Vehicle Road Test Simulation

A thesis submitted to the

Division of Research and Advanced Studies

of the University of Cincinnati

in partial fulfillment of the

requirements for the degree of

Master of Science

in the Department of Mechanical, Industrial and Nuclear Engineering

of the College of Engineering

2005

by

Udayan N. Godbole

Committee Chair: Dr. Randall J. Allemang

Abstract

There has been considerable interest among researchers in being able to recreate desired

time or frequency domain responses on a vehicle using a Road Simulator. Accurate

simulation of road tests facilitates an in depth study of a test vehicle’s dynamic response

characteristics and paves the way for future research studying non-linear system

responses.

Currently available commercial software can successfully simulate most collocated and

some non-collocated input-output configurations, but there is no means of modifying the

system modeling techniques used. Future research would aim to develop complex system

models that would either enhance the capabilities of the existing linear system models

that are in use or completely replace them. This research provides a Matlab based

research platform by way of a Road Simulator software interface, which performs most

of the functions that a commercial Road Simulator would, and provides complete access

to the system development process for custom modifications required by future

researchers. The capabilities of the developed simulator are examined through tests

conducted on a vehicle chassis, and the groundwork for future research, particularly

involving non-linear systems, is laid out.

Acknowledgement

I would like to thank my research advisor, Dr. Randall J. Allemang for his constant and

timely support throughout the duration of this research. The successful completion of this

work would not have been possible without his considerable involvement in the same.

His vast experience in the field of Structural Dynamics has been extremely useful to me

throughout my academic life at the SDRL.

I thank Dr. David L. Brown and Dr. Allyn W. Phillips for serving on my thesis

committee, for their appraisal of my work, and also for their support and encouragement.

To all my colleagues at the SDRL, particularly Ray Martell and Mike Spottswood, I

express my gratitude for the valuable discussions and constructive inputs towards my

thesis research. I also greatly appreciate Dave Breheim’s help with the timely setup and

maintenance of the test rig used for the research and Larry Schartman’s assistance with

system maintenance issues.

Amit, Aniruddha, Matin and Rohit have been very supportive throughout my effort, and I

cannot thank them enough for it.

Last, but not the least, I am grateful to my family for being there for me throughout my

life, and I owe my success to their blessings and support.

Table of Contents 1 INTRODUCTION ...................................................................................................... 6

1.1 Overview of the MTS 320 Road Simulator ........................................................ 7 1.1.1 Actuators ..................................................................................................... 7 1.1.2 Hydraulic Pump .......................................................................................... 8 1.1.3 320 Road Simulator Controller................................................................... 8

1.2 Benefits of using a Road Simulator .................................................................... 9 1.2.1 Instrumentation ........................................................................................... 9 1.2.2 Repeatability ............................................................................................... 9 1.2.3 Safety aspects............................................................................................ 10 1.2.4 Deterministic Testing................................................................................ 10 1.2.5 Flexibility.................................................................................................. 10

1.3 Disadvantages of using a Road Simulator ........................................................ 11 1.3.1 Effects of Vehicle Dynamics ................................................................... 11 1.3.2 Tire-Road surface contact ......................................................................... 11 1.3.3 Input Axes................................................................................................. 11

1.4 Research Goals.................................................................................................. 12

2 BACKGROUND INFORMATION ......................................................................... 13

2.1 Simulation Process Overview........................................................................... 16 2.2 RPC III File Format .......................................................................................... 21

2.2.1 Header ....................................................................................................... 21 2.2.2 Data ........................................................................................................... 21

3 THEORY .................................................................................................................. 22

3.1 Generating White Noise Drive File .................................................................. 22 3.1.1 White Noise Parameters............................................................................ 22

3.2 Developing System Matrices ............................................................................ 23 3.2.1 Data Processing......................................................................................... 24

3.2.1.1 Block Size ............................................................................................. 24 3.2.1.2 Window functions................................................................................. 24 3.2.1.3 Cyclic Signal Averaging ....................................................................... 26

3.2.2 FRF Estimation Algorithms...................................................................... 34 3.2.3 Ordinary Coherence Function................................................................... 38 3.2.4 Multiple Coherence Function ................................................................... 38 3.2.5 Inverse System Model............................................................................... 40

3.3 Desired Responses ............................................................................................ 42 3.3.1 Time Domain ............................................................................................ 42 3.3.2 Frequency Domain.................................................................................... 42

3.4 Convolution with Overlap Adding.................................................................... 42 3.4.1 Estimate Inverse FRF................................................................................ 42 3.4.2 Convolution with overlap adding.............................................................. 43

4 TEST SETUP............................................................................................................ 54

4.1 Wheel Input with Collocated Output ................................................................ 55

1

4.2 Wheel Input with Non-collocated Output......................................................... 56

5 RESULTS ................................................................................................................. 57

5.1 Collocated ......................................................................................................... 57 5.1.1 Time Domain Desired Response .............................................................. 57 5.1.2 Frequency Domain Desired Response ...................................................... 64

5.2 Non-collocated.................................................................................................. 73 5.2.1 Time Domain Desired Response .............................................................. 73

6 CONCLUSIONS AND FUTURE WORK ............................................................... 79

6.1 Conclusions....................................................................................................... 79 6.2 Future Work ...................................................................................................... 80

Appendix A RPC III FILE FORMAT........................................................................................... 83

A.1 Header ............................................................................................................... 83 A.2 Data ................................................................................................................... 85

B MATLAB FUNCTIONS .......................................................................................... 87

B.1 fnHlin: Calculates FRF, Coherence, Power Spectra......................................... 87 B.2 convOver: Convolution with Overlap Addition ............................................... 92 B.3 bandpass : Bandpass filter................................................................................. 93

C WHITE NOISE GENERATOR GUI........................................................................ 94

C.1 DRV File Creator.............................................................................................. 94 C.2 DRV File Creator Options ................................................................................ 94

C.2.1 Header Data .............................................................................................. 94 C.2.2 Channel Data............................................................................................. 95

D ROAD SIMULATOR GUI....................................................................................... 97

D.1 Project ............................................................................................................... 97 D.1.1 New Project: Create a New Project .......................................................... 97 D.1.2 Open Project: Open Existing Project ........................................................ 97 D.1.3 Save Project: Save Current Project ........................................................... 97

D.2 Develop System Model..................................................................................... 98 D.2.1 White Noise Generator ............................................................................. 98 D.2.2 System Model ........................................................................................... 98 D.2.3 Display System Model.............................................................................. 98

D.3 Desired Response.............................................................................................. 99 D.3.1 Select Desired Response ........................................................................... 99 D.3.2 Plot Desired Response .............................................................................. 99

D.4 Generate Drive File......................................................................................... 100 D.4.1 Calculate Drive ....................................................................................... 100 D.4.2 Write Drive File ...................................................................................... 100

D.5 View Results ................................................................................................... 101 D.5.1 Time Domain .......................................................................................... 101

2

D.5.2 Frequency Domain.................................................................................. 101 D.5.3 Plot Error................................................................................................. 101 D.5.4 Plot Trend................................................................................................ 101 D.5.5 Document Results ................................................................................... 101

3

Table of Figures

Figure 1 : MTS 320 Road Simulator ........................................................................... 7 Figure 2 : Simulation Process Overview................................................................... 16 Figure 3 : Flowchart – Generating System Matrices................................................. 19 Figure 4 : Flowchart – Iteratively Generating Drive Files ........................................ 20 Figure 5 : Hanning Window and its effect ................................................................ 25 Figure 6 : Flowchart - Cyclic Averaging Process ..................................................... 27 Figure 7 : Raw Time Domain Data ........................................................................... 28 Figure 8 : Windowed Time Data – No Overlap ........................................................ 29 Figure 9 : Cyclic Averaged Time Signal................................................................... 30 Figure 10 : Frequency Domain Signal after Cyclic and Asynchronous Averaging .... 30 Figure 11 : Raw Time Domain Data ........................................................................... 31 Figure 12 : Windowed Time Data – 50% Overlap of Time Signal between averages 32 Figure 13 : Cyclic Averaged Time Signal................................................................... 33 Figure 14 : Frequency Domain Signal after Cyclic and Asynchronous Averaging .... 33 Figure 15 : No Input Noise FRF Model: H1(ω) Algorithm......................................... 34 Figure 16 : Plot of H1(ω) FRF Estimate ...................................................................... 37 Figure 17 : Plot of Multiple Coherence....................................................................... 39 Figure 18 : Plot of Inverse System Model................................................................... 41 Figure 19 : Flowchart - Modifying System Matrix for Convolution........................... 44 Figure 20 : Flowchart - Convolution with Overlap Adding ........................................ 45 Figure 21 : Impulse Responses obtained from System FRFs...................................... 46 Figure 22 : Impulse Responses with Half Frame Circular Shift and Zero Padding .... 46 Figure 23 : Modified FRFs .......................................................................................... 47 Figure 24 : Desired Responses – single frame of data ................................................ 47 Figure 25 : Desired Responses – single frame with zero padding .............................. 48 Figure 26 : Desired Responses – Frequency Domain ................................................. 48 Figure 27 : Estimated Input Signals – Frequency Domain.......................................... 49 Figure 28 : Estimated Input Signals – Time Domain .................................................. 49 Figure 29 : Convolution Results for 5 Data Frames for Channel 1............................. 50 Figure 30 : Convolution Results for 5 Data Frames for Channel 2............................. 50 Figure 31 : Convolution Results for 5 Data Frames for Channel 3............................. 51 Figure 32 : Convolution Results for 5 Data Frames for Channel 4............................. 51 Figure 33 : Convolution Results.................................................................................. 52 Figure 34 : Calculated Input Signals after Overlap Addition of Convolved Time

Signals shown for all 4 Channels.............................................................. 52 Figure 35 : Input Signals after Ignoring First and Last Half Frames .......................... 53 Figure 36 : Accelerometer location for Collocated Configuration.............................. 55 Figure 37 : Accelerometer location for Non-Collocated Configuration ..................... 56 Figure 38 : Variation of Standard Deviation of Error with subsequent Iterations ...... 59 Figure 39 : Measured Response – Collocated Configuration: Initial Estimate ........... 60 Figure 40 : Measured Response – Collocated Configuration: Iteration 5 ................... 61 Figure 41 : Collocated – Initial Error .......................................................................... 62

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Figure 42 : Collocated – Final error ............................................................................ 63 Figure 43 : Desired Power Spectral Profile................................................................. 65 Figure 44 : Random Signal with Unit Amplitude ....................................................... 66 Figure 45 : Modified Desired Response Spectra......................................................... 67 Figure 46 : Variation of Standard Deviation of Error with subsequent Iterations ...... 68 Figure 47 : Measured Response – Collocated Configuration: Initial Estimate ........... 69 Figure 48 : Measured Response – Collocated Configuration: Iteration 5 ................... 70 Figure 49 : Collocated – Initial Error .......................................................................... 71 Figure 50 : Collocated – Final Error............................................................................ 72 Figure 51 : Variation of Standard Deviation of Error with subsequent Iterations ...... 74 Figure 52 : Measured Response – Non-Collocated Configuration: Initial Estimate... 75 Figure 53 : Measured Response – Non-Collocated Configuration: Iteration 5........... 76 Figure 54 : Non-collocated – Initial Error................................................................... 77 Figure 55 : Non-collocated – Final Error .................................................................... 78 Figure 56 : Four Channel Time Data........................................................................... 85 Figure 57 : Four Channel Demultiplexed Time Data.................................................. 86 Figure 58 : White Noise Generator GUI ..................................................................... 96 Figure 59 : Road Simulator GUI ............................................................................... 102

5

1 Introduction

There has been considerable interest among researchers in the simulation of vehicle road

tests on a road simulator. Knowing the responses measured on the vehicle during an

actual road test, it is desirable to recreate the same responses on the vehicle using the

road simulator. It is also useful to have the capability of creating custom drive signals to

be input to the simulator that would generate the desired time domain or frequency

domain responses at predetermined locations on the vehicle. Road test simulation has

potential advantages over actual road tests which will be discussed later. The Non-linear

Dynamics Test Facility, a part of the University of Cincinnati’s Structural Dynamics

Research Laboratory (UC-SDRL), has the MTS 320 Road Simulator, which is capable of

simulating road tests on vehicles, and was used for this particular research.

