UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI Date: 04/14/06 I, Suresh Babu Chennagowni, hereby submit this work as part of the requirements for the degree of: Master of Science in: Mechanical Engineering It is entitled: Study of the effect of Mass Distribution, Path of Energy and Dynamic Coupling on Combined Coherence (A Non-linerarity Detection Method) This work and its defense approved by: Chair: _Dr. Randall J. Allemang Dr. Allyn W. Phillips Dr. Ronald L. Huston

2 Study of the Effect of Mass Distribution, Path of Energy and Dynamic Coupling on Combined Coherence (A Non-linearity Detection Method) 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 Engineering of the College of Engineering 2006 by Suresh Babu Chennagowni Bachelor of Engineering, Osmania University, Hyderabad, India, 2002 Committee Chair: Dr. Randall J. Allemang

3 ABSTRACT Almost all practical systems are non-linear to some extent with the non-linearity being caused by one or a combination of factors. If the system is non-linear, errors are introduced in the data analysis and are observed during the modal tests of a structure. For example, high forcing levels may cause the frequency response function estimates to show non-coherent behavior over certain frequency bands. A new coherence function (Combined Coherence) provides a method to separate the effects of structural nonlinearities and the digital signal processing errors. Thomas Roscher [1] applied the combined coherence formulation to theoretical data generated from a lumped parameter (M, K, C) with static coupling. The results showed improvement in the combined coherence function over the ordinary coherence, but when Doug Coombs [2] applied combined coherence to a real world structure, it did not show improvement. In this thesis, as an extension of previous work, study is done on theoretical data generated from a lumped mass model with dynamic coupling. The effects of mass distribution, spatial density, forcing level, location of forcing function, path of energy and the dynamic coupling on the combined coherence are studied. The testing cases include SIMO and MIMO cases for a MDOF simulink model with a cubic hardening type of nonlinearity applied at different locations. Combined Coherence is calculated for a non-linear model and effects on the combined coherence are studied for the following cases. Effect of varying the force input Effect of dynamic coupling

4 Effect of location of input and path of energy Effect of mass distribution Effect of spatial density of masses Effect of scaling of motions

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6 ACKNOWLEDGEMENTS I would like to express my gratitude towards all who were involved in the completion of this thesis. First of all, I would like to thank Dr. Randall Allemang for providing me with this opportunity to work under his guidance. I would also like to thank Dr. Allyn Phillips who helped me out through the research. Dr. Randy and Dr. Allyn have always been a source of support and encouragement. Their inputs and advice have contributed substantially to the completion of my work. I would like to thank Dr. Ronald Huston for serving as member on my thesis committee. I express my thanks to my colleagues at Structural Dynamics Research Laboratory for their helpful discussions in various matters during the course of this work. I would also like to thank all those who helped me with the style and grammar of the writing. Finally, I would like to thank my parents and family for constantly supporting my academic pursuits.

7 Table of Contents 1. Introduction 1 2. Theoretical Background Linear Model SDOF Mechanical System Frequency Response Function Theory of Ordinary and Multiple Coherence Excitation Techniques Overview of Non-Linearity Non-Linearity Detection Techniques Non-Linear Detection Method (Combined Coherence Function) Theory of Combined Coherence Development of Combined Coherence Application of Combined Coherence to Rocher Analytical Model Application of Combined Coherence to Real world system Theoretical Model used to study the Combined Coherence Application of Combined coherence to Analytical Model Effect of Varying the Force Input SIMO situations with Dynamic Coupling MIMO situations with Dynamic Coupling Effect of Dynamic Coupling on Combined Coherence Effect of Location of Input and Path of Energy on Combined Coherence Effect of Mass Distribution on Combined Coherence.66 I

8 4.7 Effect of Spatial Density of Masses on Combined Coherence Effect of Scaling of Motions of DOF on Combined Coherence Conclusions Future Work References Appendix Simulink Model when the non-linearity is between DOFs 1 and Simulink Model when the non-linearity is between DOFs 1 and Simulink Model when the non-linearity is between DOFs 1 and Simulink Model when the non-linearity is between DOFs 2 and Simulink Model when the non-linearity is between DOFs 2 and Simulink Model when the non-linearity is between DOFs 3 and II

9 LIST OF FIGURES Figure 2-1: SDOF 4 Figure 2-2: Single Input System..6 Figure 2-3: Cubic Stiffness 11 Figure 2-4: FRF and Coherence of nonlinear system Figure 3-1: a) Lumped mass structure system b) Force system Figure 3-2: 2 DOF model with rotary inertia Figure 3-3: Roscher Theoretical Model Figure 3-4: FRF s and Coherences for Case Figure 3-5: Comparison of Coherence and CCOH for Case Figure 3-6: FRF s and Coherences for Case Figure 3-7: Comparison of Coherence and MCCOH for Case Figure 3-8: Line diagram of Doug Coombs model...23 Figure 3-9: Theoretical 4 DOF lumped model...25 Figure 3-10: Comparison of Analytical and Simulation Results...28 Figure 4-1: FRF s, Coherences and MCCOH for Case Figure 4-2: FRF s, Coherences and MCCOH for Case Figure 4-3: FRF s and Coherences of Case Figure 4-4: Coherence and CCOH of Case Figure 4-5: FRF s, Coherences and MCCOH for Case Figure 4-6: FRF s, Coherences and MCCOH for Case Figure 4-7: FRF s, Coherences and MCCOH for Case Figure 4-8: FRF s, Coherences and MCCOH for Case III

