Rod Resonant Oscillations Considering Material Relaxation Properties

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ScienceDirect Procedia Engineering 176 (2017) 226 – 236

Dynamics and Vibroacoustics of Machines (DVM2016)

Rod resonant oscillations considering material relaxation properties V.A. Kudinova*, A.V. Eremina, I.V. Kudinova, A.I. Dovgallob a

Samara State Technical University, Molodogvardeyskaya, 244, Samara, 443100, Russia b Samara State Aerospace University, Moskovskoye sh., 34, Samara, 443086, Russia

Abstract The mathematical model of oscillations of an elastic rod under external harmonic load considering the material relaxation properties and the resistance of the rod towards the process of its deformation has been developed. The model differential equation is derived considering the dependence on resistance time and deformations according to the formula for Hooke’s law brought to Maxwell’s and Kelvin-Voight’s elaborated models. The results of numerical analysis of the model show that the coincidence of the rod oscillation frequency with the external load oscillation frequency results in resonance followed by the unlimited growth of the amplitude oscillation (providing that there is no environment resistance). If the environment resistance and material relaxation behavior are considered, and the rod oscillation frequency coincides with the oscillation frequency of the internal load (resonant oscillations), the types of the oscillation process may be as follows. The oscillations damp in time (at low values of the relaxation factors and high values of the environment resistance factors). The oscillations become stabilized reaching some permanent state of the amplitude (undamped oscillations). At some high values of the relaxation factors and low values of the resistance factors the resonant frequencies are followed by the bifurcation resonance effects in the undamped oscillation processes, and the non-resonant frequencies are followed by beating in the damped oscillation processes. At some higher values of the resistance factor of the rod material, regardless of the resonant frequencies, the restore of an unbalanced rod takes place practically without the oscillation process – only the internal load oscillations remain constant. © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the Dynamics and Vibroacoustics of Machines (DVM2016). Peer-review under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of Machines Keywords: rod oscillations, hyperbolic equation, numerical solution, stress-strain relaxa-tion, relaxation factor, resistance factor, resonant oscillations.

*Corresponding author. Tel.: +7-486-332-4235. E-mail address: [email protected]

1877-7058 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of Machines

doi:10.1016/j.proeng.2017.02.292

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V.A. Kudinov et al. / Procedia Engineering 176 (2017) 226 – 236

1. Introduction Elastic deformation of a solid body caused by some disturbance propagates at the speed which is dependent on the environment properties. However, the wave process of the environment oscillation is not followed by a substance movement. The equations presenting these processes are of a hyperbolic type. In relation to technical devices there is a common case when the free oscillation process initiated by some initial displacement is followed by a load applied to the loose end of the rod which functions according to a specific law. The matter of a particular interest is determination of an increase in resonance of the oscillation amplitude, when the free oscillation frequency coincides with the oscillation frequency of the load applied to the rod [1, 2]. Derivation of a differential equation of the rod oscillations is based on Hooke’s law σ  EU / x ,

(1)

and Newton’s second law presented as a motion equation σ / x  ρ 2U / t 2 .

(2)

where  is normal stress, N / m 2 ; U is motion, m ; x is a coordinate, m ; t is time, sec ;  is density, kg / m 3 ; E is the elastic modulus (Young’s modulus), Pa ;   U / x is deformation, m . If we substitute the formula (1) into the formula (2), we will find [2, 3]  2U ( x, t )  2U ( x, t )  e2 , 2 t x 2

(3)

where e  E /  is a speed of the longitudinal disturbance propagation, m / sec . The equation (3) is a wave hyperbolic equation presenting undamped oscillations of elastic bodies. The absence of damping is due to the absence of the summand in the equation which considers the internal resistance of the environment affected by the mechanic load causing elastic motions. To consider the environment resistance we shall accept that Fс which stands for the resistance force is proportional to displacement velocity in time Fс  rU / t ,

(4)

where r is a resistance coefficient, kg / sec . The minus in the formula (4) means that the resistance force has the direction which is opposite to the displacement velocity. If we substitute the formula (4) in the equation of Newton’s second law, we will find F  ma  m

dυ d 2U d 2U  m 2  Sx 2 , dt dt dt

(5)

considering that the resistance force Fс relates to the volume forces, we will find ρ

d 2U dσ r dU   , dx V dt dt 2

(6)

where F is a force affecting the body, kg  m / sec 2 ; m is a mass of the body, kg ; a  dυ / dt is acceleration, m / sec2 ; υ  dU / dt is velocity, m / sec ; S is a cross section area of the body, m 2 ; x is the length of a surface element, m ; V is volume, m 3 .