6

1.1 Overview of the MTS 320 Road Simulator

The MTS 320 Road simulator is shown below.

Figure 1 : MTS 320 Road Simulator

The simulator essentially consists of the following components:

• Actuators

• Hydraulic Pump

• 320 Road Simulator Controller

1.1.1 Actuators

There are four hydraulic actuators, one for each wheel of the vehicle. The hydraulic

actuators can impart motion to the vehicle tires in the vertical direction, similar to road

inputs to the vehicle, and have a maximum stroke of ±3 inches from the mean position.

7

The displacements correspond to voltage signals between ±10 volts. Controlling the

voltage signals controls the position of the actuator in time, and can be used to generate

various drive signals to the vehicle. The drive signal is digitized into 216 data levels (16

bit ADC). The actuators cannot simulate very low frequencies, below 2 Hz, due to the

limited stroke of the wheel pan and piston assembly. On the higher side, the maximum

allowable frequency is around 80 Hz. This is limited by the natural frequency of the

actuator and the oil column in the actuator.

1.1.2 Hydraulic Pump

The MTS 506 Hydraulic Power Supply controls the actuators. The specifications for the

power supply in the UC-SDRL facility are as follows:

Model Number 506.52C

Supply Rating 460 V, 60 Hz, 3 Phase

Working Pressure 3000 psi, 207 bar

Flow 55 gpm, 209 lpm

1.1.3 320 Road Simulator Controller

It consists of the MTS 498.22 Test Processor and the MTS 497.05 Hydraulic Control

Unit. The test rig is defined as the four hydraulic actuators with the test vehicle mounted

on it. The Test Processor handles the actual control of this rig, the digitizing of the drive

signals, communications between the controller and the test rig, and the data acquisition

system. The Hydraulic Control handles the hydraulic lines and flow valves. A software

user interface (Flex Test II) provided with the controller enables control of the test rig

using a commercially available PC.

8

1.2 Benefits of using a Road Simulator

The aim of conducting a vehicle road test is to understand the dynamic behavior of the

vehicle, and to study and characterize its vibration response to road inputs, during actual

operating conditions. With this aim in mind, simulating road tests offers certain

advantages over conducting actual road tests and they are as follows:

1.2.1 Instrumentation

During an actual road test, only a limited amount of instrumentation of the vehicle may

be possible due to constraints imposed by available measurement locations and the

portability of the data acquisition system used. The larger the number of response

sensors, the larger the data acquisition channels required, and consequently, the bulkier

the data acquisition system. These limitations can be overcome in a laboratory test, where

a more comprehensive setup can be used, as the data acquisition system does not have to

be portable. Also, more response measurement locations may be available, thus enabling

collection of spatial response data across non-linearities to better characterize the system.

1.2.2 Repeatability

Each test on a test track will differ from the previous one, due to changes in the test

conditions, namely changes in vehicle driving speeds, road surface conditions, road track

and driver errors. These can be eliminated in a laboratory setup, where the difference

between two tests using the same input drive signals can be negligible, and the test is

practically repeatable.

9

1.2.3 Safety aspects

Certain tests may involve risks of failure of vehicle components. It would be undesirable

to have a test driver perform such tests, and a road simulator would be well utilized in

such cases.

1.2.4 Deterministic Testing

Certain modes of vibration may be of particular interest to the test engineer, and it may

be possible to excite that particular mode using suitable drive signals on the simulator.

This would particularly find applications in the troubleshooting of problems caused by

excessive vibration, or in the study of the suspension system characteristics at specific

frequencies.

1.2.5 Flexibility

The inputs to the vehicle can be changed at will, which can enable testing of the vehicle

by uniformly increasing or decreasing the input amplitudes of the drive signal, or by

varying individual channels (tire inputs) while keeping the others the same, and other

desired variations. This is obviously not possible on a test track, and thus simulation

offers more flexibility as regards the inputs that can be given to the vehicle.

10

1.3 Disadvantages of using a Road Simulator

Simulation on such a test rig has certain limitations which need to be understood while

interpreting the results obtained for a simulated test. They are as follows:

1.3.1 Effects of Vehicle Dynamics

Since the vehicle has no driving velocity on the test rig, any effects that the vehicle drive

speed has on its dynamic behavior are not simulated, for example the conditions of

sudden acceleration or braking may not be accurately represented. The engine and the

wheels are idle, and the inputs to the vehicle due to their operation (namely engine

vibrations and tire rotations) are not simulated.

1.3.2 Tire-Road surface contact

The tires are assumed to be in contact with the road at all times. This assumption would

normally be true on a highway drive. However, there may be times during the actual road

test where the tire may momentarily loose contact with the road. The tires are tied down

to the actuators, so the loss of contact case is never simulated.

1.3.3 Input Axes

The inputs to the vehicle on a road simulator are always vertical. Other angles are not

possible, so oblique inputs to the vehicle cannot be simulated.

Despite these limitations, the road simulator can be considered to be sufficiently accurate

for the purpose of simulating various highway driving conditions and for many dynamic

simulation goals.

11

1.4 Research Goals

To summarize, the research goals may be stated as follows:

• To understand the current road test simulation techniques used

• To develop a Matlab based interface capable of handling the complete simulation

process

• To identify possible modifications to the system development process to enable

simulation of non-collocated and non-linear systems

• To be better able to generate and analyze input signals for non-linear testing

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2 Background Information

Although commercially available software packages (example RPC III [4] by MTS

Systems Corporation, SIMTEST [5] by Simulation Techniques Inc.) are currently able to

achieve a road test simulation with a high degree of accuracy, these packages are “black

box” models, which give the engineer a general idea of their inner workings, but are

rarely configurable for advanced applications. Most software packages assume a linear

system model and achieve the desired response through successive iterations of the drive

signal. It is quite obvious that as most practical systems have varying degrees of non-

linearities, this approach would have practical limitations as to the variety of

configurations that it can successfully simulate. With advances in non-linear modeling

techniques [10-12], it may be possible to develop complex system models, thereby

enabling the simulation of complicated configurations while reducing the number of

iterations required to reach a satisfactory result. This would require access to the system

development process and modifications would have to be made to it for handling the

additional system dynamics. The goal of this thesis is to provide the researcher with a

clear understanding of the currently used method, evaluate its validity for some practical

configurations, review its limitations and provide a well documented platform to build

further research work on.

Simulation essentially means recreating desired responses at predetermined locations on

the vehicle. These locations may be either collocated or non-collocated with respect to

the input locations. Collocated systems can be conveniently modeled by the prevalent

method of using a linear estimate of the system, and successive iterations of the drive file

are needed to achieve the desired response. However, as the configuration used becomes

13

a non-collocated one, this approach may not yield the desired results, chiefly due to the

effects of the increased system dynamics on the measured response. As the path between

the response measurement location and the input location gets more complicated, the

accuracy of the simulation correspondingly decreases. There may be a point beyond

which the additional system dynamics play such a big role in the response measurement

that any practical simulation based on linear system estimates becomes impossible.

Future research work would try to develop a more robust method to characterize the

complex system dynamics involved, thus making it possible to simulate far more

complicated configurations than those currently possible. The Non-linear Input Feedback

of the Outputs (NIFO) [11] technique is one of the suggested approaches for handling

non-linear system identification and characterization.

The standard configurations used have four response locations corresponding to four

inputs to the vehicle at the wheels. The input locations are thus, fixed. For the collocated

case, the response locations are near the inputs (e.g. the axles), whereas for the non-

collocated case, the response locations may be elsewhere on the vehicle (e.g. the chassis

frame).

To estimate the drive signals required at the input locations corresponding to a particular

known response, an estimate of the system’s dynamic characteristics is first required. The

goal here is to determine a frequency domain relationship, between input drive signals (in

the form of input voltage levels) and the output acceleration response (measured by the

accelerometers), over the frequency band of interest. The estimate of the system’s

dynamic characteristics takes the form of the Frequency Response Function (FRF) [2],

which is a linear estimate of the system. The drive signals required to generate a specific

14

response can be estimated using a convolution of the known (desired) response signals

with the inverse system model, and transforming the calculated drive signals back into

the time domain.

Commercially available software [4-5] was studied to get a clear understanding of the

current industrial practices in simulating road tests, and for the sake of consistency, a

similar approach was used while developing the Road Simulator software.

15

2.1 Simulation Process Overview

A schematic diagram of the control algorithm is shown below [5], [13]

Desired Response

Xd

System Model Measured Response +

Differential Drive Error -

H-1 Xa

k Gain

Test Rig and Control Hardware

+

+ Fn

Previous Drive

Fn+1

New Drive

where: Xd = Desired Response Xa = Measured Response H-1 = System Model k = Gain Fn = Previous Drive Fn+1 = New (Corrected) Drive

Figure 2 : Simulation Process Overview

16

The procedure for simulation of desired responses can be outlined as follows:

• Determine the system matrices using a white noise excitation over the frequency

range of interest. As a linear system matrix is to be derived, a low level excitation

is employed.

• Measure system response to the white noise excitation, at the desired response

locations, either collocated or non-collocated with respect to the inputs.

• Generate system matrices based on the known input voltages (Drive) and

measured output accelerations (Response). These system matrices are linear

estimates of the system.

• Using the desired response at the measurement locations and the generated system

matrices, calculate the drive signal required. This is the initial estimate of the

drive signal.

• Measure the system response to the initial drive signal estimate. For a purely

linear system, the measured response should exactly match the desired response.

However, as most practical systems have varying degrees of nonlinearities, this is

almost never true, and the desired response can only be obtained through

successive iterations of the drive signal.

• Compare measured response with the desired response. The difference is the

error, which needs to be reduced iteratively. Calculate the differential drive

required to minimize the error, and add this to the initial estimate of the drive, to

obtain a modified/corrected drive signal.

• Measure the system response to this modified drive signal.

17

• Repeat until the error lies within acceptable limits. The acceptable error may be a

specified allowable maximum error over the entire measurement time for the time

domain, or over the frequency range of interest for the frequency domain.

• Write out the final drive signal to file. This file contains the input signals required

to be given to the system, in order to get the desired response. Input signals are in

the form of time base input voltage levels that are to be fed to the hydraulic

actuators of the MTS 320 Road Simulator. Since this file can be played out any

number of times, it can serve as a repeatable input to the system.

18

Flowchart: Generating System Matrices

Generate White Noise Drive File

Send to MTS 320 Road Simulator

Measure Response to Drive Signal

Calculate a linear Frequency Response Function relating the

following two data sets:

Input: Generated White noise Drive (Voltages)

Output: Measured Response

(Acceleration)

Are System Matrices Satisfactory?