10 Figure 4-9: FRF s, Coherences and MCCOH for Case Figure 4-10: FRF s, Coherences and MCCOH of Case Figure 4-11: FRF s and Coherences of Case Figure 4-12: FRF s, Coherences and MCCOH of Case Figure 4-13: FRF s, Coherences and MCCOH of Case Figure 4-14: FRF s, Coherences and MCCOH of Case Figure 4-15: FRF s, Coherences and MCCOH of Case Figure 4-16: FRF s, Coherences and MCCOH of Case Figure 4-17: FRF s, Coherences and MCCOH of Case Figure 4-18: FRF s, Coherences and MCCOH of Case Figure 4-19: FRF s, Coherences and MCCOH of Case Figure 4-20: FRF s, Coherences and MCCOH of Case Figure 4-21: FRF s, Coherences and MCCOH of Case Figure 4-22: FRF s, Coherences and MCCOH of Case Figure 4-23: FRF s, Coherences and MCCOH of Case Figure 4-24: FRF s, Coherences and MCCOH of Case Figure 4-25: FRF s, Coherences and MCCOH of Case Figure 4-26: FRF s, Coherences and MCCOH of Case Figure 4-27: FRF s, Coherences and MCCOH of Case Figure 4-28: FRF s, Coherences and MCCOH of Case Figure 4-29: FRF s, Coherences and MCCOH of Case Figure 4-29: FRF s, Coherences and MCCOH of Case Figure 4-30: FRF s, Coherences and MCCOH of Case IV

11 Figure 4-31: FRF s, Coherences and MCCOH of Case Figure 4-32: FRF s, Coherences and MCCOH of Case Figure 4-33: FRF s, Coherences and MCCOH of Case Figure 4-34: FRF s, Coherences and MCCOH of Case Figure 4-35: FRF s, Coherences and MCCOH of Case Figure 4-36: FRF s, Coherences and MCCOH of Case Figure 4-37: FRF s, Coherences and MCCOH of Case Figure 4-38: FRF s, Coherences and MCCOH of Case Figure 4-39: FRF s, Coherences and MCCOH of Case Figure 4-40: FRF s, Coherences and MCCOH of Case Figure 4-41: FRF s, Coherences and MCCOH of Case Figure 4-42: FRF s, Coherences and MCCOH of Case Figure 4-43: FRF s, Coherences and MCCOH of Case V

12 LIST OF TABLES Table 1-1: Sample test cases of combined coherence applied to Roscher model..19 Table 4-1: MIMO situations for different force exciting levels 30 Table 4-2: System with Dynamic Coupling SIMO situations...33 Table 4-3: MIMO situations of system with Dynamic Coupling..43 Table 4-4: MIMO situations of system with no Dynamic Coupling.51 Table 4-5: MIMO situations to study effect of Path of Energy.59 Table 4-6: MIMO situations to study effect of Mass Distribution 66 Table 4-7: MIMO situations to study effect of Spatial Densities of Masses.74 Table 4-8: MIMO situations to study effect of Scaling of Motions..82 VI

13 NOMENCLATURE NOTATION m M k K c C q p H pq Mass Mass Matrix Stiffness Stiffness Matrix Viscous Damping Damping Matrix Input Location Output Location Frequency Response Function at output p and input q.. ( x t) Acceleration. ( x t) Velocity x (t) Displacement F Force input in frequency domain f (t) Force input in time domain λ 1,2 Eigen Value ω η ν X`(ω) F`(ω) Circular Frequency Noise on output Noise on input Measured input of the system Measured output of the system 2 γ pq ( ω) Coherence Function GFX qp (ω) Cross Power Spectrum of Input q and output p GFF qq (ω) Input Power Spectrum at input q GXX pp (ω) Output Power Spectrum at output p ε Non-linear Scaling Factor VII

14 r j t Rotary Inertia Radius Sample time ABBREVIATION DOF SIMO MIMO SDOF COH MCOH CCOH MCCOH Degree of Freedom Single Input Multiple Output Multiple Input Multiple Output Single Degree of Freedom Coherence Function Multiple Coherence Function Combined Coherence Multiple Combined Coherence VIII

15 1. Introduction Experimental modal analysis is often used for checking the accuracy of an analytical approach such as finite element analysis and verification/correction of the results of the analytical approach (model updating). During the modal analysis procedure, there are four basic assumptions (linearity, time invariance, reciprocity and observability) made concerning any structure. Because these assumptions are assumed to be valid, errors accumulate at the modal parameter estimation phase. Among these errors are the errors due to nonlinearities in the structure and the errors due to digital signal processing. The errors due to nonlinearities are visible in the measured data as slight distortions in the frequency response function (FRF) plots, but they are also responsible for significant discrepancies in the modal analysis process. Some of the algorithms used to extract modal parameters can be surprisingly sensitive to the small deviations (from linear characteristics), which accompany the presence of slightly nonlinear elements in the structures. Understanding these effects and detecting their presence, means that alternative test procedures can be used so that the nonlinear effects are not only prevented from contaminating the measurement and analysis processes but can actually be quantified and included in the model. In this thesis, a further study is done on the Combined Coherence Method, which is a frequency domain method of detecting structural nonlinearities. It is the method of detecting the presence of nonlinearities between degrees of freedom by separating the errors due to digital signal processing and nonlinearities. It is a quick and efficient method to detect structural nonlinearities between the degrees of freedom from the data taken during the modal test. Thomas 1

16 Roscher applied combined coherence to the theoretical data generated from lumped mass model [1] and Doug Coombs applied combined coherence to the data measured from a practical nonlinear structure [2]. Combined coherence was able to locate the nonlinearities in the first case for theoretical lumped model whereas in the second case it could not locate the nonlinearities spatially. In this thesis, further study is done on theoretical data generated from a lumped model with dynamic coupling similar to the real world system used by Doug Coombs. A study is done on how different parameters such as mass distribution, spatial density, forcing level, location of forcing function, path of energy and the dynamic coupling effects the combined coherence. The second chapter in this thesis gives an introduction to non-linear vibration and methods in detecting the nonlinearities. Chapter 3 gives introduction to combined coherence method, its derivation and the previous work of Roscher and Coombs. In Chapter 4 combined coherence is applied to a non-linear model and effects on combined coherence are discussed for the following cases. Effect of varying the force input. Effect of dynamic coupling Effect of location of input and path of energy Effect of mass distribution Effect of spatial density of masses Effect of scaling of motions Summary and conclusions are given in Chapter 5. 2

