Sep 30, 2017 - The coupled motion provided by the bolt ensures dependence of the ... Operating manual ... Preloading the bolt in a bolted connection would.
Dynamic Characterization of the Bolted Connections
By – Rajesh Kumar Under supervision of Dr. SR Kale
Department of Mechanical Engineering Indian Institute of Technology Delhi 30 September 2017
Team 1-C. Report 3
Abstract Bolted connections are found in every machinery in the world. Bolts allowing easy assembly and disassembly of the system are an integral part of any mechanical design. But the presence of a bolted connection is known to affect the static as well as the dynamic properties of the system. The coupled motion provided by the bolt ensures dependence of the mechanical systems on one another as well as on the properties as well as the dynamic properties of the bolt. It is necessary to understand the behavior of the complete bolted system. The properties derived from the static as well as the dynamic behavior of the system provide us with the crucial system parameters that can be extended to prevention of the failure of the bolted assemblies. The “Goodness” of the bolted connection is needed to be established in order to ensure the optimal configuration of the bolted connections is used. We intend to analyze the response of the bolted connections on a scale lower than those used in actual machineries. The idea is to understand the behaviour of simple bolted assemblies under dynamic loading. We shall correlate the acceleration response of the assembly to the amount of loosening of the bolt or the “goodness” of the bolted connection. As the blot slips, it leads to the changing stiffness of the bolted assembly as the two connected members begin to slowly get decoupled. It is a correlation that we hypothesize and shall try to correlate to the dynamic response of the system. Other failure methodologies shall also be predicted from the experimentation. Keywords: Bolted Connections, Dynamic Failure, Vibration Response Analyses, Bolt Slippage.
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Contents Page no. 1.
Introduction
3
2.
Objective
5
3.
Experimental set-up
8
4.
Experimental procedure
22
5.
Results
30
6.
Discussion
45
7.
Conclusions
49
8.
Acknowledgements
50
References
51
Annexure 3-A
Engineering drawings
53
Annexure 3-B
Video of set-up assembly
62
Annexure 3-C
Pre-test uncertainty analysis
63
Annexure 3-D
Video of performing an experiment
66
Annexure 3-E
Post-test uncertainty analysis
67
Annexure 3-F
MATLAB Codes
73
Operating manual
Separate
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1. Introduction 1.1
Background Bolted connections are widely used in almost every mechanical and structural system
due to the added flexibility of assembly and disassembly of sub-systems or in machines. A bolted connection often constitutes the weakest link in the design in many cases; the bolted connection can be responsible for determining the overall reliability and safety of an entire system. A simple bolted connection may just include a threaded fastener (bolt/screw/stud), a nut or a tapped hole, and the parts that would be clamped together by preloading the bolt mostly by tightening the head or the nut. Preloading the bolt in a bolted connection would allow the transfer of various service loads through the clamped connection either directly or through increased frictional resistance at the interface surfaces of the joint. Several problems are associated with the bolted connections[1]. Bolted connections undergo failure condition under subjecting to different conditions. Failure of Bolted Connections can be classified broadly into two types: Failure of Bolt or Connector materials and Failure of connecting parts. To characterize the behaviour, we prepare an experimental setup to understand the behaviour at a smaller scale. The vibration input is given by the motor with an unbalance. The two sheets connected to each other via the connecting plate replicates a bolted connection. The concept of the experiment is shown in figure 1.1.1.
Figure 1.1.1: Experimental Concept. The two sheets connected by bolts via a connecting plate. The motor on the connecting plate provides the vibrations which need to pass through the bolts. The vibrations may cause slipping of the bolts leading to subsequent loosening. The figure is a schematic made. 3
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1.2
Motivation Bolted connections are widely used in almost every mechanical and structural system due to the added flexibility of assembly and disassembly of sub-systems or in machines. But the presence of a bolted connection is known to affect the static as well as the dynamic properties of the system. The coupled motion provided by the bolt ensures dependence of the mechanical systems on one another as well as on the properties as well as the dynamic properties of the bolt. The joint represents a discontinuity in the structure and results in stresses that often initiate joint failure. The design of structural systems involves elements that are connected through bolts, rivets, and pins. Joints and fasteners are used to transfer loads from one structural element to another. Joints and fasteners are used to transfer loads from one structural element to another. Bolted joints are the dominant fastening mechanism used in joining of primary structural parts for advanced composites. The complex behavior of connecting elements plays an important role in the overall dynamic characteristics, such as natural frequencies, mode shapes, and non-linear response characteristics to external excitations which are very important factors that have to be taken into account while construction of buildings. The bolted connection can be responsible for determining the overall reliability and safety of an entire system. The properties derived from the static as well as the dynamic behavior of the system provide us with the crucial system parameters that can be extended to prevention of the failure of the bolted assemblies. Hence nowadays the study of the Goodness of the bolted connection is needed hugely to be established in order to ensure the optimal configuration of the bolted connections is used.
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2. Objective 2.1
Expected outcome We want to define the optimum configuration of the bolted connection by defining the
the number of bolts that are required, the optimum size of the bolts and the optimal pre-load that should be applied to the bolts (the torque required). To define the best configuration of the bolted connection, we will compare the time varying stiffness or some other property of the complete system. We shall compare the properties at various configurations as a function of time when assembly is under impact to describe the “Goodness” of the assembly. The static analysis is also done, which suggests the static response of the assembly to the input load. For the definition, we shall describe the dynamic response of the bolted connections as well as the stress and the strain analysis of the complete system. The expected outcome of the experiment is to establish a relationship between different configurations of bolted connections and the failure conditions for them, under different loading cycles, static as well as dynamic. From our experiment, we aim to establish the aforementioned relationships. The data that we would collect from the experiment would be: Accelerations response of the assembly under dynamic loading and Strain in the assembly under static loading.
2.2
Objective of the experiment
From the experiment, it is required from us to come out with the “Goodness” of the bolted connection. We need to compare different configurations of the bolted connections and conclude with the best configuration of the bolted connection. It is expected to us that we analyse the dynamic parameters of the bolted assembly in achieving the said conclusion and also shall do the static stress analyses in order to compare the configurations of the bolted connections. By suggesting “the configuration of the connection”, it is meant to say the parameters like the “no. of bolts required”, “the size of the bolts” etc. are optimized. We also wish to compare the systems with the amount of pre-load given to the bolts (torqueing). From the mandate, we gather that the experiment needs to be such that some relation between the configurations of the bolted connection with respect to the loading can be established. An emphasis needs to be laid on the failure conditions of each of the configurations. The scope the experiment can be realised from the fact that a similar system of bolted connections is seen throughout the Indian Railway Network where two rail pieces are connected by a joint similar to the one that has been taken in consideration here. This experiment requires
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prerequisite knowledge of Solid Mechanics, specifically the failure of materials. Stress Analysis Pre-Stressing the bolts.
2.3
Intended benefit The design of structural systems involves elements that are connected through bolts,
rivets, and pins. Joints and fasteners are used to transfer loads from one structural element to another. Bolted joints are the dominant fastening mechanism used in joining of primary structural parts for advanced composites. The complex behavior of connecting elements plays an important role in the overall dynamic characteristics, such as natural frequencies, mode shapes, and non-linear response characteristics to external excitations which are very important factors that have to be taken into account while construction of buildings. The joint represents a discontinuity in the structure and results in stresses that often initiate joint failure. So by knowing characteristic of bolted joints we can beforehand know the critical stress value and the load that can be applied on to the joint. There have been catastrophic failures in the past for the same. For e.g during drilling operations in the Gulf of Mexico (GOM), leaks were detected which identified failure, severe stress corrosion cracking fracture of bolts on the lower marine riser package (LMRP) [1]. The characterization of the dynamic behavior of the bolted connections can help us to prevent the failure. Figures 2.3.1-2.3.3 shows us the examples of failures due to failure of the bolted connections.
Figure 2.3.1: The failure due to slipping of the bolt can cause structural catastrophe [1].
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Figure 2.3.2: Failure of the railway line because the bolts in the fish plate loosen out [1]
Figure 2.3.3: Offshore bolt failures leading to major crisis has been recently documented in the film DEEP WATER HORIZON [1].
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3. Experimental set-up 3.1
Design concept We needed to characterize the properties of the bolted connections under dynamic
loading. By dynamic loading, we mean to say that the cyclic loading any machine element experiences in operation. To simulate similar conditions the bolts present in machines operating at very high frequencies, we came out with few design concepts out of which the best is presented here. The comparison of the concepts can be seen in the previous reports. The characterization was done at a smaller scale than that present in the actual machineries. The dynamic load was provided by the motor with an unbalance. The nature of the input force due to the unbalance is sinusoidal and is widely known as a source of vibration with the frequency of loading being equal to the rpm of the motor. The amount of force applied to the system can be quantified by determining the radius of the center of mass of the unbalance and the value of the unbalance. So, the experimental setup was needed to provide space for the dynamic analyses (figure 1.1.1). From the experiment we basically wish to characterize the goodness of the bolted connections under dynamic loading conditions. The dynamic loading is being provided by the motor kept under the assembly with an unbalance. As the motor rotates, a sinusoidal varying force is applied on the connection. The force applied on the assembly is determined as the sinusoidal varying input of magnitude as shown in the equation. As the motor has its selfweight and hence a static load is always applied on the system, the net force input is basically a combination of a static and a dynamic load. Such an attribute of the force can be related to the loading conditions a set of bolted joints in trains or other automobiles also experience. So, we try to examine the goodness of a set of bolted connections for those applications. The dynamic response of the system is determined by the acceleration response by the accelerometers, which inherently suggest the stiffness and other dynamic characteristics of the complete assembly. When the dynamic load begins to be applied on the assembly, the bolts hold all the three system together and the acceleration because of the bolt tightly holds the complete system making the complete assembly a fixed-fixed structure. However, due to repetitive loading on the structure, the bolts start to loosen up and then the behaviour of the two beams joined by the bolts, start becoming independent of one-another leading to the reduction in the stiffness of the complete system and hence, an increase in the acceleration is predicted. The simplified model of the experimental structure is shown in figure 3.1.1.
