. Frequency response ........................................................................................ 41 3.2. Theory of the aerostatic lubrication ......................................................................................... 42
3.2.1. Theory of mass flow through porous medium ............................................................... 45 3.2.2. Modified Reynolds equation .......................................................................................... 47
4. CASE STUDY
51
5. STATIC ANALYZES
54
5.1. Models of the rotor assemble.................................................................................................. 54 5.2. Finite element models ............................................................................................................. 56 5.2.1. Boundary conditions ...................................................................................................... 56 5.2.2. Generation and mesh quality ........................................................................................ 58 5.3. Stiffness of the aerostatic bearing........................................................................................... 64 5.4. Structure and proceedings of the static analyzes ................................................................... 64 5.5. Results of the static analyzes ................................................................................................. 65 5.5.1. Static analyzes of Model A ............................................................................................ 66 5.5.2. Static analyzes of Model B ............................................................................................ 68 5.5.3. Static analyzes of Model C ............................................................................................ 69
6. DYNAMIC ANALYZES
71
6.1. Modal analyzes of the free shaft ............................................................................................. 71 6.2. Analyzes of the dynamic behavior of the rotor assembly ....................................................... 75 6.2.1. Dynamic analyzes of the rotor assembly – Model A ..................................................... 77 6.2.2. Dynamic analyzes of the rotor assembly – Model B ..................................................... 79 6.2.3. Dynamic analyzes of the rotor assembly – Model C ..................................................... 81 6.3. Analyzes of the dynamics behavior of the aerostatic ceramic air bearing .............................. 83 6.3.1. Dynamic coefficients and natural frequency of the aerostatic bearing ......................... 83 6.4. Dynamic analyzes of the rotor assembly with the aerostatic ceramic porous bearing ........... 91 6.5. Shaft supported by aerostatic ceramic porous bearings. ....................................................... 96
7. CONCLUSIONS
100
APPENDIX A: MATLAB® programs
101
A.1 Frequency response functions
101
A.2 Campbell diagram
109
REFERENCES
118
1. INTRODUCTION
The constant demand for devices and machines for the manufacture of products destined to the technology areas, such as microelectronics, industrial automation and even consumer goods, promote the improvement of the design of machines. The technical requirements for these machines include the choice of geometries and materials for their mechanical and structural elements, speed and position monitoring and control devices, which ensure repeatability in obtaining final dimensions on micro and nanometric scales. Linear and rotational precisions as well as bearing stiffness and bearing capacity, especially spindles, are essential requirements for ultra-precision machine design. These characteristics are directly associated with the tribological pair of shaft-bearings, which must guarantee functional stability during the manufacturing process. Slocum (1992) presents a brief history about the first studies carried out, for the investigation and applications of different types of bearings in machine tools. Technical characteristics of design and operation, such as ability to produce repetitive motions, adequate dynamic stiffness for high rotational speeds, low friction and heat generation, due to non-contact between surfaces indicate that the choice of aerostatic bearings is suitable for use in ultraprecision. Aerostatic bearings are the design choice for the construction of ultraprecision spindle, exemplified by diamond machining, machine tool tables, lithographs with their optical cutters, dental drills, optical surface-generating devices, machine tools, nuclear power plants and metrology devices, such as shape meters. In Silveira et al. (2006) and Silveira, et al. (2010) the porosity and the manufacture of the ceramic material (Al2O3) were studied in order to obtain a structure with permeability for application in aerostatic radial bearings, as well as a statistical investigation on the dispersion of the coefficients of viscous and inertial permeability, for a set of samples manufactured with the same porous agent concentration. The permeability equations (Darcy and Forchheimer theories) were inserted into the Reynolds equation, obtaining a dimensionless equation to obtain factors related to the porous matrix, stiffness and damping under conditions of operational stability (Nicoletti et al, 2008). This work is part of a line of research focused on the manufacture and theoretical-experimental study of porous ceramic aerostatic bearings.
2
1. Introduction
1.1. Objectives
With the intention of investigating the static and dynamic behavior of the radial aerostatic ceramic porous bearing developed by Silveira et.al. (2006) and modeled by Nicoletti et.al. (2008), an experimental set-up was designed consisting of a shaft supported by rolling element bearings and driven by an air-driven turbine (Friedel, 2011). The present work intends to develop models using the finite element method and finite difference method (Nicoletti et al., 2008) to estimate the static and dynamic characteristics of the experimental set-up and the aerostatic bearing respectively, in order to compare and analyze the possibility of construction and use the designed set-up.
The comparison between the stiffness of the shaft and the bearing will be done in order to study the possibility of the configuration of the experimental set-up to study the static stiffness and damping of the aerostatic bearing. The estimates of the FRFs (frequency response functions) of the experimental set-up will be made in order to verify if there will be overlapping of the shaft vibrations and the study of the dynamic characteristics of the bearings.
3
2. LITERATURE REVIEW
This chapter presents a review of the literature on the development of research with lubricated bearings and rolling element bearings in the areas of rotor dynamics and machine designs.
2.1. Considerations of ultra-precision machines
The current demands for components in micro and nanometric scales use precision machinery that are increasingly robust and can adapt and redesigning their structure.
Precision machining machines normally meet a working condition in which there is a need for high rotations and relatively small loads. This need directly reflects the choice of bearings, which play an important role in the quality and accuracy of the final piece. Figure 2.1 shows a schematic drawing of an ultra-precision lathe highlighting some of the general characteristics of these machines.
Spindle Air bearing
Silicon wafer Diamond tool Tool holder
Guide z
Guide x
Marble bed
Figure 2.1: General characteristics of ultra-precision machines (Zong,2010).
4
2. Literature review
Ultra-precision equipment often uses air-lubricated bearings whose main characteristics are low friction, minimum wear and high precision due to low amplitude of vibration (10 nm or less) as Ikawa and Shimada (1986) state.
The factors that differentiate these bearings in terms of stiffness and damping are mainly the air film pressure and the flow restrictors, and porous restrictors are often used which limit and distribute the air flow.
2.2. Definition and types of bearings
The term bearing can be used to define two surfaces in contact when in relative motion with each other. According to Norton (2012), lubrication is required in any bearing to reduce friction and remove heat. The bearings have rolling movement, slip or both movements. Due to these movements and applications in engineering, the bearings can be classified according to the form that this contact occurs.
In rolling bearings, rolling elements (balls, rollers, needles) are held between tracks or between contact surfaces. This type of bearing offers very low contact friction and can withstand radial, axial and combined loads.
A flat or sliding bearing is formed by two materials that slide between each other, such as a cylinder about an axis or a flat surface under a sliding part. For flat bearings, one of the parts is movable made of steel, cast iron or some material that has adequate mechanical strength and hardness. The bearing parts of the plain bearings are made of more ductile material such as brass, babbitt1 or polymer (Norton, 2012). A radial bearing usually has a cylindrical geometry mounted on the shaft (journal). The sliding bearings may be of dry friction in which the solid surfaces are in direct contact with each other, generally these surfaces are coated a layer of sintered bronze. This layer can be lubricated by some type of polymer, being common the use of a layer of Teflon® (Purquerio, 1990). Another variation of the sliding bearings are the fluid film bearings. These bearings do not have a fluid pressurizing mechanism, which operates at ambient pressure.
1
Babbitt is a low-melting alloy made of tin, antimony, copper and sometimes also lead.
Chiarelli, L.R.
5
The type of lubrication that can occur in a bearing can be: full film, mixed film and contour lubrication (Norton, 2012). These three types of lubrication are determined by the relationship between the friction and the relative speed of the bearing surfaces. Three mechanisms can create full-film lubrication: hydrostatic, hydrodynamic and elastohydrodynamic lubrication.
In the pressurized bearings, called hydrostatic (oil) and aerostatic (air or gas) bearings it occurs the type of hydrostatic lubrication, characterized by the continuous supply of a lubricant flow. Usually, there is a set of external devices such as pumps and reservoirs to maintain the pressure in the fluid film. By maintaining the fluid film, the friction is very low (around 0.002 to 0.010).
Still, in fluid film bearings called hydrodynamic (oil) or aerodynamic (air) bearings, lubrication is termed hydrodynamics. In this case, there is only sufficient fluid supply to the sliding interface to allow the relative velocity of the surfaces to pump the lubricant into the bearing gap and to become a dynamic fluid film between the surfaces.
In elastohydrodynamic lubrication the contact surfaces are non-conforming, such as the surfaces described by gear teeth, cam and rolling bearings. The formation of a complete fluid film is difficult and the fluid tends to be expelled from the contact when it should be confined. At low speeds, the type of lubrication is of boundary, which can result in high wear rates. Thus, the load creates a contact area by the elastic deflections on the surface, and this small contact area can provide a sufficient size of flat surface to allow the formation of a complete hydrodynamic film if the slip speed is high. This condition promotes elastohydrodynamic lubrication (EHD) because it depends on the elastic deflections of the surfaces and due to localized pressures raised within the contact zone greatly increase the viscosity of the fluid.
Figure 2.2 shows the operating scheme of some of the main types of bearings.
6
2. Literature review Hydraulic system
HYDRODYNAMIC BEARING
HYDROSTATIC BEARING
Pressured air/gas system
AEROSTATIC BEARING
ROLLING BEARING
Figure 2.2: Widely used bearings (Panzera, 2007)
For applications in precision and ultra precision machines the most used bearings are:
Bearings with mechanical contact between the elements (rolling elements);
Bearings without mechanical contact between the elements, with or without external pressure (aerodynamic or aerostatic).
These types of bearings offer repeatability of movement, accuracy and high rotational speeds with low friction and heat generation.
The comparisons between the different bearing configurations are exemplified in Table 2.1.
Chiarelli, L.R.
Bearing type
7
Rolling element
Dry friction
Porous metal
Operational type High temp. Low temp. Vibrations Space Dirt/dust Humidity conditions Operational costs Production costs Radial mov. Precision Stiff/size Load capac/size Damping High speed Central control Temp. increasing Durability Maintenance Starting torque Operation torque Noise Stopped starts External dimensions Easy design Easy manufacturi ng Availability of normalized parts Environment contaminati on Vacuum Variations in direction of rotation Lubrication Radiation
Hydrodynamics
External pressurized
Liquid
Aerodynamic
Aerostatic
Hydrostatic
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Table 2.1: Comparison of bearings (Panzera, 2007).
8
2. Literature review
2.2.1. Rolling element bearings
According to Norton (2012), the use of roller for object movement has been common since ancient times, and there is evidence of the use of ball bearings in the first century BC. The development of new alloys, new material discoveries and increased manufacturing technology have enabled the production of precise rolling element bearings. It is important to note that ball and roller bearings are globally normalized either metric sizes (International System) or inches (Unified System).
Rolling element bearings are commonly used as connecting elements between housings and shafts in various industrial and automotive applications. Some examples of automotive applications are gearboxes, power trains and differential boxes (Nonato, 2009).
The bearings must be sealed, protected and lubricated because the precision of operation of these bearings depends on the quality of their surfaces. Eliminating the causes of external defects, the life of the bearings depends only on the fatigue of the material caused by the stresses and rotations of the bearing (Panzera, 2007).
Figure 2.3: Radial ball bearing example (Nonato, 2009).
Radial ball bearings are more common in precision machines than roller bearings, except when high loads are required when using roller bearings. The radial
Chiarelli, L.R.
9
total error amplitude in radial contact bearings may be of the order of 0.025 to 0.5 μm for high precision machining machines (Slocum, 1992).
In applications requiring high precision and equally high speeds, angular contact bearings are a design option (Slocum, 1992). Angular contact ball bearings have conductive tracks offset relative to one another in the direction of the rolling axis. This type of bearing accommodates axial and radial loads acting simultaneously. The axial load capacity increases as the contact angle becomes larger (SKF, 2013). Figure 2.4 shows configurations of angular contact bearings.
(a)
(b)
Figure 2.4: Four-point contact ball bearing (a); single row ball bearing and angular contact (b) (SKF, 2013)
2.2.2. Aerostatic bearings
According to Purquerio (1990), the aerostatic bearings have between the surfaces a pressurized gas (air) film from an external source of supply. This type of bearing has the advantage of supporting loads even when stopped. There are basically two types of bearings: externally pressurized that operate in the viscous flow range and the “Beams” bearing that operates in the impact zone as shown in figure 2.5. “Beams” are used on very high rotating machines such as ultracentrifuges.
10
2. Literature review
Viscous flow
Bernoulli flux
Jet impact
Figure 2.5: Flow regime of operation of externally pressurized bearings (Purquério, 1990).
Aerostatic bearings can be chosen according to the following technical requirements (Tsukamoto, 2003): • Null starting friction and very small viscous friction; • Small heat generation even at high speeds; • No wear, as there is no contact between the parts; • Small average bearing error (eccentricity) due to the small thickness of the air film; • Relative independence of bearing operation relative to ambient temperature; because the viscosity variation of the air is very small with temperature; • Vibration-free when compared with other bearings; • Can be used where contamination of materials should be avoided; • There is no need for collection and return equipment for later use.
The operating principle of aerostatic bearings is the use of the viscosity of the fluid flowing between their surfaces. The control of the air film stiffness is made by flow restrictors placed between the external power supply (constant pressure) and the bearing gap, that is, if there is no flow restriction the bearing pressure is constant, as Figure 2.6 (Purquério, 1990).
Chiarelli, L.R.
11 Rigidez Increasing restriction Decreasing restriction No restriction
Restricto
(a)
(b)
Figura 2.6: Diagram of force (w) X gap (h) (a); feeding orifice (b) (Purquerio,1990).
A rather simplified estimate of the stiffness of the aerostatic bearing can be obtained by deriving the function of the estimated load capacity (w) with respect to the bearing gap (h) as shown in equation (2.1).
(2.1)
The major differences in performance are related to the compressibility effects of the air film. Aircraft bearings are smaller than hydrostatic, but their quality in the production of their surfaces and the tolerances involved must be strictly controlled.
2.2.3. Constructive types of bearings and flow restrictors
Munday (1971) classifies the aerostatic bearings into five basic types: • Cylindrical; • Axial with circular movements; • Axial with linear movements; • Spherical; • Conical.
12
2. Literature review
Regardless of type and shape, the basic operating principle of the bearings remains the same. The air film is pressurized through an external power source, usually a compressor. Air from the external source passes through the flow restrictors that distribute this pressurized air along the bearing surfaces. After distribution, air flows into the atmosphere through the outer edges of the bearing (Powell, 1970).
Figure 2.7 shows the types of restrictors being their main characteristics and effects on the mechanical system according to Munday (1971) and Balestrero (1997):
a-) Single hole
b-) Annular orifice
c-) Elastic orifice
d-) Groove
e-) Pocket-type hole
f-) Porous orifice
g-) Partially porous orifice
h-) External flow control
Figure 2.7: Restrictors used in aerostatic bearings (Panzera, 2007).