No Yes

Store System Matrices for later use

Figure 3 : Flowchart – Generating System Matrices

19

Flowchart: Iteratively Generating Drive Files

x1NNx N-1

x1N oioi [X] [H] [F] =

Select Desired Response

Calculate Initial Drive based on Desired Response and Calculated Frequency Response Function using the following formula:

Where: [H]-1 = Inverse System FRF X = Desired Response F = Required Drive No = Outputs Ni = Inputs

Send to MTS 320 Road Simulator

Calculate the Differential

Drive required to correct the error and add it to the Initial Estimate

to get the corrected Drive

signal

Measure Response to Drive Signal

Compare Measured Response with Desiredresponse and Calculate

Error

Is Error within acceptable

limits? No Yes

Write out Final Drive File

Figure 4 : Flowchart – Iteratively Generating Drive Files

20

2.2 RPC III File Format

RPC is an acronym for Remote Parameter Control. For a detailed look at the header and

data content of an RPC III file, refer to the Appendix A. This format is needed in order to

send the data in the proper form to the MTS Flex Test II software which then drives the

actuators.

A standard RPC III Format file consists of the following two parts:

• Header

• Data

2.2.1 Header

The header part contains ASCII format keyword-value pairs, which specify details about

the data, namely its type, the length of data, the number of channels, the time interval

between consecutive data points, the scaling and all other information pertaining to

correct interpretation of the stored data values.

2.2.2 Data

RPC III places the data in time history files in groups, and it divides the groups into

frames. Frame sizes can range from 256 to 16384 data points, while group sizes can

range from 2048 to 16384 data points. This grouping is referred to as demultiplexed

blocks. The data stored is generally in the little Endian binary format, which means that

the low-order byte of the number is stored in the memory at the lowest address.

21

3 Theory

3.1 Generating White Noise Drive File

White noise is a signal having a frequency spectrum that is continuous and uniform over

the entire frequency range. A band limited white noise signal has a continuous and

uniform frequency spectrum over the specified frequency band. For a detailed

mathematical understanding of white noise, refer [14].

A white noise signal can be generated by using a random number generator, and applying

a bandpass filter to the generated signal that specifies the allowable frequency content of

the signal, and filters out the rest.

3.1.1 White Noise Parameters

For the MTS 320 Road simulator, the input drive file contains a series of voltage levels

that drive the hydraulic actuators. While generating white noise, the Matlab ‘randn’

function is used to generate a white noise signal. A ‘bandpass’ filter, [Appendix B.3] is

applied to it, which allows all frequencies lying within the specified frequency band, and

filters out the rest. The signal is then scaled to the specified amplitude, and the

corresponding voltage levels are stored in an RPC III .DRV file. Typical parameters for a

white noise signal would be:

Parameter Units Sample Value Allowable Values

Amplitude inch 0.1 0-3 inch

Lower Frequency Hz 0 ≥ 0 Hz

Upper Frequency Hz 60 ≤ 80 Hz

Offset inch 0 0-3 inch

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3.2 Developing System Matrices

The system is characterized by the Frequency Response Function (FRF). The inverse of

the FRF serves as the inverse system model used for the simulation. The FRF is a

frequency by frequency relationship between the output response and the input force. For

the current simulation process, the FRF relates the measured acceleration response to the

input voltage signal, in the frequency domain. The data stored by the data acquisition

system is raw time domain data. This data has to be processed before performing a

Fourier transform to convert it to the frequency domain. This is a necessary step, and

failure to do so would result in substantial errors in the estimate of the FRF. The Fast

Fourier Transform (FFT) is used for the time-frequency conversion of the data. Certain

requirements have to be satisfied by the time domain data in order to obtain valid results

after using the FFT. They are as follows:

• The time signal should be a totally observed transient, with respect to the time

period of observation OR

• The signal must be composed only of harmonics of the time period of observation

Bias errors (leakage) will result in the frequency domain, if one of these conditions is not

achieved. A complete description of the bias errors and ways to minimize them can be

found in many texts [1-2]. The following data processing techniques try to minimize

leakage errors in the frequency domain estimates by assuring that the input and output

time histories satisfy at least one of the above conditions.

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3.2.1 Data Processing

3.2.1.1 Block Size

The block size [1] determines the number of spectral lines in the baseband measurement.

Increasing the block size gives a finer frequency resolution, consequently resulting in a

reduction of the errors during the Fourier transformation of data to the frequency domain.

However, a larger number of spectral lines increases both the observation time and the

signal processing time and a compromise has to be reached. Generally, for road

simulation, 512 to 1024 lines should provide sufficient accuracy without compromising

the speed of calculations.

3.2.1.2 Window functions

Weighting functions [1], or windows, are commonly used with time data, to ensure that

the time signal appears as a totally observed transient before being processed by the FFT.

Although the use of windows generally gives good results, there may be other methods of

ensuring the observability of the signal, which may lead to superior results, and these

should be looked into first.

24

The popular Hanning window and its effect on raw time domain random data is shown

below.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1Hanning Window

Am

plitu

de

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.1

-0.05

0

0.05

0.1Raw Time Data

Am

plitu

de

Time (s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.1

-0.05

0

0.05

0.1

Time (s)

Windowed Time Data

Am

plitu

de

Figure 5 : Hanning Window and its effect

25

3.2.1.3 Cyclic Signal Averaging

Cyclic signal averaging [1], [9], and [14] is a special case of linear averaging, with the

added constraint that the digitization is coherent between averages. The exact time

between each average is used to enhance the signal averaging process. The cyclic

averaging process acts as a comb filter, with the teeth spaced at frequency increments that

are integer multiples of the ∆f = 1/T relationship. Thus it enhances the frequencies that

are integer multiples of ∆f and averages out the others. The cyclic averaged signal is

better suited for the FFT operation and yields good results.

26

The following flowchart should clarify the cyclic averaging process

aver ata

Figure 10

Asynchronous averaging of the Cyclic aged blocks to get final averaged time d

Transform to frequency domain by employing the FFT

algorithm

Figure 7

Figure 10

Figure 9

Figure 8

No Yes

Select next block, with or

without overlap, of

length (Nc*bs)

Any more data blocks?

Perform Nc cyclic averages to get one Cyclic averaged block

of time T

Apply Hanning Window to the block

Select first block of data of length (Nc*bs)

Number of Cyclic Averages (Nc)

Block Size (bs)

Raw Time Domain data

Figure 6 : Flowchart - Cyclic Averaging Process

27

The following series of figures represents the cyclic averaging process, with frequency

domain asynchronous averaging.

Parameter Value

White Noise Signal Amplitude 0.1 units

White Noise Signal Frequency Range 5-204.8 Hz

Sampling Frequency (Fs) 512 Hz

Block Size (bs) 512

Number of Cyclic Averages (Nc) 5

Percentage Overlap of Time signal 0 %

Number of Asynchronous Averages (Na) 2

Raw Time Data

0 1 2 3 4 5 6 7 8 9 10-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1Raw Time Data

Time (s)

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Figure 7 : Raw Time Domain Data

28

Windowed Time Data (no overlap between data sets)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.1

0

0.1Raw Time Data

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.1

0

0.1Windowed Time Data with 0% Overlap between Records

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.1

0

0.1

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1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9-0.1

0

0.1

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2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9-0.1

0

0.1

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3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9-0.1

0

0.1

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4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9-0.1

0

0.1

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Time (s)

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0

0.1Raw Time Data

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0

0.1Windowed Time Data with 0% Overlap between Records

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5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9-0.1

0

0.1

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6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9-0.1

0

0.1

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0

0.1

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0

0.1

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9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10-0.1

0

0.1

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Time (s)

Figure 8 : Windowed Time Data – No Overlap

29

Cyclic Averaged Time Data – 2 blocks, each of T = 1 second time duration

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.03

-0.02

-0.01

0

0.01

0.02

0.03Time Domain : Cyclic Averaged Data Set # 1

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-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025Time Domain : Cyclic Averaged Data Set # 2

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Time (s)

Figure 9 : Cyclic Averaged Time Signal

Frequency Domain data after Cyclic Averaging and Asynchronous Averaging

-300 -200 -100 0 100 200 30010-8

10-6

10-4

10-2

100

102Frequency Domain: Cyclic Averaged Data Set # 1

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10-6

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100

102Frequency Domain: Cyclic Averaged Data Set # 2

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-300 -200 -100 0 100 200 30010-10

10-8

10-6

10-4

10-2

100Data After 2 Asynchronous Averages

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Frequency (Hz)

Figure 10 : Frequency Domain Signal after Cyclic and Asynchronous Averaging

30

The following series of figures represents the cyclic averaging process followed by

asynchronous averaging of a raw time signal for the following parameters

Parameter Value

White Noise Signal Amplitude 0.1 units

White Noise Signal Frequency Range 5-204.8 Hz

Sampling Frequency (Fs) 512 Hz

Block Size (bs) 512

Number of Cyclic Averages (Nc) 5

Percentage Overlap of Time signal 50 %

Number of Asynchronous Averages (Na) 2

Raw Time Data

0 1 2 3 4 5 6 7-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1Raw Time Data

Time (s)

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Figure 11 : Raw Time Domain Data

31

Windowed Time Data (50% overlap)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.1

0

0.1Raw Time Data

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.1

0

0.1Windowed Time Data with 50% Overlap between Records

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.1

0

0.1

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1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9-0.1

0

0.1

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0

0.1

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0

0.1

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0

0.1

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Time (s)

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0

0.1Raw Time Data

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0

0.1Windowed Time Data with 50% Overlap between Records

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2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4-0.1

0

0.1

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3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4-0.1

0

0.1

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4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4-0.1

0

0.1

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0

0.1

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6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5-0.1

0

0.1

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Time (s)

Figure 12 : Windowed Time Data – 50% Overlap of Time Signal between

averages

32

Cyclic Averaged Time Signal – 2 blocks, each of T = 1 second time duration

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.03

-0.02

-0.01

0

0.01

0.02

0.03Time Domain : Cyclic Averaged Data Set # 1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025Time Domain : Cyclic Averaged Data Set # 2

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Time (s)

Figure 13 : Cyclic Averaged Time Signal

Frequency Domain data after Cyclic Averaging and Asynchronous Averaging

-300 -200 -100 0 100 200 30010-8

10-6

10-4

10-2

100

102Frequency Domain: Cyclic Averaged Data Set # 1

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10-6

10-4

10-2

100

102Frequency Domain: Cyclic Averaged Data Set # 2

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-300 -200 -100 0 100 200 30010-8

10-6

10-4

10-2

100Data After 2 Asynchronous Averages

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Frequency (Hz)

Figure 14 : Frequency Domain Signal after Cyclic and Asynchronous Averaging

33

3.2.2 FRF Estimation Algorithms

Frequency response functions [1,2,8] can be estimated by using various models which

assume the location of noise in the measurement and try to minimize it. The H1 algorithm

assumes that all the noise exists at the output, and that there is no input noise. The H2

algorithm assumes that all the noise is on the input signal, with no noise on the output

signal. In the case considered here, the drive signals are completely known, and therefore

it is safe to assume that there is no noise on the inputs. Any noise would have to be

present on the measured outputs, and hence the H1 algorithm is preferred here.