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Figure 3.1.1: The setup constitutes two aluminium sheets with two fixed fixed ends and a load at the center. When the bolts are tightened the structure replicates a fixed fixed structure and as bolts start two loosen, the decoupling renders it to a set of two cantilevers. 3.2
Theoretical basis The experimental setup applies a time varying load to the set of bolted sheets (figure
1.1.1) via an unbalance on the motor. The motor unbalance is known to provide a time varying harmonic force to the system. The figure 3.1.1 shows a simplification of the system. The basis of the experiment is to understand the dynamic behaviour of the bolted connections under dynamic loading and to relate the dynamic behaviour to the acceleration characteristics. The acceleration measurements and the motor rpm measurements are the two parameters that are the outputs from the experimental setup. From the two set of parameters we try to characterize the “Goodness” of the bolted connection. We use the single DOF substitute for the whole system in order to determine the net dynamic properties explaining the whole system. Though, the frequency response coupling method has been widely used for mathematical modelling, defining the system as multiple degree of freedom systems; but the definition of net system parameters for such definitions is difficult. In order to determine the complete system parameters, we need to define the complete system as a single degree of freedom system [2]. Optimum literature is available on the single dof modelling of the bolted system. We use the methodology given in for the determination of the parameters. A single dof manifestation of the complete system is given in equation 3.2.1. ̈( ) where ( ̇
( )
( ̇
)
( )
(3.2.1)
) represents the system dynamics. The feature of the system dynamics is usually
needs to be understood by visualizing the net response from the system only, but the nonlinear dynamics of a linear lap joint is usually in the form of equation 3.2.2.
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( ̇
)
̇( )
(3.2.2)
The manifestation is an initial definition of the dynamics of the system based on the experiences of the other experiments on the dynamics of the bolted connections and hence we use it as a starting model. If the experimental response of the system don‟t show coherence with the dynamics form we shall need to revert to other methodologies of the presentation of the dynamics of a lap bolted connection or a connection in shear [3]. As the mathematical model is a non linear ordinary differential equation (ODE) in time, it needs to be computationally solved. But, to determine the system parameters required (stated earlier), we can use a very simple algebraic manifestation of the ODE, as from the experimentation, we shall know the states at many instances of time. Considering, we have “n” instances of state values from the experiments then we have the set of equations (3.2.3). ̈( )
( )
̇( )
( )
( )
̈( )
( )
̇( )
( )
( )
( )
( )
. . . ̈( )
( )
̇( )
(3.2.3)
So, we have a set of “n” ODE‟s. As, during experimentation, we shall know the acceleration states from the accelerometers, we need to define the position and the velocity states of the system. As, we know from the existing literature that any system‟s dynamic states can be represented as a combination of sinusoids, we use the sinusoid formulation to assume the nature of the position and velocity state functions. The acceleration state points from the accelerometer data is needed to be fitted in the following curve at minimum error parameters given in equation 3.2.4 ̈
∑
(
(
)
(
))
(3.2.4)
The equation can be integrated in order to determine the position and the velocity states of the system. Hence the system of ODE‟s are easily converted to a system of linear equations, with the solution given in equation 3.2.5. (
)
( ) ( ( )
( ) ( )
( ) ̇( ) ) ( ( ) ̇( )
̈( ) ̈ ( ))
10
(3.2.5)
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where „+‟ represents the nth order Moore-Penrose Pseudo Inverse of the rectangular matrix. Hence, we know all the required dynamic parameters to ensure the dynamic characterization of the bolts. The net bolted connection, when bolted afresh, that is, when the dynamic load is just started to be applied, the complete system shall behave as a fixed fixed system, showing lower acceleration values than after being hit by the periodic load for a period of time. The time varying load due to the mass unbalance is given by the equation 3.2.6. (
)
where “m” is the mass of the unbalance,
(3.2.6) is the angular velocity of the motor and “r” is the
radius of the unbalance. To consider the fact whether the change in the acceleration response of the bolted sheets is due to the slipping and loosening of the bots or due to fatigue of the sheets. So, we need to determine the time varying stresses in the sheets as well. As the acceleration response, subsequently filtered and integrated can give us the displacement response. The approximation of each sheet as an cantilever allows us to check the system for the worst case scenario. We are considering each sheet to be a cantilever as it is known that for a given loading condition, the maximum stress generated is in a cantilever beam. Also, the system presented changes its character from a fixed fixed beam (when fully bolted) to a cantilever (when all the bolts fail). So, for a fatigue testing, considering the worst case scenario seems a viable option. Equations 3.2.7-3.2.9 are the equations used to determine the time varying maximum stress in each of the sheets [7]. (3.2.7) (3.2.8) (3.2.9)
3.3
Pre-test uncertainty analysis In the dynamic characterization of the system, we determine the time varying
dynamic properties of the bolted connections. As, for example, the stiffness (
̇ ̈
), were (
̇ ̈ ) are the position, velocity and acceleration states of the system
and Q is the forcing function (the external load). So, we need to determine the uncertainties in all the parameters to get the final pre-test uncertainty.
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For the dynamic considerations, we use accelerometers for measurement of acceleration. The position and velocity states of the structure are to be found using curve fitting techniques and the subsequent integration of the defined curves. As, we yet do not know about the nature of the output acceleration from the data, it is, hence, difficult to get the approximation of the error due to the curve fits. However, we shall put the errors based on the experiences of other experimentalists. The accelerometers used configuration details suggests the basis of only capabilities of the system. The accelerometer to be used (ADXL 327) gives the uncertainty in acceleration it provides. So, we know the uncertainty values of the acceleration points. As, the position and velocity states of the system are a function of the acceleration points for the system, we shall use the same error factors. The uncertainty factors for the acceleration in the accelerometer are given by equation 3.3.1. (on the basis of the accuracy of the instrument). Equation 3.3.1 suggests uncertainty acceleration. ̈
(3.3.1)
̈
The position and velocity states have the same errors as that due to acceleration, but the error due to curve fit needs to be included. We shall aim for a regression of the acceleration upto the root mean square error of 0.5%. Let us assume the regression error to be ε. Equations 3.3.2 – 3.3.3 state the uncertainty in position and velocity. ̇
̈
( ̇)
( ̈)
( )
( ̈)
(3.3.2)
̈
(3.3.3)
So, the net error in the stiffness as well as the natural frequency and the damping coefficient is given in equation 3.3.4-3.3.6. (
)
(
)
( )
̇
( ̇)
(
)
̈
( ̈)
(3.3.4)
The uncertainty in the stress in the sheets is also given by So, ( ) ( )
3.4
( ( )
)
( (
)
(
)
( )
)
(3.3.5) (3.3.6)
Post-test uncertainty analysis In the dynamic characterization of the system, we determine the time varying
dynamic properties of the bolted connections. As, for example, the stiffness
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(
̇ ̈
), were (
̇ ̈ ) are the position, velocity and acceleration states of the system
and Q is the forcing function (the external load). So, we need to determine the uncertainties in all the parameters to get the final pre-test uncertainty. We use the method to calculate the uncertainty given in [8]. For the dynamic considerations, we use accelerometers for measurement of acceleration. The position and velocity states of the structure are to be found using curve fitting techniques and the subsequent integration of the defined curves. As, we yet do not know about the nature of the output acceleration from the data, it is, hence, difficult to get the approximation of the error due to the curve fits. However, we shall put the errors based on the experiences of other experimentalists. The accelerometers used configuration details suggests the basis of only capabilities of the system. The accelerometer to be used (ADXL 327) gives the uncertainty in acceleration it provides. So, we know the uncertainty values of the acceleration points. As, the position and velocity states of the system are a function of the acceleration points for the system, we shall use the same error factors. As the dynamic load put up due to the unbalance of the motor is given by equation 3.2.6, we shall use it for the uncertainty in Q. So, the load measured has the uncertainty of the two components, the uncertainty in the measured current and the uncertainty in the correspondence from the charts. 3.4.1 Uncertainty in the Acceleration measurements We use the accelerometer ADXL345 with 16g measurement range. The acceleration response of the accelerometer is based on the uncertainity in the measurement of the acceleration as 1.4%. However, the accelerometer output is analog in nature and hence needs to be converted to digital. The ADC converter used has a 12-bit resolution. The 11 bit is uncertain for the measurement range of 2.5 V input. So, as the calibration factor or the sensitivity of the accelerometer = 256 LSB/g. That means at least first 8 bits are needed to be put into use. The 10 bits need to completely determine the output, so 1LSB = . As the resolution is 256LSB/g, so the uncertainty in measurement = . Hence the uncertainty in the measurement of acceleration for the accelerometer, means an uncertainty of 1.4%. It is known the acceleration from the accelerometer ADXL 345 varies due to the temperature difference as well. The error in the measurement of the accelerometer due to the temperature difference is around 0.2% of the total.
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The random uncertainty in the measurement of the accelerometer was based on holding the accelerometer at a place and measuring the acceleration response for a period of time and then finding the standard deviation and hence the uncertainty in the measurements. Figure 3.4.1 suggests that the The random uncertainty value is given by – 3 %.The total uncertainty is given by – 3.3% in the acceleration.
Figure 3.4.1: The random uncertainty in the measurement is based on the standard deviation of the data in the figure.
The position and velocity states have the same errors as that due to acceleration, but the error due to curve fit needs to be included. We shall aim for a regression of the acceleration upto the root mean square error of 0.5%. Let us assume the regression error to be ε. Equations 3.3.2 – 3.3.3 state the uncertainty in position and velocity. The errors in the position as well as the velocity measurements is given by the error in the acceleration measurements as well as the error in the subsequent filtering techniques as well as the error in the numerical integration methodology. As we use the trapezoidal integration technique to calculate the velocity and then the position characteristics of the response. The error in the trapezoidal integration is given by – (
)
(
)
which is very small comparable to the other parts of the
uncertainty so, we neglect this very small value. So, the net error in the stiffness as well as the natural frequency and the damping coefficient is given in equation 3.4.1-3.4.4 as well as in equations 3.3.2-3.3.3. (
)
(
(
)
̇
( )
( ̇) ̇
)
(
)
( )
( ̇)
( )
(
)
(
( )
)
( (
) )
̈
( ̈) ̈
( ̈)
(3.4.1) (3.4.2) (3.4.3)
̈
(3.4.4)
̈
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We aim for the root mean square error in the regression to be 0.5%. So,
= 0.0005.
The net uncertainty is given in equations 3.4.5 and 3.4.6. ̇
( ̇)
(3.4.5)
( )
(3.4.6)
The uncertainty in the force values is given by equation 3.4.7. ( )
3.5
(
)
(
)
( )
(3.4.7)
Detailed engineering design The experimental setup consists of a base frame made up of aluminium frames and
two aluminium sheets bolted via a motor holder. The motor with the unbalance provides the dynamic loads to the bolted sheets. The accelerometers take up the acceleration values. The actual setup is shown in figure 3.5.1.