(a) Single hole: extremely common, provides great stiffness. Its major disadvantage is the tendency to exhibit static instability or instability of pneumatic hammer (discussed below). (b) Annular orifice: In most cases this system is free of static instability and is usually the simplest form of construction. (c) Elastic orifice: whereas the operation of the quite dependent on the ratio of flow and air pressure of the restrictors, many have been carried out with the aim of improving the efficiency in this relationship.
Chiarelli, L.R.
13
(d) Groove: although this is a common geometry in hydrostatic beams, the use of of capillary restraints in aerostatic bearings has presented problems of static instability. (e) Pocket-type orifice: it has as main advantage the ease of construction, however this system has a low stiffness. (f) Porous orifice: this type of orifice has two functions, such as air restrictor and surface of the bearing itself. Because the air pressure is wide and high load capacities can be achieved with this system. Several techniques have been developed for the production of materials porosity, aiming the control of open pores and permeability. (g) Partially porous orifice: uses grafts of distributed porous materials homogeneously along the bearing surface. It shows a decrease load capacity. (h) External flow control: due to the difficulties encountered in the various types restraints to achieve high stiffness and stability, it is possible to control the air pressure by externally monitoring the air film thickness through sensors. This method presents relevant results in static systems, not presenting improvements in dynamic systems.
Slocum (1992) states that the phenomenon of pneumatic hammer instability is caused by gas compressibility and the consequent delay between changes in bearing clearance and the response to such change through variations in pressure in the bore or pocket. If the pocket volume is too large and the response time is too long, the resulting pressure in the pocket may increase excessively. This causes an increase in the clearance between the bearing surfaces. The increase in slack reduces the pressure in the pocket and the gap between the surfaces decreases again.
The reduction in the gap between the two surfaces increases the resistance to the gas flow, thus increasing the pressure in the gas. As a consequence of the increase in pressure occurs the increase in the gap between the two surfaces and the above cycle repeats. The instability that occurs is due to very low damping in a compressible fluid and is called instability of pneumatic hammer or just pneumatic hammer. Pneumatic hammer is often found in association with axial faces. There are
14
2. Literature review
two ways to overcome this problem: one is to reduce the depth and diameter that reduces the total volume of the hole or pocket. The second method is to use compensated holes and accept the loss in load capacity.
2.2.4. Aeorstatic bearings of porous materials
The ideal aerostatic bearing design seeks to uniformly supply the entire surface with air and this can be achieved with the use of a porous medium. Porous restrained bearings or porous bearings may have greater rigidity and load bearing capacity than other types of bearings. In addition to this, this type of bearing generally has no pneumatic instability and porous matrices such as graphite have high mechanical strength (Slocum, 1992). Figure 2.8 shows a sample of porous materials.
Figure 2.8: Porous sample in aqueous medium (New Way,2013).
2.2.4.1. Development and structure of aerostatic bearings
The first reports of porous radial bearings were presented by Montgomery and Sterry (1955), whose axis rotated at 258,000 rpm.
Sneck and Yen (1964) analyzed the characteristics and stability of externally pressurized radial bearings. In the analysis, only the radial air flow was considered and then the dynamic characteristics were calculated, through perturbation. The results were compared with the experimental results of Mori et al. (1965) who investigated theoretically and experimentally the characteristics of porous radial
Chiarelli, L.R.
15
bearings. In addition, Mori et al. (1965) calculated the static characteristics in large eccentricities by considering a circumferential flow of air in the porous material.
Sun (1975) also analyzed the static characteristics and stability of gaslubricated radial bearings. The rotational and pneumatic hammer instabilities were discussed in aerostatic radial bearings taking into account the effects of the eccentricity ratio, material permeability and feed pressure.
Gargiulio (1979) investigated the dynamic characteristics of porous-walled gas-lubricated bearings. The results were compared with experimental results of static and dynamic characteristics that included rotor instability.
Miyatake, Yoshimoto and Sato (2005) argue that porous bearing causes instability (pneumatic hammer) and that the reduction of permeability using a restriction layer on the porous surface is a way to avoid instability.
Mori and Yabe (1973) obtained static characteristics of a bearing with a restricted metallic porous surface. The equivalent gap model was modified to consider the effect of the restriction layer and compared the theoretical results with the experimental results (porous sintered bronze was used). Figure 2.9 shows the image of a radial porous restrictor bearing.
Porous material (graphite)
Metal structure
Figure 2.9: Image of the form of porous radial bearings (Miyatake, 2006).
16
2. Literature review
The permeability (the ease that air encounters in passing through a porous medium) is strongly related to the porosity or the average pore size of the material. The difficulty of controlling these parameters is extremely high and ends up being a problem originated from the production and processing of porous materials. The clogging of the pores during the machining process in cases of ductile materials results in the partial or total closure of the pores on the surface. This drastically influences the flow restriction behavior of the porous medium. With this, the fragile ceramic materials are indicated for this type of application (Polone and Gorez, 1980).
The high damping rate and simplicity of construction make porous bearings justify the research and development of new porous materials such as ceramics (Tsukamoto, 2003).
Kwan (1996) developed a porous circular beam with hot pressed alumina particles (1750°C) under a pressure of 100MPa over the 1 hour interval. The sintered particles had a range of 7 to 400μm and the measured viscous permeability coefficient was 4,75 x 10-15 m2.
Panzera (2007) developed a Portland cement ceramic composite material and silica microparticles for application as a porous axial bearing, aiming at a low production cost. The composite was fabricated with angular particles and compaction pressure of 10MPa.
Silveira et.al. (2010) developed an alumina matrix using sucrose as a pore forming agent. A series of samples of ceramic structures was obtained using different concentrations of pore forming agent (sucrose). There was homogeneity in the distribution of the pores whose mean value was 0.14 mm, with porosity around 28%, presenting a small standard deviation. There is great difficulty in controlling the pores in both the metal and ceramic aerostatic bearings, limiting the performance and studies of these elements, and Kawashima et al. (1990) studied the static characteristics of ceramic aerostatic bearings.
Chiarelli, L.R.
17
Kwan (1996) studied the behavior of fluid flow in porous alumina processed at high temperature and isostatic pressure for application in porous aerostatic axial bearings.
Aerostatic bearings in which the restriction is a graphite porous surface were studied theoretically and experimentally by Yoshimoto (2003) and Miyatake (2006), where it was concluded that such restriction increases the static stiffness and stability of the rotor at high speeds due to the low surface permeability porous.
Nicolletti et.al. (2008) present a non-dimensional modeling using the modified Reynolds equation considering the Forchheimer equation for porous media with turbulent regime.
Partially porous ceramic axial bearings were studied considering even the deflection of the ceramic material which, although small, has influence on the static stiffness of the bearing according to Fujii (2008). The literature on porous surface theory for bearing use gained significant prominence in the late 1970s. Notable examples were published by Rao and Majumdar (1978). Theoretical and experimental studies performed in Japan on rectangular bearings were discussed by Shih and Yang (1990), Yoshimoto (1996), Nakamura and Yoshimoto (1996), and Yoshimoto et al. (1999). Aspects such as load capacity, linear motion, permeability anisotropy and slip velocity were investigated.
Kwan (1996), after sintering a porous circular beam with hot pressed alumina particles, machined the pellet and tested it on a test bench for porous bearings. Their tests showed that two operating regions exhibited greater pneumatic stability: working air film below 5μm or work load below 25N. Under air pressures operating below 1.5bar no pneumatic instability was observed. The design of the experimental bench and the static performance of the porous bearing on various loads and air pressures can be visualized in figures 2.10 and 2.11 respectively.
18
2. Literature review
Loading cilynder Load cell Load shaft Aerostatic journal bearing
Aerostatic porous thrust bearing
Displacement transducer (x4)
Reference plate
Pressure sensor
Load (N)
Figure 2.10: Experimental set-up used in Cranfield (Kwan, 1996).
Air film thickness (µm)
Figure 2.11: Static performance of axial porous bearings under various loads and different feed pressures (Kwan, 1996).
Chiarelli, L.R.
19
Partially porous axial bearings have some properties and characteristics similar to porous bearings. Fujii et al. (2008) developed a device to analyze the fundamental static characteristics of partially porous ceramic thrust bearings, taking into account pore size, bearing flow and deflection. The experiment consisted of a 30mm piston installed on the tested ceramic body. The piston is supported by a guide to prevent radial displacement. The compressed air reaches the lower surface of the piston through the ceramic body. The air supply pressure remains constant while the ratio between the applied load and the gap is measured. The clearance designates the mean of the gaps measured with the displacements of sensors 1 and 2 to correct the slope generated by the imbalance of the applied load. The specimens consisted of untreated (non-plated) ceramics and treated ceramics (plated) through chemical coating baths. Figures 2.12 and 2.13 show the schematic of the experiment and the data obtained of the stiffnesses (non-dimensional).
Load Displacement sensor 1
Displacement sensor 3
Displacement sensor 2
Guide
Piston 2 Ceramics specimen t
Clearance
Piston 1
Air
Figure 2.12: Experimental scheme for obtaining characteristics of partially porous aerostatic bearings (Fujii, 2008).
2. Literature review
(a)
Rigidez adimensional stiffness Non-dimensional
Non-dimensional stiffness
20
Minimum clearance (μm)
(b)
Minimum Folga minima clearance (μm) (μm)
Figure 2.13: Dimensionless stiffness calculated by subtracting the minimum clearance from the applied load; thickness of the test piece (a) 3,5mm and (b) 4,5mm (Fujii, 2008).
Yoshimoto (2003) applied an epoxy-formed restraint layer on the surface of a radial porous aerostatic bearing of graphite matrix. In the experiment, the static characteristics for two types of imposed loads were measured. Figure 2.14 shows the experimental apparatus for symmetrical and cross loads. The axis shows the parallel displacement for symmetrical loads and the angular displacement for cross loads. The shaft was suspended by a thin rope, connected in the center of gravity of the rod. Shaft displacements were measured by two non-contact probes located at the top and bottom of the shaft. Figure 2.15 shows the results of the load capacity (nondimensional) as a function of the eccentricity ratio (relation between bearing length, shaft displacement and average clearance between bearing surfaces).
Chiarelli, L.R.
Pully
21
String
Shaft
Shaft
String
Pully
Distance probe Distance probe
Weights
Test bearing
Test bearing Weights
(b) Symmetrical loads
(a) Coupled loads
Figure 2.14: Experimental apparatus; (a) with symmetrical loads and (b) cross loads (Yoshimoto, 2003).
The maximum load capacity was obtained in the 8μm bearing gap and reached a dimensionless value of 0.6. The experimental results for the 11 and 14 μm clearances showed good agreement with the theoretical predictions, but a discrepancy to the 8μm bearing clearance was observed. The cause of this discrepancy is unclear, but geometric imprecision and non-uniformity of permeability of the porous material may have an influence on the load capacity.
Figure 2.15: Relation between load and eccentricity ratio (Yoshimoto,2003).
22
2. Literature review
Miyatake (2006) developed an experimental set-up to evaluate the rotational instability of a rotor supported by a resilient metal matrix radial aerostatic bearing. In the experiment, one rotor was supported by two radial hot air bearings with almost equal surface restraint ratios. The weight of the rotor was supported by an axial bearing. The rotor was driven by air jets in the turbine housings located at the top and bottom of the rotor. The rotor amplitude was measured by two displacement probes installed in directions perpendicular to each other as shown in Figure 2.16.
Distance probe
Rotor Housin g
Test bearings Air feed ports Optical fibre sensor
Turbine nozzle and bucket
Aerostatic thrust bearing
Figure 2.16: Experimental apparatus for instability test of aerostatic radial bearings with metallic matrices (Miyatake, 2006).
The critical speed of the rotor has the optimum play and reaches around 150000rpm at a feed pressure of 0.39MPa.
Friedel (2011) provides a proposal for an experimental bench model for testing porous ceramic bearings made by Silveira et. al. (2010), in the work was estimated the rigidity of the axis using the principle of superposition to obtain the deflection of this as shown in figure 2.17:
Chiarelli, L.R.
23
Figure 2.17: Method used to estimate the stiffness of the porous aerostatic bearing test bench (Friedel, 2011).O resultado da rigidez do eixo foi de 37,9N/μm.
24
2. Literature review
Friedel (2011) uses the Ocvirck model to estimate the stiffness of the porous aerostatic bearing. The model is very simplified and does not take into account the bearing pressure and the compressibility of the fluid. The estimated stiffness of the porous aerostatic bearing was approximately 7.8N/μm.
The advantages of this type of bearing are many, but its manufacture, maintenance and operation require a high financial cost, making it necessary to study the static and dynamic behavior of these bearings. The concepts and equations of the dynamics of rotating shafts and aerostatic bearings will be presented in detail in chapter 3.
25
3. THEORETICAL CONCEPTS
The purpose of this chapter is to show the theories on which the work was based, through basic concepts and numerical and analytical models for rotors.
3.1. Considerations about rotor dynamics
According to Pereira (2003), the most common rotary machines, also called rotors, can be of the most varied types, for example: turbochargers, aircraft turbines, steam turbines for the production of electric energy, etc.
The rotors are subjected to loads due to the inertia of their components and potential vibration and instability problems. The prediction of rotor behavior through mathematical models is relatively successful when compared with experimental measurements. In analyzes of the dynamic behavior of rotors, the most frequently performed studies are: • Critical velocity prediction: Velocities at which vibration due to unbalance of the rotor is maximum; • Design Modifications to Change Critical Speeds: When it is necessary to change the rotor operating speed, modifications to the rotor design are necessary to change critical speeds; • Predict natural frequencies of torsional vibrations: When multiple axes are coupled (eg gearbox) and these axes are excited by the motor pulses during startup; • Calculate correction masses and their locations from vibration data: rotor balancing; • Predict vibration amplitudes caused by rotor unbalance; • Predicting the frequencies of vibration in dynamic instabilities: Not always simple to achieve, given that not all destabilizing forces are known; • Project modifications to eliminate dynamic instabilities.
According to Nelson (2007), the first rotor model was proposed by Föppl (1895) and consisted of a single disc located in the center of a circular axis, without damping. Supercritical operation was shown to be stable. Unfortunately, Föppl (1985)
26
3. Theoretical concepts
published his work in a German civil engineering journal rarely read by the rotor dynamics community at the time.
Jeffcott (1919) conceived the same model by considering cushioning and published his work in a frequently-read English scientific journal. As a result, in the UK and USA a single disc rotor is called a Jeffcott rotor (Figure 3.1). Over time many variations of the Jeffcott rotor (1919) have been studied, but their most frequent construction is with a single hard disk mounted on a flexible shaft, which is supported by bearings at each end.