17 2. Theoretical Background 2.1 Linear Systems A clear understanding of the concept of a degree of freedom is required for understanding the concept of modal analysis. The number of degrees of freedom is the minimum number of coordinates required to specify completely the motion of a mechanical system. There exist six degrees of freedom at each point, the motion in each direction and the rotational motion of each axis. A mechanical system has an infinite number of degrees of freedom, because the system is continuous. The observed degrees of freedom are in reality, of course, a finite number, limited by different physical causes. The following parameters reduce the effective number of degrees of freedom: the frequency range of interest and physical points of interest. There are four assumptions made during the modal analysis procedure [11]. The first basic assumption is that the structure is linear. This means that the structure obeys the superposition principle, which states that the response of the system to a combination of forces applied simultaneously is equal to the sum of the responses due to the individual forces. The second assumption is that the structure is time invariant. This means that the properties of the system such as mass, stiffness and damping do not change with time (i.e., they remain unchanged for any two different testing times). The third assumption is that the structure obeys Maxwell reciprocity. This principle states that the response of the function at a degree of freedom q due to the input at p is equal to the response at p due to the input at q i.e., H pq = H qp. The fourth assumption is that the structure is 3

18 observable. The response points are chosen such that the complete structure is observed. For example, structures with loose components, which have degrees of freedom that cannot be measured, are not completely observable. 2.2 SDOF Mechanical System Simple systems can be modeled as a mass-damper-spring system at a single point in a single direction. These are referred as single degree of freedom (SDOF) systems. A SDOF mechanical system is described by Newton s equation as shown in equation below.... m x( t) + c x( t) + kx( t) = f ( t) (2.1) Figure 2-1: SDOF 4

19 This equation has two solutions, a transient solution and a steady state solution. The equation can be solved using the Laplace transform, assuming initial conditions to be zero. The equation of the above system can be written as Ms 2 X(s) + csx(s) + kx(s) = F(s) (2.2) where: s is a complex-valued frequency variable (Laplace variable). The above equation can be rewritten as 1/(ms 2 +cs+k)=x(s)/f(s) = H(s) (2.3) and finally the homogeneous equation is solved using f (t) = 0 which yields the natural frequencies as c c 2 k λ 1,2 = ± ( ) (2.4) 2m 2m m 2.3 Frequency Response Function The relation between the input to the system and its response is determined by the frequency response of the system, which is a characteristic feature of the system. The response of a system to an output is completely determined by its frequency response function. 5

20 Consider a single input system shown below: Figure 2-2: Single Input System Ideally the frequency response of the system is calculated by H (ω) = X (ω)/f (ω) (2.5) where: H (ω) = Frequency response function of the system F (ω) = Frequency Domain information of the input signal with no noise on signal X (ω) = Frequency Domain information of the output signal with no noise on the signal But, due to measurement errors, the actual frequency response function is given by X` (ω) - η = (F` (ω) -ν) H (ω) (2.6) where: ν = Noise on the input signal. η = Noise on the output signal. F` (ω) = Measured input of the system. X` (ω) = Measured output of the system. 6

21 The three most common types of frequency response algorithms are based on the least squares model: the H 1 algorithm which minimizes the noise on the output, the H 2 algorithm which minimizes the noise on the input and H v algorithm which minimizes the noise on both the input and output. In this thesis, the H 1 algorithm is used. 2.4 THEORY OF COHERENCE Ordinary and Multiple Coherence: The ordinary coherence function (COH) is computed as [11]: 2 GXF ( ) ( ) ( ) 2 pq ω GXFpq ω GFX qp ω COH pq ( ω) = γ pq ( ω) = = (2.7) GFF ( ω) GXX ( ω) GFF ( ω) GXX ( ω) qq This function is frequency dependent and is a real value between zero and one. The value 1 indicates that the measured response power is totally correlated with the measured input power. The value zero indicates the output is totally correlated with the sources other than the measured input. A coherence value less than unity at any frequency is due to variance and bias errors. The low coherence due to a variance error like random noise can be significant provided sufficient averaging had occurred. Since coherence is a statistical indicator, the more ensembles averaged, the more reliable is the result (smaller standard deviation). The bias errors can be broadly classified into two categories, digital signal processing errors and the errors due to nonlinearities. All errors causing drops in the coherence fall into one of these two categories. The frequencies where the coherence is low are often the same frequencies where the FRF is maxima in magnitude (resonance) or minima in magnitude (anti-resonance), which may be an indication of leakage. The drop pp qq pp 7

22 in coherence at any other frequency is more clearly due to other errors such as noise or nonlinearities. Multiple inputs are often desired during testing so that the energy is more evenly distributed throughout a structure and as a result the vibratory amplitudes across the structure will be more uniform, with a consequent decrease in the effect of nonlinearities. Coherence is not an appropriate measure of linear dependency between input and output when there is more than one input. The multiple coherence function (MCOH) that determines the linear dependency of input and output is computed as [12] MCOH p N = i N * i H pq ( ω) GFFqt ( ω) H pt ( ω) ( ω ) (2.6) GXX ( ω) q= 1 t= 1 The value of MCOH varies between zero and one. A value of one indicates an output is correlated with all known inputs, while a value less than unity indicates unknown contributions such as measurement noise and nonlinearities. pp 2.5 Excitation Techniques For a linear system the dynamic characteristics will not vary according to the choice of the excitation technique used to measure them. However, the effects of most kinds of nonlinearities, encountered in structural dynamics are generally found to vary with the external excitation. Hence, the first problem of a nonlinearity investigation is to decide the type of excitation so that the nonlinearity is exposed and identified. There are currently many types of excitation methods widely used in vibration study practice. These excitation techniques are broadly classified as sinusoidal, transient and random excitation. Sinusoidal excitation is widely regarded as the best excitation technique for the identification of nonlinearities. The advantage of a sinusoidal excitation is, it is easy 8