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If we substitute the formula (1) into (6), we will find a wave equation which presents damping oscillations [4] 2 U  2U 2  U , e γ  2 2 t x t

(7)

where γ  r /(ρV ) is a resistance coefficient of 1 / sec dimension. Derivation of the equation (6) was performed by applying Hooke’s law where there are no cause and effect relationships between phenomena. The cause (the effective force) is deformation ε  U / x , and the effect is stress σ . The absence of a time variable in the Hooke’s law formula shows that the cause and the effect is not separated in time in this case, and that is why, when the cause is changed, the effect follows it immediately (discontinuously). But potential propagation velocities of any physical fields cannot equal infinite values. The process of their changes in a real body is followed by a time delay according to relaxational properties of a material considered by the relaxation coefficients. To consider the relaxation properties of a material we shall present the formula (1) of Hooke’s law as a linear combination of stress derivatives and time deformation in a product with the corresponding relaxation coefficients τ1 and τ 2   τ1

 U    2  2U  3U  τ12 2  L  E   τ2  τ 22  L  , 2 t xt t xt  x 

(8)

where τ1 , τ 2 are stress and deformation coefficients, sec . Limited by two first terms in the right- and left-hand sides, the relation (8) is reduced to  U  2U  σ   τ1 . σ  E   τ2   x  x  t t  

(9)

The relation (9) completely coincides with the standard models of a viscoelastic body known as Maxwell’s models, Kelvin-Voight’s models and real body models [5]. It should be noted that the relation (9) corresponds to Maxwell’s and Kelvin-Voight’s elaborated models where there is the third element added – a string which is parallel to Hooke’s and Newton’s bodies connected in sequence (in Maxwell’s model) and a damper which is parallel to Hooke’s and Newton’s bodies (in Kelvin-Voight’s model). Maxwell’s and Kelvin-Voight’s models are different only in the formulas for the relaxation coefficients τ1 and τ 2 . The physical significance of the models is consideration of time dependence of stresses and deformations and their mutual influence on each other. Coincidence of the model (9) with Maxwell’s and Kelvin-Voight’s models and a real body model (up to constants) shows an application of the same initial principles. It should be stated that well-known references don’t give certain examples of derivation of elastic body oscillation equations using Maxwell’s and Kelvin-Voight’s models. The formula (9) may be also derived from a system offered by A.V. Lykov which is a system of Onzager’s differential equations of the type [6 – 11] J i  L(i r )

N J i X k   ,   Li k X k  Lik t k 1  t 



(10)

where J i is a flow of a substance (heat, mass, momentum, etc.); X k is effective forces (in this case X k  ε  u / x ); Lir  , Li k , Li k are constant. If we take Lir    τ1 ; Lirk  E ; Lik  E τ 2 ; J i  σ ; X k  u / x , the relation (10) is reduced to the formula (9). The agreement of the formula (9) with the formula (10) proves that the formula (9) considers cross effects related to simultaneous consideration of the time and space non-equilibrium and their mutual influence in the non-local process of momentum transfer.

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To derivate a differential equation which considers stress and deformation change in time according to the Hooke’s law formula we shall substitute (9) into the motion equation (6) 

 2U  2U  3U   σ  r U E  E  τ1     τ . 2 t  x  V t t 2 x 2 x 2 t

If we substitute the relation of the value  / x in the latter equation for the value of the equation (2), we will find τ1

2  3U  2U  3U U 2  U 2   e  e τ γ . 2 3 2 2 2 t t t x x t

(11)

It is obvious that if τ1  τ 2  γ  0 , the equation (11) is reduced to the undamped oscillation equation (3). We shall solve a boundary value problem for oscillations of the rod, where one of the ends is rigidly fixed, and another one is affected by the force F per unit area which changes according to the Cosine Rule. F U ( x, t )  cos(ωt ) ,  ES x

where S is the rod cross section area;   2 , 1 / с is angular frequency. 2. The mathematical formulation of the problem At the initial instant the rod is deformed according to the linear law which says that maximum displacement is a property of the loose end of the rod. In that case the mathematical formulation of the problem is as follows: τ1