No Input Noise (H1(ω)) Model [8]

F

H V

N X

F = Noise Free Input Spectrum V = Noise Free Output Spectrum N = Output Noise Spectrum X = Output Spectrum H = System Matrix

Figure 15 : No Input Noise FRF Model: H1(ω) Algorithm

34

The assumptions made while deriving this system model are as follows:

• There is no input noise

• Output noise does not correlate with the system input

The response can be written as follows:

{ } [ ] { } { } 11 )()()()( ××× += NoNiNN NFHXio

ωωωω

Post multiplying both sides by : { }HF )(ω

[ ] [ ] [ ] [ ] NiNoNFNiNiFFNiNoNiNoXF GGHG ×××× += )()()()( ωωωω

where the Power Spectra are as follows:

• [ ] NiNoXFG ×)(ω : Cross Power Spectrum of the Input F and the Output X

• [ ] NiNiFFG ×)(ω : Auto Power Spectrum of the Input F

• [ ] NiNoNFG ×)(ω : Cross Power Spectrum of the Input F and the Noise N

There exists no correlation between the noise N and the input F, so [ ])(ωNFG equals zero.

The equation therefore reduces to:

[ ] [ ] [ ] NiNiFFNiNoNiNoXF GHG ××× = )()()( ωωω

Or

[ ] [ ] NiNiFFNiNoXFNiNo

GGH ×−

××

∧

=

1

1 )()()( ωωω

This is the mathematical form of the [H1(ω)] FRF estimate

Normally, the existence of the matrix inverse [ ] 1)( −ωFFG would be a cause for concern,

when the input forces are obtained through measurement. Any correlation of the input

forces would cause the input power spectral matrix to be singular, and its inverse would

35

not exist. However, in the simulation algorithm, the input force spectrum consists of

generated input voltages and not measured input forces. Since the input voltage signals

have been generated in such a way that they are independent of each other, the question

of their correlation and the subsequent singularity of the power spectral matrix does not

arise. The [ can be calculated by simply inverting the diagonal elements. This

would generally hold if a sufficient number of averages are carried out. The off-diagonal

terms then reduce to several orders of magnitude as compared to the diagonal terms, and

may be ignored. This is currently not being done in the developed code. The off-diagonal

terms are being used in the simulation.

] 1)( −ωFFG

The FRF is calculated over the frequency range between the specified lower and upper

frequency limits, and while inverting the matrix, only this frequency range of interest is

used. Although the calculated FRFs are currently being used, it may be desirable to use

some kind of smoothing of the FRFs before performing the inversion. Also note that the

FRFs may not always be plotted to the same scale, and the diagonal elements are

generally prominent.

36

A sample FRF for a 4 input 4 output case is shown here

10 20 30 40 50 60

101

Output: 1 Input: 1

Frequency (Hz)

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10 20 30 40 50 60100

101

Output: 1 Input: 2

Frequency (Hz)

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10 20 30 40 50 60

10-1

100

101

Output: 1 Input: 3

Frequency (Hz)

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10 20 30 40 50 60

10-1

100

101

Output: 1 Input: 4

Frequency (Hz)

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10 20 30 40 50 60

100

101

Output: 2 Input: 1

Frequency (Hz)

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10 20 30 40 50 60

101

Output: 2 Input: 2

Frequency (Hz)

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10 20 30 40 50 60

10-1

100

101

Output: 2 Input: 3

Frequency (Hz)

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10 20 30 40 50 6010-1

100

101

Output: 2 Input: 4

Frequency (Hz)

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10 20 30 40 50 60

10-1

100

101

Output: 3 Input: 1

Frequency (Hz)

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10 20 30 40 50 60

10-1

100

101

Output: 3 Input: 2

Frequency (Hz)

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10 20 30 40 50 60

101

Output: 3 Input: 3

Frequency (Hz)

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10 20 30 40 50 60

100

101

Output: 3 Input: 4

Frequency (Hz)

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10 20 30 40 50 60

10-1

100

101

Output: 4 Input: 1

Frequency (Hz)

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10 20 30 40 50 60

100

101

Output: 4 Input: 2

Frequency (Hz)

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10 20 30 40 50 60

10-1

100

101

Output: 4 Input: 3

Frequency (Hz)

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10 20 30 40 50 60

101

Output: 4 Input: 4

Frequency (Hz)

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System Matrix (H)

Figure 16 : Plot of H1(ω) FRF Estimate

37

3.2.3 Ordinary Coherence Function

Ordinary coherence function [8] is defined as the measure of the linear relationship

between two signals with the presence of other signals. Considering the ith output and the

jth input:

),(),(),( 2

2

iiGjjGjiG

XXFF

XFij =γ

If the value of coherence is one, the signals are completely related. Ordinary coherence

functions are generally used only for a single input single output configuration, due to the

obvious fact that for a multiple input configuration, the ordinary coherence, by definition,

would be less than unity. The multiple coherence function is used in that case to

determine the linear nature of the response relating to all measured inputs.

3.2.4 Multiple Coherence Function

Multiple coherence function [8] describes the linear relationship between one output

signal and all known input signals. It is defined as the ratio between the following two

power spectra:

• Cross power spectra between the output and all known inputs

• Auto power spectrum of the output

Considering the ith output and the jth input:

ii

ji

iXX

Ni

jFXij

FX G

GH∑== 12γ

(Where, the Hij term is determined by the H1 algorithm derived before).

If the multiple coherence value is unity, then the response is entirely due to all the input

forces to the system. If the multiple coherence value drops below unity, it means that the

38

response is not entirely due to the system input forces, and additional unmeasured sources

of excitation may be present, or the system response may be non-linear.

The Plot of the Multiple Coherence is shown below.

5 10 15 20 25 30 35 40 45 50 55 600

0.2

0.4

0.6

0.8

Output: 1

Frequency (Hz)

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5 10 15 20 25 30 35 40 45 50 55 600

0.2

0.4

0.6

0.8

Output: 2

Frequency (Hz)

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5 10 15 20 25 30 35 40 45 50 55 600

0.2

0.4

0.6

0.8

Output: 3

Frequency (Hz)

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5 10 15 20 25 30 35 40 45 50 55 600

0.2

0.4

0.6

0.8

Output: 4

Frequency (Hz)

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Multiple Coherence

Figure 17 : Plot of Multiple Coherence

39

3.2.5 Inverse System Model

The inverse of the FRF matrix serves as the Inverse System Model. The FRF is a linear

estimate of the system; hence it underestimates the response caused by an input force. For

a linear system, increasing the input signal amplitude would result in an increase in the

output signal amplitude which is a constant factor times the input signal level. The output

to input relation thus follows a straight line relationship and is therefore linear. For a non

linear system, this is not true. The output signal amplitude would increase by a factor

which is dependant on the input signal amplitude. For example, assuming a cubic

relationship between the output and input signals, increasing the input amplitude by a

factor k would cause the output amplitude to increase by a factor of k3. For a linear

system, however, the output would be k times the input. Assuming a linear system model

for a non linear system would mean that the calculated input signal estimated to cause a

desired response would be 1/k times, instead of 1/k3 times, the desired response. It is

clear that this would cause an error in the estimate of the required input signals, with the

estimated input signal being a lot higher that the one actually required for a particular

response signal. Similarly, the inverse of the FRF results in an overestimate of the input

signal required to get a particular desired response. The estimated input signals have to be

multiplied by a scaling factor between 0 and 1 to determine the actual signal that should

be applied to the actuator to get the desired response. It is advisable to look at the

calculated estimate of the input signal amplitude before sending it out to the system as a

precautionary measure.

40

The plot of the Inverse System Model is shown below

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10-1

Output: 1 Input: 1

Frequency (Hz)

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10 20 30 40 50 60

10-2

10-1

Output: 1 Input: 2

Frequency (Hz)

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10-2

10-1

Output: 1 Input: 3

Frequency (Hz)

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10 20 30 40 50 6010-3

10-2

10-1

Output: 1 Input: 4

Frequency (Hz)

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10-2

10-1

Output: 2 Input: 1

Frequency (Hz)

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10-1

Output: 2 Input: 2

Frequency (Hz)

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10 20 30 40 50 60

10-3

10-2

10-1

Output: 2 Input: 3

Frequency (Hz)

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10-3

10-2

10-1

Output: 2 Input: 4

Frequency (Hz)

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10-3

10-2

10-1

Output: 3 Input: 1

Frequency (Hz)

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10 20 30 40 50 6010-3

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10-1

Output: 3 Input: 2

Frequency (Hz)

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10-1

Output: 3 Input: 3

Frequency (Hz)

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Output: 3 Input: 4

Frequency (Hz)

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10 20 30 40 50 60

10-3

10-2

10-1

Output: 4 Input: 1

Frequency (Hz)

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10 20 30 40 50 6010-3

10-2

10-1

Output: 4 Input: 2

Frequency (Hz)

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10-2

10-1

Output: 4 Input: 3

Frequency (Hz)

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10-1

Output: 4 Input: 4

Frequency (Hz)

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Inverse System Matrix (invH)

Figure 18 : Plot of Inverse System Model

41

3.3 Desired Responses

3.3.1 Time Domain

Time domain desired responses are in the form of acceleration values at various times,

for each measurement location. The responses may be those measured during an actual

road test performed on the vehicle.

3.3.2 Frequency Domain

Frequency domain desired responses are in the form of output auto power spectra. They

may be power spectra measured during an actual road test or custom generated by the test

engineer, by describing the desired profile of the spectra in the frequency domain.

3.4 Convolution with Overlap Adding

The procedure used for estimating the forces required to obtain a desired response is a

convolution of the inverse system model and the desired response, with overlap adding

[4]. The procedure is explained as follows. It can be divided into two stages:

3.4.1 Estimate Inverse FRF

• Invert the element to time domain

• Perform a half frame circular shift

• Add a frame of zeros at the end (zero pad)

• Transform back to frequency domain

42

3.4.2 Convolution with overlap adding

• Take first frame of desired response time history for all channels and add a frame

of zeros to it

• Fourier transform it to the frequency domain

• {F(ω)} = [H(ω)]-1*{X(ω)} gives 2 frames of Force data

• Invert the calculated force to time domain. If the original time signal was from 0

to T, the calculated signal will be from –T/2 to 3T/2

• Repeat for next time frame of desired response. This will yield a force signal for

the time T/2 to 5T/2, which overlaps the previously calculated signal in the region

T/2 to 3T/2, and has to be added to it in that region

• Continue till the desired response time history is exhausted.

The above procedure is performed by the ‘convOver’ [B.2] Matlab function. For details

refer to the Appendix. The following flowcharts and figures explain the procedure.

43

Preparing System Matrices for Convolution

Figure 23

Figure 22

Figure 21

Store Modified System Matrix (FRF) [H(ω)]

Perform FFT to obtain modified FRFs for each

element

Add a Frame of Zeros at the end (Zero Padding)

Perform a Half Frame Circular

Shift

Perform Inverse FFT to obtain time domain

Impulse Responses for each element

System Matrix (FRF)

Figure 19 : Flowchart - Modifying System Matrix for Convolution

44

Convolution with Overlap Adding

Figure 34

Figure 27

Figure 26

Figure 35

Yes

Figure 28

Figure 25

Figure 24

Perform time Overlap Addition of Drive Signals

Write out Drive Signals

Any more Frames?