The dynamics of a rotor are not exactly the same dynamics as a mass-spring system. The reference of the mass-spring system is stationary, whereas of the rotor the reference is rotational. However, they share the same natural frequencies and modes of vibration.
a-) Jeffcott rotor
b-) Coordinate system Figure 3.1: Jeffcott rotor example - flexible shaft and rigid bearings (Yoon,2012).
Chiarelli, L.R.
27
The Jeffcott model presented in figure 3.1 considers the deformation of the shaft due to the flexion caused by the unbalanced mass represented in point G of Figure 3.1. This model disregards the elastic stiffness of the bearings, and in it the orbit C of the shaft is circular. Further details on considerations, bearing assumptions, axes, and spin and inertia effects will be discussed in the following chapters.
3.1.1. Whirl, synchronous and asynchronous motion
The precession movement, also known as secondary rotation or "whirling" is defined as the rotation of the center line of the rotative shaft, relative to the line joining the bearings. This phenomenon is mainly due to mass imbalance, however, other factors contribute to this effect, such as gyroscopic forces. The precession movement can occur in the same direction of the rotation speed, then it is called forward whirl, or it can occur in the direction opposite to the speed of rotation, then called backward whirl. The speed of rotation of the precession may be equal to the speed of rotation of the shaft, in this way called the synchronous; otherwise, the moviment is asynchronous (Mesquita, 2004).
3.1.2. Basic modeling of rotor dynamics
The dynamic equations for rotors are obtained by applying Newton's law for the motion of the disc. With the assumption that the shaft is devoid of mass, the forces acting on the disc are: the inertia force, stiffness and damping forces generated by the lateral deformation of the shaft. The equations of lateral motion on the x and y axes are: ̈ ̈
̇ ̇
(3.1a) (3.1b)
The stiffness coefficient at lateral flexion in the axial center of a uniform axis is represented by ks. In addition, it is assumed that there is a very low damping (given by cs) due to the combination of the structural damping of the shaft, damping due to the flow in machines and the damping originating from the bearings (Yoon, 2012).
28
3. Theoretical concepts
3.1.3. Undamped free vibrations
As in the work of Yoon (2012), the analysis of free-vibration does not deal with the vibration of the rotor, in the case where the unbalance (eu=0) and damping (cs=0) is neglected. The equations of motion (3.1a), (3.1b) are simplified to: ̈
(3.2a)
̈
(3.2b)
The solution for this second-order homogeneous system takes the form of:
(3.3a) (3.3b)
For some complex constant s. The values of the constants of Ax and Ay are obtained from the initial conditions of the rotor disc. Substituting the solution of equations (3.3a), (3.3b) into equations (3.2a), (3.2b) gives: (
)
(3.4a)
(
)
(3.4b)
The above equations are true for any value of Ax and Ay if the undamped characteristic is assumed.
(3.5)
Solving Eq. (3.5) with the solution given by Eq. (3.6) and considering S as a complex constant, it is concluded:
(3.6) In which, ωn is the natural undamped frequency of the axis defined as:
Chiarelli, L.R.
29
√
(3.7)
Thus, the solutions are non-damped oscillatory functions with frequency ± ±ωn. The undamped critical speed of the system is defined through equality (3.8): (3.8)
In that the positive component +ωn corresponds to the forward whirl and the negative component -ωn corresponds to the backward whirl. 3.1.4. The finite element method (FEM)
All physical systems are continuous by nature. The mathematical models of continuous systems result in partial differential equations usually described by variables dependent on time and space over a specific domain and boundary conditions. Analytical solutions are not always possible or are not feasible in many engineering problems. The discretization of a continuous system implies in a set of approximations, being used in the first moment the method of the finite elements to convert a continuous model inside a discrete model.
As a result of these approximations and discretization, a physical model represented by partial differential equations is transformed into a mathematical model driven by a set of ordinary differential equations described by variables, which are functions of time (Fish, 2009).
3.1.4.1. Euler-Bernoulli theory
In rotor dynamics, problems can be addressed in different ways such as using classic physical formulas or FEM. Although it is more difficult to define, the FEM is a very effective method to solve dynamics problems (Nassis, 2010).
30
3. Theoretical concepts
The development of the model was based on Nelson (1976), which is based on the Euler-Bernoulli beam theory which, in turn, considers that the neutral line of normal stresses passes through the geometric center of the axis (constant shear stress along the axis ) and disregards the damping effects generated by the deformation of the shaft. This theory is adequate when the slenderness index (ratio between length and diameter) is greater than 10. The model of axis element can be observed in Figure 3.2.
Z
Y
Z1 α1 Y1 β1
Z2
α2
Y2
s
l
β2
Ω
X
Figure 3.2: Shaft element and their respective degrees of freedom.
Where s is the axial position in the element. According to Nelson (1976) the vector containing the displacements is:
, -
(3.9)
[
]
The matrix containing the displacement shape functions is given by:
Chiarelli, L.R.
31
, -
[
]
(3.10)
In this case, the individual functions represent the modes of static displacement of a point of coordinates with all others reduced to zero. These functions are:
. / [
. / . / [ . /
. /
(3.11a)
. / ]
(3.11b)
. /
(3.11c)
. / ]
(3.11d)
The rotation shape functions matrix is composed of the derivatives of the above functions with respect to s.
, -
[
]
(3.12)
The translational mass matrices [M], rotational mass [N] and stiffness due to flexion [K] are:
, -
∫ , - , -
(3.13) [
]
Where μ is the mass per unit length of the element and the abbreviation SYM, indicates symmetry of the matrix.
32
3. Theoretical concepts
, -
∫
, - , -
(3.14) [
]
Where Id corresponds to the diametrical inertia of the element.
, -
∫
,
- ,
-
(3.15) [
]
Where E is the Young's modulus and I is the moment of inertia of the crosssection of the beam element. The gyroscopic matrix [G] is, in turn, antisymmetric (SKEW SYM):
, -
(3.16) [
]
In the assembly of the system matrices (global matrices), the translational [M] and rotational mass matrices [N] must be summed.
3.1.4.2. Timoshenko theory
The content of this chapter was based on Nelson (1980) and Rao (1996). Timoshenko's theory considers the effects of the transverse shear Φ and the
Chiarelli, L.R.
33
matrices of the elements are inversely proportional to the term (1+ Φ) 2. The matrices and the term composing this theory are described:
(3.17)
Where Ae is the cross-sectional area of the element. The stiffness matrix [K], as well as all other matrices (mass and gyroscopic) is composed of a sum of "sub-matrices" such as:
,
-
(
)
(3.18) [
,
-
(
]
)
(3.19) [
, -
,
]
-
,
-
(3.20)
The matrix [K] is the stiffness matrix to be used in the assembly of the global matrix. The translational [M] and rotational mass matrices [N] are:
34
3. Theoretical concepts
,
-
(
)
(3.21) [
,
-
(
]
)
(3.22) [
,
-
(
]
)
(3.23) [
]
, -
,
-
(
,
-
,
-
,
-
(3.24)
)
(3.25) [
]
Chiarelli, L.R.
,
35
-
(
)
(3.26) [
,
-
(
]
)
(3.27) [
, -
,
-
]
,
-
,
-
(3.28)
The gyroscopic matrix [G] is given by:
,
-
(
)
(3.29) [
,
-
(
]
(3.30)
) [
]
36
3. Theoretical concepts
,
-
(
)
(3.31) [
, -
,
]
-
,
-
,
-
(3.32)
3.1.4.3. Rolling bearings matrices
If the boundary conditions at the ends of a rotor are flexible then the dynamic coefficients of the rolling element bearing are added to the nodes corresponding to the rotor supports (Nassis, 2010). The nodal stiffness matrix of the bearing [Kbn] is:
,
-
[
]
(3.33)
The coupled terms (outside the main diagonal) are allowed to be negligible and the upper index n indicates that the matrix is nodal. The damping matrix is described as follows:
,
-
[
]
(3.34)
The inclusion of the bearing matrices in the model tends to decrease the natural frequencies of the system.
Chiarelli, L.R.
37
3.1.4.4. Disc matrices
One method of representing rotor coupled elements (gears, turbine blades, pulleys, etc.) is the use of hard drives that add inertia to the respective nodes in which they are positioned. The translational inertia matrices [Md], rotational inertia [Nd] and gyroscopic inertia [Gd] are described:
,
-
[
]
(3.35)
Where md is the mass of the disc.
,
-
[
]
(3.36)
Id is the diametral inertia of the disc.
,
-
[
]
(3.37)
Where Ip is the polar inertia of the disc. For constant thickness Ip=2 Id. 3.1.4.5. Assembling the matrices (global matrices)
Using the finite element method, a beam or a rotor is considered a system that can easily be described with one or more elements. The array of matrices can be developed with elements that overlap each other on the common nodes, as shown in Figure 3.3. The rotor can be divided into several shaft elements (there is no need for equality between lengths and radius of these elements) that connects two nodes (Nassis, 2010).
38
3. Theoretical concepts
Z
Node 1
Node 3
Node 2
Node 4
X Y Element 3
Element 2
Element 1
Figure 3.3: Example of shaft divided into three elements and supported on bearings at its ends.
The matrices of the elements must be positioned according to the nodes in which they are. For example, the translational mass matrix [Me] of element 1 must be positioned in the rows and columns corresponding to degrees of freedom of index 1 and 2, and the translational mass matrix [Md] of element 2 must be positioned on the lines and columns corresponding to the degrees of freedom of index 2 and 3. Thus, the terms located in degrees of index 2 must be added together. To make the reader clearer, Table 3.1 below, which contains some of the terms of the translational matrices Me (i, j) and Md (i, j) of the axis elements 1 and 2 respectively, is shown.
Y1
Z1
β1
α1
Y2
Y1
Me(1,1)
Me(1,2)
Me(1,3)
Me(1,4)
Me(1,5)
Me(1,6)
Me(1,7)
Me(1,8)
Z1
Me(2,1)
Me(2,2)
Me(2,3)
Me(2,4)
Me(2,5)
Me(2,6)
Me(2,7)
Me(2,8)
β1
Me(3,1)
Me(3,2)
Me(3,3)
Me(3,4)
Me(3,5)
Me(3,6)
Me(3,7)
Me(3,8)
α1
Me(4,1)
Me(4,2)
Me(4,3)
Me(4,4)
Me(4,5)
Me(4,6)
Me(4,7)
Me(4,8)
Y2
Me(5,1)
Me(5,2)
Me(5,3)
Me(5,4)
Me(5,5)+Md(1,1)
Me(5,6)+Md(1,2)
Me(5,7)+Md(1,3)
Me(5,8)+Md(1,4)
Z2
Me(6,1)
Me(6,2)
Me(6,3)
Me(6,4)
Me(6,5)+Md(2,1)
Me(6,6)+Md(2,2)
Me(6,7)+Md(2,3)
Me(6,8)+Md(2,4)
β2
Me(7,1)
Me(7,2)
Me(7,3)
Me(7,4)
Me(7,5)+Md(3,1)
Me(7,6)+Md(3,2)
Me(7,7)+Md(3,3)
Me(7,8)+Md(3,4)
α2
Me(8,1)
Me(8,2)
Me(8,3)
Me(8,4)
Me(8,5)+Md(4,1)
Me(8,6)+Md(4,2)
Me(8,7)+Md(4,3)
Me(8,8)+Md(4,4)
Z2
β2
α2
Table 3.1: Example of assembly of part of the global translational mass matrix.
Chiarelli, L.R.
39
3.1.4.6. Eigenvalues and eigenvectors
According to Tisseur (2001), every dynamic system can be described according to Newton's second law: [
]* ̈ +
,
-* ̇ +
,
-* +
(3.38)
Where [Mg] is the global mass matrix, [Kg] the global stiffness matrix and * ̈ + * ̇ + * + are the acceleration, velocity and displacement vectors respectively. [Dg] is a matrix consisting of: [
]
,
-
,
-
(3.39)
Where [Cg] is the global damping matrix and [Gg] is the global matrix representing the gyroscopic effect.
After finding the global equation, it can be observed that there is a problem involving matrices and differential equations and to facilitate the obtaining of results, the differential equation of second order is reduced to a first degree equation through state model.
[
, - , ] * ̇+ , , -
, [ , -
[ ,
] ]* + -
* +
(3.40)
Where:
* ̇+ * +
̈ 2 3 ̇ ̇ 2 3
Eq. (3.40) can also be represented as:
(3.41a) (3.41b)
40
3. Theoretical concepts
[
,
, -
, ,
] * ̇+
-
, - [ ] [ ]* + , - , -
* +
(3.42)
Where [ ] is an identity matrix and [0] is a matrix composed of zeros. Assuming the first matrix as - [Sg] and the second as [Rg], equation (3.42) can be rewritten as: [ ]* ̇ +
,
-* +
(3.43)
Assuming as solution for the Eq. (3.43) is:
(3.44a) (3.44b)
The problem becomes an eigenvalue problem and equation (3.43) becomes: |
|
(3.45a)
|
|
(3.45b)
|
(3.45c)
|
The matrix of eigenvalues λ contains the natural frequencies in rad/s and can be extracted directly from matrix [Ag], provided that: ,
-
, - ,
-
(3.46)
Considering that the global matrices are of nxn dimensions, the reduction of an order of equation (3.38) from the origin to the matrix of eigenvectors of dimensions 2n x 2n, where the n first lines correspond to the displacements of the respective degrees of freedom and the last n lines correspond to speeds. The eigenvalue matrix is also of order 2n x 2n and the eigenvalues are doubled, but one with the negative sign. This is due to the fact that eigenvalues are complex conjugates and always have an equal and negative value.
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41
3.1.4.7. Frequency response
Both in rotors used in industry and in rotors used for academic purposes, it is common to have excitation elements such as contact gears, chain gears, shakers, etc. These elements produce forces that can be decomposed and represented in the directions corresponding to the degrees of freedom of the system. This chapter aims to show how the evaluation of the responses of the various degrees of freedom in the most varied excitations is based on Ewins (1984) and Nicoletti (2012).
At first it will be assumed that Eq. (3.38) is nonzero and when expressed in the matrix notation becomes: [
]* +̈
[
]* ̇ +
[
]* +
* +
(3.47)
Where {f} is the vector containing the excitation forces. Assuming that the excitation occurs at a given frequency, the vector {f} and the solution proposal for the displacement vector {x} are respectively: * +
* +
(3.48a)
* +
* +
(3.48b)
Substituting the excitation function of Eq. (3.48a), the displacement function of Eq. (3.48b) and its derivatives in Eq. (3.47) yields: [
]* +
[
]* +
[
]* +
* +
(3.49)
Isolating the above equation according to the displacement amplitudes {X}:
* +
0
[
]* +
[
]* +
[
]1
* +
(3.50)
Assuming that the amplitudes of force {F} are unitary, the amplitudes of the responses are obtained by unit of force and are:
42
3. Theoretical concepts
* +
0
[
]* +
[
]* +
[
]1
(3.51)
In that the matrix {H}jk contains the answer in the "jth " degree of freedom to an excitation in the "kth" degree. As an example, Table 3.2 showing the vertical response of node 1 (Z1) to a horizontal excitation at node 2 (Y 2) can be obtained through the coordinate (j, k) corresponding to the indicated degrees.