Figure 3.5.1: Actual experimental setup developed There are four main blocks to the system (as presented in the plan view as well (present in Annexure 3-A). First is the main experimental setup where all the experimentation needs to be done. The power to the motor driver comes from the AC to DC converter, which itself takes in the power from the UPS AC input which also gives power to the signal
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processing unit which processes the signals from the accelerometers and sends it to the system via a USB. The unit also gives the power to the accelerometers. The Arduino gets the power from the computer system itself through the USB which was used to code the Arduino UNO. Another view of the experimental setup is shown in figure 3.5.2.
Figure 3.5.2: The main parts are the motor holder plate and the plate specimens joined by the bolts and clamped to the base frame. Two accelerometers are placed therein.
Figure 3.5.3 and figure 3.5.4 shows the details of the experimental setup. The reader should get an idea of the procedure to assemble the mechanical structure as well as the electrical connections needed to assemble the complete system.
Figure 3.5.3: The Motor Driver takes in the power from the DC input and the arduino basically acts as the switch for using the PWM signal.
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Figure 3.5.4: The connections of the motor driver and the arduino board 3.6
Instrumentation and Data Acquisition Systems We have used the following instruments and softwares and built the MATLAB codes for the data acquisition and subsequent analysis – 1. Two accelerometers to measure the acceleration response of the system, 2. DAQ system to convert the acceleration response from the accelerometers ADXL345 (which is an analog device). The DAQ system converts the analog to a digital output and subsequently outputs it to the computer system. 3. The DAQ software built in the DAQ PCs allow us to take the acceleration response. 4. The data needs to be manually saved. 5. The data needs to be parsed in .mat format in MATLAB software. 6. Each set of data needs to be processed through two sets of MATLAB functions to get the results which can be used by the user. (see Annexure for the MATLAB codes) In our experiment, we have used data acquisition systems for collecting the
acceleration data at certain points of the setup for further analysis. The accelerometers used have a maximum acceleration measuring limit of 16g. The software used for the data acquisition is iQ Data Acquisition Software. This software can vary the sampling rate from
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10 Hz to 5 KHz. An Arduino board has been used to vary the power supplied to the motor, to change the motor rpm and hence control the vibrations given to the bolted connection system. Due to a constraint on the number of ports available, we had to connect the power sources of two accelerometers in parallel, to accommodate the accelerometers. No open ports are available on the current data acquisition hardware. In case more accelerometers are needed, external data acquisition systems will be needed. We are also currently using an Arduino board, which could be used for the above purpose. The data acquisition system is used to determine the accelerations at two parts of the structure, at either of the two sheets. We use the 16G accelerometers to determine the accelerations. The debugging was based on the system, where a
unbalance was
used in the structure. The debugging was based on various trials of the accelerometers input and different configurations of the input frequency was stored. An initial static shift was observed in all the systems which shall be removed on filtering as the system is just based on defining a static shift. The two accelerometers‟ input frequencies were optimized on the basis of the static shift observed. The test runs on the systems were done and the raw data was saved for various cases at different frequencies of sampling (figure 3.6.1 – 3.6.3). TEST RUN CONFIGURATION – Bolt Type – M4 No. of Bolts – 4 Motor Rpm ~ 2930 Static shift (DC component errors) observed in ACC 1, the first of the two accelerometers we are using at 1000 Hz. A nice change in acceleration observed on switching on the motor. Hence the sampling rate holds promise. The data in pictorial form is given in figure 3.6.1. However, no signification rationale can be developed on the basis of the output from sampling at 10Hz (figure 3.6.2). Hence, we use sampling rates above 100 Hz.
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Figure 3.6.1: Acceleration response output from the accelerometers – Unfiltered.
Figure 3.6.2: Unfiltered reponse at sampling rate of 10 Hz. A beat phenomenon observed because may be the piezoelectric vibration frequency was corresponding for the sampling rate at 5000 Hz (figure 3.6.3). Hence, we use the sampling at 1000 Hz only.
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Figure 3.6.3: Acceleration output response at 5000 Hz. The figure shows the acceleration data stored for both the accelerometers
As the motor rpm is around 3000 rpm in the operating range, so, the input vibration frequency to the system is around 50 Hz. By the Nyquist principle (ref.), the sampling rate need to be more than twice the input frequency, so the sampling rate cannot be 100 Hz. SNR for the selected sampling frequency – We have used the MATLAB inbuilt function to calculate the SNR of the selected system. The system a good net repeatability with the SNR of 292 dB, which is a highly acceptable value of the SNR. The standard deviation or the errors in measurement of the acceleration is difficult to calculate as it is impossible to get the same exact two states of the system. So, we can‟t predict the random uncertainty in the acceleration measurements. For the analysis of DATA, we have created two MATLAB functions – MATLAB Function 1 – runner1 (Code in Appendix – 3-F) 1. Name of the function – runner1 2. Input Variables – The accelerometer 4 channel output from the DAQ stored in the computer system and parsed in MATLAB and the base name needed to be given to all the figures that are the output of the function developed. 3. Processing – The MATLAB code built performs the following functions – a. It parses the acceleration data from the two accelerometers. b. Filters the acceleration output.
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c. Plots the filtered response and creates appropriate envelopes of the data to study the behaviour of the system with respect to time. d. The envelopes are plotted without the uncertainty band and with the uncertainty band respectively. The plots are subsequently saved in the “Current Directory”. e. Integrates the acceleration response to determine the velocity and then the displacement. All the curves developed are plotted and saves f. Performs the required calculations based on the inputs from the acceleration data. g. The stiffness values (time varying) are plotted and normalized to the maximum stiffness to bring everything into one place. h. The uncertainty band is added to the stiffness curves and then saved to the current directory. i. All the files saved are both in the “.jpeg” format as well as the “.fig” format. MATLAB function 2 – fatigurecalc (Code in Appendix 3-F) 1. Name of the function – fatigurecalc 2. Input Variables - The accelerometer 4 channel output from the DAQ stored in the computer system and parsed in MATLAB and the base name needed to be given to all the figures that are the output of the function developed. 3. Processing – The MATLAB code built performs the following functions – a. It plots the standard SN curve for aluminium for fatigue analysis. b. It takes in the acceleration response from the accelerometers placed at either of the two sheets and calculates subsequent displacements at the end. c. The time varying cyclic stress is calculated for the sheets with the uncertainty band and a comparison plot is prepared and saved in the current directory in the “.jpeg” as well as the “.fig” format.
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4. Experimental procedure 4.1
Apparatus assembly
Figure 4.1.1: Exploded assembly of the structural setup To assemble the complete setup, initially the base frame was needed to be set up (figure 4.1.1). It was a combination of 40×40 cm aluminium blocks. The complete structure formed was cuboidal. So, once the base structure was formed. Different specimens were needed to be attached to the frame in order to do experimentation with different structures. The two sheets needs to be connected to the connector plate tightly. It is needed to be ensured by the assembler that the motor is completely tightened into place in the connector plate. Once, the specimen is constrained into position, the whole specimen is needed to be put into place on the base frame and clamped at both the ends. We used a single clamp on either side to fully constrain the system, but if required multiple clamps can be used. The motor connections to the motor driver were soldered into place. The motor driver subsequently connected to the arduino which was later connected to the PC using a USB cable. So, the mechanical specimen needs to be assembled according to the said procedure. The electrical connections of the system is based on the figure 4.1.2.
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Figure 4.1.2: The electrical connections of the experimental setup 4.2
Operation of the set-up Before starting the setup, ensure that all the bolts in the base frame are tightened.
After checking each of the connection, put the specimen on the base frame and clamp it. Switch ON the UPS and the computer system. Then open the DAQ software in the PC and switch ON the “Signal Processing Unit”. In the DAQ software, connect the accelerometers and select the sampling rate as 1KHz. Select the calibration settings in the software and appropriately select “16g” calibration table for channels 1,2 and 3. Observe the static readings of the accelerometers, if the data holds promise, start the motor from the AC to DC converter. The motor starts running and the acceleration data starts coming up on the DAQ software. Wait for 10 mins and observe the proceedings and be careful on any undue happenings. Be prompt to shut down the system in case of any mishappening by switching
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off the “UPS”, it shuts down everything. If everything goes well, shut down the motor from the AC to Dc converter and click on “STOP” in the DAQ software. Save the readings in appropriate “.csv” file. Transfer the data to MATLAB and parse it in “.mat” format as an array. Process the readings through the two functions “runner1” and “fatigurecalc” whose source codes are presented in the annexure. Your all results shall be saved in the current directory of MATLAB in both “.jpeg” as well as “.fig” format. Retrieve them and analyse the data. 4.3
Methods for conducting an experiment The experiment is to quantify the “Goodness of the bolted connection”. To use the
setup in order to get the acceleration data from the accelerometers the following steps need to be followed – 1. Ensure that all the bolts of the base structure are tightened. 2. Ensure all the power inputs to the experimental setup is switch off. 3. There are three specimen that can be tested in the setup – the one containing the M4 holes, the one containing the M6 holes and the third one containing the M8 bolts. Take any one of the specimen. 4. Put the motor into the specimen and tighten the complete system ensuring no room is available to the motor to vibrate. 5. Check the soldered connections of the motor. Due to lack of flux the soldering has been the weakest part of the setup and comes off due the vibration input. In case you feel the soldering is coming off, resolder the connections on the motor. 6. Place the two accelerometers at the required palces, preferably one at each sheet and near the end of the sheet. Such a system allows to understand the response of each of the two sheets under vibration. 7. Once the accelerometers are placed, ensure all the power connections follow the path according to the wiring diagram as shown in the figure. 8. Switch on the Power Supply. 9. Switch on the UPS and then the CPU. 10. Let the computer system “ON” and open the DAQ software for the accelerometer. Connect the DAQ software to the DAQ system. 11. Select the sampling frequency to be 1KHz. 12. Select the required calibration files “16 g” for the accelerometer.
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13. The required vertical channels are CH1 and CH3. Ensure that the two channels show viable output. If the viable output response is not visible check the electrical connections. 14. If the channels CH1 and CH3 don‟t show up the acceleration data, recheck the connections. 15. If the accelerometers seem to work well, tighten the bolts of the specimen with the desired torque using a torque wrench. 16. Click on “Acquire” on the DAQ software. 17. Start the AC to DC converter which connects to the motor. Select an appropriate voltage on the AC to DC converter. As the Arduino + PWM based motor driver sends in a lower voltage to the motor than that prescribed at the AC to Dc converter. So, the voltage upto 13.5 volts works well with the 12 V DC motor. 18. As the motor start, notice a sudden significant variation in the accelerometer responses (CH1 and CH3). If no change is noticed, shut down the motor and recheck the connections. 19. If the acceleration response is visibly stored, use a tachometer to measure the RPM of the motor. 20. Wait for around 10 mins for the acceleration data to be stored. 21. After 10 mins, shut down the AC to DC converter. 22. Save the accelerometer data in “.csv” format. 23. Import the data to a MATLAB file by parsing the “.csv” file and delimiting the spaces and the commas. 24. Name the data file in the appropriate manner. 25. Process the data files via the function “runner1” and “fatigurecalc”. The appropriate required figures are saved in the current directory of MATLAB. 26. Open the figures either in “.jpeg” format or the “.fig” format. 27. Analyse the developed figures. 4.4
Test plan We had manufactured three sets of specimens to work with, one with the M4 bolts, the next with the M6 bolts and the third with the M8 bolts. For each set of specimen, we plan to do the analysis based on the set of tables 4.5.1 – 4.5.3.