23 to accurately control the input signal level and hence, enables a high input force to be fed into the structure. However, the drawback of this type of excitation is, it is relatively slow compared to many of the other techniques used in practice. Since the excitation is performed frequency by frequency and at each step, time is required for the system to settle to its steady-state value, sinusoidal methods are very time consuming. On the other hand, with the random excitation technique, the system can be excited at every frequency simultaneously within the range of interest. This wide frequency band excitation enables it to be much faster than the sinusoidal excitation. Also, random excitation in general linearizes the nonlinear structure due to randomness of input force amplitude. This technique is the best match for modal analysis, as most of the modal parameter estimation methods are based on linearity. Due to the above stated factors, random excitation is very commonly used in actual testing conditions. Hence, test engineers need a nonlinear detection method that is compatible with normal modal analysis methods employing random excitation. Therefore, in this thesis, random excitation is used in detecting the structural nonlinearities using combined coherence method. 2.6 Overview of Non-linearity Most practical engineering structures exhibit a certain degree of nonlinearity due to nonlinear dynamic characteristics of structural joints, nonlinear boundary conditions and nonlinear material properties. For practical purposes, in many cases, they are regarded as linear structures because the degree of nonlinearity is small and therefore, insignificant in the response range of interest. Most theories, upon which structural dynamic analysis is 9

24 founded, rely heavily on this assumption of linearity (superposition principle). The superposition principle states that, the deflection due to two or more simultaneously applied loads is equal to the sum of the deflections caused, when the loads are applied individually. But for some cases, the effect of nonlinearity may become so significant that it has to be taken into account in the analysis of dynamic characteristics of the structure. The present thesis focuses on the location of nonlinearity based on the measurement of input and output using combined coherence function. Nonlinear structures are often divided into three main types: zero memory, finite memory and infinite memory systems. The zero memory type of system is the most simple of the three types, as it only applies the nonlinear operator at system input, whereas the infinite memory type of system applies nonlinearity to the system response as well. A typical infinite memory type of system for a MDOF system can be written as [12] [3] [M] x(t) +[C] x(t) +[K] x(t) +[K n ] x 3 (t) = f (t) (2.7) The common types of nonlinearities are displacement type nonlinearities (hardening, softening, hardening/softening and dead zone) and velocity related nonlinearities (quadratic damping, softening/hardening damping and coulomb friction). In this study the effect of a cubic stiffness non-linearity on the combined coherence is studied by applying it to the MDOF system. The mathematical model of a cubic stiffness element can be expressed as f(x) = k( x+εx 3 ) (2.8) 10

25 where the coefficient k represents spring stiffness, and the coefficient ε represents the degree of nonlinearity. The Figure 2.3 below represents both the linear and the nonlinear behavior of a cubic stiffness element. It can be seen that the overall stiffness changes with the displacement x, while the stiffness coefficients k and ε remain constant. Figure 2-3: Cubic Stiffness Cubic stiffness is applied to the simulation model used in this study to observe its nonlinear characteristics by exciting the system at five forcing levels. The FRF and coherences of a nonlinear system can be seen in the Figure 2.4. It can be seen from the FRF and COH function plots that the anti-resonances and resonances are changed as the excitation force level changes and thus it can be assumed that the system is non-linear. 11

26 Figure 2-4: FRF and Coherence of nonlinear system 2.7 Non-linear detection techniques A linear time-invariant system is relatively well understood and theoretically well developed. The same is not true for the case of a nonlinear system. In most of the situations, it is necessary to first detect the presence of nonlinearity. A lot of work is done in this direction and quite a number of procedures are suggested. A brief review of some of the detection methods is presented here. M. Simon and G. R. Tomlinson [4] proposed a Hilbert transform technique to detect and quantify structural nonlinearities. The basis that the Hilbert transform technique can be used to identify nonlinearity is due to the fact that for a linear structure, the real and imaginary parts of a measured FRF constitute a Hilbert transform pair, whereas for the FRF of a nonlinear structure, the Hilbert transform relationships do not hold. By calculating the Hilbert transform of the real part (or the imaginary part) of a measured 12

27 FRF and comparing it with the corresponding imaginary part (or real part), the existence of nonlinearity can be identified based on the difference of the transform pair. M. Mertens, H.Vander, P. Vanherck, R. Snoeys [6] proposed a complex stiffness method, which is based on the mapping of different estimates of stiffness and damping for each measured frequency as a function of magnitude of displacement and the velocity respectively. The equivalent stiffness and damping of a linear system are constant while for a nonlinear system stiffness and/or damping vary. This method gives an idea of degree and type of nonlinearity. He J. and D.J. Ewins [7] proposed Inverse Receptance method in which nonlinearity is detected as whether it exists in the stiffness or damping, by displaying the FRF data in inverse form. For a linear system a plot of real part of inverse FRF against ω 2 and the imaginary part against ω yields a straight lines while for non-linear systems the plots are not straight lines. The nonlinearities associated with stiffness show up in the real part while in the imaginary part the nonlinearities due to damping show up. Vanhoenacker K., T. Dobrowiecki, J. Schouskens [8] proposed a multisine excitation method to detect nonlinearities. In this method, the system is excited at only a few chosen set of frequency lines. It is shown that by exciting the system only at a selected set of frequency lines, the even nonlinear disturbances can be determined at the even frequency lines while at unexcited odd frequency lines the odd nonlinear distortions can be determined. 13

28 Kim W-J and Y-S Park [10] proposed non-causal power ratio (NPR) method. It is a causality check method that quantifies the non-linearity. The NPR value grows with the increase in nonlinearity and is a function of excitation amplitude. NPR function detects the non-linearity and also the type of nonlinearity by examining the variation of the NPR values with excitation force. The advantages of this method are 1. It takes less computation time 2. This method does not require prior information of the system 3. It can be applied without any limitations to the nonlinearities 14

29 3. Non-linear Detection Method (Combined Coherence Function) In this chapter the theory of the combined coherence is discussed and mathematical equations for both the ordinary and multiple combined coherence (MCCOH) are derived. 3.1 THEORY OF COMBINED COHERENCE In general structures are represented by assuming lumped masses as node elements with mass and no stiffness, and are connected by stiffness and damping terms. The distribution of mass is important in dynamic analysis. The general representation of the structure and the force system is shown in figure below [1]. Figure 3-1: a) Lumped mass structure system b) Force system At any node point if Newton s law is applied and an equation of motion is developed, then the acceleration is the sum of both the internal force terms caused by stiffness and 15