Y1
Z1
β1
α1
Y2
...
Y1
H11
H12
H13
H14
H15
H1k
Z1
H21
H22
H23
H24
H25
H2k
β1
H31
H32
H33
H34
H35
H3k
α1
H41
H42
H43
H44
H45
H4k
...
Hj1
Hj2
Hj3
Hj4
Hj5
Hjk
Table 3.2: Frequency response matrix check example {H}jk. 3.2. Theory of the aerostatic lubrication
Most cases of bearing flow are laminar and pressure losses occur due to the viscous shear in the gas film. This reason makes the understanding of the laminar flow of gases between plates of great importance and some assumptions must be made (Powell, 1970):
i. The inertia due to acceleration can be neglected when compared to the frictional force due to the viscous shear; ii. The laminar flow conditions exist for all points of the gas film; iii. The pressure is constant in any section normal to the flow direction; iv. There is no slippage in the contours (borders) between the fluid and the plates.
Figure 3.4 shows the flow in the direction of x because the pressure P1 is greater than P2. The velocity distribution along the y-direction follows a curve and in contact with the surfaces is stationary. The velocity and pressure of the gas at any point are given by u and P respectively, and the viscosity of the gas is η.
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43
Using the nomenclature defined above and applying the boundary conditions for the well-known Navier Stokes equations the resulting expression is:
(3.52)
P
P1
𝑥
P1
𝑎̅
𝑦
P2
P2
𝛿𝑦
𝑢
𝑢
𝛿𝑢
𝑥 𝑙 Figure 3.4: Distribuition of the fluid speeds in plates (Powell, 1970).
Integrating Eq. (3.52) it is obtained:
(3.53)
Where A1 is the integration constant; integrating again the sides of the Eq.(3.53) we have: (3.54)
Considering that h is the gap between the plates and remembering that in contact with the surfaces the gas is stationary, then the boundary conditions can be accepted as u = 0 at y = 0 and at y = h. Substituting in Eq. (3.54) gives the following:
44
3. Theoretical concepts
(3.55)
(3.56)
By replacing A1 and B in Eq. (3.54) it can be concluded that:
(
)
(3.57)
Eq. (3.57) gives the gas velocity at any position of the film section and it can be noted that the velocity distribution is parabolic. The velocity at the center of the gap is maximal and determined when y = h / 2. The flow between the plates can be obtained by:
̇
̅ ∫
(3.58)
Where ̇ and ρ are the mass flow rate and the gas density respectively. The representation of the distance along the z-axis is given by ̅ as shown in Figure 3.4. Substituting (3.57) into Eq. (3.58) it becomes:
̇
̅
∫ (
)
(3.59)
Integrating the term and rearranging it is obtained:
̇
̅
(3.60)
̇ ̅
(3.61)
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45
Eq. (3.61) shows the relationship between the mass flow ratio and the pressure gradient between the plates in the direction of flow. The density ρ is so far assumed as constant in the y direction and Eq. (3.61) is valid for liquids and gases. However, the density of a gas depends on the pressure and as it varies in the xdirection, Eq. (3.61) can not be integrated to provide the pressure distribution in the x-direction until some relation between density and pressure is established.
It can be assumed that the gas behavior is isothermal as long as the heat generated in the gas film is small and the bearing walls are of metal containing high thermal conductivity. Considering isothermal conditions, we have:
(3.62)
Where R and T are the gas constant and the absolute temperature respectively. Isolating the density ρ in equation (3.61) and integrating the two sides of the equation gives: ̇
(3.63)
̅ ̇ ̅
(3.64)
Eq. (3.64) expresses the pressure variation along the aerostatic bearing in terms of flow, gas properties and dimensions. 3.2.1. Theory of mass flow through porous medium
The study presented in this item is based on Nicoletti et. al. (2008), permeability is a fundamental physical property in the development and design of aerostatic bearings because it represents the ability of a fluid to pass through a porous medium. From the equation of Darcy (1856) it is shown that:
46
3. Theoretical concepts
(3.65)
Where
represents the average velocity of the fluid through the porous
medium along the dimension s (porous matrix thickness), η is the dynamic viscosity of the fluid, k1 is known as the viscous permeability coefficient or the Darcy coefficient, and dp/ds is a variation of pressure along dimension s.
(3.66)
Assuming that the fluid (air) is an ideal gas, which is generally acceptable under pressures below 106N/m2 as put by Cieslicki (1994), and the fluid is under isothermal conditions, it is possible to be considered that the fluid density in the porous medium is the average density between inlet and outlet, as follows:
(3.67)
The supply pressure is represented by Ps and the pressure of the air film by the letter p. Substituting equations (3.66) and (3.67) into equation (3.65), we obtain after integration:
(
)
(3.68)
A figura 3.5 mostra o esquema para melhor visualização e entendimento das equações citadas acima.
Chiarelli, L.R.
47
Figure 3.5: Schematic representation of the porous medium with cross-sectional area and constant thickness (Silveira, 2008).
Eq. (3.68) can be manipulated to obtain the mass flow rate through the Darcy hypothesis (1856), thus: (
)
(3.69)
3.2.2. Modified Reynolds equation
Pressurized air at pressure Ps is injected into the gap h of the bearing through the porous matrix that builds the aerostatic bearing, thus forming a pressure distribution along the bearing. In order to calculate the pressure distribution in the bearing gap it is assumed that the fluid is Newtonian, compressible and operates in laminar flow (Nicoletti et.al., 2008).
Porous bearing r
Figure 3.6: Aerostatic porous bearing diagram (Silveira, 2008).
48
3. Theoretical concepts
By simplifying the Navier-Stokes equations by rewriting it on the reference axis (x, y, z), fixed on the bearing slip surface and admitting zero fluid flow on the bearing and rotor shaft surfaces, the boundary conditions are represented in Figure 3.7 and in equation (3.70):
Rotor surface
Aerostatic bearing
Figure 3.7: Kinematics of fluid flow, velocity profiles (Nicoletti, 2008). (
)
(
)
( (
) )
(
)
(
)
(3.70)
Where Vinj is the air injection velocity, rotor radius, (w, v, u) are the velocities in the directions of (x, y, z) respectively, t is the time, U is the speed of the rotor surface and V is the translation speed of the rotor in the gap. By integrating the NavierStokes equations subject to the boundary conditions shown in Eq. (3.70), the expressions for the fluid velocity distributions are obtained:
( ) ( )
( (
( )
(
)
(3.71a)
)
(3.71b) )
(3.71c)
Chiarelli, L.R.
49
By inserting the expressions for the fluid velocity profiles (Eq. (3.71)) into the continuity equation, integrating between the limits [0, h] and considering the fluid under isothermal conditions, it is obtained:
(
) ̅
( ̅̅ ̅
̅
)
̅
( ̅̅
̅ ) ̅
( ̅ ̅) ̅
( ̅ ̅)
̅( ̅
)
(3.72)
Equation (3.72) corresponds to the non-dimensional form of the modified Reynolds equation for porous aerostatic bearings as a function of the Vinj injection velocity on the bearing surface. Table 3.3 and Eq. (3.73) define the terms used and their corresponding functions for a better understanding of the reader.
(3.73a)
(3.73b)
. /
(3.73c)
Eq. (3.72) is a partial differential equation and can be solved by numerical methods such as the finite difference method or the finite element method.
Chapters 4 and 5 detail the case study, modeling, and results obtained from a shaft and bearings using the theories described in this chapter.
50
3. Theoretical concepts
Termo
Definição
Equivalência
Non-dimensional parameter of the velocity
Eq. 3.73a
Non-dimensional parameter of excitation
Eq. 3.73b
Non-dimensional parameter of the porous matrix ̅
Non-dimensional coordinate
̅
Non-dimensional coordinate ̅
Non-dimensional pressure
̅
Eq. 3.73c
Non-dimensional gap Non-dimensional time Assemble clearance
-
Excitation frequency (angular velocity)
-
Bearing length
-
Bearing perimeter
̅
Internal radius of the bearing
-
External radius of the bearing
-
Non-dimensional stiffness Stiffness
Table 3.3: Definition of the terms used in the Reynolds equation.
-
51
4. CASE STUDY
This work proposes the static and dynamic analysis of the shaft assembly, rolling element bearings and drive turbine, to investigate the static and dynamic behavior of a porous ceramic aerostatic bearing. In Friedel (2011) the shaft and geometry of the porous ceramic aerospace bearing was simplified, based on the Ocvirck short bearing model (Norton, 2012). A first geometric configuration was also proposed, in order to obtain the mentioned parameters. The calculation of the axis deflection is an initial and fundamental step, to be included in the dynamic analysis of the system. According to Norton (2012), the shaft stiffness should be significantly (10 times) higher than the bearing stiffness, to avoid future overlaps of responses such as vibration amplitudes and FRFs (frequency response functions). Figure 4.1 presents the steps developed for the research with porous ceramic aerostatic bearings, started in 2006 at the Tribology Laboratory, Department of Mechanical Engineering, EESC-USP.
Design of the bearing
Mathematic model of the bearing
Design of the rotor configuration
Nummerical models of the system
Build experimental set-up
Experimental validation of the models
Figure 4.1: Structure of the research line of porous ceramic aerostatic bearing.
52
4. Case study
The bearing is constituted by a ceramic bush attached to a metal jacket through adhesive glue. The air enters the metal jacket through a feed tube and penetrates between the pores of the ceramic acting as a lubricant between the shaft and the bearing. The diametral clearance between the bearing and the shaft is 16 μm. Figure 4.2 clearly illustrates the design and construction of the bearing to be studied. Air feed
SAE 4340
φ11
φ14.2
φ20
SAE 4340
Alumina
Alumina 17
(A)
(B)
Figure 4.2: (A) technical drawing of the bearing; (B) manufactured bearing.
The design of the experimental set-up was elaborated by Friedel (2011) and based on Carter (2009), who proposed an assembly for static and dynamic investigation of a segmented bearing. The set-up layout is shown in Figure 4.3.
The shaft will be driven by the air-driven turbine and the aerostatic bearing will be positioned between the rolling bearings, which will increase the stiffness of the shaft. The design parameters such as dimensions, bearing materials and bearing dimensions were defined in Friedel (2011). The operating range of the seat shall be set according to the estimation of the natural frequencies of the bearings and the shaft. The maximum bearing capacity supported by the bearing will be adopted in 1kg or 10N.
Chiarelli, L.R.
53
Housing of the bearing Rolling bearing
Porous matrix
Bearing housing
Rolling bearing housing
Rolling bearing Safety ring Air feed Turbine housing
Safety rings
Shaft Fixing screw
Screw Base
Figure 4.3: Experimental set-up scheme (Friedel, 2011).
In the project, experiments designed for static analysis will be performed by applying vertical loads (up to 10N or 1kg) on the bearing housing. Obtaining the shaft stiffness will be obtained by dividing the load applied by the bearing gap after the load application. In order to avoid interference of the experimental set-up in obtaining the static stiffness of the bearings, the static shaft and bearing rigidities must be estimated in order to assess the set-up viability (Chapter 5).
The dynamic experiments will be conducted using a shaker placed in the housing of the aerostatic bearing so that it applies a vertical load as a function of time in order to obtain the dynamic coefficients of the bearing. In order to estimate the natural frequencies of the rotating set, we will use frequency response functions that will be obtained by simulating a vertical load varying over time on the node in which the bearing is located and analyzing the vibration amplitudes in the same node and same direction of charge (Chapter 6). It must be ensured that the natural frequencies of the axle do not match the natural frequencies of the aerostatic bearing so that there is no interference of the seat in obtaining the dynamic coefficients of the bearing.
54
5. STATIC ANALYZES
This chapter presents the static analysis of a preliminary configuration for an experimental set-up of a porous ceramic aerostatic bearing. The use of ceramic material as a restrictor can improve bearing performance with respect to wear resistance, good thermal stability and adequate stiffness, for conditions requiring high rotational speeds (over 10,000rpm), as well as small radial clearances in the bearings (40μm). These design features are required for ultra-precision machine heads. Thus, two verification procedures were performed for the proposed configuration, one of them being the static shaft dimensioning, for the estimates of the shaft stiffness and shaft deflection estimation, as well as estimates of geometry, stiffness and stresses in the ceramic porous bearing; the other procedure is the development of a finite element model of the entire experimental set-up, considering different stiffness values based on the literature for the rolling bearings and the stiffness of the ceramic porous bearing, based on its permeability matrix. 5.1. Models of the rotor assemble
In the present work different models of the test bench were proposed. The reason for this action is to provide alternative model proposals and to compare them. Model A consists of limiting the analysis of the axis between the rolling bearings. Model B considers the extension of the shaft that attaches to the turbine, but does not consider any effect of the turbine on the system. Model C considers the effects of the turbine (T) at the end of the axis, which is modeled through a disk. Figure 5.1 shows the models described.
The bearings (B) are represented only by their internal diameters and their stiffnesses are applied in the respective positions in which the bearings are located on the rotor shaft. The force (F) passes through the center of the thicker section of the shaft, where the hot air bearing will be positioned.
Chiarelli, L.R.
55
Model A
Ø 17
F
Ø 11
Ø 10
9
B
B
22 28.75
28.75 Model B
Ø 17
F
Ø 11
Ø 10
9
B
B
22 28.75
28.75
33.50
Ø 17
F
Ø 11
Ø 10
9
Ø 26
Model C
B
T
B
22 28.75
28.75
25.50
8
Figure 5.1: Proposed test models with dimensions in millimeters; the inner rings of the bearings are marked with the letter "B" and the turbine disc with the letter "T".
56
5. Static analyzes
5.2. Finite element models
The finite element rotor models were developed in ANSYS®, Autodesk Simulator® and MATLAB® software. The simulations were done without the aerostatic bearing in order to obtain the stiffness of the components of the experimental set-up and to compare it with the stiffness of the aerostatic porous ceramic bearing.
5.2.1. Boundary conditions
The rolling bearing stiffnesses were extracted from Yi Guo (2012) for singlerow ball bearing (57x106N/m) and Hagiu (1997) for angular contact bearings (50x106N/m). They were also considered high stiffness (10x1010N/m). The contour conditions representing the bearings are shown in Figure 5.2:
Point of application of the force
Supports with elastic stiffness in the center and ends of the internal diameters of the rolling bearings
Figure 5.2: Boundary conditions applied to the Axis subset of Models A and B using ANSYS® software.
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57
In both ANSYS® and Autodesk Simulator® software, the elastic supports impose stiffness on translational displacements along the three axes (x, y, z).