  2U ( x, t ) U ( x, t )  3U ( x, t )   3U ( x, t )  2U ( x, t ) ,  τ2  e2   γ 3 2 2 2 x x t t    t t  

(t  0; 0  x  δ)

(12)

U ( x, 0)  b(δ  x) ,

(13)

U ( x, 0) / t  0 ,

(14)

 2U ( x, 0) / t 2  0 ,

(15)

U (0, t ) / x  cos( ωt ) ,

(16)

U (δ , t )  0 ,

(17)

where δ is the rod length, m ; b is a coefficient considering the rod initial displacement. The initial condition (13) shows that if t  0 , the rod displacement is linearly dependent on the x coordinate, taking the maximum value U (0 ; 0)  U 0  bδ at the x  0 point and the minimum one U (δ, 0)  0 at the x  δ point. We shall include the non-dimensional variables and parameters as follows: Θ

U , U0

ξ

x , δ

Fo 

еt , δ

F1 

еτ1 , δ

F2 

еτ 2 , δ

F3 

δγ , e

(18)

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where Θ is non-dimensional displacement; ξ is a non-dimensional coordinate; Fo is a Fourier number (nondimensional time); U 0  bδ ; F1 , F2 are non-dimensional relaxation coefficients; F3 is a non-dimensional environment resistance coefficient. Considering the relations (18) the problem (12) – (17) is written as F3

Θ(ξ, Fo)  3 Θ(ξ, Fo)  2 Θ(ξ, Fo)  2 Θ(ξ , Fo)  3 Θ(ξ, Fo)  F1    F2 , 3 2 2 Fo Fo Fo ξ ξ 2 Fo

(Fo  0 ; 0  ξ  1)

(19)

Θ (ξ ,0)  1  ξ ,

(20)

Θ(ξ ,0)  0, Fo

(21)

 2 Θ(ξ ,0) 0, Fo 2

(22)

Θ(0, Fo)  F4 cos( F5 Fo) , ξ

(23)

Θ(1, Fo)  0 ,

(24)

where F4  δ/U 0 ; F5  ωδ/e . 3. Results of the calculations To solve the problem (19) – (24) by using a finite difference method we shall include a spatial mesh in the region under consideration. The mesh has the steps ξ  0,005 , Fo  0,005 accordingly in the variables ξ and Fo ξ k  k ξ ,

k  0, K ;

Fo i  i Fo ,

i  0, I ,

(25)

where K  200 , I  50000 are a number of steps in the coordinates ξ , Fo . Mesh functions Θ ik  Θ(ξ k , Fo i ) are substituted into the mesh (25). If we use a conventional scheme of differential operator approximation, the problem (19) – (24) is written as F3

 ik1  ik  i 1  3 ik  3 ik1  ik2 Θ ik1  2Θ ik  Θ ik1  F1 k   Fo ΔFo 2 Fo3



 Θ ik 1  2Θ ik  Θ ik 1 Θ ik11  2Θ ik1  Θ ik11  Θ ik 1  2Θ ik  Θ ik 1  ,   F 2   Δξ 2 Δξ 2 Fo Δξ 2 Fo  

Θ1k  Θ 0k  0, ΔFo

Θ 0k  2 Θ1k   2k 0, Δ

Θ1i  Θ i0  F4 cos( F5 Fo i ) , Δξ

Θ k0  1  ξ k ,

ΘiK  0 .

The results of the solutions obtained are given in Fig. 1 – 10. If F1  F2  F3  0 , the oscillations are undamped. If F3  0,3 and F1  F2  F4  F5  0 , the oscillations become damped with the exponentially decreasing amplitude (Fig. 1a). If F1  F2  F3  0 and F4  0,5 , F5  1,575 , the non-dimensional free oscillation frequency


  1,575 of the rod coincides with the frequency of the forced oscillations caused by the internal load of the type (23). Simultaneously the unlimited growth of the oscillation amplitude is observed (Fig. 1b).