No

Inverse FFT to Time domain to get 2 Frames of Time Data (-T/2 to 3T/2)

{F(ω)} = [H(ω)]-1*{X(ω)}gives 2 Frames of Drive

Signal Data in Frequency Domain

Perform FFT to get Frequency Domain Data

{X(ω)}

Add a Frame of Zeros at the end (Zero Padding)

Select first Data Frame for all Channels (0 to T)

Select Desired Response

Figure 20 : Flowchart - Convolution with Overlap Adding

45

Original Impulse Responses

0 1 2 3 4 5-0.05

0

0.05Output: 1 Input: 1

Time (s)

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0

0.05Output: 1 Input: 2

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 1 Input: 3

Time (s)

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0

0.05Output: 1 Input: 4

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 2 Input: 1

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 2 Input: 2

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 2 Input: 3

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 2 Input: 4

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 3 Input: 1

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 3 Input: 2

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 3 Input: 3

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 3 Input: 4

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 4 Input: 1

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 4 Input: 2

Time (s)

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0 1 2 3 4 5-0.05

0

0.05Output: 4 Input: 3

Time (s)A

mpl

itude

0 1 2 3 4 5-0.05

0

0.05Output: 4 Input: 4

Time (s)

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Impulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response FunctionImpulse Response Function

Figure 21 : Impulse Responses obtained from System FRFs

Impulse Responses with half frame circular shifting and zero padding

0 2 4 6 8 10-0.05

0

0.05Output: 1 Input: 1

Time (s)

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0

0.05Output: 1 Input: 2

Time (s)

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0

0.05Output: 1 Input: 3

Time (s)

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0 2 4 6 8 10-0.05

0

0.05Output: 1 Input: 4

Time (s)

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0 2 4 6 8 10-0.05

0

0.05Output: 2 Input: 1

Time (s)

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0 2 4 6 8 10-0.05

0

0.05Output: 2 Input: 2

Time (s)

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0

0.05Output: 2 Input: 3

Time (s)

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0

0.05Output: 2 Input: 4

Time (s)

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0 2 4 6 8 10-0.05

0

0.05Output: 3 Input: 1

Time (s)

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0 2 4 6 8 10-0.05

0

0.05Output: 3 Input: 2

Time (s)

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0 2 4 6 8 10-0.05

0

0.05Output: 3 Input: 3

Time (s)

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0 2 4 6 8 10-0.05

0

0.05Output: 3 Input: 4

Time (s)

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0 2 4 6 8 10-0.05

0

0.05Output: 4 Input: 1

Time (s)

Am

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0 2 4 6 8 10-0.05

0

0.05Output: 4 Input: 2

Time (s)

Am

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0 2 4 6 8 10-0.05

0

0.05Output: 4 Input: 3

Time (s)

Am

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0 2 4 6 8 10-0.05

0

0.05Output: 4 Input: 4

Time (s)

Am

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Impulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero PaddingImpulse Response Function With Half Circular Shift and Zero Padding

Figure 22 : Impulse Responses with Half Frame Circular Shift and Zero Padding

46

FRFs obtained after FFT operation on modified impulse responses

-60 -40 -20 0 20 40 6010-2

10-1

100Output: 1 Input: 1

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 1 Input: 2

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 1 Input: 3

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 1 Input: 4

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 2 Input: 1

Frequency (Hz)

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 2 Input: 2

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 2 Input: 3

Frequency (Hz)

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 2 Input: 4

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 3 Input: 1

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 3 Input: 2

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 3 Input: 3

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 3 Input: 4

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 4 Input: 1

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 4 Input: 2

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 4 Input: 3

Frequency (Hz)

Am

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-60 -40 -20 0 20 40 6010-2

10-1

100Output: 4 Input: 4

Frequency (Hz)

Am

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Modified Frequency Response FunctionModified Frequency Response FunctionModified Frequency Response FunctionModified Frequency Response Function

Figure 23 : Modified FRFs

Single frame of Desired Response signals

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3Output: 1

Time (s)

Am

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3Output: 2

Time (s)

Am

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3Output: 3

Time (s)

Am

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3Output: 4

Time (s)

Am

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Desired Response - Single FrameDesired Response - Single FrameDesired Response - Single FrameDesired Response - Single Frame

Figure 24 : Desired Responses – single frame of data

47

Single Frame of Desired Response with Zero Padding

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3Output: 1

Time (s)

Am

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0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3Output: 2

Time (s)

Am

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0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3Output: 3

Time (s)

Am

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0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3Output: 4

Time (s)

Am

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Desired Response With Zero PaddingDesired Response With Zero PaddingDesired Response With Zero PaddingDesired Response With Zero Padding

Figure 25 : Desired Responses – single frame with zero padding

Desired Response in Frequency Domain

5 10 15 20 25 30 35 40 45 50 55

100

101

Channel 1

Frequency

Am

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5 10 15 20 25 30 35 40 45 50 55

10-1

100

101

Channel 2

Frequency

Am

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5 10 15 20 25 30 35 40 45 50 55

10-1

100

101

Channel 3

Frequency

Am

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5 10 15 20 25 30 35 40 45 50 55

10-1

100

101

Channel 4

Frequency

Am

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Desired Response - Frequency Domain

Figure 26 : Desired Responses – Frequency Domain

48

Estimated Input Signal in Frequency Domain

5 10 15 20 25 30 35 40 45 50 55

10-10

100Channel 1

Frequency

Am

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5 10 15 20 25 30 35 40 45 50 55

10-10

100Channel 2

Frequency

Am

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5 10 15 20 25 30 35 40 45 50 55

10-10

100Channel 3

Frequency

Am

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5 10 15 20 25 30 35 40 45 50 55

10-10

100Channel 4

Frequency

Am

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Estimated Input signal -Frequency Domain

Figure 27 : Estimated Input Signals – Frequency Domain

Estimated Input Signals – Time Domain

-2 -1 0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1Channel 1

Time

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-2 -1 0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1Channel 2

Time

Am

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-2 -1 0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1Channel 3

Time

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-2 -1 0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1Channel 4

Time

Am

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Estimated Input Signals -Time Domain

Figure 28 : Estimated Input Signals – Time Domain

49

Thus each frame of the Desired Response signal yields two frames of Input signals as

shown below. Each subsequent calculated Input signal is added with the previous signal,

taking into consideration the overlap in the time domain, to obtain the final Input signal.

The process is shown for five frames (one frame = 1024 points) of 4-channel data.

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 1

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 2

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 3

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 4

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 5

Convolution - Channel 1

Figure 29 : Convolution Results for 5 Data Frames for Channel 1

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 1

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 2

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

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Frame 3

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

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Frame 4

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 5

Convolution - Channel 2

Figure 30 : Convolution Results for 5 Data Frames for Channel 2

50

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 1

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

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Frame 3

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

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Frame 4

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 5

Convolution - Channel 3

Figure 31 : Convolution Results for 5 Data Frames for Channel 3

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 1

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 2

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 3

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

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Frame 4

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Frame 5

Convolution - Channel 4

Figure 32 : Convolution Results for 5 Data Frames for Channel 4

51

The same signals are shown below, for 12 frames of 4 channel data, with the overlap

between successively calculated signals clearly visible.

0 10 20 30 40 50 60-0.1

-0.05

0

0.05

0.1Channel 1

Time

Am

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0 10 20 30 40 50 60-0.1

-0.05

0

0.05

0.1Channel 2

Time

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0 10 20 30 40 50 60-0.1

-0.05

0

0.05

0.1Channel 3

Time

Am

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0 10 20 30 40 50 60-0.1

-0.05

0

0.05

0.1Channel 4

Time

Am

plitu

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Figure 33 : Convolution Results

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Channel 1

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Channel 2

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Channel 3

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Channel 4

Time Signal After Overlap AdditionTime Signal After Overlap AdditionTime Signal After Overlap AdditionTime Signal After Overlap Addition

Figure 34 : Calculated Input Signals after Overlap Addition of Convolved Time

Signals shown for all 4 Channels

52

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Channel 1

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Channel 2

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Channel 3

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (s)

Am

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Channel 4

Final Time SignalFinal Time SignalFinal Time SignalFinal Time Signal

Figure 35 : Input Signals after Ignoring First and Last Half Frames

(Note: Since the Desired response Signal goes from 0 to 60 seconds, only the input signal

corresponding to that time period is required, and the rest can be ignored.)

53

4 Test Setup

For all the tests conducted, four response measurements were made, corresponding to the

four inputs to the vehicle at the wheels. The accelerometers used were PCB Model 308

B02. Specifications of the accelerometer are as follows:

Type: High Sensitivity ICP Accelerometer

Parameter Value

Mass 65 g

Sensitivity 1V/g

Frequency Range 2.5 – 3000 Hz

Connector Type Side 10-32 Connector

Seal Type Hermetically Sealed

Mounting Type 10-32 mount

54

4.1 Wheel Input with Collocated Output

The output locations are collocated with respect to the inputs. The accelerometers

measuring the acceleration response are located on the wheel axles. The following image

shows the collocated sensor location for the right rear wheel. The dot indicates the

location of the sensor, oriented vertically. Since the input signal is applied at the wheel

patch, the sensor is technically non-collocated with respect to the input. The axle/spindle

is where motion is normally measured in operation, and hence the location of the

response sensor on the axle/spindle is as close to the collocated configuration as

practically possible, and was used.

Figure 36 : Accelerometer location for Collocated Configuration

55

4.2 Wheel Input with Non-collocated Output

The output locations are not collocated with respect to the inputs in this example. The

accelerometers measuring the acceleration response are located on the vehicle

body/chassis, oriented vertically. In this case, the additional dynamics of the suspension

system and the non collocated configuration are expected to play a part in the systems

response to the input force. The coherence may not be very good for such a case, and it

may not be possible to simulate such configurations using existing methods. The

following image shows the Non-collocated measurement location for the right rear wheel.

The dot indicates the location of the accelerometer.

Figure 37 : Accelerometer location for Non-Collocated Configuration

56

5 Results

5.1 Collocated

5.1.1 Time Domain Desired Response

The time domain desired response may be obtained in different ways. They are as

follows:

• Time domain acceleration response measured during an actual road test

performed on the vehicle

• Acceleration response measured for user generated drive signals, created by using

the time history generator (.DRV File Creator)

For some cases, the easiest way to obtain a desired response may be to put accelerometers

at the chosen measurement locations and to actually drive the vehicle on a test track,

while measuring the response. The measured acceleration would serve as the desired

response signal. However, if it is not practicable to locate sensors on a vehicle during an

actual road test, the user may generate response measurements to custom drive signals

and treat them as the desired response signals. The .DRV File Creator GUI enables the

creation of a wide variety of drive signals for each of the four input channels. The test

vehicles response to these drive signals can then be used as the desired response signal

for the simulation.

Since actual test data were not available for the vehicle being tested (the vehicle is not

operational), the desired responses used for the simulation were response signals

measured for the vehicle in response to randomly generated drive signals for each of the

four wheels.

57

Desired Response Parameters:

For the configuration with outputs collocated with respect to the inputs, the following

parameters were used to generate the drive signals:

Parameter Value

Amplitude (in) 0.1 (for each channel)

Frequency Range (Hz) 5 – 50

Time (s) 60

Signal Type Random (band limited between 5-50 Hz)

The parameters were chosen so as to be representative of an actual road test, where the

vehicle would be driven on a highway. The road surface irregularities on a highway

would not generally be more than 0.1 inch, and for a vehicle doing around 60 mph, the

maximum frequency of the input signal would not generally exceed 50 Hz. Time

response of one minute (60 s) was chosen to reduce the test time required for each

iteration, thus enabling a wider variety of tests to be carried out.

The simulation converges, and is stopped after 5 iterations. The results for the standard

deviation of the error signal are tabulated for the iterations, along with plots for the

measured responses and calculated errors for the initial and final drive signals.