Point of application of the force
Supports with elastic stiffness in the center and ends of the internal
diameters of the rolling bearings
Figure 5.3: Boundary conditions applied to the Axis subset of Model A and B using Autodesk Simulator ® software.
The turbine, represented by a disc at the end of the shaft, has its weight modeled as a force applied to the center of the disc in ANSYS®.
Place of application of the weight generated by the turbine.
Figure 5.4: Modeling the turbine of Model C using ANSYS®.
In MATLAB®, the effects of the turbine mass are modeled through the moment of inertia according to the theory of chapter 3.1.4.4.
58
5. Static analyzes
5.2.2. Generation and mesh quality Meshes are generated with three-dimensional tetrahedra-like elements in Autodesk Simulator®, hexahedra and tetrahedra in ANSYS®.
Figure 5.5: Tetrahedral mesh (a) and hexahedral mesh (b) – ANSYS®.
Figure 5.6: Tetrahedral mesh using Autodesk Simulator®.
It was also generated meshes containing one-dimensional elements in ANSYS® and MATLAB® as shown in Figures 5.7 and 5.8.
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Figure 5.7: Mesh composed of one-dimensional elements in ANSYS®.
There are several forms of mesh quality control that can be evaluated (Bakker, 2002; MIDAS, 2013). Some of these mesh quality criteria were applied to meshes with three-dimensional hexahedral and tetrahedron elements in the ANSYS® program environment. In Figures 5.9, 5.10 and 5.11 three approaches are described and analyzed to verify the quality of the mesh generated:
60
5. Static analyzes
Mesh generated in Model A
Mesh generated in Model B
Mesh generated in Model C
Figure 5.8: One-dimensional meshes generated in MATLAB®.
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Aspect Ratio: is defined as the ratio between the longest and longest edges. Using the ANSYS® recommendation, aspect ratios lower than 500 were
Percentage of mesh volume
considered.
da malha doofvolume Percentual mesh volume Percentage
Aspect Ratio (A)
Aspect Ratio (B)
Figure 5.9: Relation between the hexahedral mesh (A) and tetrahedral (B) X aspect ratio
It can be observed that the values obtained from Aspect Ratio are well below the maximum allowable limits, in the mesh with three-dimensional elements of the type hexahedrons and tetrahedra.
Skewness: is an indicator of symmetry that can be measured in several ways and the most common one is shown in Eq. (5.1):
62
5. Static analyzes
(5.1)
The skewness values range from 0 (minimum) to 1 (maximum), and the quality of the element increases as the skewness value approaches zero. As an example of
Number of elements
skewness 0 has the square, in parallel the rectangle has skewness greater than 0.
elements ofelementos Número Numberde
Skewness (A)
Skewness (B)
Figure 5.10: Relation between the hexahedral mesh (A) and tetrahedral (B) X skewness
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The number of elements with skewness greater than 0.5 is small relative to the total number of elements in the two meshes generated (68937 of the mesh with hexahedral elements and 48263 of the mesh with tetrahedral elements).
Jacobian Ratio: the calculation of this criterion is made at the integration points of the elements commonly known as Gaussian points. At each integration point a Jacobian determinant is calculated and the Jacobian Ratio is the ratio between the maximum and minimum determinant within the element. A ratio equal to -100
mesh volume Percentagedoofvolume da malha Percentual
is sought (MIDAS, 2013).
Percentage of mesh volume
Jacobian Ratio (A)
Jacobian Ratio (B)
Figure 5.11: Relation between the hexahedral mesh (A) and tetrahedral (B) X Jacobian Ratio
64
5. Static analyzes
The results show that the meshes of the ANSYS® models are homogeneous enough to proceed with the analyzes.
5.3. Stiffness of the aerostatic bearing
Friedel (2011) makes an analytical estimate of the stiffness of the aerostatic bearing described in chapter 2.2.4.1. In the present work, the bearing stiffness is estimated by the finite difference method implemented in MATLAB® by Nicoletti (2008). The results obtained through the methods were:
Method
Stiffness (106 N/m)
Analytical
7,7903
Finite differences
2,5666
Table 5.1: Static stiffness obtained by different methods.
The dynamic and static bearing coefficients will be discussed in detail in Chapter 6.
5.4. Structure and proceedings of the static analyzes
This work proposes the analytical-numerical study of the experimental test bench presented in chapter 5. The flowchart shown in Figure 5.12 describes the steps developed.
The results of the models using Tymoshenko's theory do not differ significantly from the results presented using Euler-Bernoulli's theory in the static analyzes, and it is not necessary to approach them separately.
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Material properties
Geometry of components
Boundary conditions
Mesh generation
Satisfactory mesh quality
No
Yes Maximum deformation
Stiffness of the shaft higher than of the aerostatic bearing
No
Yes Acceptable physical configuration for the experimental set-up
Comparison of results based on the different models
Figure 5.12: Strategy adopted for the evaluation of numerical models.
5.5. Results of the static analyzes
The calculation of the shaft deflection is an extremely important stage for investigation of the system. The stiffness of the assembly should be significantly (10 times) higher than the bearing stiffness, to avoid overlapping of dynamic responses.
66
5. Static analyzes
For the calculation of the stiffness of the bench a force of 10N was applied in the place where the geometric center of the aerostatic bearing would be, as shown in chapter 5.2.1 illustrating the boundary conditions. 5.5.1. Static analyzes of Model A Analyzes were performed from the initial design of the test bench. In the first model, rolling bearings were considered extremely high stiff supports (10 10N/m) and the results of the finite element models are compared with the analytical model described in chapter 2.2.4.1.
Model
Mesh
Rotor stiffness (106 N/m)
Analytical
-
37,93
Autodesk Simulator®
Tetrahedral
1,16
Tetrahedral
88,03
Hexahedral
85,20
One-dimensional
134,70
Euler-Bernoulli
250,00
ANSYS®
MATLAB®
Table 5.2: Results and comparisons in the initial design with rigid rolling element bearings.
The models presented in Table 5.2 with rigid rolling element bearings are very different from one another, making necessary changes in the boundary conditions to converge with the analytical model. This is probably due to the fact that the axle analytical model generated by Friedel (2011) is a bi-supported beam in which only degrees of freedom are removed in the vertical direction and not a bi-coupled beam in which the degrees of freedom are drawn in the direction vertical and slope (Beer, 2012). The high stiffness has different influence on each of the models, making it necessary to emphasize that each software uses elements and functions differently for different degrees of freedom (Fish, 2009). Models considering the stiffness of the ball bearing (B) and angular contact bearing (AC) shows other values of Model A. It can be seen from Table 5.3 that the models that take into consideration the stiffness of the rolling bearings present values very close to the analytical model.
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Model
Mesh
Analytical
Tetrahedral
ANSYS
Hexahedral
One-dimensional
MATLAB
Euler-Bernoulli
Support
Rotor stiffness
bearings
(106 N/m)
-
37,93
AC
33,52
B
34,67
AC
32,28
B
33,48
AC
36,53
B
38,03
AC
42,19
B
44,44
Table 5.3: Results of Model A considering the stiffness of the bearings.
The results show that although they are smaller than the stiffness shown in Table 5.2, the obtained stiffness are still much higher than the stiffness of the aerostatic porous ceramic bearing.
Figure 5.13: Deformations of Model A.
68
5. Static analyzes
5.5.2. Static analyzes of Model B
This model has a slightly larger dimension than Model A (figure 5.1), a condition that reduces stiffness a bit. Their stiffnesses are estimated according to Table 5.4: Model
Mesh
Analytical
Tetrahedral
ANSYS
Hexahedral
One-dimensional
MATLAB
Euler-Bernoulli
Support
Rotor stiffness
bearings
(106 N/m)
-
27,71
AC
28,60
B
29,01
AC
29,98
B
30,51
AC
33,83
B
38,47
AC
42,01
B
27,71
Table 5.4: Stiffness of Model B with variations in rolling element bearings.
Figure 5.14: Deformations of Model B.
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The analyzes show that the Model B is a proposal to be considered, knowing that its less stiff model presents stiffness considerably higher than the greater estimate of stiffness of the aerostatic ceramic porous bearing.
5.5.3. Static analyzes of Model C
In this configuration the dimensions and the weight of the turbine interfere in the deflection of the shaft and the results are observed in Table 5.5: Model
Mesh
Tetrahedral
ANSYS
Hexahedral
One-dimensional
MATLAB
Euler-Bernoulli
Support
Rotor stiffness
bearings
(106 N/m)
AC
21,92
B
22,47
AC
20,10
B
20,65
AC
32,10
B
33,14
AC
38,47
B
42,01
Table 5.5: Stiffness of Model C with variations in rolling element bearings.
Figure 5.15: Deformations of Model C.
70
5. Static analyzes
Model C presents relatively low stiffness when compared to the other models, but it is the model that takes a greater number of variables under consideration. Figure 5.15 shows the distribution of deformations originated along the Model C due to the load applied to the bearing and force weight due to the mass of the turbine.
The dynamic bearing and the rotor assemble models are shown in detail in Chapter 6.
71
6. DYNAMIC ANALYZES
In this chapter, studies will be carried out on the dynamic behavior of the shaft through different models and theories using the concepts and theories described in chapter 3. Also studies will be done on the characteristics and dynamic coefficients of aerostatic ceramic porous bearing. The survey of the natural frequencies will be done to evaluate the working frequency ranges of the test bench. Comparisons between the results of different theories, models and methods will be made along the chapter.
6.1. Modal analyzes of the free shaft
The modal analysis of the free shaft seeks to find the natural frequencies of the axis while not supported. The models that use the finite element method (MEF) in MATLAB® environment are based on theories of Euler-Bernoulli and Timoshenko. The total length of the supported shaft is 110mm. Figure 6.1 shows the shaft design and ANSYS® models with hexagonal and tetragonal element meshes.
Figure 6.1: Models with hexahedral and tetrahedral meshes respectively.
72
6. Dynamic analyses
Mesh
1st Natural frequency (Hz)
2nd Natural frequency (Hz)
Tetrahedral
3708,9
9387,8
Hexahedral
3710,7
9391,8
Model ANSYS®
Table 6.1: First natural frequencies of the detailed shaft using diferente meshes.
The models shown in figure 6.1 take into account all details of the shaft, including the keyway and the chamfers at the ends. For modal analysis of the axis these models are well elaborated. Knowing that the future analyzes will involve more complex models with discs and bearings, a simplified model of the shaft (Model D) is
10
elaborated considering only the length of 110mm and average radius of 5mm.
110
Figure 6.2: Model D and the generated mesh in MATLAB® (dimensions in mm).
The mesh generated in MATLAB® has eleven elements containing 10mm each. The results of the first natural frequencies obtained by Model D are shown in Table 6.2:
Model
ANSYS®
FEM Analytical
1st Natural frequency
2nd Natural frequency
(Hz)
(Hz)
Tetrahedral
3636,2
9670,4
Hexahedral
3619,5
9633,8
Euler/Bernoulli
3757,5
10207,3
Timoshenko
3726,0
9918,1
-
3809,7
10493,8
Mesh
Table 6.2: Natural frequencies of Model D obtained through different theories.
The results presented in Table 6.2 differ in less than 3% of the results obtained with the detailed shaft model, showing that Model D is acceptable. The
Chiarelli, L.R.
73
comparison between the first three natural frequencies obtained through the models and methods described so far is shown in figure 6.3: 2.5E+04
Frequency (Hz)
2.0E+04
1.5E+04
1.0E+04
5.0E+03
0.0E+00 0
1
2
3
4
Natural frequency number Euler-Bernoulli - MATLAB Analytical Hexaedral - ANSYS Hexahedral - design
Timoshenko - MATLAB Tetrahedral - ANSYS Tetrahedral - design
Figure 6.3: Comparison between natural frequencies using different methods.
The results for the first natural frequency are very similar between the methods used, and from the second frequency they begin to differ significantly, especially among the models analyzed in ANSYS®. In order to study the results obtained through the theories described in chapter 3, a comparison was made between the analytical method and the MEF with theories of Euler-Bernoulli and Timoshenko in relation to the shaft slenderness index. In this comparison a gradual increase of 10mm (corresponding to an element) in the shaft is simulated, keeping the radius of 5mm constant so that there is variation in the slenderness index.
74
6. Dynamic analyses 5
5
Comparação
x 10
4.5 FEM - Timoshenko Analytical Analítico FEM - Euler / Bernoulli
Natural frequency (Hz)
Freqëncia natural (Hz)
4 3.5 3 2.5 2 1.5 1 0.5 0
1
2
3
4
5
6 L/d
7
8
9
10
11
(A)
Comparação percentual 50 45
FEM - Timoshenko FEM - Euler / Bernoulli
(Hz) (%) frequency Natural Diferença percentual
40 35 30 25 20 15 10 5 0
1
2
3
4
5
6 L/d
7
8
9
10
11
(B)
Figure 6.4: First natural frequency as a function of the slenderness ratio through different methods (A); Percentage difference between FE and analytical method (B).
The numerical model of the shaft that uses the theory of Timoshenko differs from the analytical method in less than 5% from a slenderness index equal to 7. The numerical model using the Euler-Bernoulli beam theory achieves an equivalent divergence, with slenderness index equal to 5.
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6.2. Analyzes of the dynamic behavior of the rotor assembly
This section presents the numerical procedure for identifying the natural frequencies of the rotor assembly (shaft, rolling element bearings and turbine) and to compare them with the natural frequencies of the porous aerostatic bearing. The models used and their respective meshes are generated using the MATLAB® program and shown in Figure 6.5: Model A Mesh generated for Model A (9 nodes)
Model B Mesh generated for Model B (10 nodes)
Model C
Mesh generated for Model C (12 nodes)
Figure 6.5: Models and meshes generated in MATLAB® environment.
It can be observed that the models are similar to those presented in chapter 5. The difference occurs because in the study of the natural frequencies of the set there is no application of force. On the other hand, in the study of the frequency response function (FRF), the same models described in figure 5.1 with application of the force of 10N at 50rad / s (3000rpm) will be considered. In the dynamic analyzes made with the aid of the ANSYS® program, the meshes used are one-dimensional as shown in
76
6. Dynamic analyses
figure 5.7. The flowchart of Figure 6.6 shows the numerical steps for the dynamic analysis of the proposed models for the rotor assembly.
Material properties
Element geometries
Generation of matrices
Assemble of global matrices
Boundary conditions
State modeling (reduction of the order of the differential equation)
Obtaining eigenvalues
Obtaining the first natural frequency of the rotor assembly
First natural frequency of the assembly hihger than the aerostatic bearing
No
Yes Comparison of results based on the different models
Figure 6.6: Strategy used in dynamic analysis of the rotating system.
Chiarelli, L.R.