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TABLE 4.5.1 : BOLT TYPE - M6 : The table needs to be used to put in the information of the files saved via DAQ and is to be handled for data processing Pre-Load S.no
Torque
Washer
Frequency
(in Present
Nm)
Input
(y/n)
Vibrations (Motor Rpm)
Torque 1 = No 1.
0.2
~2300
Nm
(approximate to very less tightening) 1.8 Nm
No
~2300
3.3 Nm
No
~2300
0.2 Nm
Yes
~2300
1.8 Nm
Yes
~2300
3.3 Nm
Yes
~2300
1.8 Nm
No
~2300
1.8 Nm
No
~3000
1.8 Nm
No
~1800
2
3.
4 5 6 7 8 9
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Table 4.5.2 – Specimen – with bolt connection M8: The table needs to be used to put in the information of the files saved via DAQ and is to be handled for data processing S.no
Pre-Load Torque
Washer
Frequency
(in Present
Nm)
Input Vibrations
(y/n)
(Motor Rpm)
Torque 1 = 0.2 No 1.
~2300
Nm (approximate to
of TEST ID
very less
tightening) 1.8 Nm
No
~2300
3.3 Nm
No
~2300
0.2 Nm
Yes
~2300
1.8 Nm
Yes
~2300
3.3 Nm
Yes
~2300
1.8 Nm
No
~2300
1.8 Nm
No
~3000
1.8 Nm
No
~1800
2
3.
4 5 6 7 8 9
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Table 4.5.3 – Specimen – with bolt connection M4: The table needs to be used to put in the information of the files saved via DAQ and is to be handled for data processing S.no
Pre-Load Torque
Washer
Frequency
(in Present
Nm)
Input Vibrations
(y/n)
(Motor Rpm)
Torque 1 = 0.2 No 1.
~2300
Nm (approximate to
of TEST ID
very less
tightening) 1.8 Nm
No
~2300
3.3 Nm
No
~2300
0.2 Nm
Yes
~2300
1.8 Nm
Yes
~2300
3.3 Nm
Yes
~2300
1.8 Nm
No
~2300
1.8 Nm
No
~3000
1.8 Nm
No
~1800
2
3.
4 5 6 7 8 9
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For each experimental setup, we defined a “state card”. The state card constitutes the following parameters based on our test matrix – TEST ID - …………………. Motor Rpm (Mean) - ……………. Type of Specimen - ……………. Time of Experimentation - ………. Washer Used (yes/no) - ………… Pre-load torque - ………………
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5. Results 5.1
Results of qualification experiments For the basis of the experiment, the variation of the “Goodness” of the bolted joint
with respect to the preload torques, each test was done thrice and the variation was the result was not unexpectedly different. As our experimentation analysis is not based on regression but is based on understandability out of the graphs developed, a traditional statistical “goodness of fit” of the data is difficult to find. However, the envelopes of the acceleration response created can be attributed to a goodness of the data. We were able to experiment on all the test cases we predicted in our test matrix. The acceleration response was measured for two accelerometers for all the test cases. We present few of the test cases results here. Although extensive data set is available, we present the selected results to keep the reader interested and evade the monotony. We present the result of one of the case to suggest how the results are to be analyzed. The results are thoroughly explained for the state card 1 and the rest shall follow suit. Case 1: State Card – TEST ID – TAKE5m6 Motor Rpm (Mean) – 2363.1 rpm Type of Specimen – M6 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 1.8 Nm Figure 5.1.1 and figure 5.1.2 are the acceleration response curves generated. Figure 5.1.1 is the acceleration data from the accelerometer. Figure 5.1.2 is the envelope of the acceleration of the data. The data is subsequently filtered and the peaks from the fourier transform is taken and plotted. The variation of the acceleration amplitudes is useful in analyzing the trend of the acceleration. Figure 5.1.3 is the acceleration growth during a given period of time. The band shows the uncertainty in the acceleration (around 33.3%). The growth needs to analyzed considering the uncertainty band only. The normalized stiffness of the system is shown in figure 5.1.4. The variation in the stiffness comes within the uncertainty band. So, no conclusion can be drawn from the uncertainty in stiffness. It is hence hypothesized that the results shall be drawn from the variation in the acceleration only and not from the stiffness variation.
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Figure 5.1.1: Acceleration Response of the M6 bolted connection according to the State Card with the TEST ID – TAKE5m6
Figure 5.1.2: Development of the Envelope of the acceleration response of one of the sheet
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Figure 5.1.3: Envelope of the Acceleration Response variation with the desired uncertainties
Figure 5.1.4: Normalized Stiffness of one of the sheets. The normalized stiffness is the ratio of the stiffness and is unit less.
The four characteristic plots for each State card needs to be analyzed. The variation of acceleration in figure is quantified using the percent increase in the acceleration of the
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sheet over the period of experimentation. The ratio (suggested as 33% in the above case) is an important comparison factor for the determination of the “Goodness” of the bolted connection. The higher is the “Acceleration Growth Factor”, poorer is the bolted connection. Figure suggests the variation in the complete stiffness of the bolted connection. Although a steady decline in the acceleration response is observed, but the strong uncertainty band around the stiffness, makes the rise uncertain. So, we shall not analyze the stiffness curves due to its high uncertainty. If the reader or the secondary experimenter is able to determine a methodology to improve on the uncertainty of the stiffness, then the system can be analyzed.
5.2
Studying the Variation of Preload Torque on various specimens Case 1: State Card – TEST ID – TAKE5m6 Motor Rpm (Mean) – 2363.1 rpm Type of Specimen – M6 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 1.8 Nm For the case, figures 5.1.1-5.1.2 show the acceleration growth rate of more than 30%.
Three takes for the same set of curves were being developed and for each of the three cases the acceleration growth factor remained in excess of 30%. We don‟t present the results here to avoid monotony with similar results. The similar curves also relate to the same fact that the growth in thee acceleration for M6 bolt remained in excess of 30%. Case 2: State Card – TEST ID – TAKE6m6 Motor Rpm (Mean) – 2347.1 rpm Type of Specimen – M6 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 3.3 Nm Figure 5.2.1 and figure 5.2.2 show the acceleration response for the M6 bolt for a much higher pre-load torque (3.3 Nm). The preload torque is higher than the standard limits of a M6 bolt. Within the acceleration band, the growth of the acceleration is around 10%, which is much lower than in the case when M6 bolt was preloaded with 1.8 Nm.
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Figure 5.2.1: Acceleration Response with Uncertainity band. The Acceleration Growth Factor ~ 10%.
Figure 5.2.2: Variation of the normalized stiffness with time. The normalized stiffness is necessarily unit less as it just represents a ratio. Case 3: State Card – TEST ID – TAKE7m6………………….
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Motor Rpm (Mean) – 2347.1 rpm Type of Specimen – M6 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 2.2 Nm Figure 5.2.3 shows the variation of the acceleration for M6 bolts without washer and a preload torque of around 2.2 Nm. It is within the range specified for an M6 bolt in the standards. For such preload along with the absence of washers leads to an increase in the acceleration of around 20% for the given period of time.
Figure 5.2.3: Acceleration Envelope with uncertainty (acceleration growth factor = ~20%) Case 4: State Card – TEST ID – TAKE8m6…………………. Motor Rpm (Mean) – 2340.1 rpm Type of Specimen – M6 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 2.2 Nm
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Figure 5.2.4: Acceleration envelope of the sheet 1 with acceleration growth factor of 22.2%
Figure 5.2.5: Acceleration Envelope of the sheet 2 with acceleration growth factor of 23.4%. Case 5 (Take9m6) consider the same case as in case 3. The acceleration growth factor in Case 5 = 18.4%. We don‟t put up the figures to avoid monotony of multiple figures for the same case, but the reader should be assured that the nature is case 5 is highly similar to case 3 and case 4. Case 6: State Card – TEST ID – TAKE13m6
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Motor Rpm (Mean) – 2340.1 rpm Type of Specimen – M6 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 0.4 Nm Figure 5.2.6 shows the variation of the acceleration for a very loosened bolt or a lightly tightened bots (for the preload torque of around 0.4 Nm). The figure 5.2.6 shows significant rise in the acceleration (around 40%) suggesting that the lower pre-load allows the bolt to loosen more in a given period of the stipulated time.
Figure 5.2.6: Acceleration Response with the acceleration growth factor = ~40%. For the figure 5.2.6, the acceleration variation is significantly higher, in this case and in the other cases as well, the rise in acceleration can be attributed to the fact that as the vibrations are passed through the bolts, repetitive change in the direction of the motion of the nut leads to subsequent reduction in the tension in the bolt. The reduction in the tension of the bolt inherently reduces the friction primarily responsible for holding the nut together. The slipping is caused when the pre-laod reduces and appropriate friction force is not applied by the bolt on the surface of the base. It is hence observed that loosing is a function of the preload.
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Team 1-C. Report 3
Case 7: State Card – TEST ID – TAKE14m8 Motor Rpm (Mean) – 2400.5 rpm Type of Specimen – M8 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 1.8 Nm Figure 5.2.7 shows the variation of the acceleration band for an M8 bolted system and the pre-load torque of around 1.8 Nm. No significant variation in the acceleration is observed.
Figure 5.2.7: Acceleration response with the growth rate of 6.1%. The other tests for M8 with 1.8 Nm (take15m8, take16m8) showed similar results with no great change in the acceleration over the period of time. The figures are not presented here to prevent the monotony of the report with the similar results.
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Team 1-C. Report 3
Case 8: State Card – TEST ID – TAKE14m8 Motor Rpm (Mean) – 2400.5 rpm Type of Specimen – M8 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 3.3 Nm Figure 5.2.8 shows the acceleration variation of an M8 bolt with pre-load torque of around 3.3 Nm which is around the range for the required standard torque for an M8 bolt. So, M8 bolt with 3.3 Nm preload torque doesn‟t only prevent loosening of bolts but also prevents the set of sheets to vibrate at accelerations it would otherwise vibrate. However, such an observation can lead to higher stresses as well.