Iteration Channel 1 Channel 2 Channel 3 Channel 4 Initial Estimate 0.1567 0.1366 0.1454 0.1492

1 0.0894 0.0895 0.0813 0.0845 2 0.0612 0.0557 0.0525 0.0562 3 0.0427 0.0391 0.0371 0.0386 4 0.0352 0.0337 0.0298 0.0313 5 0.0304 0.0299 0.0254 0.0262

58

0 1 2 3 4 50

0.05

0.1

0.15

0.2Channel 1

0 1 2 3 4 50

0.05

0.1

0.15

0.2Channel 2

0 1 2 3 4 50

0.05

0.1

0.15

0.2Channel 3

0 1 2 3 4 5 0

0.05

0.1

0.15

0.2

Am

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------

>

Iteration Number -------->

Channel 4

Standard Deviation of the Error

Figure 38 : Variation of Standard Deviation of Error with subsequent Iterations

59

Response Measured for Initial Estimate of Drive Signal

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 2

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 3

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 4

Time (s) -------->

Am

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des

(V) -

------

->

MeasuredDesired

MeasuredDesired

MeasuredDesired

MeasuredDesired

Desired vs Measured Responses - Initial Estimate

Figure 39 : Measured Response – Collocated Configuration: Initial Estimate

60

Response Measured for Final Estimate of Drive Signal

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 2

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 3

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 4

Time (s) -------->

Am

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des

(V) -

------

->

MeasuredDesired

MeasuredDesired

MeasuredDesired

MeasuredDesired

Measured vs Desired Responses - Iteration 5

Figure 40 : Measured Response – Collocated Configuration: Iteration 5

61

Error between Measured Response and Desired Response

Initial Drive

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 2

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 3

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 4

Am

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(V) -

------

->

Time (s) -------->

Error - Initial Estimate

Figure 41 : Collocated – Initial Error

62

Error between Measured Response and Desired Response

Final Drive

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 2

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 3

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 4

Time (s) -------->

Am

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(V) -

------

->

Error - Iteration 5

Figure 42 : Collocated – Final error

63

5.1.2 Frequency Domain Desired Response

Time domain desired response may not always be readily available. However, the desired

response may be available in the form of frequency domain power spectra, and it may be

required that these particular power spectra need to be recreated at the measurement

locations. Alternately, the test engineer may want to obtain certain predefined power

spectral profiles at the measurement locations, and may want to define these desired

spectra as the desired responses. The Road Simulator GUI enables such an input, where

the user can draw the desired power spectral profiles in the frequency domain. The

required profile is then convolved with a band limited random signal of unit amplitude,

over the desired test time period, to obtain the acceleration response signals for the

measurement locations, so that this desired response in the time domain has the required

profile in the frequency domain. This generated signal is subsequently used for

simulation.

Desired Response Parameters:

The following parameters were used while generating the desired response signals in the

frequency domain:

Parameter Value

Maximum Power Spectral Amplitude 104 (for each channel)

Frequency Range (Hz) 10 – 50

Time (s) 60

Signal Type Desired Power Spectra

64

The desired power spectral profiles for the four channels are as shown below. These are

convolved with random signals with unit amplitudes, to obtain the modified desired

responses, against which the simulation is carried out.

10 15 20 25 30 35 40 45 50102

103

104

105Channel 1

10 15 20 25 30 35 40 45 50102

103

104

105Channel 2

10 15 20 25 30 35 40 45 50102

103

104

105Channel 3

10 15 20 25 30 35 40 45 50102

103

104

105Channel 4

Frequency (w) -------->

Am

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des

------

-->

Desired Power Spectra

Figure 43 : Desired Power Spectral Profile

65

The following figure shows the random signal with unit amplitude

0 10 20 30 40 50 60-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Am

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------

>

Time (s) -------->

Random Signal with Unit Amplitude

Figure 44 : Random Signal with Unit Amplitude

66

Convolution of the desired frequency spectra with the random signal gives the modified

desired power spectra as shown. The targeted frequency range is between 10 and 50 Hz.

Although there seems to be some difference between the desired profile and the actual

profile, it is expected that with a larger number of averages, the two profiles will match to

a large extent.

5 10 15 20 25 30 35 40 45 50 55

100

Channel : 1

Am

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------

> 5 10 15 20 25 30 35 40 45 50 55

100

Channel : 2

5 10 15 20 25 30 35 40 45 50 55

100

Channel : 3

5 10 15 20 25 30 35 40 45 50 55

100

Channel : 4

Frequency (Hz) -------->

Simulation ResponseDesired Response

Simulation ResponseDesired Response

Simulation ResponseDesired Response

Simulation ResponseDesired Response

Response used for Simulation compared with Actual Desired Response

Figure 45 : Modified Desired Response Spectra

67

The simulation converges, and is stopped after 5 iterations. The results for the standard

deviation of the error signal are tabulated for the iterations, along with plots for the

measured responses and calculated errors for the initial and final drive signals.

Iteration Channel 1 Channel 2 Channel 3 Channel 4 Initial Estimate 0.3451 0.2592 0.3557 0.3271

1 0.2019 0.1541 0.1909 0.2226 2 0.1466 0.1187 0.1459 0.1547 3 0.1050 0.0867 0.1059 0.1094 4 0.0972 0.0841 0.0867 0.1116 5 0.0842 0.0813 0.0952 0.0935

0 1 2 3 4 50

0.1

0.2

0.3

0.4Channel 1

0 1 2 3 4 50

0.1

0.2

0.3

0.4Channel 2

0 1 2 3 4 50

0.1

0.2

0.3

0.4Channel 3

0 1 2 3 4 50

0.1

0.2

0.3

0.4Channel 4

Iteration Number -------->

Am

plitu

de --

------

>

Standard Deviation of the Error

Figure 46 : Variation of Standard Deviation of Error with subsequent Iterations

68

Response Measured for Initial Estimate of Drive Signal

5 10 15 20 25 30 35 40 45 50 55

105Channel : 1

Am

plitu

de --

------

>

Measured ResponseDesired Response

5 10 15 20 25 30 35 40 45 50 55

105Channel : 2

Measured ResponseDesired Response

5 10 15 20 25 30 35 40 45 50 55

105Channel : 3

Measured ResponseDesired Response

5 10 15 20 25 30 35 40 45 50 55

105Channel : 4

Frequency (Hz) -------->

Measured ResponseDesired Response

Measured vs Desired Response - Initial Estimate

Figure 47 : Measured Response – Collocated Configuration: Initial Estimate

69

Response Measured for Final Estimate of Drive Signal

5 10 15 20 25 30 35 40 45 50 55

105Channel : 1

Am

plitu

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------

>

Measured ResponseDesired Response

5 10 15 20 25 30 35 40 45 50 55

105Channel : 2

Measured ResponseDesired Response

5 10 15 20 25 30 35 40 45 50 55

105Channel : 3

Measured ResponseDesired Response

5 10 15 20 25 30 35 40 45 50 55

105Channel : 4

Frequency (Hz) -------->

Measured ResponseDesired Response

Measured vs Desired Response - Iteration 5

Figure 48 : Measured Response – Collocated Configuration: Iteration 5

70

Error between Measured Response and Desired Response (shown in time domain for

better visualization)

Initial Drive

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 2

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 3

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 4

Time (s) -------->

Am

plitu

des

------

-->

Error - Initial Estimate

Figure 49 : Collocated – Initial Error

71

Error between Measured Response and Desired Response

Final Drive

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 1

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 2

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 3

0 10 20 30 40 50 60-1

-0.5

0

0.5

1Error - Channel : 4

Time (s) -------->

Am

plitu

des

(V) -

------

->

Error - Iteration 5

Figure 50 : Collocated – Final Error

72

5.2 Non-collocated

5.2.1 Time Domain Desired Response

The considerations while defining the desired time domain response for the non-

collocated configuration are similar to those for the collocated configuration.

Desired Response Parameters:

For the configuration with outputs non-collocated with respect to the inputs, the

following parameters were used in generating the desired time domain responses:

Parameter Value

Amplitude (in) 0.1 (for each channel)

Frequency Range (Hz) 10 – 40

Time (s) 60

Signal Type Random (band limited between 10-40 Hz)

The error between the desired and measured response converges, and the simulation is

stopped after 5 iterations. It is expected that the rate of convergence is slower for the non-

collocated configuration, and more iterations may need to be performed to obtain

satisfactory results.

Iteration Channel 1 Channel 2 Channel 3 Channel 4 Initial Estimate 0.2721 0.2985 0.1446 0.1287

1 0.1952 0.2227 0.1102 0.0912 2 0.1694 0.1954 0.0980 0.0775 3 0.1573 0.1843 0.0928 0.0713 4 0.1524 0.1779 0.0898 0.0681 5 0.1534 0.1772 0.0893 0.0676

73

0 1 2 3 4 5

0.2

0.25

0.3

0.35Channel 1

0 1 2 3 4 5

0.2

0.25

0.3

0.35Channel 2

0 1 2 3 4 50.08

0.1

0.12

0.14

0.16Channel 3

0 1 2 3 4 50.06

0.08

0.1

0.12

0.14

0.16

Iteration Number -------->

Am

plitu

de --

------

>

Channel 4

Standard Deviation of the Error

Figure 51 : Variation of Standard Deviation of Error with subsequent Iterations

74

Response Measured for Initial Estimate of the Drive Signal

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 1

MeasuredDesired

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 2

MeasuredDesired

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 3

MeasuredDesired

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 4

MeasuredDesired

Figure 52 : Measured Response – Non-Collocated Configuration: Initial Estimate

75

Response Measured for Final Estimate of the Drive Signal

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 1

MeasuredDesired

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 2

MeasuredDesired

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 3

MeasuredDesired

0 10 20 30 40 50 60-1

-0.5

0

0.5

1x 104 Response - Channel : 4

MeasuredDesired

Figure 53 : Measured Response – Non-Collocated Configuration: Iteration 5

76

Error between Measured Response and Desired Response

Initial Drive

0 10 20 30 40 50 60-2

-1

0

1

2Error - Channel : 1

0 10 20 30 40 50 60-2

-1

0

1

2Error - Channel : 2

0 10 20 30 40 50 60-2

-1

0

1

2Error - Channel : 3

0 10 20 30 40 50 60-2

-1

0

1

2Error - Channel : 4

Figure 54 : Non-collocated – Initial Error

77

Error between Measured Response and Desired Response

Final Drive

0 10 20 30 40 50 60

-2

-1

0

1

2

Error - Channel : 1

0 10 20 30 40 50 60

-2

-1

0

1

2

Error - Channel : 2

0 10 20 30 40 50 60

-2

-1

0

1

2

Error - Channel : 3

0 10 20 30 40 50 60

-2

-1

0

1

2

Error - Channel : 4

Figure 55 : Non-collocated – Final Error

78

6 Conclusions and Future Work

6.1 Conclusions

It is seen from the results that the simulation algorithm works well for the collocated

case, where the response measurements are made on the vehicle wheel axles. The

measured response rapidly converges to the desired response, and after around 5

iterations, the simulation may be stopped as satisfactory results are obtained. For the non-

collocated configuration, the simulation does converge, but not as rapidly as the previous

case, and a larger number of iterations may be necessary to arrive at a satisfactory result.

As the path between the input and output locations becomes progressively complex, the

number of iterations required rises significantly, and there may be a point beyond which

using the current simulation algorithm becomes unfeasible. An advanced system

modeling technique would be needed to replace the existing linear model, to achieve the

same results with a fewer number of iterations. To summarize the results, the following

conclusions can be drawn:

• The current system works well for the collocated configuration.