Fig. 1. Change of the rod displacement in time: (a) damped oscillations ( F1  F2  F4  F5  0 ; F3  0,3 ); (b) resonant frequencies ( F1  F2  F3  0 ; F4  0,5 ; F5  1,575 )

When the environment resistance coefficient F3 increases, the rod oscillation amplitude decreases (Fig. 2a). If the value of the coefficient becomes higher ( F3  100 ), the restore of the rod to the initial position occurs practically without the process of oscillation of the rod internal points (critical damping) at a constant oscillation amplitude of the internal load (Fig. 2b). Starting with some value of F3 , resonant oscillations are not observed (Fig. 2).

Fig. 2. Change of the rod displacement in time at the resonant frequencies: (a) F1  F2  0 ; F3  20 ; F4  1 ; F5  1,575 ; (b) F1  F2  0 ; F3  100 ; F4  1 ; F5  1,575

Fig. 3 shows the results of calculations for the case of resonant oscillations ( F5  1,575 ) at F1  F2  0,1 and F3  0,3 . Their analysis demonstrates that the oscillation amplitude decreases exponentially within the range of 0  Fo  9 . Then within the range of 9  Fo  25 it increases and becomes stabilized at the point A  0,4 in the process of undamped oscillation in time.

231

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Fig. 4 shows the results of calculations for F1  F2  0,1 ; F3  0,3 at the load oscillation frequency F5  0,5 which does not coincide with the free oscillation frequency of the rod that equals   1,575 , i.e. the oscillations occur at the non-resonant frequencies. Their analysis demonstrates that the oscillations damp exponentially within the range of 0  Fo  18 . If the number Fo increases, the oscillation process gets stabilized at the similar deviation in the region of the positive and negative values of displacement Θ  0,1 , i.e. the oscillations are undamped.

Fig. 3. Change of the rod displacement in time at the resonant frequencies, when F1  F2  0,1 ; F3  0,3 ; F4  0,1 ; F5  1,575

Fig. 4. Change of the rod displacement in time at the non-resonant frequencies, when F1  F2  0,1 ; F3  0,3 ; F4  0,1 ; F5  0,5

The results of calculations for non-resonant oscillations ( F4  0,1 ; F5  0,1 ) at some higher values of the relaxation coefficients F1  F2  10 are given in Fig. 5. Their analysis demonstrates that the rod oscillations occur sequentially in the region of the positive and negative values of displacement relating to the initial (undisturbed) position of the rod in the process of oscillation undamped in time.


The results of calculations for resonant oscillations F1  F2  2 ; F3  1 ; F4  0,1 ; F5  1,575 are given in Fig. 6, 7. Their analysis demonstrates that if the relaxation coefficients F1 and F2 increase at other equal conditions, the undamped oscillation stabilization occurs at higher amplitude A  0,4 .

Fig. 5. Change of the rod displacement in time at the non-resonant frequencies, when F1  F2  10 ; F3  0,3 ; F4  0,1 ; F5  0,1

Fig. 6. Change of the rod displacement in time at the resonant frequencies, when F1  F2  2 ; F3  1 ; F4  0,1 ; F5  1,575

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Fig. 7. Change of the rod displacement in time at the resonant frequencies, when F1  F2  2 ; F3  1 ; F4  0,1 ; F5  1,575

If the relaxation coefficients ( F1  F2  10 ; F3  0,3 ; F4  0,1 ; F5  1,575 ) increase in the process of undamped oscillation, the bifurcation resonance is observed. This resonance is followed by the periodic increase of the amplitude (See Fig. 8).

Fig. 8. Change of the rod displacement in time at the resonant frequencies, when F1  F2  10 ; F3  0,3 ; F4  0,1 ; F5  1,575

It should be noted that bifurcation change of the oscillation amplitude is also observed at the oscillation frequencies of the internal load close to the frequency of free oscillations of the rod but not absolutely equal to it. Such type of the oscillations is referred as beats. The results of calculations for the type of investigation at F1  F2  10 ; F3  0,3 ; F4  0,1 ; F5  1,5 are given in Fig. 9,10. Their analysis demonstrates that the process of oscillation stabilization in time is not observed in that case, i.e. the process of oscillation is damped.