Figure 5.2.8: Acceleration growth factor ~1%. Also the acceleration values on the y axis is much smaller than the other cases. Case 9: State Card – TEST ID – TAKE14m8 Motor Rpm (Mean) – 2400.5 rpm Type of Specimen – M8
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Team 1-C. Report 3
Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 0.02 Nm Figure 5.2.9 shows the variation in the acceleration for the M8 bolt for a very small preload torque. The acceleration was splendidly higher than normal. Infact the acceleration grew over 200% over the period of stipulated time. Infact, one of the bolt was visibly loosened as shown in figure 5.2.10.
Figure 5.2.9: The variation of acceleration envelope for an approximately loosened M8 bolted connection. More than 200% rise in acceleration. Infact, a bolt came out.
Figure 5.2.10: The 4th bolt came out, the complete video in the experiment DVD.
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Team 1-C. Report 3
CASE 10: TEST ID – TAKE26m4 Motor Rpm (Mean) – 2400.5 rpm Type of Specimen – M4 Time of Experimentation – 10 min Washer used (yes/no) - No Pre-load torque – 1.8 Nm Figure 5.2.11 shows the M4 bolt at 1.8 Nm. No significant Variation in the acceleration observed over time as the preload torque 1.8 Nm is the standard preload torque for an M4 bolt and hence the stiffness for the system is maximum for such a case.
Figure 5.2.11: Acceleration response for M4 bolt without washer at 1.8 Nm torque.
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5.3
Variation bolted connection on the basis of the absence of presence of the washer We needed to quantify the effect of washers on the “goodness of the bolted
connections”. We studied the effect of washers on each of the specimen. One of the cases for each of the specimen is plotted here. CASE 11: TEST ID – TAKE28m4 Motor Rpm (Mean) – 2400.5 rpm Type of Specimen – M4 Time of Experimentation – 10 min Washer used (yes/no) – Yes Pre-load torque – 1.8 Nm The effect of washers is clearly visible on the acceleration response. Figure 5.3.1 shows the variation of the acceleration response for the same with not much difference.
Figure 5.3.1: Acceleration Envelope of M4 bolted connection with washers. No effective rise in the acceleration is observed CASE 12: TEST ID – TAKE30m8 Motor Rpm (Mean) – 2431.2 rpm
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Team 1-C. Report 3
Type of Specimen – M8 Time of Experimentation – 10 min Washer used (yes/no) – Yes Pre-load torque – 1.8 Nm The figure 5.3.2 shows the variation of acceleration for an M8 bolt with washer at the preload of 1.8 Nm. No significant change in the acceleration was observed.
Figure 5.3.2: Acceleration response for M8 bolt with washer at the preload torque of 1.8 Nm. CASE 13: TEST ID – TAKE22m6 Motor Rpm (Mean) – 2401.2 rpm Type of Specimen – M6 Time of Experimentation – 10 min Washer used (yes/no) – Yes Pre-load torque – 1.8 Nm A small growth pattern observed for the M6 bolt. The acceleration growth was less than 10% as shown in figure 5.3.3. The slope of the growth is much lesser than that without the washers ((to be compared with figure 5.2.4).
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Team 1-C. Report 3
Figure 5.3.3: Acceleration Envelope for the M6 bolt at 1.8 Nm preload torque.
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6. Discussion 6.1
Effect of Pre-load Torque for various specimen. The pre-load torque was varied on the three specimen (M4,M6 and M8). We have
used the three preload torques for experimentation. The standard pre-load torque for M4 bolted connection (1.8 Nm) and 3.3 Nm and 0.4 Nm, with the latter representing non completely bolted states. We claim that the variation of acceleration for the set of bolted connections with respect to time has to be due to the loosening of the bolts. We claim that the time varying acceleration is not due to the fatigue failure. The matlab code “fatigurecalc” allows us to state whether the variation in the acceleration is due to the change in the acceleration or is it just fatigue. Its visible that for the complete experiment the fatigue failure does not interfere with the loosening of the bolts (figure 6.1.1). Figure 6.1.1 is the result for the TEST ID – take7m6
Figure 6.1.1: The comparison of the maximum cyclic stress generated in the aluminium sheets with respect to the standard aluminium SN curve. Figure 6.1.2 shows another variation of the stress in the aluminium sheet (M8) and the fatigue curve. Its visible that for the complete experiment the fatigue failure doesn‟t interfere with the loosening of the bolts. The figure 6.1.2 is the result for the TEST ID – take18m8.
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Team 1-C. Report 3
Figure 6.1.2: The comparison of the maximum cyclic stress generated in the aluminium sheets with respect to the standard aluminium SN curve. So, the fatigue failure doesnot interfere with the loosening of the bolts. Hence, we can claim that the gradual rise in the acceleration for the figures 5.2.1 – 5.2.11 has been because of the gradual slipping of the bolts and hence the decoupling of the two aluminium sheets. For the case of M6 bolts, two of the three preload torque values (i.e. 0.4 Nm and 1.8 Nm) is below the required pre-loading standard for an M6 bolted connection. For the preload torque of 0.4 Nm, the acceleration of the sheets rises by around 40% for a time span of 600 s, where as it is around 20% for 1.8 Nm preload torque and around 10% for 3.3 Nm torque (see figures 5.2.1-5.2.6). So, the acceleration growth factor is a function of the preload applied on the bolted connection. In case of the M8 bolted connection, unless the preload torque gets very low, it is able to constrain the two bolted sheets comprehensively. For the preload torque of 1.8 Nm and 3.3 Nm, the growth in acceleration over the period of 600 s is below 10%. However for highly loosened bolts, the acceleration rise was significant (around thrice) and infact a bolt was visibly coming out (figure 5.2.6 – 5.2.10). For the m4 bolted connection however, the effect of pre-load torque higher than 1.8 Nm is negligible as 1.8 Nm is the standard preload torque
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Team 1-C. Report 3
which is known to tighten the bolt completely. However, for lesser preload torque, the acceleration rise is significant and infact, visible rotation of the bolts is present (video present in the auxiliary DVD). 6.2
Comparison with theory/numerical model Although, we were unable to characterize the variation in the stiffness of the complete
assembly as the uncertainty band was significant. So, the comparison of stiffness can‟t be done based on the difference between a fixed fixed beam and a cantilever of approximately half a length. We wish to compare the two cases as when the assembly was completely tightened, it could represent a fix-fix beam and after loosening of the bolts, due to deoupling the bolted sheet assembly starts behaving more like a combination of two cantilever beams of half the length. For an approximation we compare the stiffness of a fix-fix beam versus a cantilever. The stiffness of a fixed-fixed beam is given by equation 6.2.1 [11] (6.2.1) where E = Elastic Modulus of the material used. I = Area moment of Inertia of the cross section. = (3L-2a)2 where a and b are according to figure 6.2.1.
Figure 6.2.1: Formula dimensions for a fixed-fixed beam The stiffness of a cantilever of half a length is given by – (6.2.2)
( )
The ratio of the stiffness of a cantilever vs a fixed –fix beam of double the length is given by (a=b=l/2) 47
Team 1-C. Report 3
( )
(6.2.3)
( )
So, the maximum ratio of decoupling is 8 as in equation 6.2.3. It can be hypothesized that the stiffness of the system affects the dynamic acceleration. In fact is known to be proportional to the square root of stiffness. So, the upper limit of the acceleration growth factor had to be
, the level upto
which we actually got (200% rise) when there was loosening of the bolt (figure 5.2.9). 6.3
Effect of washers on the dynamic response The washers, in all the cases be it M4, M6 or M8, have helped in improving the
acceleration response. For the same pre-load, the presence of washers improve the contact stiffness between the bolt and the surface, so the chances of slipping are much less than the cases without the washers. For the cases in figure 5.3.1 and 5.3.2, where no significant rise in the acceleration was observed for a preload of 1.8 Nm. However, in case of M6 (1.8 Nm), the slope of growth of the acceleration is reduced and the acceleration growth factor is also reduced.
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7. Conclusions 7.1
Major conclusions of this work The major conclusions of the work area) The pre-load torque on the bolts play a major role in determining the amount of loosening of the bolt under vibrations. b) The effect of pre-load is different for different size of the bolts. c) The standard loading torques (according to the ASTM standard) should be followed, as the bolts work best for those preloads or greater than that. d) Unoptimally tightened bolts can loosen during vibrations no matter the size of the bolts. e) The bolts change the stiffness of the assembly. f) The dynamic change in acceleration response of a bolted system is not only due to the properties of the material but the nature of the bolted conncetion should also be considered. g) The washers have a positive effect on the behaviour of bolted connections.
7.2
Recommendations for further work The team for the secondary experiment can work in the range of 0.1-0.5 Nm preload
torque to understand how unoptimally bolted bolts behave for various input frequency. The resonance conditions of the complete assembly can also be characterized for the different cases. The team shall also see the behavior of “randomly tightened” bolts and how the system behaves. The presence/absence of washers can be more strictly understoppod by defining more test cases or state cards.
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8. Acknowledgements We would like to acknowledge all the help of all those without whom the project would not have been completed. First of all, we would like to thank Prof. SR Kale for his continuous inputs, advices and his unvaluable critique. We would also like to thank Prof.Naresh Varma Datla, our facilitator who have been helped us in developing the concept of the experiment and executing it. We would also like to acknowledge the help of Mr. Jitendra Kumar, the lab assistant of ME Core Lab who has helped us with all the instruments that we were novice to. We were also like to thank the staff and the Lab assistants of the Central Workshop who helped us to manufacture the setup. Also we would like to thank our teaching assistants of the course whose regular inputs have helped us to put this project to successful completion.
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References [1]
Kumar, Prof. S.R.Satish Kumar and Prof. A.R.Santha. "NPTEL." nptel.ac.in. n.d. (accessed 7 24, 2017).
[2]
Vasudev Upadhye, Mira Mitra, Sauvik Banerjee, Dynamic Characterization of Connections in Plane Frames Using SFFEM, Procedia Engineering, Volume 144, 2016,
Pages
156-161,
ISSN
1877-7058,
http://dx.doi.org/10.1016/j.proeng.2016.05.019. [3]
Hassan Jalali, Hamid Ahmadian, John E. Mottershead, Identification of nonlinear bolted lap-joint parameters by force-state mapping, International Journal of Solids and Structures, Volume 44, Issue 25, 2007, Pages 8087-8105, ISSN 0020-7683, DOI: 10.1016/j.ijsolstr.2007.06.003.