• Non collocated configurations require a larger number of iterations to converge

satisfactorily.

• Highly non-collocated or non-linear inputs cannot be simulated.

Using the same algorithm for the non-collocated configuration gives a slower

convergence of the desired and measured responses, and a substantially larger number of

iterations would be required to obtain satisfactory results. The linear estimate of the

system does not take into account the additional system dynamics that come into play for

the non-collocated case, namely the suspension system, and the longer path between the

79

input and the output due to the non-collocated location. As the path between the input

location and the measurement location increases, the number of iterations required to

achieve an accurate solution increases, and there may be a point beyond which the

current algorithm cannot simulate the desired response.

6.2 Future Work

Future research will try to address these issues by developing complex system models

that take into account the additional system dynamics involved along with non-linearities

thus enhancing the capabilities of the simulation techniques currently used.

The Non-linear Input Feedback of the Outputs (NIFO) technique was developed by

Adams and Allemang [10-12] to characterize a general non-linear system. The possibility

of using the NIFO technique to identify and characterize system nonlinearities and the

ability to develop a non-linear system model needs to be researched further. The NIFO

technique is able to identify the non-linearities in the system, but the parameters

characterizing them depend on the type of non-linear model chosen [12]. Tests

conducted indicate that in the non-linear case, different models may be used to describe

the same system and the actual results obtained are highly dependent on the choice of

non-linear model. Of the available models, choosing the one that best describes the

system being analyzed, therefore, becomes a key issue.

80

References

1 Allemang, R.J , Vibrations: Experimental Modal Analysis, University of

Cincinnati, Structural Dynamics Research Lab, 1999,

2 Phillips, Allyn W., Allemang, R.J., An overview of MIMO-FRF

excitation/averaging/processing techniques, 2002

3 MTS Systems Corporation Training Manual, RPC III Operation, v 4.2,

1997

4 MTS Systems Corporation Software Manual : RPC III, v 4.2, Vol. 1, 1996

5 Simulation Techniques: SIMTEST User Guide, v 4.0, 2000

6 Matlab Reference Guide: Creating Graphical User Interfaces, v 6, 2000

7 Matlab Reference Guide: Using Matlab Graphics, v 6, 2002

8 He, Jimin, Fu, Zhi-Fang, Modal Analysis, Butterworth-Heinemann, 2001

9 Phillips, Allyn W., Zucker, Andrew T., Allemang, R.J., Frequency

resolution effects on FRF estimation: Cyclic Averaging vs. Large block

size, Proceedings of the IMAC XVII, Vol. 2, 1999

10 Adams D.E., A spatial approach to nonlinear vibration analysis, 2000

11 Adams, D.E., Allemang, R.J., A new derivation of the Frequency Response

Function Matrix for Vibrating Non-linear Systems, 1999

12 Martel, R., Spottswood, M., Godbole, U., Allemang, R.J., Dahling, T,

Application of a Non-linear Feedback method for determining local

system damage, 2005

13 Zak, Stanislaw, H., Systems and Control, Oxford University Press, 2003

81

14 Taylor, Fred, Mellott, Jon, Hands on Digital Signal Processing, McGraw-

Hill, 1998

15 Kuo, Hui-Hsiung, White Noise Distribution Theory, CRC Press, 1996

82

Appendix A RPC III File Format

The RPC III File Format [3] consists of an ASCII header followed by binary data.

A.1 Header

Keyword Description Possible Values BYPASS_FILTER Turns A/D Filtering ON or

OFF Options: 0 : Uses the filter 1: Does not use the filter

CHANNELS Number of channels in the file

Range of Channels is 1 to 128

DATA_TYPE Type of Data contained in the file

Types: SHORT_INTEGER FLOATING_POINT

DATE Date and Time the file was created

NTFS Date and Time Example : 22-Feb-2000 10:20:11 dd-mmm-yyyy hh:mm:ss

DELTA_T Time interval between consecutive points

Real Number Example: 4.882812E-03 Format : E8.6

DESC.CHAN_n ASCII Description of the specified channel. Repeated for each channel in the file

Maximum number of characters allowed is 96 for RPC III File

FILE_TYPE Type of Data File TIME_HISTORY FORMAT Format in which Data is

stored BINARY_IEEE_LITTLE_END BINARY_IEEE_BIG_END BINARY ASCII

FRAMES Number of frames of data per channel. 1 frame = 5 seconds of data

Range 2 to any integer that is a power of 2, maximum 8192

HALF_FRAMES Specifies whether a half frame is present at the beginning and end of the data, to facilitate mathematical processing

Options 0 : No Half frame added 1: Half Frames added

INT_FULL_SCALE Maximum 16 bit integer value of the data

Default : 32752 ( 216-16)

LOWER_LIMIT.CHAN_n Lower limit for Channel n Any Valid number MAP.CHAN_n Physical Channel to

which logical channel is mapped

Any valid physical channel number

NUM_HEADER_BLOCKS Numbes of 512 character blocks used by the header information

Upto 256 blocks

NUM_PARAMS Total number of parameters Maximum = 1024 (256*4) OPERATION Name of Program that

created file RPC III Software Program or operation

PARENT_k File from which this file Any valid NTFS filename

83

was created PART.CHAN_n First channel assigned to

partition n Any valid integer. Default : 1

PART.NCHAN_n Number of consecutive channels assigned to partition n

Default :Value in keyword CHANNELS

PARTITIONS The number of groups of channels wanted

Range : 1 to 128 partitions Default : 1

PTS_PER_FRAME Number of points per frame of stored data. Must be a power of 2

Options: 256, 512, 1024, 2048 PTS_PER_FRAME ≤ PTS_PER_GROUP

PTS_PER_GROUP Total number of data points in the group

Integer sum of all points

REPEATS Number of times the frame is identically repeated in the file

Typically 1 repeat

SCALE.CHAN_n Value used to multiply the binary 16-bit converter value to find the Engineering Units scale value

Example 10V/32752.0

TIME_TYPE Type of time history 1 DRIVE 2 RESPONSE 3 MULT_DRIVE 4 MULT_RESP 5 CONFIG_DRIVE 6 CONFIG_RESP 7 PEAK_PICK

UNITS.CHAN_n ASCII description of the Engineering Units associated with the channel

Any valid unit. Example “INCH” Maximum 96 characters allowed for RPC III File

UPPER_LIMIT.CHAN_n Upper limit value defined for channel n

Any valid number

84

A.2 Data

Consider 4 channels of data, each channel having 4 frames, i.e. 20 seconds of data. The

data for the four channels is as shown below.

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.05

0

0.05

0.1

Am

p (in

)

Time Data : Channel 1

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.05

0

0.05

0.1

Am

p (in

)

Time Data : Channel 2

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.05

0

0.05

0.1

Am

p (in

)

Time Data : Channel 3

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.05

0

0.05

0.1Time Data : Channel 4

Am

p (in

)

Time (s) -------->

Figure 56 : Four Channel Time Data

Assuming that the points per group are 2048 (corresponding to the first 10 seconds of

data, sampled at 204.8 Hz), the data is stored in demultiplexed blocks as shown below.

The first group of 2048 points of Channel 1 is stored, followed by the first group of 2048

points of the second channel and so on through Channel 4. Then the next group of points

for Channel 1 is stored, followed by the second group of points for the second channel

85

and so on through Channel 4, till all the data groups are exhausted. The resulting data

arrangement is as shown below.

0 10 20 30 40 50 60 70 80

-0.1

-0.05

0

0.05

0.1

0.15

Am

p (in

)

Time (s) -------->

Demultiplexed data

Channel 2 Channel 3 Channel 4Channel 1 Channel 4Channel 1 Channel 2 Channel 3

Figure 57 : Four Channel Demultiplexed Time Data

86

B Matlab Functions

B.1 fnHlin: Calculates FRF, Coherence, Power Spectra

% Function fnHlin determines the Linear FRF Matrix with Hanning Window and cyclic averaging % Syntax : [h,c,mc,w,GFF,GXX,GFX,GXF,Fmax] = fnHlin(Ft,Xt,bs,Nc,overlap,window (1 = Hanning),Estimator type (1 = H1 , 2 = H2)) function [h1,c1,mc1,w,GFF1,GXX1,GFX1,GXF1,Fmax] = fnHlin(Ft,Xt,bs,Nc,ov,win,type,lf,hf) try % Number of Inputs ni = size(Ft); ni = ni(1); % Number of Outputs no = size(Xt); no = no(1); % Block Size bs = fix(bs); % Number of Data Blocks Ncycblk = 1; cnt2 = 1; cnt3 = 1; while cnt3 < length(Ft)-(Nc*bs) cnt2 = cnt2 + (1-ov/100)*(Nc*bs); cnt3 = cnt2 + Nc*bs-1; Ncycblk = Ncycblk + 1; end shft = fix((1-ov/100)*(bs)); % Sampling Frequency Fs = 204.8; % dt dt = 1/Fs; % Fmax Fmax = 1/(2*dt); % Omega om = linspace(0,Fmax,bs/2); % om = Fmax*(0:(bs/2)-1)/(bs); % Window if win == 1 windcyc = hann(Nc*bs).'; else windcyc = ones(1,Nc*bs); end % Cyclic Averaging of Data Xcyc = zeros(no,Nc*bs); Fcyc = zeros(ni,Nc*bs); Xavg = zeros(no,Ncycblk*bs); Favg = zeros(ni,Ncycblk*bs);

87

hhhh = waitbar(0,'Cyclic Averaging Response Data.....'); for ii = 1:no indx1 = 1; indx3 = 1; for jj = 1:Ncycblk indx2 = 1; Xcyc(ii,:) = windcyc.*Xt(ii,indx1:indx1+Nc*bs-1); indx1 = indx1 + shft; Xtemp = zeros(1,bs); for kk = 1:Nc Xtemp(1,:) = Xtemp(1,:) + Xcyc(ii,indx2:indx2+bs-1); indx2 = indx2 + bs; end Xavg(ii,indx3:indx3+bs-1) = Xtemp./Nc; indx3 = indx3 + bs; end waitbar(ii/no,hhhh); end close(hhhh) hhhh = waitbar(0,'Cyclic Averaging Force Data.....'); for ii = 1:ni indx1 = 1; indx3 = 1; for jj = 1:Ncycblk indx2 = 1; Fcyc(ii,:) = windcyc.*Ft(ii,indx1:indx1+Nc*bs-1); indx1 = indx1 + shft ; Ftemp = zeros(1,bs); for kk = 1:Nc Ftemp(1,:) = Ftemp(1,:) + Fcyc(ii,indx2:indx2+bs-1); indx2 = indx2 + bs; end Favg(ii,indx3:indx3+bs-1) = Ftemp./Nc; indx3 = indx3 + bs; end waitbar(ii/ni,hhhh); end close(hhhh) clear indx1 indx2 indx3 Ftemp Xtemp Fcyc Xcyc windcyc GFF = zeros(ni,ni,bs); GXX = zeros(no,no,bs); GXF = zeros(no,ni,bs); GFX = zeros(ni,no,bs); H1 = zeros(no,ni,bs); H2 = zeros(no,ni,bs); % Computing Power Spectra for FRF Calculation indx = 1;

