The Fundamentals of Modal Testing

Transcription

1 The Fundaentals of Modal Testing Application Note Η(ω) = Σ n r=1 φ φ i j / ( ω n - ω ) + (2ξωωn)

2 Preface Modal analysis is defined as the study of the dynaic characteristics of a echanical structure. This application note ephasizes experiental odal techniques, specifically the ethod known as frequency response function testing. Other areas are treated in a general sense to introduce their eleentary concepts and relationships to one another. Although odal techniques are atheatical in nature, the discussion is inclined toward practical application. Theory is presented as needed to enhance the logical developent of ideas. The reader will gain a sound physical understanding of odal analysis and be able to carry out an effective odal survey with confidence. Chapter 1 provides a brief overview of structural dynaics theory. Chapter 2 and 3 which is the bulk of the note describes the easureent process for acquiring frequency response data. Chapter 4 describes the paraeter estiation ethods for extracting odal properties. Chapter 5 provides an overview of analytical techniques of structural analysis and their relation to experiental odal testing. 2

3 Table of Contents Preface 2 Chapter 1 Structural Dynaics Background 4 Introduction 4 Structural Dynaics of a Single Degree of Freedo (SDOF) Syste 5 Presentation and Characteristics of Frequency Response Functions 6 Structural Dynaics for a Multiple Degree of Freedo (MDOF) Syste 9 Daping Mechanis and Daping Model 11 Frequency Response Function and Transfer Function Relationship 12 Syste Assuptions 13 Chapter 2 Frequency Response Measureents 14 Introduction 14 General Test Syste Configurations 15 Supporting the Structure 16 Exciting the Structure 18 Shaker Testing 19 Ipact Testing 22 Transduction 25 Measureent Interpretation 29 Chapter 3 Iproving Measureent Accuracy 30 Measureent Averaging 30 Windowing Tie Data 31 Increasing Measureent Resolution 32 Coplete Survey 34 Chapter 4 Modal Paraeter Estiation 38 Introduction 38 Modal Paraeters 39 Curve Fitting Methods 40 Single Mode Methods 41 Concept of Residual Ters 43 Multiple Mode-Methods 45 Concept of Real and Coplex Modes 47 Chapter 5 Structural Analysis Methods 48 Introduction 48 Structural Modification 49 Finite Eleent Correlation 50 Substructure Coupling Analysis 52 Forced Response Siulation 53 Bibliography 54 3

4 Chapter 1 Structural Dynaics Background Introduction A basic understanding of structural dynaics is necessary for successful odal testing. Specifically, it is iportant to have a good grasp of the relationships between frequency response functions and their individual odal paraeters illustrated in Figure 1.1. This understanding is of value in both the easureent and analysis phases of the survey. Knowing the various fors and trends of frequency response functions will lead to ore accuracy during the easureent phase. During the analysis phase, knowing how equations relate to frequency responses leads to ore accurate estiation of odal paraeters. The basic equations and their various fors will be presented conceptually to give insight into the relationships between the dynaic characteristics of the structure and the corresponding frequency response function easureents. Although practical systes are ultiple degree of freedo (MDOF) and have soe degree of nonlinearity, they can generally be represented as a superposition of single degree of freedo (SDOF) linear odels and will be developed in this anner. Figure 1.1 Phases of a odal test Test Structure Frequency Response Measureents Curve Fit Representation Modal Paraeters n r=1 Η ij(ω) = Σ φ ir φjr r ( ω 2 r - ω 2 + j2 ζωω r) ω Frequency ζ Daping { φ} Mode Shape First, the basics of an SDOF linear dynaic syste are presented to gain insight into the single ode concepts that are the basis of soe paraeter estiation techniques. Second, the presentation and properties of various fors of the frequency response function are exained to understand the trends and their usefulness in the easureent process. Finally, these concepts are extended into MDOF systes, since this is the type of behavior ost physical structures exhibit. Also, useful concepts associated with daping echaniss and linear syste assuptions are discussed. 4