[4]
https://gradeup.co/strain-gages-and-rosettes-i-c7920a83-bea7-11e5-82f3359c139f6ef2, (accessed 8, 1, 2017).
[5]
Nagy, G., "A Review of the “Pugh” Methodology for Design Concept Selection," SAE Technical Paper 940887, 1994, doi:10.4271/940887.
[6]
Menachem P. Weiss, Amihud Hari, Extension of the Pahl & Beitz Systematic Method for Conceptual Design of a New Product, Procedia CIRP, Volume 36, 2015, Pages 254-260, ISSN 2212-8271, DOI: 10.1016/j.procir.2015.03.010.
[7]
Common
Beam
Formulas,
“http://structsource.com/analysis/types/beam.htm”,
accessed 7,31,2017. [8]
Kale,
SR.
"Pre-test
uncertainty
analysis
case
studies."
UNCERTAINTY
ANALYSISPart - VI, 08 2017: 13. [9]
http://en-us.fluke.com/products/power-quality-analyzers/fluke-345-clamp-meter.html, accessed 08,05,2017”.
[10]
http://www.slideserve.com/tamarr/concept-evaluation , accessed (08,02,2017)
[11]
"Shigley's Mechanical Engineering Design", Budynas and Nisbett , 8th ed. ,McGraw Hill, Ch-8, pp - 410-451
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Annexure-I
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Annexure 3-A Engineering drawings
3-A Detailed engineering design (all drawings by Rajesh Kumar)
Figure 3-A-1: The complete assembly drawing of the complete experimental setup made. It excludes the computes systems and other power units and the signal processing units. The drawing is in the units of mm.
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Figure 3-A-2: The exploded labeled assembly of the complete experimental structure excluding the computer setup and the signal processing units
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Figure 3-A-3: Approximate plan view of the complete experimental setup. The complete setup is placed on a table with all the four major structures placed on it.
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Figure 3-A-4: The wiring diagram of all the electrical connections made in the system. The blue colored wires are the signal processing wires and the red colored wires are the power systems except the main power. The main power excites the UPS and all the auxillary systems draw power from the UPS.
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PART DRAWINGS
Figure 3-A-5: The base frame structure to be made of Aluminium cuboidal blocks of primarily two lengths. 4 larger blocks and 8 smaller ones are required. The aluminum structures were joint using the L-brackets.
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Figure 3-A-6: Drawing of the plate specimen. Two such plates are to be required for one specimen. All dimensions in mm.
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Figure 3-A-7: The drawing of the motor holder. It holds the motor together with the sheet specimen and also connects the two sheets via the bolted connections. All dimensions in mm.
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Team 1-C. Report 3
BILL OF MATERIALS . ITEM NO.
Table 3-A-1: The bill of materials for the complete system PART
DESCRIPTION
1.
12 V DC Motor
Buy
2.
240mm
3
300mm
1
Aluminum Aluminium
40*40 blocks
QUANTITY
–
8
–
4
Manufacture
Aluminum Aluminium
40*40 blocks
Manufacture
4
T-bolts
Available in Lab
32
5
L-Brackets
Available in Lab
8
6
C-Clamps
Available in the lab
4
7
Arduino-Uno Board
Available in the lab
1
8
Motor Driver
Available in the lab
1
9
Bolts
M4-M10
4 each specimen
(different specimens have
different
connections) 10
Accelerometer+DAQ
Available in the lab
2
11
Dial Guage
Available in the lab
1
12
Torque Wrench
To tighten the bolts
1
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Team 1-C. Report 3
13
Specimen Plates
Part of the specimen 2 for each specimen (to be cut out) Material – Aluminum
14
Motor Holding Plate
Material – Mild Steel
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1 for each specimen
Team 1-C. Report 3
Annexure 3-B Video of set-up assembly
The experimental setup consists of a base frame made up of aluminium frames and two aluminium sheets bolted via a motor holder. Motor holder plate and the plate specimens joined by the bolts and clamped to the base frame. Two accelerometers are placed thereinThe motor with the unbalance provides the dynamic loads to the bolted sheets. The accelerometers take up the acceleration values.The power to the motor driver comes from the AC to DC converter, which itself takes in the power from the UPS AC input which also gives power to the signal processing unit which processes the signals from the accelerometers and sends it to the system via a USB. The unit also gives the power to the accelerometers. The Arduino gets the power from the computer system itself through the USB which was used to code the Arduino UNO.We have taken 3 different sizes of bolts M4,M6,M8 and prepared a setup
so that it can be clamped and preloaded torque is applied to the assemblyVibration response comes from motor. Motor rotates a unbalanced mass which transmit‟s its vibrations to the system and these vibrations are characterized by acceleration graphs through accelerometers
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Team 1-C. Report 3
Annexure 3-C Pre-test uncertainty analysis In the dynamic characterization of the system, we determine the time varying dynamic properties of the bolted connections. As, for example, the stiffness (
̇ ̈
), were (
̇ ̈ ) are the position, velocity and acceleration states of the system
and Q is the forcing function (the external load). So, we need to determine the uncertainties in all the parameters to get the final pre-test uncertainty. For the dynamic considerations, we use accelerometers for measurement of acceleration. The position and velocity states of the structure are to be found using curve fitting techniques and the subsequent integration of the defined curves. As, we yet do not know about the nature of the output acceleration from the data, it is, hence, difficult to get the approximation of the error due to the curve fits. However, we shall put the errors based on the experiences of other experimentalists. The accelerometers used configuration details suggests the basis of only capabilities of the system. The accelerometer to be used (ADXL 327) gives the uncertainty in acceleration it provides. So, we know the uncertainty values of the acceleration points. The position and velocity states have the same errors as that due to acceleration, but the error due to curve fit needs to be included. We shall aim for a regression of the acceleration upto the root mean square error of 0.5%. Let us assume the regression error to be ε. Uncertainty equations are given in 3.3.1-3.3.6We aim for the root mean square error in the regression to be 0.5%. So, ̇
= 0.0005.
( ̇)
(3-C-1)
( )
(3-C-2)
The uncertainty in the force shall be a combination of the random uncertainty and the standard uncertainty. a) Mass of the Unbalance – Table 3-C-1: Random uncertainty measurements for the mass of the unbalance attached to the motor. S.No
Mass (in g) 2.701
1
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Team 1-C. Report 3
2.751 2 2.732 3 2.750 4 2.733 Mean 0.023 g ̅
(Random
Uuncertainity)
Least count of the Weighing Machine = 0.001 g. So, the systematic uncertainty = 0.001 g. Total Uncertainty = 2.3×10-5 Kg. b) Angular Velocity Measurements – We use the tachometer EAPL DT-2001B to measure the angular velocity of the system. The least count of the tachometer = 0.1 rpm. For the random uncertainty, we use the following measurement table – Table 3-C-2: Random uncertainty measurements for the angular velocity of the motor. S.No
Angular Velocity (in rpm) 2920.1
1 2930.1 2 2930.4 3 2930.2 4 2930.2 5
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Team 1-C. Report 3
2928.2
Mean
̅
4.5
(Random
Uuncertainity) rpm
Least count of the Tachometer = 0.1 rpm So, the systematic uncertainty = 0.1 rpm.
The uncertainty in the measurement of the radius is dependent only on the systematic uncertainty as no difference in the multiple observations in the dimensions was observed once the photograph was taken. So, the uncertainty in the measurement of “r” = 0.0001 m. So, the net uncertainty in the force measurements for a test case= ( )
(
( )
(
(
)
( )
) (
( ) )
( ) = 0.927×10-4 N.
)
at 95% uncertainty level.
- (3-C-3)
So, the uncertainty in the stiffness is around 10% which is a little high due to the crude measurement of the external load. But we cannot put the load cells for the same because of their slower response with respect to the duration of the impact. However, as we wish to know the time taken by the bolted connection to go below a threshold fraction of the stiffness, the error of 10% shall not affect the results on the “Goodness” of the bolted connection.
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Team 1-C. Report 3
Annexure 3-D Video of performing an experiment
From the experiment we basically wish to characterize the goodness of the bolted connections (4M,6M,8M) -with and without washers] under both static as well as the dynamic loading conditions. The dynamic loading is being provided by the motor kept over the assembly with an unbalance. As the motor rotates, a sinusoidal varying force is applied on the connection. The force applied on the assembly is determined as the sinusoidal varying input .As the motor has its self-weight and hence a static load is always applied on the system, the net force input is basically a combination of a static and a dynamic load.Such an attribute of the force can be related to the loading conditions a set of bolted joints in trains or other automobiles also experience. So, we try to examine the goodness of a set of bolted connections for those applicationsThe dynamic response of the system is determined by the acceleration response by the accelerometers, which inherently suggest the stiffness and other dynamic characteristics of the complete assembly.When the dynamic load begins to be applied on the assembly, the bolts hold all the three system together and the acceleration because of the bolt tightly holds the complete system making the complete assembly a fixed-fixed structure.However, due to repetitive loading on the structure, the bolts start to loosen up and then the behaviour of the two beams joined by the bolts, start becoming independent of one-another leading to the reduction in the stiffness of the complete system and hence, an increase in the acceleration is predicted. The experimental setup applies a time varying load to the set of bolted sheets via an unbalance on the motor. The motor unbalance is known to provide a time varying harmonic force to the system.We have to ensure that correct amount of pre-loaded torque is applied on the bolts through torque-wrench. We have to check the bolts of supporting structure so that they remain tightened and unnecessary vibration transmission is avoided. Rpm of the motor is checked through tachometer. Rpm can be increased or decreased though voltage regulation. Sampling rate in DAQ is fixed to be 1Khz and in calibration setting 5g acceleration settings are applied
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Team 1-C. Report 3
Annexure 3-E Post-test uncertainty analysis
In the dynamic characterization of the system, we determine the time varying dynamic properties of the bolted connections. As, for example, the stiffness (
̇ ̈
), were (
̇ ̈ ) are the position, velocity and acceleration states of the system
and Q is the forcing function (the external load). So, we need to determine the uncertainties in all the parameters to get the final pre-test uncertainty. We use the method to calculate the uncertainty given in [8]. For the dynamic considerations, we use accelerometers for measurement of acceleration. The position and velocity states of the structure are to be found using curve fitting techniques and the subsequent integration of the defined curves. As, we yet do not know about the nature of the output acceleration from the data, it is, hence, difficult to get the approximation of the error due to the curve fits. However, we shall put the errors based on the experiences of other experimentalists. The accelerometers used configuration details suggests the basis of only capabilities of the system. The accelerometer to be used (ADXL 327) gives the uncertainty in acceleration it provides. So, we know the uncertainty values of the acceleration points. As, the position and velocity states of the system are a function of the acceleration points for the system, we shall use the same error factors. As the dynamic load put up due to the unbalance of the motor is given by equation 3.2.6, we shall use it for the uncertainty in Q. So, the load measured has the uncertainty of the two components, the uncertainty in the measured current and the uncertainty in the correspondence from the charts. Uncertainty in the Acceleration measurements We use the accelerometer ADXL345 with 16g measurement range. The acceleration response of the accelerometer is based on the uncertainity in the measurement of the acceleration as 1.4%. However, the accelerometer output is analog in nature and hence needs to be converted to digital. The ADC converter used has a 12-bit resolution. The 11 bit is uncertain for the measurement range of 2.5 V input. So, as the calibration factor or the sensitivity of the accelerometer = 256 LSB/g. That means at least first 8 bits are needed to be put into use. The 10 bits need to completely determine the output, so 1LSB = . As the resolution is 256LSB/g, so the uncertainty in measurement =
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Team 1-C. Report 3
. Hence the uncertainty in the measurement of acceleration for the accelerometer, means an uncertainty of 1.4%. It is known the acceleration from the accelerometer ADXL 345 varies due to the temperature difference as well. The error in the measurement of the accelerometer due to the temperature difference is around 0.2% of the total. The random uncertainty in the measurement of the accelerometer was based on holding the accelerometer at a place and measuring the acceleration response for a period of time and then finding the standard deviation and hence the uncertainty in the measurements.