4. Conclusion 1. If the non-dimensional frequency of free oscillations of the rod (   1,575 ) coincides with the oscillation frequency of the internal load ( F5  1,575 ), when F1  F2  F3  0 , there is resonance which is followed by the unlimited growth of the oscillation amplitude (Fig. 1b). 2. Under conditions of resonant oscillation ( F5  1,575 ), when F1  F2  0,1 and F3  0,3 , at first, the oscillation amplitude decreases (within the range of 0  Fo  9 ), and then (if Fo  9 ) increases and becomes stabilized at the point A  0,4 (if Fo  25 ) in the process of oscillation undamped in time (Fig. 3). 3. If the frequency of free oscillations of the rod   1,575 does not coincide with the frequency of the internal load F5  0,5 , if F1  F2  0,1 and F3  0,3 within the range of 0  Fo  18 , the oscillation frequency of the rod

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decreases exponentially, and then, when the number Fo increases, it becomes stabilized at the point A  0,15 in the process of undamped oscillation (Fig. 4). 4. At some higher values of the non-dimensional relaxation coefficients F1  F2  2 of the resonant ( F5  1,575 ) frequencies bifurcation resonance is observed in the processes of undamped oscillation (See Fig. 8). If the frequencies are non-resonant ( F5  1,5 ), the phenomena of beating are observed in the processes of damped oscillation (See Fig. 9, 10). Acknowledgements The study has been carried out with financial support of RFBR as a part of the scientific project № 16–38–00059 mol_a. References [1] I.M. Babakov. Teoriya kolebaniy (in Russian) [Theory of oscillations] Moscow, Drofa Publ, 2004. 592 p. [2] A.N. Tikhonov, A.A. Samarskiy. Uravneniya matematicheskoy fiziki (in Russian) [Equations of mathematical Physics] Moscow, MGU Publ., 1999. 798 p. [3] K.S. Kabisov, T.F. Kamalov, V.A. Lour’ye. Kolebaniya i volnoviye protzessi: Teoriya. Zadachi s resheniyami (in Russian) [Oscillations and wave processes: Theory. Problems and solutions] Moscow, KomKniga Publ., 2010. 360 p. [4] I.V. Kudinov, V.A. Kudinov. Exact closed-form solution of the hyperbolic equation of string vibrations with material relaxation properties taken into account. Mechanics of Solids, 2014, Vol. 49, № 5, pp. 531 – 542. [5] A.P. Filin. Prikladnaya mekhanika tverdogo deformiruyemogo tela (in Russian) [Applied mecha nics of solid body deformation] V. 1. Moscow, Nauka Publ., 1975. 832 p. [6] A.V.Lykov. Primemeniye metodov termodinamiki neobratimykh protzessov k issledovaniyu teplo – i massoobmena (in Russian) [Application of the methods of thermodynamics of irreversible processes to the study of heat transfer and mass transfer] Inzhenerno-fizicheskiy zhurnal [Engineering and Physics Magazine], 1965, vol. 9, no. 3, pp. 287 – 304. [7] I.V. Kudinov, V.A. Kudinov. Study of the exact analytical solution of the equation of longitudinal waves in a liquid with account of its relaxation properties. Journal of Engineering Physics and Thermophysics (2013): Vol. 86, № 5, p. 1191-1201. [8] I.V. Kudinov, V.A. Kudinov. Determination of the dynamic stresses in an infinite plate on the basis of an exact analytical solution of the hyperbolic heat-conduction equation for it. Journal of Engineering Physics and Thermophysics (2015): Vol. 88, № 2, pp. 398-405 [9] I.V. Kudinov, V.A. Kudinov. Mathematical simulation of the locally nonequilibrium heat transfer in a body with account for its nonlocality in space and time. Journal of Engineering Physics and Thermophysics (2015): Vol. 88, № 2, pp. 406-422. [10] I.V. Kudinov, V.A. Kudinov. Problems of Dynamic Thermoelasticity on the Basis of an Analytical Solution of the Hyperbolic Heat Conduction Equation. High Temperature, 2015, Vol. 53, No. 4, pp 521 – 525. [11] A.V. Eremin, V.A. Kudinov, I.V. Kudinov. Mathematical Model of Heat Transfer in a Fluid with Account for Its Relaxation Properties. Fluid Dynamics, Vol. 51, No. 1, pp. 33 − 44.