77
6.2.1. Dynamic analyzes of the rotor assembly – Model A
Shorter compared to others, this model has high natural frequencies (due to its high stiffness) by limiting the length of the shaft between the rolling element bearings. Table 6.3 and figure 6.7 show the results obtained considering angular contact (AC) rolling element bearings, radial ball bearings (B) and considering rigid support (according to the boundary conditions described in chapter 5.2.1).
Rotor configuration
Model A EDH (CA) Model A EDH (B) Model A Rigid
MATLAB - Euler / Bernoulli theory (E/B) Resonance Frequencies (Hz)
ANSYS – Onedimensional (A) Resonance Frequencies (Hz)
MATLAB - Timoshenko theory (T) Resonance Frequencies (Hz)
1ª (E/B)
2ª (E/B)
3ª (E/B)
1ª (T)
2ª (T)
3ª (T)
1ª (A)
2ª (A)
3ª (A)
5075,3
11918,8
15302,1
4996,8
11627,2
15216,8
6311,6
12238,0
13349,0
5217,5
12576,6
16143,2
5134,3
12224,3
16049,2
6480,7
12238,0
14156,0
14721,5
42201,3
97957,1
11138,1
32692,4
70437,3
12238,0
15169,0
29431,0
Table 6.3: Natural frequencies of Model A obtained through different theories.
Natural frequency (Hz)
5.E+04 4.E+04 4.E+04 3.E+04 3.E+04 2.E+04 2.E+04 1.E+04 5.E+03 0.E+00 Model A - EDH (CA)
Model A - EDH (B)
Model A - Rigid
1ª natural frequency (E/B)
2ª natural frequency (E/B)
1ª natural frequency (T)
2ª natural frequency (T)
1ª natural frequency (A)
2ª natural frequency (A)
Figura 6.7: Comparação entre os resultados do Quadro 6.3.
Figure 6.8 shows the frequency responses (FRFs) of the rotating set obtained from Model A through FEM using different theories.
78
6. Dynamic analyses FRFs - Modelo (a) A - Euler/Bernoulli 0
CA CA B E Rígido Rigid
-50
Amplitude [db m/N]
Amplitude [db m/N]
-100
-150
-200
-250
-300
0
1
2
3 Frequência [Hz] Frequency [Hz]
4
5
6 4
x 10
FRFs - Modelo(b) A - Timoshenko -80
CACA B E Rígido Rigid
-100
-120
Amplitude [db m/N]
Amplitude [db m/N]
-140
-160
-180
-200
-220
-240
-260
0
1
2
3 Frequência [Hz] Frequency [Hz]
4
5
6 4
x 10
Figure 6.8: Frequency responses obtained through FEM of model A; Theory of EulerBernoulli(A); Theory of Timoshenko (B).
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79
6.2.2. Dynamic analyzes of the rotor assembly – Model B
This model has lower stiffness than Model A because it takes into account the length of one end of the shaft. Thus the values of the natural frequencies of Model B are smaller when compared to those of Model A. The values of the frequencies of the rotor assemble obtained through this model are visualized in Table 6.4 and compared in Figure 6.9.
Rotor configuration
MATLAB - Euler / Bernoulli theory (E/B) Resonance Frequencies (Hz)
Model B EDH (CA) Model B EDH (B) Model B Rigid
ANSYS – Onedimensional (A) Resonance Frequencies (Hz)
MATLAB - Timoshenko theory (T) Resonance Frequencies (Hz)
1ª (E/B)
2ª (E/B)
3ª (E/B)
1ª (T)
2ª (T)
3ª (T)
1ª (A)
2ª (A)
3ª (A)
4069,1
7492,9
13264,1
4016,4
7296,3
13180,8
3990,4
7258,9
11262,0
4147,5
7802,6
13963,8
4091,8
7576,3
13867,7
4065,0
7534,2
11262,0
8327,4
18805,9
50986,3
7671,8
15411,3
38809,4
7557,1
11262,0
15187,0
Table 6.4: Natural frequencies of Model B obtained through different theories.
2.E+04 Natural frequency (Hz)
2.E+04 2.E+04 1.E+04
1.E+04 1.E+04 8.E+03 6.E+03 4.E+03
2.E+03 0.E+00 Model B - EDH (CA)
Model B - EDH (B)
Model B - Rigid
1ª natural frequency (E/B)
2ª natural frequency (E/B)
1ª natural frequency (T)
2ª natural frequency (T)
1ª natural frequency (A)
2ª natural frequency (A)
Figure 6.9: Comparison between the results of the Table 6.4.
80
6. Dynamic analyses FRFs - Modelo (a) B - Euler/Bernoulli -80
CACA B E Rígido Rigid
-100
-120
Amplitude [db m/N]
Amplitude [db m/N]
-140 -160
-180
-200 -220
-240
-260
0
1
2
3 Frequência [Hz]
4
5
6 4
Frequency [Hz]
x 10
(b) FRFs - Modelo B - Timoshenko -50
CACA B E Rígido Rigid
Amplitude [db m/N]
Amplitude [db m/N]
-100
-150
-200
-250
-300
0
1
2
3 Frequência [Hz]
4
5
Frequency [Hz]
6 4
x 10
Figure 6.10: Frequency responses obtained through FEM of model B; Theory of Euler-Bernoulli(A); Theory of Timoshenko (B).
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81
6.2.3. Dynamic analyzes of the rotor assembly – Model C
The most complete model, takes into consideration the entire rotor assembly including the turbine. The effects caused by the moment of inertia of the representative turbine disc significantly decrease the values of the natural frequencies. The values obtained assuming extremely rigid bearings are very similar to those of Model B as shown in Table 6.5 and Figure 6.11.
Rotor configuration
Model C EDH (CA) Model C EDH (B) Model C Rigid
MATLAB - Euler / Bernoulli theory (E/B) Resonance Frequencies (Hz)
MATLAB - Timoshenko theory (T) Resonance Frequencies (Hz)
ANSYS – Onedimensional (A) Resonance Frequencies (Hz)
1ª (E/B)
2ª (E/B)
3ª (E/B)
1ª (T)
2ª (T)
3ª (T)
1ª (A)
2ª (A)
3ª (A)
1365,9
7205,4
8752,7
1345,3
7014,5
8568,6
1960,1
7018,2
10872,0
1387,2
7461,2
8960,7
1365,8
7251,9
8742,1
1990,8
7149,4
11214,0
2296,3
14719,2
18806,1
2155,8
12347,3
15418,0
3194,0
7149,4
15169,0
Table 6.5: Natural frequencies of Model C obtained through different theories.
2.E+04
Freqüência natural (Hz)
1.E+04 1.E+04 1.E+04
8.E+03 6.E+03 4.E+03 2.E+03 0.E+00 Model C- EDH (CA)
Model C- EDH (B)
Model C - Rigid
1ª natural frequency (E/B)
2ª natural frequency (E/B)
1ª natural frequency (T)
2ª natural frequency (T)
1ª natural frequency (A)
2ª natural frequency (A)
Figure 6.11: Comparison between the results of the Table 6.5.
82
6. Dynamic analyses FRFs - Modelo(a) C - Euler/Bernoulli -80
CA CA B E Rígido Rigid
-100
-120
Amplitude [db m/N]
Amplitude [db m/N]
-140
-160 -180
-200
-220
-240
-260
0
1
2
3
4
5
Frequência[Hz] [Hz] Frequency
6 4
x 10
(b) FRFs - Modelo C - Timoshenko -80
CA CA BE Rígido Rigid
-100 -120
Amplitude [db m/N]
Amplitude [db m/N]
-140 -160 -180 -200 -220 -240 -260 -280
0
1
2
3 Frequência [Hz]
4
5
Frequency [Hz]
6 4
x 10
Figure 6.12: Frequency responses obtained through FEM of model C; Theory of Euler-Bernoulli(A); Theory of Timoshenko (B).
Chiarelli, L.R.
83
6.3. Analyzes of the dynamic behavior of the aerostatic ceramic air bearing
This study was performed using the finite difference method. For this analysis was used the program developed by Nicoletti et.al. (2008). The analysis consists of generating a mesh containing 41 nodes in the axial direction and 50 nodes in the tangential direction on the inner surface of the bearing so that the pressure distribution in the porous radial airspace bearing can be obtained numerically.
-3
x 10 6 4
Z(m)
2 0 -2 -4 -6 0.02 0.015
0.01 0.005
0.01 0
0.005 X(m)
-0.005 0
-0.01
Y(m)
Figure 6.13: Mesh used in the finite difference method.
6.3.1. Dynamic coefficients and natural frequency of the aerostatic bearing
The coefficients of stiffness (K) and damping (D) of the bearing vary according to a number of factors, such as bearing geometry, bearing and shaft clearance, feed pressure, rotational speed, excitation frequency, applied forces and permeability of the porous matrix. In this work, the dynamic coefficients were obtained at a frequency of excitation of 40Hz (2400rpm), rotation of 15000rpm (250Hz), permeability of the
84
6. Dynamic analyses
porous matrix of 10-12m2 and viscosity of the fluid (air) 1,8 * 10-5 Ns/m2. The reference plane adopted refers to the cross-section of the bearing, with the y and z axes corresponding to the horizontal and vertical directions, respectively.
Porous matrix
Figure 6.14: Aerostatic ceramic porous bearing scheme (Silveira et al., 2010).
Physical properties and geometry
Calculation of equilibrium position
Calculation of the air flow
Calculation of dynamic coefficients
Figure 6.15: Basic algorithm used to obtain the dynamic coefficients of the aerostatic ceramic porous bearing.
The results obtained for stiffness and damping are shown as a function of the relationship between the supply pressure (Ps) and the ambient pressure (Pa) in Tables 6.6 and 6.7:
Chiarelli, L.R.
85
Coefficient
Aerostatic bearing stiffness (N/m)
Ps/Pa = 5 Ps/Pa = 10 Ps/Pa = 15 Ps/Pa = 20 Kyy 6,44E+05 1,29E+06 1,93E+06 2,58E+06 Kzz 6,11E+05 1,22E+06 1,83E+06 2,44E+06 Kyz 1,06E+03 1,06E+03 1,06E+03 1,06E+03 Kzy -5,48E+02 -1,38E+02 2,72E+02 6,82E+02 Table 6.6: Stiffness coefficients obtained at different supply pressures.
Coefficient
Aerostatic bearing damping (Ns/m) Ps/Pa = 5
Ps/Pa = 10
Ps/Pa = 15
Ps/Pa = 20
Dyy
1,2939E+04
1,2939E+04
1,2939E+04
1,2939E+04
Dzz
1,2918E+04
1,2918E+04
1,2918E+04
1,2918E+04
Dyz
3,27E-02
1,64E-02
1,09E-02
8,24E-03
Dzy
3,27E-02
1,64E-02
1,09E-02
8,26E-03
Table 6.7: Damping coefficients obtained at different supply pressures.
It can be observed that the terms of coupled directions (subscripts yz and zy) have relatively small values when compared to terms of normal directions (subscripts yy and zz), this occurs in both the stiffness coefficients and the damping coefficients. The crossover terms are so small that they can be overlooked.
The results show that there is great influence of the supply pressure variation on the stiffness coefficients and the damping coefficients in the coupled directions, but the same does not occur with the damping coefficients in the normal directions.
Figure 6.16 allows a better comparison between the results.
86
6. Dynamic analyses 1.E+03
3.0E+06
1.E+03 8.E+02 Stiffness (N/m)
Stiffness (N/m)
2.5E+06 2.0E+06 1.5E+06 1.0E+06
6.E+02 4.E+02 2.E+02 0.E+00 -2.E+02 -4.E+02
5.0E+05
-6.E+02 -8.E+02
Ps/Pa = 5
Ps/Pa = 10
Ps/Pa = 15
Ps/Pa = 20
Kyz 1.1E+03
1.1E+03
1.1E+03
1.1E+03
Kzy -5.5E+02 -1.4E+02 2.7E+02
6.8E+02
0.0E+00 Kyy
Ps/Pa Ps/Pa Ps/Pa Ps/Pa = 5 = 10 = 15 = 20
Kzz
(a)
(b)
1.2945E+04
3.5E-02 3.0E-02
1.2935E+04 Damping (Ns/m)
Damping (Ns/m)
1.2940E+04
1.2930E+04 1.2925E+04 1.2920E+04 1.2915E+04
2.5E-02 2.0E-02 Dyz
1.5E-02
Dzy
1.0E-02
1.2910E+04 5.0E-03
1.2905E+04
Dyy
Ps/Pa Ps/Pa Ps/Pa Ps/Pa = 5 = 10 = 15 = 20
Dzz
(c)
0.0E+00 Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa = 5 10 15 20
(d)
Figure 6.16: Normal stiffness coefficients (a); Crossed stiffness coefficients (b); Normal damping coefficients (c); Crossed damping coefficients (d). As already mentioned, the dynamic coefficients also vary according to the speed of rotation of the shaft. The study of this behavior is made from figures 6.17, 6.18, 6.19, 6.20.
Chiarelli, L.R.
87 x 10-2
Rigidez adimensional
0.035 3.501421 0.035 3.501416 0.035 3.501412
Kxx*
Kyy*
0.035 3.501407
3.501403 0.035 3.501398 0.035 3.501394 0.035 3.501389 0.035
Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa =
0.5 1.0 1.5 2.0
3
4
3.501385
0.035
1
2
5
6
7
8
Velocidade(Rpm) Velocity [Rpm]
x 10-2
9 4
x 10
(B)(a) adimensional Rigidez
0.0332 3.32273 0.0332 3.32272 0.0332 3.32270
0.5 1.0 1.5 2.0
3
4 5 6 Velocidade(Rpm) Velocity [Rpm]
3.32268 0.0332
Kyy*
Kzz*
0.0332 3.32269
Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa =
3.32267 0.0332
3.32266 0.0332 3.32264 0.0332 3.32263 0.0332 3.32262 0.0332
1
2
7
8
9 4
x 10
(b)
Figure 6.17: Variation of the stiffness as a function of the shaft speed; Kyy*(a); Kzz*(b).
88
6. Dynamic analyses -3
2.5
Rigidez adimensional
x 10
2 0.5 1.0 1.5 2.0
3
4 5 6 Velocidade(Rpm) Velocity [Rpm]
Kyz*
Kxy*
1.5
Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa =
1
0.5
0
1
2
8
9 4
x 10
(a)(B) Rigidez adimensional
-3
0
7
x 10
-0.2 -0.4 -0.6
Kyx* Kzy*
-0.8 -1 -1.2 -1.4 -1.6
Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa =
0.5 1.0 1.5 2.0
3
4 5 6 Velocidade(Rpm) Velocity [Rpm]
-1.8 -2
1
2
7
8
9 4
x 10
(b)
Figure 6.18: Variation of the stiffness as a function of the shaft speed; Kyz*(a); Kzy*(b).
Chiarelli, L.R.