5 Structural Dynaics of a Single Degree of Freedo (SDOF) Syste Although ost physical structures are continuous, their behavior can usually be represented by a discrete paraeter odel as illustrated in Figure 1.2. The idealized eleents are called ass, spring, daper and excitation. The first three eleents describe the physical syste. Energy is stored by the syste in the ass and the spring in the for of kinetic and potential energy, respectively. Energy enters the syste through excitation and is dissipated through daping. Figure 1.2 SDOF discrete paraeter odel Response Displaceent (x) Spring k Mass Daper c Excitation Force (f) The idealized eleents of the physical syste can be described by the equation of otion shown in Figure 1.3. This equation relates the effects of the ass, stiffness and daping in a way that leads to the calculation of natural frequency and daping factor of the syste. This coputation is often facilitated by the use of the definitions shown in Figure 1.3 that lead directly to the natural frequency and daping factor. The natural frequency, ω, is in units of radians per second (rad/s). The typical units displayed on a digital signal analyzer, however, are in Hertz (Hz). The daping factor can also be represented as a percent of critical daping the daping level at which the syste experiences no oscillation. This is the ore coon understanding of odal daping. Although there are three distinct daping cases, only the underdaped case (ζ< 1) is iportant for structural dynaics applications. Figure 1.3 Equation of otion odal definitions Figure 1.4 Coplex roots of SDOF equation Figure 1.5 SDOF ipulse response/ free decay... x + cx + kx = f(t) ωn 2 = k, 2ζω n = s 1,2 = - σ + jωd c or ζ = c 2k σ Daping Rate ωζ Daped Natural Frequency e -σ t ωd 5

6 When there is no excitation, the roots of the equation are as shown in Figure 1.4. Each root has two parts: the real part or decay rate, which defines daping in the syste and the iaginary part, or oscillatory rate, which defines the daped natural frequency, wd. This free vibration response is illustrated in Figure 1.5. When excitation is applied, the equation of otion leads to the frequency response of the syste. The frequency response is a coplex quantity and contains both real and iaginary parts (rectangular coordinates). It can be presented in polar coordinates as agnitude and phase, as well. Figure 1.6 Frequency response polar coordinates Magnitude ω d H( ω) = 1/ ( ω 2 n - ω 2 ) 2 + (2ζωω n ) 2 Presentation and Characteristics of Frequency Response Functions Because it is a coplex quantity, the frequency response function cannot be fully displayed on a single twodiensional plot. It can, however, be presented in several forats, each of which has its own uses. Although the response variable for the previous discussion was displaceent, it could also be velocity or acceleration. Acceleration is currently the accepted ethod of easuring odal response. Phase θω ( ) = tan -1 2ξωωn 2 ωn - 2 ω One ethod of presenting the data is to plot the polar coordinates, agnitude and phase versus frequency as illustrated in Figure 1.6. At resonance, when ω = ωn, the agnitude is a axiu and is liited only by the aount of daping in the syste. The phase ranges fro 0 to 180 and the response lags the input by 90 at resonance. ω d 6

7 Another ethod of presenting the data is to plot the rectangular coordinates, the real part and the iaginary part versus frequency. For a proportionally daped syste, the iaginary part is axiu at resonance and the real part is 0, as shown in Figure 1.7. Figure 1.7 Frequency response rectangular coordinates A third ethod of presenting the frequency response is to plot the real part versus the iaginary part. This is often called a Nyquist plot or a vector response plot. This display ephasizes the area of frequency response at resonance and traces out a circle, as shown in Figure 1.8. Real H( ω) = 2 2 ωn - ω ( ωn - ω ) + (2 ζωωn ) By plotting the agnitude in decibels vs logarithic (log) frequency, it is possible to cover a wider frequency range and conveniently display the range of aplitude. This type of plot, often known as a Bode plot, also has soe useful paraeter characteristics which are described in the following plots. When ω << ωn the frequency response is approxiately equal to the asyptote shown in Figure 1.9. This asyptote is called the stiffness line and has a slope of 0, 1 or 2 for displaceent, velocity and acceleration responses, respectively. When ω >> ωn the frequency response is approxiately equal to the asyptote also shown in Figure 1.9. This asyptote is called the ass line and has a slope of -2, -1 or 0 for displaceent velocity or acceleration responses, respectively. Figure 1.8 Nyquist plot of frequency response Iaginary H( ω) = -2ξωωn 2 ( n - 2 ) 2 ω ω + (2 ζωωn ) 2 θω ( ) H( ω) 7