30 damping terms, and the external force terms. Considering a 2 DOF model shown in the Figure 3-2, the equations of motion can be written as Figure 3-2: 2 DOF model with rotary inertia j2 / r2 ) x1 ( j2 / r2 ) x2 = ( c01 + c12 + c20 ) x1+ c12 x 2 ( k01 + k12 + k20 ) ( m x + k x + f (3.1) j2 / r2 ) x2 ( j2 / r2 ) x1 = ( c12 + c20 ) x 2 + c12 x1 ( k12 + k20 ) ( m + x + k x + f (3.2) When motions of two degrees of freedom are combined under the condition of equal mass i.e., (m 1 = m 2 ), the contribution of motions due to internal forces between degrees of freedom will disappear. The equation obtained by combining the motions of DOF is x + x = / m[ c x 2 c x1] + 1/ m[ k x k x ] + [ f f ]/ m (3.3) If a coherence function is calculated for a virtual coordinate created by combining the motions between these DOF s, the drops in coherence due to non-linearity would go away but the low coherence values due to digital signal processing errors would not improve. The critical condition for this method is the equality of masses between the degrees of freedom between which the motions are combined. If the masses are not equal, 16

31 the detection method can still be applied, if the motions are scaled according to the mass ratio. In this thesis, a test case is run to see if scaling the masses would improve the combined coherence in detecting the nonlinearities. 3.2 Development of Ordinary and Multiple Combined Coherence functions [2]: The standard equation for ordinary coherence function is given by 2 GXF ( ) ( ) ( ) 2 pq ω GXFpq ω GFX qp ω COH pq ( ω) = γ pq ( ω) = = (3.4) GFF ( ω) GXX ( ω) GFF ( ω) GXX ( ω) qq Since the CCOH function is based on the sum of the motion between two DOF s. Substituting X p + X r for X p we get pp qq pp CCOH ( pr) q * * ( X r + X p ) Fq ( X r + X p ) Fq = (3.5) * F F ( X + X )( X + X ) q * q r p r p CCOH CCOH ( pr) q ( pr) q * * * ( X r Fq + X p Fq )( X r Fq + X p Fq ) = (3.6) * * * * * F F ( X X + X X + X X + X X ) q q r r r p p r 2 GXFpq + GXFrq = (3.7) GFF ( GXX + GXX + GXX + GXX ) qq rr rp pr p p pp The standard equation for multiple coherence function is given by [11] MCOH p N = i N * i H pq ( ω) GFFqt ( ω) H pt ( ω) ( ω ) (3.8) GXX ( ω) q= 1 t= 1 after following the similar steps as for CCOH, MCCOH can be derived as MCCOH p+ r N = i Ni ps rs qt pt rt s= 1 t= 1 GXX pp ( ω) + GXX pr ( ω) + GXX rp ( ω) + GXX rr ( ω) pp ( H ( ω) + H ( ω)) GFF ( ω)( H ( ω) + H ( ω)) * ( ω ) (3.9) 17

32 3.3 Applying CCOH formulation to the Roscher Theoretical Model Roscher applied the combined coherence function to the data generated from the theoretical lumped parameter (M, K, C) model with static coupling. The model used by the Roscher is shown in the Figure 3.3. Roscher had applied the combined coherence formulation for various testing conditions for different types of displacement and velocity related nonlinearities. There was complete improvement in the combined coherence for some of the cases and in some cases, for some frequency ranges, the combined coherence did not show improvement. Only a few cases were tested for different kinds of nonlinearities. The mass distribution, which is a critical parameter for combined coherence in determining the nonlinearities, was not extensively studied. In this thesis, a study is done on how the mass distribution affects the combined coherence by simulating cases with mass equality between the DOF s. 18

33 Figure 3-3: Roscher Theoretical Model A few cases simulated by Roscher are shown in the Table 1-1. As it can be seen from the combined coherence (CCOH) plot, it is not improved completely. There is a drop in CCOH in the range of 16 to 18 Hz and this can be due to the nonlinear motion entering through other paths. This drop in CCOH still needs to be studied, before CCOH can be applied to any real world structure. Case Location of Force M 1, M 2, M 3 ε Non-Linearity and M 4 (Kg) 1 1 and 3 F 3 = 30 N 12, 7, 9 and and 3 F 1 = 50 N and F 3 = 50 N 12, 7, 9 and Table 1-1: Sample test cases of combined coherence applied to Roscher model 19

34 Figure 3-4: FRF and Coherence for Case 1 20

35 Figure 3-5: Comparison of Coherence and CCOH for Case 1 Figure 3-6: FRF and Coherence for Case 2 21

36 Figure 3-7: Comparison of Coherence and MCCOH for Case Application of CCOH to Real world structure Doug Coombs applied combined coherence to a real world structure. The system consisted of an H-frame with (2x6x0.25) with another square frame (2x2x0.125 ) steel tubing. These two frames were connected at 4 discrete points giving various options for linear/non-linear conditions. Two shakers were connected in a skew direction at an angle of 45 0 in order to get energy in all three directions. The following testing scenarios were examined to check the ability of combined coherence to spatially locate nonlinearities. The line diagram of testing structure is shown below in Figure

37 Figure 3-8: Line diagram of Doug Coombs model Different testing cases such as Cases with and without leakage errors Varying the number of spectral averages Reducing the number of nonlinear paths Varying the input force locations Changing the spatial density of the responses on combined coherence were studied. For a nominal linear connection between the connections, the improvement in combined coherence was near the resonances instead at the anti-resonances raising a question if leakage is affecting the combined coherence. For many of the testing cases the improvement in the combined coherence was small when compared to multiple coherence. In one case, when the combined coherence is examined 23

38 by changing the location of input to the square frame, the previous large improvements in the combined coherence away from the input locations were not seen. In this thesis, a study is done on a theoretical model with dynamic coupling similar to the real world system used by Coombs to study the behavior of combined coherence for various testing conditions. 3.5 Theoretical Model used to study Combined Coherence A 4 DOF model with rotary inertia is used to study combined coherence. Figure 3.9 shows a near real time 4 DOF model, similar to that used by Doug Coombs, which is dynamically coupled. The m i, c ij, and k ij variables denote the mass, linear viscous damping, and linear stiffness parameters; the f i variables denote the applied external forces. The independent coordinates, x i, are defined with respect to an absolute coordinate system. The idea of this type of model is to study the effect on combined coherence when the path of energy is across the boundary and to get dynamic coupling between the degrees of freedom. As can be seen from the equations of motion, degrees of freedom 1, 2 and 3, 4 are dynamically coupled. 24