88

hhhh = waitbar(0,'Calculating Power Spectra.....'); for asyncavg = 1:Ncycblk % Input Auto Power Spectrum for ii = 1:ni F1 = fft(Favg(ii,indx:indx+bs-1)); for jj = 1:ni F2 = fft(Favg(jj,indx:indx+bs-1)); gff = F1.*conj(F2)/(bs); for kk = 1:bs GFF(ii,jj,kk) = GFF(ii,jj,kk) + gff(1,kk); end nd e end % Cross Power Spectrum GXF for ii = 1:no X = fft(Xavg(ii,indx:indx+bs-1)); for jj = 1:ni F = fft(Favg(jj,indx:indx+bs-1)); gxf = X.*conj(F)/(bs); for kk = 1:bs GXF(ii,jj,kk) = GXF(ii,jj,kk) + gxf(1,kk); end end end % Cross Power Spectrum GFX for ii = 1:no X = fft(Xavg(ii,indx:indx+bs-1)); for jj = 1:ni F = fft(Favg(jj,indx:indx+bs-1)); gfx = F.*conj(X)/(bs); for kk = 1:bs GFX(jj,ii,kk) = GFX(jj,ii,kk) + gfx(1,kk); nd e end end % Output Auto Power Spectrum for ii = 1:no X1 = fft(Xavg(ii,indx:indx+bs-1)); for jj = 1:no X2 = fft(Xavg(jj,indx:indx+bs-1)); gxx = X1.*conj(X2)/(bs); for kk = 1:bs GXX(ii,jj,kk) = GXX(ii,jj,kk) + gxx(1,kk); end nd e end indx = indx + bs; waitbar(asyncavg/Ncycblk,hhhh); end

89

GFF = GFF./Ncycblk; GXX = GXX./Ncycblk; GFX = GFX./Ncycblk; GXF = GXF./Ncycblk; close(hhhh) % H1 H2 Algorithms hhhh = waitbar(0,'Calculating FRF.....'); for kk = 1:bs Gff = GFF(:,:,kk); Gxf = GXF(:,:,kk); Gfx = GFX(:,:,kk); Gxx = GXX(:,:,kk); if type == 1 h1 = Gxf*pinv(Gff); H1(:,:,kk) = h1; else h2 = Gxx*pinv(Gfx); H2(:,:,kk) = h2; end waitbar(kk/bs,hhhh) end close(hhhh) hhhh = waitbar(0,'Calculating Ordinary Coherence.....'); for kk = 1:bs for ii = 1:ni for jj = 1:no Gff = GFF(ii,ii,kk); Gxf = GXF(jj,ii,kk); Gfx = GFX(ii,jj,kk); Gxx = GXX(jj,jj,kk); coh(jj,ii,kk) = (Gxf*Gfx)/(Gff*Gxx); end end waitbar(kk/bs,hhhh) end close(hhhh) hhhh = waitbar(0,'Calculating Multiple Coherence.....'); for kk = 1:bs for jj = 1:no mat = horzcat(GXX(jj,jj,kk),GXF(jj,:,kk)); mat2 = horzcat(GFX(:,jj,kk),GFF(:,:,kk)); nmat = vertcat(mat,mat2); mcoh(jj,:,kk) = 1 - abs(det(nmat))/(GXX(jj,jj,kk) * abs(det(GFF(:,:,kk)))); end waitbar(kk/bs,hhhh)

90

end close(hhhh) if type == 1 h = H1; else h = H2; end c = coh; w = om; mc = mcoh; ll = fix((lf*bs/2)/Fmax); hl = fix((hf*bs/2)/Fmax); h1 = zeros(no,ni,bs); c1 = zeros(no,ni,bs); mc1 = zeros(no,1,bs); GFF1 = zeros(ni,ni,bs); GXX1 = zeros(no,no,bs); GXF1 = zeros(no,ni,bs); GFX1 = zeros(ni,no,bs); for kk = ll:hl h1(:,:,kk) = h(:,:,kk); c1(:,:,kk) = c(:,:,kk); mc1(:,:,kk) = mc(:,:,kk); GFF1(:,:,kk) = GFF(:,:,kk); GXX1(:,:,kk) = GXX(:,:,kk); GXF1(:,:,kk) = GXF(:,:,kk); GFX1(:,:,kk) = GFX(:,:,kk); end for kk = bs-hl:bs-ll h1(:,:,kk) = h(:,:,kk); c1(:,:,kk) = c(:,:,kk); mc1(:,:,kk) = mc(:,:,kk); GFF1(:,:,kk) = GFF(:,:,kk); GXX1(:,:,kk) = GXX(:,:,kk); GXF1(:,:,kk) = GXF(:,:,kk); GFX1(:,:,kk) = GFX(:,:,kk); end msgbox('System Matrices Calculated!') catch msgbox('Error Calculating System Matrices! Try again by Varying Parameters') end

91

B.2 convOver: Convolution with Overlap Addition

Performs convolution with overlap addition of inverse system model with desired

responses, to get time domain desired force signals

% Syntax: F = convOver(invH(system model), dr (desired response)) function F = convOver(invH,dr) s = size(invH); ni = s(1); no = s(2); bs = s(3); for jj = 1:no for ii = 1:ni tinvH(jj,ii,:) = real(ifft(invH(jj,ii,:))); dum = [squeeze(tinvH(jj,ii,bs/2+1:bs)).' , squeeze(tinvH(jj,ii,1:bs/2)).',zeros(1,bs)]; convH(jj,ii,:) = fft(dum); end end pos = 1; tF = zeros(ni,length(dr)+ bs); for ii = 1:length(dr)/(bs) for jj = 1:no fx(jj,:) = fft([dr(jj,pos:pos+bs-1),zeros(1,bs)]); end for kk = 1:2*bs prod(:,kk) = convH(:,:,kk)*fx(:,kk); end for pp = 1:ni tprod(pp,:) = real(ifft(prod(pp,:))); end tF(:,pos:pos+2*bs-1) = tF(:,pos:pos+2*bs-1) + tprod; pos = pos + bs; end F = tF(:,bs/2+1:length(tF)-bs/2);

92

B.3 bandpass : Bandpass filter

Acts as a bandpass filter to filter out unwanted frequency content from a pure random

white noise signal

% Function: bandpass % Syntax: filtered_wht = bandpass(original_wht, Sampling Frequency, Lower Frequency, Higher frequency); function f = bandpass(x,sfreq,lfreq,hfreq) % Butterworth Filter % Filtering out frequencies above specified higher frequency [b,a] = butter(4,ufreq/(freq/2)); f = filter(b,a,wht); % Filtering out frequencies below specified lower frequency if lfreq ~= 0 [d,c] = butter(4,lfreq/(freq/2),'high'); wht = filter(d,c,wht); end

93

C White Noise Generator GUI

C.1 DRV File Creator

The DRV File creator GUI [6] enables the creation of RPC III Format .DRV files which

can be input to the MTS 320 Road Simulator through the Flex Test II software interface.

The RPC III format .DRV file essentially contains a header section, which contains

information about the different drive channels, the scaling, the maximum and minimum

limits, the units of the data, and other identifying information, followed by the data

section which contains the voltage signals stored in a binary format. The DRV file

generator uses user inputs to generate the drive signals, and can thus be used to create a

variety of input excitations. These are explained below:

C.2 DRV File Creator Options

C.2.1 Header Data

In this section, the following information for the drive file is given:

Format: It defines the format that the current drive file is to be stored in. Currently,

Binary IEEE Big Endian, or Binary formats can be used.

File Type: This determines the type of file being created. Since the data stored is time

domain drive or response data, the file type is Time History.

Frames: This determines the number of frames of data. The stored data is divided into

frames of data, with each frame corresponding to 5 seconds of time data. A file can have

a maximum of 8192 frames.

Sampling Frequency: Specifies the Sampling Frequency in Hz.

Bypass A/D Filter: Options Yes/No. Yes bypasses the filter, while No uses the filter.

94

C.2.2 Channel Data

Checkbox: Checked – Select Channel, Unchecked – Ignore channel

Signal: Signal Type – Sine, Square, White Noise (band limited random), Non-Linear

(stepped band limited random)

Amplitude: Signal Amplitude in Inches

Offset: Offset position of Actuator from the Mean position, in Inches

Single Frequency: Selectable for Sine/Square wave signals, value in Hz

Frequency Range: Selectable for White Noise/Non-Linear signals, values in Hz

95

Figure 58 : White Noise Generator GUI

96

D Road Simulator GUI

The Road Simulator GUI [6] performs the complete simulation of the road test. The

Simulator menus are divided into 3 parts based on function.

• The first menu is the Project menu, in which the user can create a new project,

open an existing project for review of continuation, and save the current project.

• The second menu is the System Model Development menu, which handles the

calculation of the FRF and the system matrices, namely the different power

spectra and the coherences, and displays the same.

• The third menu is the Generate Drive File menu, which handles the choice of

desired responses, and calculates the drive signal required to obtain that desired

response, and also plots the errors between the measured and calculated

responses.

The Road Simulator menus and their functions are detailed below:

D.1 Project

D.1.1 New Project: Create a New Project

This menu creates a new .mat file containing project information, namely the project title

and description, the location of the project file, the number of inputs, the number of

outputs, and the sampling frequency used for data collection.

D.1.2 Open Project: Open Existing Project

This menu opens an existing .mat project file for continuing a previously saved project.

D.1.3 Save Project: Save Current Project

This menu saves all current project data to the specified file.

97

D.2 Develop System Model

D.2.1 White Noise Generator

• Launch White Noise Generator: Launches the .DRV File Creator GUI

• Display Created Drive File: Displays File created by the .DRV File Creator

D.2.2 System Model

• Calculate H: Calculates the system FRF, the inverse FRF, the coherence functions

and the input and output auto and cross power spectra

• Recalculate H: Recalculates the system matrices for different parameters, namely

block size and number of averages.

D.2.3 Display System Model

• Plot H: Plots System FRF as a function of frequency

• Plot invH: Plots Inverse System FRF as a function of frequency

• Plot Ordinary Coherence: Plots Ordinary Coherence functions

• Plot Multiple Coherence: Plots Multiple Coherence Functions

• Plot GFF: Plots Input Auto Power Spectra

• Plot GXX: Plots Output Auto Power Spectra

• Plot GXF: Plots Cross Power Spectra between inputs F and outputs X

• Plot GFX: Plots Cross Power Spectra between inputs F and outputs X

98

D.3 Desired Response

D.3.1 Select Desired Response

• Time Domain:

From File: Select Time Domain Desired Response from File

User Defined: User Defined Time Domain Desired Response

• Frequency Domain:

From File: Select Frequency Domain Desired Response from File

User Defined: User Defined Frequency Domain Desired Response

D.3.2 Plot Desired Response

• Plots the Desired Response, either in Time domain or Frequency domain,

depending on the type of analysis

99

D.4 Generate Drive File

D.4.1 Calculate Drive

• Calculates drive signals required to generate the desired response signals

D.4.2 Write Drive File

• This menu writes out an RPC III .drv file containing the drive signals required for

the simulation of desired response.

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D.5 View Results

D.5.1 Time Domain

• Plots measured response versus desired response in the time domain

D.5.2 Frequency Domain

• Plots measured response versus desired response in the frequency domain.

D.5.3 Plot Error

• Plots the Error between Measured Response and Desired Response for each

Iteration

D.5.4 Plot Trend

• Plots the Trend in Maximum value of the Standard Deviation of the Error, as a

function of the Iteration number

D.5.5 Document Results

• Documents the Standard Deviation of the Error for each Iteration in a .txt File for

future retrieval

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Figure 59 : Road Simulator GUI

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