89 4 4
2.0217 1.29389371
x x1010
Amortecimento
2.0217 1.29389368
0.5 1.0 1.5 2.0
3
4
1.29389360 2.0217 Dxx
Dyy (Ns/m)
2.0217 1.29389364
Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa =
1.29389356 2.0217 1.29389353 2.0217 1.29389349 2.0217 1.29389345 1
2
5
6
7
8
Velocidade(Rpm) Velocity [Rpm] x x1010
2.0185 1.2918081 2.0185 1.2918080 2.0185 1.2918078
Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa =
0.5 1.0 1.5 2.0
3
4 5 6 Velocidade(Rpm)
1.2918077 2.0185 Dyy
Dzz (Ns/m)
x 10
(B) (a) Amortecimento
4 4
2.0185 1.2918083
9 4
1.2918075 2.0185 1.2918074 2.0185 1.2918072 2.0185 1.2918071 2.0185 1.2918069 2.0185 1.2918068 2.0184
1
2
Velocity [Rpm]
7
8
9 4
x 10
(b)
Figure 6.19: Variation of the damping as a function of the shaft speed; Dyy(a); Dzz(b).
90
6. Dynamic analyses Amortecimento 2 1.8 1.6
Dyz Dxy(Ns/m)
1.4
Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa =
0.5 1.0 1.5 2.0
3
4 5 6 Velocidade(Rpm) Velocity [Rpm]
1.2 1 0.8 0.6 0.4 0.2 0
1
2
7
8
9 4
x 10
(B)
(a) Amortecimento 2 1.8
Dzy (Ns/m) Dyx
1.6 1.4
Ps/Pa = Ps/Pa = Ps/Pa = Ps/Pa =
0.5 1.0 1.5 2.0
3
4 5 6 Velocidade(Rpm)
1.2 1 0.8 0.6 0.4 0.2 0
1
2
Velocity [Rpm]
7
8
9 4
x 10
(b)
Figure 6.20: Variation of the damping as a function of the shaft speed; Dyz(a); Dzy(b).
Chiarelli, L.R.
91
Small variations in the coefficients can be observed, showing that the speed of rotation does not influence as much as the supply pressure. Knowing the stiffnesses, it is possible to estimate the natural frequency of the bearing as shown in Chiarelli et al. (2015), knowing that the mass of the bearing with the housing on the support is 0.103 kg, the value of the highest value stiffness (Kyy) in Eq. 3.7 is substituted. The first natural frequency of the bearing as a function of the supply pressure is shown in Table 6.8: Natural frequency (Hz)
Ps/Pa = 5
Ps/Pa = 10
Ps/Pa = 15
Ps/Pa = 20
388
548
671
775
Table 6.8: Natural frequencies of the bearing as a function of supply pressure.
6.4. Dynamic analyzes of the rotor assembly with the aerostatic ceramic porous bearing
These dynamic analyzes consider the influence of the porous ceramic aerostatic bearing on the rotating assembly. For this study, the models described in figure 4.8 were used, and the characteristics of the porous ceramic aerostatic bearing were represented at the same force application node. The method used to perform the analyzes was FEM with Thymoshenko beam theory. The frequency of rotation used was the same as the analyzes performed to obtain the stiffness and damping coefficients shown in Tables 6.6 and 6.7 respectively (15000rpm or 250Hz).
MATLAB - FEM –Timoshenko theory
Model
st
1 natural frequency (Hz) Ps/Pa = 5
Ps/Pa = 10
Ps/Pa = 15
Ps/Pa = 20
Model A - EDH (CA)
10469,33
13382,04
15120,98
16322,04
Model A - EDH (B)
11001,41
13946,72
15673,45
16854,12
Model A – Rigid
24370,58
24389,48
24394,43
24395,41
Model B - EDH (CA)
4703,23
5254,74
5531,94
5719,28
Model B - EDH (B)
4817,34
5346,91
5617,97
5803,88
Model B – Rigid
7671,97
7680,30
7683,10
7684,51
Model C- EDH (CA)
1490,44
1281,11
1134,68
1031,49
Model C- EDH (B)
1513,86
1296,53
1147,94
1043,95
Model C - Rigid
2156,63
1729,61
1484,67
1320,94
Table 6.9: First natural frequency of the system with all bearings.
92
6. Dynamic analyses
The porous aerostatic bearing has great damping, which reduces the amplitude of the vibration. Figures 6.21, 6.22 and 6.23 present the FRFs of the rotor assembly with aerostatic bearing, comparing the amplitudes of damped and undamped vibrations. FRFs com aerostático - Modelo A - Euler/Bernoulli - CA
FRFs com aerostático - Modelo A - Euler/Bernoulli - CA
-140
-100 -120
Amplitude [db m/N]
-180
Amplitude [db m/N]
Amplitude [db m/N]
Amplitude [db m/N]
-160
-200
-220
-140 -160 -180 -200 -220
-240
-240 -260
0
1
2
3 Frequência [Hz]
4
5
-260
6
0
Frequency [Hz]
3
4
5
6 4
x 10
(a) FRFs com aerostático - Modelo A - Euler/Bernoulli - E
-140
-100 -120
[db Amplitude m/N]m/N] Amplitude [db
-160
[db Amplitude Amplitude [db m/N]m/N]
2
Frequência [Hz] Frequency [Hz]
FRFs com aerostático - Modelo A - Euler/Bernoulli - E
-180
-200
-220
-240
-260
-280
1
4
x 10
-140 -160 -180 -200 -220 -240 -260
0
1
2
3 Frequência [Hz]
4
5
-280
6
0
1
2
4
Frequency [Hz]
3 Frequência [Hz]
4
5
6 4
Frequency [Hz]
x 10
x 10
(b)
FRFs com aerostático - Modelo A - Euler/Bernoulli - Rígido
FRFs com aerostático - Modelo A - Euler/Bernoulli - Rígido
-160
-100
-170
-120
-180
-140
Amplitude [db m/N]
Amplitude [db m/N]
-190 -200 -210 -220
-160 -180 -200
-230
-220 -240
-240
-250 -260
0
1
2
3 Frequência [Hz]
4
5
-260
6 4
x 10
0
1
2
3 Frequência [Hz]
4
5
6 4
x 10
(c) Figure 6.21: Comparison between damped amplitudes (left) and undamped amplitudes (right) of Model A; angular contact rolling element bearings (a); ball bearing (b); rigid (c);Ps/Pa = 5 (-), Ps/Pa = 10(--), Ps/Pa = 15 (- -) e Ps/Pa = 20 (--).
Chiarelli, L.R.
93
FRFs com aerostático - Modelo B - Euler/Bernoulli - CA
FRFs com aerostático - Modelo B - Euler/Bernoulli - CA
-140
-100 -120
Amplitude [db m/N]
-180
Amplitude [db m/N]
Amplitude [db m/N]
Amplitude [db m/N]
-160
-200
-220
-240
-140 -160 -180 -200 -220 -240
-260
-280
-260
0
1
2
3 Frequência [Hz]
4
5
-280
6
0
1
Frequency [Hz]
2
3
4
5
Frequência [Hz] Frequency [Hz]
4
x 10
6 4
x 10
(a) FRFs com aerostático - Modelo B - Euler/Bernoulli - E
FRFs com aerostático - Modelo B - Euler/Bernoulli - E
-150
-100
-170
Amplitude [db m/N]
-180
Amplitude [db m/N]
Amplitude [db m/N]
Amplitude [db m/N]
-160
-190 -200 -210 -220
-150
-200
-230 -240 -250
0
1
2
3 Frequência [Hz]
4
5
-250
6
0
1
2
4
x 10
Frequency [Hz]
3 Frequência [Hz]
4
5
(b)
6 4
x 10
Frequency [Hz] FRFs com aerostático - Modelo B - Euler/Bernoulli - Rígido
FRFs com aerostático - Modelo B - Euler/Bernoulli - Rígido -120
-160
-140
[db m/N] Amplitude Amplitude [db m/N]
[db m/N] Amplitude Amplitude [db m/N]
-180
-200
-220
-240
-160 -180 -200 -220 -240
-260 -260
-280
0
1
2
3 Frequência [Hz]
4
Frequency [Hz]
5
-280
6 4
x 10
(c)
0
1
2
3 Frequência [Hz]
4
Frequency [Hz]
5
6 4
x 10
Figure 6.22: Comparison between damped amplitudes (left) and undamped amplitudes (right) of Model B; angular contact rolling element bearings (a); ball bearing (b); rigid (c); Ps/Pa = 5 (-), Ps/Pa = 10(--), Ps/Pa = 15 (- -) e Ps/Pa = 20 (--).
94
6. Dynamic analyses FRFs com aerostático - Modelo C - Euler/Bernoulli - CA
FRFs com aerostático - Modelo C - Euler/Bernoulli - CA
-150
-100
-170
Amplitude [db m/N]
-180
Amplitude [db m/N]
[db m/N] Amplitude Amplitude [db m/N]
-160
-190 -200 -210 -220
-150
-200
-230 -240 -250
0
1
2
3 Frequência [Hz]
4
5
-250
6
0
1
Frequency [Hz]
2
3
4
5
Frequência [Hz] Frequency [Hz]
4
x 10
6 4
x 10
(a) FRFs com aerostático - Modelo C - Euler/Bernoulli - E
FRFs com aerostático - Modelo C - Euler/Bernoulli - E
-140
-100 -120
[db Amplitude m/N]m/N] Amplitude [db
[db Amplitude Amplitude [db m/N]m/N]
-160
-180
-200
-220
-140 -160 -180 -200 -220
-240
-240 -260
0
1
2
3 Frequência [Hz]
4
5
-260
6
0
1
Frequency [Hz]
2
3
4
5
Frequência [Hz] Frequency [Hz]
4
x 10
6 4
x 10
(b) FRFs com aerostático - Modelo C - Euler/Bernoulli - Rígido
[Digite uma citação do
-160
FRFs com aerostático - Modelo C - Euler/Bernoulli - Rígido [Digite uma citação do
-120 -140
m/N] m/N] Amplitude [db[db Amplitude
Amplitude [db[db m/N] m/N] Amplitude
-180
-200
-220
-240
-160 -180 -200 -220 -240
-260
-260 -280
0
1
2
3 Frequência [Hz]
4
Frequency [Hz]
5
-280
6 4
x 10
(c)
0
1
2
3 Frequência [Hz]
4
Frequency [Hz]
5
6 4
x 10
Figure 6.23: Comparison between damped amplitudes (left) and undamped amplitudes (right) of Model C; angular contact rolling element bearings (a); ball bearing (b); rigid (c); Ps/Pa = 5 (-), Ps/Pa = 10(--), Ps/Pa = 15 (- -) e Ps/Pa = 20 (--).
Chiarelli, L.R.
95
As noted in chapter 6.3.1 the dynamic coefficients of the aerostatic bearing also vary according to the speed of rotation, so when it is placed in the rotating assembly the natural frequencies also vary according to the rotation. Figures 6.24, 6.25 and 6.26 express this variation using FEM with Thimoshenko theory and 2MPa feed pressure.
4
x 10
frequency Natural Freqüência natural [Hz] [Hz]
2.5
Diagrama de Campbell - Modelo A - Timoshenko
2
CA B
1.5
1
1
2
3
4 5 6 Freqüência [rpm] Frequency [Rpm]
7
8
9 4
x 10
Figure 6.24: Campbell diagram of Model A. 4
1.6
x 10
Diagrama de Campbell - Modelo B - Timoshenko
frequency Natural Freqüência natural[Hz] [Hz]
1.4
1.2
CA E
1
0.8
0.6
0.4
1
2
3
4 5 6 Freqüência [rpm]
Frequency [Rpm]
7
8
9 4
x 10
Figure 6.25: Campbell diagram of Model B.
96
6. Dynamic analyses Diagrama de Campbell - Modelo C - Timoshenko
frequency [Hz] Natural Freqüência natural [Hz]
15000
CA E 10000
5000
0 1
2
3
4 5 6 Freqüência [rpm] Frequency [Rpm]
7
8
9 4
x 10
Figure 6.26: Campbell diagram of Model C.
6.5. Shaft supported by aerostatic ceramic porous bearings
So far we have studied the configurations of the supported axle on rolling element bearings. This chapter studies the behavior of the same models, but now supported by aerostatic bearings. Although the rolling element bearings have greater rigidity, the length of the porous ceramic aerosol bearings is 17 mm, while the rolling element bearings are only 9 mm long, thus the contact region of the ceramic porous ceramic bearings is relatively larger when compared to rolling element bearings. Matlab - FEM - Timoshenko beam theory - 1st Natural frequency (Hz) - undamped Ps/Pa = 5 Ps/Pa = 10 Ps/Pa = 15 Ps/Pa = 20 1104,20 1909,30 2688,50 3448,90 Model A 898,91 1495,10 1988,20 2376,90 Model B 385,33 489,87 529,12 535,73 Model C Table 6.10: Undamped natural frequencies of the shaft supported by aerostatic Model
ceramic porous bearings.
Chiarelli, L.R.
97 Model AA A - Timoshenko Modelo FRFs sobre aerostático - Modelo -100
[db N/m] Amplitude Amplitude [db m/N]
-120
-140
-160
-180
-200
-220
0
1
2
3
4
5
Frequência [Hz] Frequency [Hz]
6 4
x 10
Modelo FRFs sobre aerostático - Modelo Model BB B - Timoshenko -100
[db N/m] Amplitude Amplitude [db m/N]
-120
-140
-160
-180
-200
-220
0
1
2
3 Frequência [Hz]
4
5
6 4
x 10
Frequency [Hz]
Model CC C - Timoshenko Modelo FRFs sobre aerostático - Modelo -100
N/m] [db [db Amplitude Amplitude m/N]
-120
-140
-160
-180
-200
-220
0
1
2
3 Frequência [Hz] Frequency [Hz]
4
5
6 4
x 10
Figure 6.27: FRFs undamped of the shaft supported by aerostatic ceramic porous bearings; Ps/Pa = 5 (-), Ps/Pa = 10(--), Ps/Pa = 15 (- -) e Ps/Pa = 20 (--).
98
6. Dynamic analyses
The undamped natural frequencies of the shaft are much smaller when compared to the shaft supported by rolling element bearings, this is due to the fact that the difference between the stifness of the aerostatic ceramic porous bearings is about 20 times smaller. However, when the damping is taken into account there occurs a great variation in the natural frequencies. FRFs sobre aerostático - Modelo A - Timoshenko -150
-155
Amplitude [db m/N]
Amplitude [db m/N]
-160
-165
-170
-175
-180
-185
0
1
2
3
4
5
Frequência[Hz] [Hz] Frequency
6 4
x 10
Figure 6.28: FRFs damped of the shaft supported by aerostatic ceramic porous bearings – Model A. FRFs sobre aerostático - Modelo B - Timoshenko -150
-155
[db[dbm/N] Amplitude Amplitude m/N]
-160
-165
-170
-175
-180
-185
0
1
2
3 Frequência [Hz]
4
Frequency [Hz]
5
6 4
x 10
Figure 6.29: FRFs damped of the shaft supported by aerostatic ceramic porous bearings – Model B.