39 Figure 3-9: Theoretical 4 DOF lumped model The equations of motion of the model are expressed in terms of a set of coordinates that are defined with respect to the unique static equilibrium point of the linear system: 25

40 26 = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( / / 0 0 / / / / 0 0 / / t f t f t f t f t x t x t x t x k k k k k k k k k k k k k k k k k k k k k k k k k k t x t x t x t x c c c c c c c c c c c c c c c c c c c c c c c c c c t x t x t x t x r j m r j r j r j m r j m r j r j r j m & & & & && && && && (3.10) Frequency response function and coherence are evaluated using the dynamic stiffness method, where the FRF matrix was computed by inverting the system impedance matrix at each frequency of interest. This provided a means of checking the simulink model, which used time domain integration to obtain the responses. It can be seen from the FRF plots below that the dynamic stiffness method results matched the results obtained through simulink model perfectly. Figure 3-10 below shows the FRF s of all 4 DOF s for an input applied at DOF 1.

41 27

42 Figure 3-10: Comparison of Analytical and Simulation Results 28

43 4. Application of Combined Coherence to Analytical model In this chapter, the simulation results obtained for various cases from a 4 DOF system MATLAB Simulink model are presented. The 5 th order fixed-step Dormand-Prince ODE method is used. The sample time, t, is set at seconds and 2 16 time steps are computed, resulting in seconds of signal for each simulation. Data is processed in the Fourier frequency domain and FRF s are determined for each simulation using the H 1 FRF calculation [11] with F jk (ω) as the input and X ik (ω) as the output. The H 1 calculation seeks to minimize noise on the output. 4.1 Effects of Varying the Force Input In this section, simulations are done to verify whether the system is linear or non-linear, by exciting the system with five different force-exciting levels. Further, the effect on the combined coherence in detecting the structural non-linearities, for different exciting levels is studied. The following MIMO cases shown in Table 4-1 are simulated. 29

44 Case Location of Force (Increased in steps M 1, M 2, M 3 & ε Non-Linearity of 10 N) M 4 (Kg) & 3 F 1 = 30 to 70 N & 12, 10, 8 & F 3 = 20 to 60 N & 4 F 1 = 30 to 70 N & 12, 10, 8 & F 3 = 20 to 60 N Table 4-1: MIMO situations for different force exciting levels It can be seen from the FRF and coherence function plots (Figures 4-1 and 4-2), that the anti-resonances and resonances are changed as the excitation force level changes. Thus, it can be assumed that the system is non-linear. The drops in coherence can be attributed to digital signal processing errors as well as to the non-linearity. For example, from the coherence plot (coherence 1) of case 4.1.1, it can seen that the drop in coherence value at 3 Hz is due to digital signal processing error and drops at 8 11 Hz, Hz, Hz are due to non-linear motion. It can be seen from the MCCOH of case (Figure 4-1), at lower forcing levels the MCCOH showed improvement while at higher forcing levels it still showed improvement but with more distortion. The distortion at a forcing level of 70 N is more when compared to a forcing level of 30 N. Hence, it can be concluded that the nature of the improvement in the MCCOH is inversely proportional to the forcing level, i.e., at lower force levels the 30

45 ability of the MCCOH to detect non-linearities is greater when compared to higher force levels. Figure 4-1: FRF s, Coherences and MCCOH for Case

46 Figure 4-2: FRF s, Coherences and MCCOH for Case

47 4.2 SIMO Situations for a system with Dynamic Coupling In this section, the following cases shown in Table 4-2 are simulated for a system close to the real world testing conditions where the mass distribution between the DOF is uneven and also, there is mass coupling between the degrees of freedom. Case Location of Non-Linearity Force M1 M2 M3 M4 ε and 2 F1 = 50 N and 3 F3 = 50 N and 4 F1 = 50 N and 3 F3 = 50 N and 4 F2 = 50 N and 4 F4 = 50 N Table 4-2: System with Dynamic Coupling SIMO situations It can be seen from the FRF plots that there are distortions at both resonances and antiresonances. From the coherence function plots, it can be seen that the drops in the coherence value can be attributed to digital signal processing errors as well as to the nonlinearity. In the coherence function plot of Case 4.2.1, one could see the drops between 5 to 6 Hz and 10 to 14 Hz which are not associated with either resonance or anti-resonance but are due to non-linearity. Also, one could see the drops at 3.5, 6 and 8 Hz that are at resonances or anti-resonances and the drops in the higher frequency range (above 20 Hz). 33

48 As a next step, the CCOH for responses 1 and 2 is compared with ordinary coherence. It can be seen that the drops in the higher frequency range and the drops associated with non-linearity (5-6 Hz) are completely eliminated and one could only see the drops at resonances. The drop in coherence over the frequency range of 10 to 14 Hz is not completely eliminated but has shown improvement over the ordinary coherence. The complete clear up of the CCOH is not seen because of one or a combination of three factors: 1. Dynamic coupling 2. Due to the non-linear motion entering the system from other paths 3. Mass difference between the DOF s between which the CCOH has been computed Similar results have been observed for all other combination of cases i.e., when the nonlinearity is located between DOF s 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4 that there are drops in coherences due to non-linearity, leakage and at higher frequencies. CCOH has shown improvement at anti-resonances but not at resonances and complete clear up the CCOH has not been registered. 34

49 Figure 4-3: FRF s and Coherences of Case

50 Figure 4-4: Coherence and CCOH of Case

51 Figure 4-5: FRF s, Coherence and CCOH for Case

52 Figure 4-6: FRF s, Coherences and MCCOH for Case

53 Figure 4-7: FRF s, Coherences and CCOH for Case

54 Figure 4-8: FRF s, Coherences and CCOH for Case

55 Figure 4-9: FRF s, Coherences and CCOH for Case

56 4.3 MIMO Situations for a system with Dynamic Coupling In this section, the MIMO situations shown in the Table 4-3 below are simulated. The testing conditions, severity of non-linearity, locations of non-linearity, mass and all other conditions are similar to that of the previous case (4.2) except for the input given at two DOF s between which the non-linearity is located. Multiple inputs determine if the structure responds in a non-linear regime. More often, most modal analysis procedures involve the application of multiple inputs in order to get more uniform energy distribution. Whereas, the SIMO situation induces non-linear behavior in the vicinity of the input location and structure might not be excited well at remote points, therefore, further study is done only for MIMO situations. 42