Figure 3.4.1: The random uncertainty in the measurement is based on the standard deviation of the data in the figure. The random uncertainty value is given by – 3 %. The total uncertainty is given by – 3.3% in the acceleration. The position and velocity states have the same errors as that due to acceleration, but the error due to curve fit needs to be included. We shall aim for a regression of the acceleration upto the root mean square error of 0.5%. Let us assume the regression error to be ε. Equations 3.3.2 and 3.3.3 state the uncertainty in position and velocity. The errors in the position as well as the velocity measurements is given by the error in the acceleration measurements as well as the error in the subsequent filtering techniques as well as the error in the numerical integration methodology. As we use the trapezoidal integration technique to calculate the velocity and then the position characteristics of the response. The error in the trapezoidal integration is given by – (
)
(
)
which is very small comparable to the other parts of the
uncertainty so, we neglect this very small value Uncertainty in individual measurements The random uncertainty of the mass of the unbalance part held on the motor
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Team 1-C. Report 3
a) Mass of the Unbalance – Table 3-E-1: Random uncertainty measurements for the mass of the unbalance attached to the motor. S.No
Mass (in g) 2.701
1 2.751 2 2.732 3 2.750 4 2.733 Mean 0.023 g ̅
(Random
Uuncertainity)
Least count of the Weighing Machine = 0.001 g. So, the systematic uncertainty = 0.001 g. Total Uncertainty = 2.3×10-5 Kg. b) Angular Velocity Measurements – We use the tachometer EAPL DT-2001B to measure the angular velocity of the system. The least count of the tachometer = 0.1 rpm. For the random uncertainty, we use the following measurement table –
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Team 1-C. Report 3
Table 3-E-2: Random uncertainty measurements for the angular velocity of the motor. S.No
Angular
Velocity
(in
rpm) 2920.1 1 2930.1 2 2930.4 3 2930.2 4 2930.2 5 2928.2
Mean
̅
4.5 rpm
(Random
Uuncertainity)
Least count of the Tachometer = 0.1 rpm So, the systematic uncertainty = 0.1 rpm.
The uncertainty in the measurement of the radius is dependent only on the systematic uncertainty as no difference in the multiple observations in the dimensions was observed once the photograph was taken. So, the uncertainty in the measurement of “r” = 0.0001 m. So, the net uncertainty in the force measurements for a test case= ( ( ) (
(
)
(
)
(
)
(
)
(
)
( ) = 0.927×10-4 N.
)
)
at 95% uncertainty level.
(3-E-1)
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Team 1-C. Report 3
So, the uncertainty in the stiffness is around 10% which is a little high due to the crude measurement of the external load. But we cannot put the load cells for the same because of their slower response with respect to the duration of the impact. However, as we wish to know the time taken by the bolted connection to go below a threshold fraction of the stiffness, the error of 10% shall not affect the results on the “Goodness” of the bolted connection. The uncertainty in the stress measurement in the rod is given by –
-(3-E-2) ( ) ( )
(
)
( )
( (
)
(
)
( )
)
- (3-E-3) - (3-E-4)
The random uncertainty in the measurements is given by for a test case is given in table 3-E-3. Table 3-E-3: Random Uncertainty of the Length of the sheet Length (in m) S.No 0.2001 1 0.2002 2 0.2001 3 0.2002 4 0.2001 Mean 5.773e-5 ̅
(Random
Uuncertainity)
The random uncertainty measurements in breadth is given by in table 3-E-4.
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Team 1-C. Report 3
Table 3-E-5: Random uncertainty measurements of the breadth of the sheet Breadth (in m) S.No 0.0501 1 0.0501 2 0.0504 3 0.0503 4 0.0502 Mean 0.00012 ̅
(Random
Uuncertainity)
The random uncertainty measurements in height is given by table 3-E-6. Table 3-E-6:Random uncertainty measurements of the breadth of the sheet Height (in m) S.No 0.0011 1 0.0010 2 0.00099 3 0.00098 4 0.00099 Mean 0.000056 ̅
(Random
Uuncertainity) The
net
uncertainty
in
the
measurement
of
the
stress
=
which is very high because of the 10% systematic uncertainty in the measurement of the thickness of the sheets.
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Team 1-C. Report 3
Annexure 3-F MATLAB CODES 1. Code 1 – Plots all the required figures and saves in the current directory. Takes the accelerometer data (should be a 5 cross 1 column matrix, with the first column being the time and the first and the third column as the two acceleration response data). The name is the string which shall be the base name for all the 10 figures to be saved for each acceleration response. The robust code developed gives all the data except the stress analysis which shall be given in the Code 2. function runner1(file,name) take9m627 = file; close all Fs = 1000; Ns = Fs/2; a = take9m627(:,4); L = length(a); t=(1:L)/Fs; f = Fs*(0:L-1)/L; kkk = figure; figure(1) plot(t, a) title('Acceleration'); xlabel('Time(s)'); ylabel('Acceleration(m/s^2)'); figname = strcat(name,'acc'); print('-f1',figname,'-djpeg') savefig(kkk,figname) %% Cumsum Integration a = take9m627(:,4); a = a-mean(a); t = take9m627(:,1); [B,A] = butter(3, 1/Ns,'high'); %% filtr a_filtr = filter(B,A,a); v_cumsum = cumsum(a_filtr*1/Fs); v_cumsum = v_cumsum - mean(v_cumsum); [C,D] = butter(3, 5/Ns, 'high'); v_cumsum = filter(C, D, v_cumsum); d_cumsum = cumtrapz(v_cumsum*1/Fs); d_cumsum = d_cumsum-mean(d_cumsum); [C,D] = butter(3, 10/Ns, 'high'); d_cumsum = filter(C, D, d_cumsum); source.signals.values = take9m627(:,4); source.signals.dimensions = 1; source.time = take9m627(:,1); [m,n] = size(a); q = d_cumsum; qdot = v_cumsum; kkk = figure (2) plot(t,qdot) title('Velocity'); xlabel('Time(s)'); ylabel('Velocity(m/s)'); figname = strcat(name,'vel'); print('-f2',figname,'-djpeg') savefig(kkk,figname) kkk = figure (3) plot(t,q)
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Team 1-C. Report 3 title('Displacement'); xlabel('Time(s)'); ylabel('Displacement(m)'); figname = strcat(name,'disp'); print('-f3',figname,'-djpeg') savefig(kkk,figname) kkk = figure (1) hold on; [qenv,lc] = findpeaks(a_filtr,1000,'MinPeakDistance',1); % qenv = envelope(a,1000); maxt = max(t') plot(lc,qenv,'r','LineWidth',2) ylabel('Acceleration Envelope (in m/s^2)') xlabel('time (in s)') figname = strcat(name,'env'); print('-f1',figname,'-djpeg') savefig(kkk,figname) kkk = figure (4) plot(lc,qenv,'r','LineWidth',2) ylabel('Acceleration Envelope (in m/s^2)') xlabel('time (in s)') figname = strcat(name,'onlyenv'); print('-f4',figname,'-djpeg') savefig(kkk,figname) kkk = figure (6) shadedErrorBar(lc,qenv,0.033.*(qenv)) ylabel('Acceleration Envelope (in m/s^2)') xlabel('time (in s)') title('Envelopes with uncertainty') figname = strcat(name,'envuncer'); print('-f6',figname,'-djpeg') savefig(kkk,figname) Mat1 = 0; Mat2 = 0; K = 0; m=m-1; % qdot = cumtrapz(source.signals.values); % q = cumtrapz(qdot(5:m)); rpm = 2900; w = rpm*6.28/60; r = 0.015; massunbalance = 0.001; for i = 1:m force(i) = w^2*r*sin(w*source.time(i)); end count = 0; t=0; for i = 1000:m-5000 Mat1 = [q(i:i+500,1), qdot(i:i+500,1)]; Mat2 = [force(1,i:i+500)'./10-a(i:i+500,1)]; psudoinvert = pinv(Mat1); res = psudoinvert*Mat2; count = count+1; t(count) = take9m627(i,1); % wn(count) = sqrt(res(1)); c(count) = res(2); K(count) = res(1); end count1=0; for i =1:2000:count-2000 count1 = count1+1; K1(count1) = max(K(i:i+2000)); time(count1) = t(i); end p = max(K1); kkk = figure(5) plot(time,K1) title('Stiffness (N/m)');