Chiarelli, L.R.
99
FRFs sobre aerostático - Modelo C - Timoshenko -150
-155
Amplitude [db m/N]
Amplitude [db m/N]
-160
-165
-170
-175
-180
-185
0
1
2
3 Frequência [Hz] Frequency [Hz]
4
5
6 4
x 10
Figure 6.30: FRFs damped of the shaft supported by aerostatic ceramic porous bearings – Model C.
The variation in the pressures has no relevance in the natural frequencies. Small amplitude peaks of 0 and 10000 Hz can be observed in the Model B and Model C, but the three models present the natural frequency of greater amplitude around 26000 Hz.
Chapter 7 shows the conclusions that can be drawn from the static and dynamic analyzes made at work.
100
7. CONCLUSIONS
In this work, analytical studies were carried out, as well as numerical analyzes of the shaft-bearing-turbine assembly, to obtain the deflection of the shaft and natural frequencies of the rotating set and aerostatic ceramic porous bearing. Este estudo é fundamental para a construção do banco experimental destinado ao estudo do comportamento estático e dinâmico do mancal aerostático cerâmico poroso. The maximum deformation (49.74μm) and maximum stiffness (44.44N/μm) of the rotor assembly allow the maximum stiffness of the ceramic porous ceramic bearing (2.56N/μm) to be evaluated by the experimental set-up. This study is relevant for the design of ultra-precision machines, once aerostatic bearings offer significant operational advantages: low friction generation, work at high rotations, among others.
The results of the dynamic analysis of the rotor assembly and the aerostatic ceramic porous bearing show different natural frequencies when comparing the different models. The lowest value of the first natural frequency obtained with the rotor assembly models (1365.9 Hz) is approximately twice as higher as the highest value of the first natural frequency of the bearing (775.00 Hz). There seems to be a significant influence of the pressure variation on the stiffness coefficients of the aerostatic bearing that the higher the pressure, the greater the stiffness (K). The damping coefficients (D) show no relevant changes when the feed pressure is changed.
The proposed physical models, based on the conceptual design proposed by Friedel (2011), presented static and dynamic characteristics, which allow the study of the dynamic behavior of aerostatic ceramic porous bearing. Recalling the importance of ensuring that the stiffness of the shaft is at least five times greater than the stiffness of the aerostatic bearing, in order to avoid overlapping of the dynamic effects.
101
APPENDIX A: MATLAB® programs
These programs shows how to implement the theory of the presented work and are not optimized.
A.1 Frequency response functions %********************************************* %* SOFTWARE TO SIMULATE THE NATURAL FREQUENCIES %* OF THE MODELS WITHOUT AEROSTATIC BEARING %* USING EULER-BERNOULLI THEORY %* %* Luis Renato Chiarelli %* 07/12/2017 %********************************************* clear all close all clc % Material properties rho = 7850; % density [kg/m3] E = 210*(10^9) ; % Young modulus [Pa] v = 0.3 ; % Poison coefficient G = E/(2*(1+v)) ; % Tensversal elasticity modulus [Pa] k = 9/10 ; % Form factor - (9/10 for circles and 6/5 for retangles) % DISC MATRICES vT = 1840.504e-9; % Turbine volume [m3] mD = vT*rho; % Disc mass [kg] dD = 26.518e-3; % Disc diameter [m3] lD = 8e-3; % Disc length [m] ID = mD*((dD/2)^2)/4; % Diametral inertia IP = 2*ID; % Polar inertia % Translational disc mass matrix MD = zeros(4); MD(1,1) = mD; MD(2,2) = mD; % Rotational disc mass matrix ND = zeros(4); ND(3,3)=ID; ND(4,4)=ID; % Disc mass matrix MTD = MD+ND; % Gyroscopic disc matrix GD = zeros(4); GD(4,3)=IP; GD(3,4)=IP;
102
Appendix A
% Loop over the models (A, B, C) for Model = [1, 2, 3] % Loop over the stiffness of the rolling bearings %(57e6 - Yi Guo ball bearing / 50e6 - Hagiu angular contact/ 10e10 rigid) for Kyy = [50e6, 57e6, 10e10] % ROLLING BEARING MATRICES % Stiffness matrix - rolling bearing Km = zeros(4); Kzz = Kyy ; % Stiffness at z direction Kbb = 0; % Stiffness around y axis Kgg = 0; % Stiffness around z axis Km(1,1) = Kyy; Km(2,2) = Kzz; Km(3,3) = Kbb; Km(4,4) = Kgg; % Damping matrix Dm = zeros(4); Dyy = 0; % Damping Dzz = 0; % Damping Dbb = 0; % Damping Dgg = 0; % Damping
rolling bearing in y direction in z direction around y axis around z axis
Dm(1,1) = Dyy; Dm(2,2) = Dzz; Dm(3,3) = Dbb; Dm(4,4) = Dgg; % ELEMENT DATA FOR EACH MODEL % Model A if Model ==1 % Lengths L1 = 0.0045; L2 = 0.0045; L3 = 0.01325; L4 = 0.011; L5 = 0.011; L6 = 0.01325; L7 = 0.0045; L8 = 0.0045; % Radius r1 = 0.0085; r2 = 0.0085; r3 = 0.005; r4 = 0.0055; r5 = 0.0055; r6 = 0.005; r7 = 0.0085; r8 = 0.0085; % Lengths and radius arrays L = [L1, L2, L3, L4, L5, L6, L7, L8]; r = [r1, r2, r3, r4, r5, L6, L7, L8]; end
Chiarelli, L.R.
103 % Model B if Model == 2 % Lengths L1 = 0.0045; L2 = 0.0045; L3 = 0.01325; L4 = 0.011; L5 = 0.011; L6 = 0.01325; L7 = 0.0045; L8 = 0.0045; L9 = 0.029; % Radius r1 = 0.0085; r2 = 0.0085; r3 = 0.005; r4 = 0.0055; r5 = 0.0055; r6 = 0.005; r7 = 0.0085; r8 = 0.0085; r9 = 0.005; % Lengths and radius arrays L = [L1, L2, L3, L4, L5, L6, L7, L8, L9]; r = [r1, r2, r3, r4, r5, r6, r7, r8, r9]; end % Model C if Model == 3 % Lengths L1 = 0.0045; L2 = 0.0045; L3 = 0.01325; L4 = 0.011; L5 = 0.011; L6 = 0.01325; L7 = 0.0045; L8 = 0.0045; L9 = 0.021; L10 = 0.004; L11 = 0.004; % Radius r1 = 0.0085; r2 = 0.0085; r3 = 0.005; r4 = 0.0055; r5 = 0.0055; r6 = 0.005; r7 = 0.0085; r8 = 0.0085; r9 = 0.005; r10 = 0.013; r11 = 0.013;
104
Appendix A % Lengths and radius arrays L = [L1, L2, L3, L4, L5, L6, L7, L8, L9, L10, L11]; r = [r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11]; end % Number of elements ne = length(L); % ELEMENT MATRICES % Loop over the elements for a=(1:1:ne)
Ma(6,6) = Ma(8,8) = Ma(5,4) = Ma(7,3) = Ma(8,5) =
% Translational mass matrix (M0 = Ma) Ma = zeros(8); Ma(1,1) = 156; Ma(2,2) = 156; 156; Ma(3,3) = 4*L(a)^2; Ma(4,4) = 4*L(a)^2; 4*L(a)^2; Ma(3,2) = -22*L(a); Ma(4,1) = 22*L(a); 13*L(a); Ma(6,2) = 54; Ma(6,3) = -13*L(a); -3*L(a)^2; Ma(7,6) = 22*L(a); Ma(8,1) = -13*L(a); -22*L(a); % Symmetry for i=1:7 for j = i+1:8 Ma(i,j) = Ma(j,i); end end
% Rotational mass matrix (N0 = Na = zeros(8); Na(1,1) = 36; Na(2,2) Na(6,6) = 36; Na(3,3) = 4*L(a)^2; Na(4,4) 4*L(a)^2; Na(8,8) = 4*L(a)^2; Na(3,2) = -3*L(a); Na(4,1) Na(5,4) = -3*L(a); Na(6,2) = -36; Na(6,3) 3*L(a); Na(7,3) = -L(a)^2; Na(7,6) = 3*L(a); Na(8,1) L(a)^2; Na(8,5) = -3*L(a); % Symmetry for i=1:7 for j = i+1:8 Na(i,j) = Na(j,i); end end
Ma(5,5) = 156; Ma(7,7) = 4*L(a)^2; Ma(5,1) = 54; Ma(7,2) = 13*L(a); Ma(8,4) = -3*L(a)^2;
Na) = 36;
Na(5,5) = 36;
= 4*L(a)^2;
Na(7,7) =
= 3*L(a);
Na(5,1) = -36;
= 3*L(a);
Na(7,2) = -
= 3*L(a);
Na(8,4) = -
% Gyroscopic matrix (G0 = Ga) Ga = zeros(8); Ga(2,1) = 36; Ga(3,1) = -3*L(a); Ga(4,3) = 4*L(a)^2; Ga(5,2) = 36; Ga(5,3) = -3*L(a); Ga(6,4) = -3*L(a);
Ga(4,2) = -3*L(a); Ga(6,1) = -36;
Chiarelli, L.R.
105
Ga(6,5) = 36; Ga(7,1) = -3*L(a); Ga(7,5) = 3*L(a); Ga(8,2) = -3*L(a); Ga(8,3) = -L(a)^2; Ga(8,7) = 4*L(a)^2; % Skew simmetry for i=1:7 for j = i+1:8 Ga(i,j) = -Ga(j,i); end end
Ka(6,6) = Ka(8,8) = Ka(5,4) = Ka(7,6) =
% Stiffness matrix (K0 = Ka) Ka = zeros(8); Ka(1,1) = 12; Ka(2,2) 12; Ka(3,3) = 4*L(a)^2; Ka(4,4) 4*L(a)^2; Ka(3,2) = -6*L(a); Ka(4,1) -6*L(a); Ka(6,2) = -12; Ka(6,3) = 6*L(a); Ka(7,2) 6*L(a); Ka(8,1) = 6*L(a); Ka(8,4) % Symmetry for i=1:7 for j = i+1:8 Ka(i,j) = Ka(j,i); end end
Ga(7,4) = L(a)^2; Ga(8,6) = 3*L(a);
= 12;
Ka(5,5) = 12;
= 4*L(a)^2;
Ka(7,7) = 4*L(a)^2;
= 6*L(a);
Ka(5,1) = -12;
= -6*L(a);
Ka(7,3) = 2*L(a)^2;
= 2*L(a)^2;
Ka(8,5) = -6*L(a);
% MULTIPLIER FACTORS % Moment of inertia I = (pi*(r(a)^4))/4; % Mass per unity of length m = pi*(r(a)^2)*rho ; % Translational mass matrix multiplier Cm = (m*L(a)) / 420; % Rotational mass matrix multiplier Cn = (m*(r(a)^2)) / (120*L(a)); % Gyroscopic matrix multiplier Cg = (m*(r(a)^2)) / (60 * L(a)); % Stiffness matrix multiplier Ck = (E*I) / (L(a)^3);
% FINAL ELEMENT MATRICES M(:,:,a) = Cm * Ma; % Translational element mass N(:,:,a) = Cn * Na;
% Rotational element mass
Gf(:,:,a) = Cg * Ga; % Gyroscopic element mass Kf(:,:,a) = Ck * Ka; % Stiffness element mass
106
Appendix A Mf = M+N; end % end elements
% ASSEMBLE GLOBAL MATRICES MG = zeros(4+(4*ne)); % Global mass matrix KG = zeros(4+(4*ne)); % Global Stiffness matrix GG = zeros(4+(4*ne)); % Gelobal gyroscopic matrix % Auxiliar array o = 0:4:(4*ne)-4; % Loop over elements for a=(1:1:ne) KG(1+o(a):8+o(a),1+o(a):8+o(a)) = KG(1+o(a):8+o(a),1+o(a):8+o(a))+Kf(:,:,a); MG(1+o(a):8+o(a),1+o(a):8+o(a)) = MG(1+o(a):8+o(a),1+o(a):8+o(a))+Mf(:,:,a); GG(1+o(a):8+o(a),1+o(a):8+o(a)) = GG(1+o(a):8+o(a),1+o(a):8+o(a))+Gf(:,:,a); end
% INSERT BEARINGS % Inserting stiffness matrices KG(1:4,1:4)=KG(1:4,1:4)+Km; KG(5:8,5:8)=KG(5:8,5:8)+Km; KG(9:12,9:12)=KG(9:12,9:12)+Km; KG(25:28,25:28)=KG(25:28,25:28)+Km; KG(29:32,29:32)=KG(29:32,29:32)+Km; KG(33:36,33:36)=KG(33:36,33:36)+Km; % Inserting damping matrices DG = zeros(4+(4*ne)); DG(1:4,1:4)=DG(1:4,1:4)+Dm; DG(5:8,5:8)=DG(5:8,5:8)+Dm; DG(9:12,9:12)=DG(9:12,9:12)+Dm; DG(25:28,25:28)=DG(25:28,25:28)+Dm; DG(29:32,29:32)=DG(29:32,29:32)+Dm; DG(33:36,33:36)=DG(33:36,33:36)+Dm; % Insert disc in Model C if Model == 3 % Insert mass MG(37:40,37:40)=MG(37:40,37:40)+MTD; MG(41:44,41:44)=MG(41:44,41:44)+MTD; MG(45:48,45:48)=MG(45:48,45:48)+MTD; % Insert gyroscopic GG(37:40,37:40)=GG(37:40,37:40)+GD;
Chiarelli, L.R.
107 GG(41:44,41:44)=GG(41:44,41:44)+GD; GG(45:48,45:48)=GG(45:48,45:48)+GD;
end % STATE MODEL % Angular speed (rad/s) Om = 50; % Identity and zeros I = eye(length(KG)); O = zeros(length(KG)); Ss = [-I,O;O,MG]; Rs = [O,I;KG,(Om*GG)+DG]; % EIGENVALUES (Wn) / EIGENVECTORS (Vn) A = (inv(Ss))*Rs; [Vn,Wn]=eig(A); Wn = Wn/(2*pi); WN = sort(diag(imag(Wn))); len = length(WN)/2; % Display first natural frequencies % disp (WN(len+1:len+8)) %====================== >>>> F-R-F % Excitation on node 2 / horizontal % Excitation on node 2 / vertical
>>>>>>>>>>>>>>>>>>>>>> END FRF