57 Case Location of Non-Linearity Force M1 M2 M3 M4 ε & 2 F1 = 50 N & F2 = 40 N & 3 F1 = 50 & F3 = 40N & 4 F1 = 50N & F4 = 40 N & 3 F2 = 50 N & F3 = 40 N & 4 F2 = 50 N & F4 = 40 N & 4 F3 = 50 N & F4 = N Table 4-3: MIMO situations of system with Dynamic Coupling The FRF estimation, the MCOH and the MCCOH obtained are as shown in the Figures (4-10 to 4-16) below. As seen from the plots below, the results obtained in this case are similar to that of the previous case. There are frequency shifts in the FRF s at resonances and anti-resonances. Also, there are low coherence values due to non-linearity and digital signal processing errors, like leakage at resonances and anti-resonances. One can see the complete improvement of the MCCOH values at higher frequencies and at anti- 43

58 resonances. At some frequencies, the MCCOH has not registered complete improvement. In the coherence plot of Case 4.3.1, one can see the drop in coherence over the frequency range of 5 7 Hz, which is at anti-resonance, is completely improved. The drop over the frequency range of Hz, which is near the resonance, is not improved. It can be concluded from this observation that the MCCOH is sensitive to anti-resonance. Similar results have been observed for all other combination of cases i.e., when the nonlinearity is located between DOFs 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4 that there are drops in coherences due to non-linearity, leakage and at higher frequencies. MCCOH has shown improvement at anti-resonances, but not at resonances, and complete improvement of the MCCOH has not been registered. It can be seen from the figures that the MCCOH has shown improvement over the MCOH in all of the above situations but for some frequency ranges the complete improvement in the MCCOH is not accomplished. As mentioned in the previous SIMO situations, this can be due to one or combinations of the three factors: 1. Dynamic coupling. 2. Due to the non-linear motion entering the system from other paths. 3. Mass difference between the DOF s between which the MCCOH has been computed. But, it is not clear from this case whether the incomplete improvement in the MCCOH is due to either dynamic coupling or due to the mass difference or because of the non-linear motion entering from other paths. 44

59 Figure 4-10: FRF s, Coherences and MCCOH of Case

60 Figure 4-11: FRF s and Coherences of Case Figure 4-12: Coherence and MCCOH of Case

61 Figure 4-13: FRF s, Coherences and MCCOH of Case

62 Figure 4-14: FRF s, Coherences and MCCOH of Case

63 Figure 4-15: FRF s, Coherences and MCCOH of Case

64 Figure 4-16: FRF s, Coherences and MCCOH of Case

65 4.4 Effect of Dynamic Coupling on Combined Coherence In the previous cases, it is seen that the complete improvement of the combined coherence is not observed and reasons for it are attributed to dynamic coupling, mass difference and/or path of energy. In this section, the following cases are simulated to study the effect of dynamic coupling on the MCCOH. The rotary inertia term has been reduced by 100 times (i.e., making the dynamic coupling between DOF s negligible.) Sl. No. Location of Force M 1, M 2, M 3 J 2, J 4 ε Non-Linearity & M and 2 F 1 = 50 N & F 2 = 40 N and 3 F 1 = 50 & F 3 = 40N and 4 F 1 = 50N & F 4 = 40 N and 3 F 2 = 50 N & F 3 = 40 N and 4 F 2 = 50 N & F 4 = 40 N and 4 F 3 = 50 N & F 4 = 40 N 12, 10, 8 &14 J 2 =M*R 2 2 /200 J 4 =M*R 2 4 /200 12, 10, 8 & 14 J 2 =M*R 2 2 /200 J 4 =M*R 2 4 /200 12, 10, 8 & 14 J 2 =M*R 2 2 /200 J 4 =M*R 2 4 /200 12, 10, 8 & 14 J 2 =M*R 2 2 /200 J 4 =M*R 2 4 /200 12, 10, 8 & 14 J 2 =M*R 2 2 /200 J 4 =M*R 2 4 /200 12, 10, 8 & 14 J 2 =M*R 2 2 /200 J 4 =M*R 2 4 / Table 4-4: MIMO situations of system with no Dynamic Coupling 51

66 The FRF estimation, the MCOH and the MCCOH obtained are shown in Figures (4-17 to 4-22) below. It is concluded from last case that dynamic coupling is one of the reasons why the MCCOH has not shown complete improvement. Therefore, it is expected from this case, that the MCCOH will show improvement, as the dynamic coupling is made negligible. It can be seen from the MCCOH plot of Case the improvement in MCCOH is complete whereas in all other cases (4.4.2 to 4.4.6) there is not complete improvement in MCCOH. In the MCCOH plot of Case 4.4.2, it can be seen that over the frequency range of Hz, the MCCOH has not shown improvement. Though, the effect of dynamic coupling is made negligible, the MCCOH has not shown complete improvement in all the cases. Therefore, it can be concluded from this case that the dynamic coupling has no effect on the MCCOH. So, the incomplete improvement of MCCOH might be due to either the mass difference or the non-linear motion entering from other paths. 52