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Team 1-C. Report 3 xlabel('Time(s)'); ylabel('Stiffness'); figname = strcat(name,'stiffness'); print('-f5',figname,'-djpeg') savefig(kkk,figname) kkk = figure(7) plot(time,K1/p) title('Normalised Stiffness'); xlabel('Time(s)'); ylabel('Normalised Stiffness'); figname = strcat(name,'normalisedstiffness'); print('-f7',figname,'-djpeg') savefig(kkk,figname) kkk = figure(8) pl = K1/p; shadedErrorBar(time,pl,0.05.*pl) title('Normalised Stiffness with uncertainty'); xlabel('Time(s)'); ylabel('Normalised Stiffness'); figname = strcat(name,'stiffness'); print('-f8',figname,'-djpeg') savefig(kkk,figname) %% close all name = strcat(name,'other') Fs = 1000; Ns = Fs/2; a = take9m627(:,2); L = length(a); t=(1:L)/Fs; f = Fs*(0:L-1)/L; kkk = figure; figure(1) plot(t,a) title('Acceleration'); xlabel('Time(s)'); ylabel('Acceleration(m/s^2)'); figname = strcat(name,'acc'); print('-f1',figname,'-djpeg') savefig(kkk,figname) %% Cumsum Integration a = take9m627(:,2); a = a-mean(a); t = take9m627(:,1); [B,A] = butter(3, 1/Ns,'high'); %% filtr a_filtr = filter(B,A,a); v_cumsum = cumsum(a_filtr*1/Fs); v_cumsum = v_cumsum - mean(v_cumsum); [C,D] = butter(3, 5/Ns, 'high'); v_cumsum = filter(C, D, v_cumsum); d_cumsum = cumtrapz(v_cumsum*1/Fs); d_cumsum = d_cumsum-mean(d_cumsum); [C,D] = butter(3, 10/Ns, 'high'); d_cumsum = filter(C, D, d_cumsum); source.signals.values = take9m627(:,2); source.signals.dimensions = 1; source.time = take9m627(:,1); [m,n] = size(a); q = d_cumsum; qdot = v_cumsum; kkk = figure (2) plot(t,qdot) title('Velocity'); xlabel('Time(s)'); ylabel('Velocity(m/s)'); figname = strcat(name,'vel');
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Team 1-C. Report 3 print('-f2',figname,'-djpeg') savefig(kkk,figname) kkk = figure (3) plot(t,q) title('Displacement'); xlabel('Time(s)'); ylabel('Displacement(m)'); figname = strcat(name,'disp'); print('-f3',figname,'-djpeg') savefig(kkk,figname) kkk = figure (1) hold on; [qenv,lc] = findpeaks(a_filtr,1000,'MinPeakDistance',1); % qenv = envelope(a,1000); maxt = max(t') plot(lc,qenv,'r','LineWidth',2) ylabel('Acceleration Envelope (in m/s^2)') xlabel('time (in s)') figname = strcat(name,'env'); print('-f1',figname,'-djpeg') savefig(kkk,figname) kkk = figure (4) plot(lc,qenv,'r','LineWidth',2) ylabel('Acceleration Envelope (in m/s^2)') xlabel('time (in s)') figname = strcat(name,'onlyenv'); print('-f4',figname,'-djpeg') savefig(kkk,figname) kkk = figure (6) shadedErrorBar(lc,qenv,0.033.*(qenv)) ylabel('Acceleration Envelope (in m/s^2)') xlabel('time (in s)') title('Envelopes with uncertainty') figname = strcat(name,'envuncer'); print('-f6',figname,'-djpeg') savefig(kkk,figname) Mat1 = 0; Mat2 = 0; K = 0; m=m-1; % qdot = cumtrapz(source.signals.values); % q = cumtrapz(qdot(5:m)); rpm = 2900; w = rpm*6.28/60; r = 0.015; massunbalance = 0.001; for i = 1:m force(i) = w^2*r*sin(w*source.time(i)); end count = 0; t=0; for i = 1000:m-5000 Mat1 = [q(i:i+500,1), qdot(i:i+500,1)]; Mat2 = [force(1,i:i+500)'./10-a(i:i+500,1)]; psudoinvert = pinv(Mat1); res = psudoinvert*Mat2; count = count+1; t(count) = take9m627(i,1); % wn(count) = sqrt(res(1)); c(count) = res(2); K(count) = res(1); end count1=0; for i =1:2000:count-2000 count1 = count1+1; K1(count1) = max(K(i:i+2000)); time(count1) = t(i); end
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Team 1-C. Report 3 p = max(K1); kkk = figure(5) plot(time,K1) title('Stiffness (N/m)'); xlabel('Time(s)'); ylabel('Stiffness'); figname = strcat(name,'stiffness'); print('-f5',figname,'-djpeg') savefig(kkk,figname) kkk = figure(7) plot(time,K1/p) title('Normalised Stiffness'); xlabel('Time(s)'); ylabel('Normalised Stiffness'); figname = strcat(name,'normalisedstiffness'); print('-f7',figname,'-djpeg') savefig(kkk,figname) kkk = figure(8) pl = K1/p; shadedErrorBar(time,pl,0.05.*pl) title('Normalised Stiffness with uncertainty'); xlabel('Time(s)'); ylabel('Normalised Stiffness'); figname = strcat(name,'stiffness'); print('-f8',figname,'-djpeg') savefig(kkk,figname)
2. CODE 2 – Gives out the stresses in the sheet for the acceleration response, takes in the acceleration response in the same way as in CODE 1, the sheet dimensions and the base name as the string. Also compares the stress with the characteristic SN curve for Aluminium. function fatigurecalc(take9m627,name,Length,b,h) close all Fs = 1000; Ns = Fs/2; a = take9m627(:,4); L = length(a); t=(1:L)/Fs; f = Fs*(0:L-1)/L; kkk = figure; I = b*h^3/12; E = 70*1e9; %% Cumsum Integration a = take9m627(:,4); a = a-mean(a); t = take9m627(:,1); [B,A] = butter(3, 1/Ns,'high'); %% filtr a_filtr = filter(B,A,a); v_cumsum = cumsum(a_filtr*1/Fs); v_cumsum = v_cumsum - mean(v_cumsum); [C,D] = butter(3, 5/Ns, 'high'); v_cumsum = filter(C, D, v_cumsum); d_cumsum = cumtrapz(v_cumsum*1/Fs); d_cumsum = d_cumsum-mean(d_cumsum); [C,D] = butter(3, 10/Ns, 'high'); d_cumsum = filter(C, D, d_cumsum); source.signals.values = take9m627(:,4); source.signals.dimensions = 1; source.time = take9m627(:,1); [m,n] = size(a);
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Team 1-C. Report 3 q = d_cumsum; qdot = v_cumsum; [qenv,lc] = findpeaks(q,1000,'MinPeakDistance',1); force = (3*E*I/(Length^3)).*qenv; MaxMoment = force.*Length; stress = MaxMoment.*(h/2)/I; stress = medfilt1(stress,1); stress = stress.*1e-6; shadedErrorBar1(lc.*50,stress,0.31.*(stress)) xlabel('No. of cycles') ylabel('Stress (in Mpa)') hold on S = 14479./sqrt(lc.*50) + 96.5; kkk = figure(1); semilogy(lc.*50,S,'linewidth',3) legend('Stress Generated (with uncertainity band)','SN curve for Aluminium') figname = strcat(name,'Stress_fatigue'); print('-f1',figname,'-djpeg') savefig(kkk,figname)
3. CODE 3 – Called by code 1 (Provided by MATLAB, not written by us, is a defined function in future MATLAB versions) function varargout=shadedErrorBar(x,y,errBar,varargin) narginchk(3,inf) params = inputParser; params.CaseSensitive = false; params.addParameter('lineProps', '-k', @(x) ischar(x) | iscell(x)); params.addParameter('transparent', true, @(x) islogical (x) || x==0 || x==1); params.parse(varargin{:}); lineProps = params.Results.lineProps; transparent = params.Results.transparent; if ~iscell(lineProps), lineProps={lineProps}; end if iscell(errBar) fun1=errBar{1}; fun2=errBar{2}; errBar=fun2(y); y=fun1(y); else y=y(:).'; end if isempty(x) x=1:length(y); else x=x(:).'; end if length(errBar)==length(errBar(:)) errBar=repmat(errBar(:)',2,1); else s=size(errBar); f=find(s==2); if isempty(f), error('errBar has the wrong size'), end if f==2, errBar=errBar'; end end if length(x) ~= length(errBar) error('length(x) must equal length(errBar)') end H.mainLine=plot(x,y,lineProps{:}); col=get(H.mainLine,'color'); edgeColor='r';
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Team 1-C. Report 3 patchSaturation=0.15; if transparent faceAlpha=patchSaturation; patchColor='r'; else faceAlpha=1; patchColor='r'; end uE=y+errBar(1,:); lE=y-errBar(2,:); holdStatus=ishold; if ~holdStatus, hold on,
end
yP=[lE,fliplr(uE)]; xP=[x,fliplr(x)]; xP(isnan(yP))=[]; yP(isnan(yP))=[]; H.patch=patch(xP,yP,1,'facecolor',patchColor, ... 'edgecolor','none', ... 'facealpha',faceAlpha); H.edge(1)=plot(x,lE,'-','color','r'); H.edge(2)=plot(x,uE,'-','color','r'); uistack(H.mainLine,'top') if ~holdStatus, hold off, end if nargout==1 varargout{1}=H; end
4. CODE 4 – Called by code 2 (Provided by MATLAB, not written by us, is a defined function in future MATLAB versions) function varargout=shadedErrorBar1(x,y,errBar,varargin) narginchk(3,inf) params = inputParser; params.CaseSensitive = false; params.addParameter('lineProps', '-k', @(x) ischar(x) | iscell(x)); params.addParameter('transparent', true, @(x) islogical (x) || x==0 || x==1); params.parse(varargin{:}); lineProps = params.Results.lineProps; transparent = params.Results.transparent; if ~iscell(lineProps), lineProps={lineProps}; end if iscell(errBar) fun1=errBar{1}; fun2=errBar{2}; errBar=fun2(y); y=fun1(y); else y=y(:).'; end
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Team 1-C. Report 3
if isempty(x) x=1:length(y); else x=x(:).'; end if length(errBar)==length(errBar(:)) errBar=repmat(errBar(:)',2,1); else s=size(errBar); f=find(s==2); if isempty(f), error('errBar has the wrong size'), end if f==2, errBar=errBar'; end end if length(x) ~= length(errBar) error('length(x) must equal length(errBar)') end H.mainLine=semilogy(x,y,lineProps{:}); col=get(H.mainLine,'color'); edgeColor='r'; patchSaturation=0.15; if transparent faceAlpha=patchSaturation; patchColor='r'; else faceAlpha=1; patchColor='r'; end uE=y+errBar(1,:); lE=y-errBar(2,:); if ~holdStatus, hold on,
end
yP=[lE,fliplr(uE)]; xP=[x,fliplr(x)]; xP(isnan(yP))=[]; yP(isnan(yP))=[]; H.patch=patch(xP,yP,1,'facecolor',patchColor, ... 'edgecolor','none', ... 'facealpha',faceAlpha); H.edge(1)=semilogy(x,lE,'-','color','r'); H.edge(2)=semilogy(x,uE,'-','color','r'); uistack(H.mainLine,'top') if ~holdStatus, hold off, end if nargout==1 varargout{1}=H; end
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