67 Figure 4-17: FRF s, Coherences and MCCOH of Case

68 Figure 4-18: FRF s, Coherences and MCCOH of Case

69 Figure 4-19: FRF s, Coherences and MCCOH of Case

70 Figure 4-20: FRF s, Coherences and MCCOH of Case

71 Figure 4-21: FRF s, Coherences and MCCOH of Case

72 Figure 4-22: FRF s, Coherences and MCCOH of Case

73 4.5 Effect of Location of Input and Path of Energy on Combined Coherence This case is simulated for the MIMO situations as above but the forcing function is not placed directly on the DOF that is associated with the non-linearity. These cases are simulated to study the ability of the combined coherence to detect the non-linearity when the energy comes from the linear path and also when input is placed some distance away from the DOF s between which non-linearity is present. Sl. No. Location of Non- Force M 1, M 2, M 3 & ε Linearity M & 2 F 3 = 50 N & 12, 10, 8 and F 4 = 40 N & 3 F 2 = 50 & 12, 10, 8 and F 4 = 40N & 4 F 2 = 50N & 12, 10, 8 and F 3 = 40 N & 3 F 1 = 50 N & 12, 10, 8 and F 4 = 40 N & 4 F 1 = 50 N & 12, 10, 8 and F 3 = 40 N & 4 F 1 = 50 N & 12, 10, 8 and F 2 = 40 N Table 4-5: MIMO situations to study effect of Path of Energy 59

74 The FRF estimation, the MCOH, and the MCCOH obtained are shown in Figures (4-23 to 4-28) below. It can be seen that for Cases and where the forcing function is away from the DOF s between which the non-linearity is located, the MCCOH has registered a drastic improvement. For these cases, when compared with Cases and respectively for the same level of excitation, the FRF s and coherence functions are not distorted as much as when the forcing function is directly placed at the DOF s where the non-linearity is located. Whereas in Cases to the result is reversed, the FRF and MCCOH are distorted more than Cases and and also there is not complete clear up of the MCCOH function. The improvement in the MCCOH in Cases and can be because the forcing function is away from the DOF where the nonlinearity is being located and the energy is entering through a more linear path. However, it has been concluded in the previous case that the dynamic coupling has no affect on the MCCOH, so from this case it can be concluded that the location of inputs and energy path are critical in determining the ability of the MCCOH in detecting the non-linearities. 60

75 Figure 4-23: FRF s, Coherences and MCCOH of Case

76 Figure 4-24: FRF s, Coherences and MCCOH of Case

77 Figure 4-25: FRF s, Coherences and MCCOH of Case

78 Figure 4-26: FRF s, Coherences and MCCOH of Case

79 Figure 4-27: FRF s, Coherences and MCCOH of Case

80 4.6 Effect of Mass Distribution on Combined Coherence In this section, MIMO situations are simulated by considering equal mass at all DOFs. This is to see how the mass difference of the DOF s between which the non-linearity is associated, affects the MCCOH. The masses at all the DOF s are made equal. The following cases have been simulated. Sl. No. Location of Force M 1, M 2, M 3 & ε Non-Linearity M & 2 F 1 = 50 N & 15,15,15 & F 2 = 40 N & 3 F 1 = 50 & 15,15,15 & F 3 = 40N & 4 F 1 = 50N & 15,15,15 & F 4 = 40 N & 3 F 2 = 50 N & 15,15,15 & F 3 = 40 N & 4 F 2 = 50 N & 15,15,15 & F 4 = 40 N & 4 F 3 = 50 N & 15,15,15 & F 4 = 40 N Table 4-6: MIMO situations to study effect of Mass Distribution 66

81 It can be seen from the plots of the MCCOH, that it has shown complete improvement in Case and in all other cases from to 4.6.5, the MCCOH is improved but still exhibits drops over some frequency ranges. For example, from the MCCOH plot of Case 4.6.2, it can be seen that over the frequency ranges of 7 8 Hz and Hz there is no complete improvement in the MCCOH. By comparing the MCCOH of Case and Case it can be concluded that the mass inequality between the DOF s can be a possibility for the MCCOH to detect non-linearities. Even though the mass difference between the degrees of freedom 3 and 4 in Case is small (6 kg, this difference is significant when compared to original masses of 14 kg and 8 kg), by eliminating this mass difference, the MCCOH has shown great improvement. In all other cases, the MCCOH has shown improvement when compared to Cases to but of much smaller values. It is expected that when the mass inequality between the degrees of freedom is eliminated, the combined coherence should show greater improvement. But from these cases, it can be concluded that besides the mass inequality, the path of energy is also critical in detecting the non-linearities. This can be seen from Cases to 4.6.5, in which the improvement in the MCCOH is not complete. 67

82 Figure 4-28: FRF s, Coherences and MCCOH of Case

83 Figure 4-29: FRF s, Coherences and MCCOH of Case

84 Figure 4-29: FRF s, Coherences and MCCOH of Case

85 Figure 4-30: FRF s, Coherences and MCCOH of Case

86 Figure 4-31: FRF s, Coherences and MCCOH of Case

87 Figure 4-32: FRF s, Coherences and MCCOH of Case

88 4.7 Effect of Spatial Density of Masses on Combined Coherence In this section, MIMO situations similar to the real world testing situations, where the system consists of more than one component, with a difference in mass densities are simulated. As an example, it can be seen from the Doug Coombs model there are two frames, one being lighter than the other. Case Location of Force M 1, M 2, M 3 and ε Non-Linearity M & 2 F 1 = 50 N & 100, 80, 10 & F 2 = 40 N & 3 F 1 = 50 & 100, 80, 10 & F 3 = 40N & 4 F 1 = 50N & 100, 80, 10 & F 4 = 40 N & 3 F 2 = 50 N & 100, 80, 10 & F 3 = 40 N & 4 F 2 = 50 N & 100, 80, 10 & F 4 = 40 N & 4 F 3 = 50 N & 100, 80, 10 & F 4 = 40 N Table 4-7: MIMO situations to study effect of Spatial Densities of Masses 74

89 It can be seen from the MCCOH plots of Cases and that the MCCOH has shown improvement where the mass difference between the DOF s for which the MCCOH is computed is negligible. In other cases, the MCCOH has not shown any improvement at all due to the huge mass difference between the DOF s. In Cases to 4.7.6, the improvement in the MCCOH is not complete due to the relative motion entering from other paths. 75

90 Figure 4-33: FRF s, Coherences and MCCOH of Case

91 Figure 4-34: FRF s, Coherences and MCCOH of Case

92 Figure 4-35: FRF s, Coherences and MCCOH of Case

93 Figure 4-36: FRF s, Coherences and MCCOH of Case

94 Figure 4-37: FRF s, Coherences and MCCOH of Case

95 Figure 4-38: FRF s, Coherences and MCCOH of Case