8 The various fors of frequency response function based on the type of response variable are also defined fro a echanical engineering viewpoint. They are soewhat intuitive and do not necessarily correspond to electrical analogies. These fors are suarized in Table 1.1. Figure 1.9 Different fors of frequency response Displaceent 1 k 1 ω 2 Table 1.1 Different fors of frequency response Definition Response Variable Frequency Copliance X Displaceent F Force Mobility V Velocity F Force Accelerance A Acceleration F Force Acceleration Velocity ω k 1 ω 1 Frequency ω 2 k Frequency 8

9 Structural Dynaics for a Multiple Degree of Freedo (MDOF) Syste The extension of SDOF concepts to a ore general MDOF syste, with n degrees of freedo, is a straightforward process. The physical syste is siply coprised of an interconnection of idealized SDOF odels, as illustrated in Figure 1.10, and is described by the atrix equations of otion as illustrated in Figure Figure 1.10 MDOF discrete paraeter odel k 1 k 3 k c k c 4 3 c 2 c 4 The solution of the equation with no excitation again leads to the odal paraeters (roots of the equation) of the syste. For the MDOF case, however, a unique displaceent vector called the ode shape exists for each distinct frequency and daping as illustrated in Figure The free vibration response is illustrated in Figure The equations of otion for the forced vibration case also lead to frequency response of the syste. It can be written as a weighted suation of SDOF systes shown in Figure Figure 1.11 Equations of otion odal definitions... []{x} + [c]{x} + [k]{x} = {f(t)} { φ} r, r = 1, n odes 3 The weighting, often called the odal participation factor, is a function of excitation and ode shape coefficients at the input and output degrees of freedo. Figure 1.12 MDOF ipulse response/ free decay Aplitude 0.0 Sec 6.0 9

10 The participation factor identifies the aount each ode contributes to the total response at a particular point. An exaple with 3 degrees of freedo showing the individual odal contributions is shown in Figure The frequency response of an MDOF syste can be presented in the sae fors as the SDOF case. There are other definitional fors and properties of frequency response functions, such as a driving point easureent, that are presented in the next chapter. These are related to specific locations of frequency response easureents and are introduced when appropriate. Figure 1.13 MDOF frequency response db Magnitude 0.0 Η(ω) = Σ n r=1 φi φj / ( ω n - ω ) + (2ξωωn) ω 1 ω 2 ω 3 Frequency Figure 1.14 SDOF odal contributions Mode 1 Mode 2 Mode 3 db Magnitude 0.0 ω 1 ω 2 ω 3 Frequency 10

11 Daping Mechanis and Daping Model Daping exists in all vibratory systes whenever there is energy dissipation. This is true for echanical structures even though ost are inherently lightly daped. For free vibration, the loss of energy fro daping in the syste results in the decay of the aplitude of otion. In forced vibration, loss of energy is balanced by the energy supplied by excitation. In either situation, the effect of daping is to reove energy fro the syste. Figure 1.15 Viscous daping energy dissipation E = πωc eq X 2 cẋ X Force vs Displaceent x In previous atheatical forulations the daping force was called viscous, since it was proportional to velocity. However, this does not iply that the physical daping echanis is viscous in nature. It is siply a odeling ethod and it is iportant to note that the physical daping echanis and the atheatical odel of that echanis are two distinctly different concepts. Figure 1.16 Syste block diagra Input Excitation G(s) Syste Output Response Most structures exhibit one or ore fors of daping echaniss, such as coulob or structural, which result fro looseness of joints, internal strain and other coplex causes. However, these echaniss can be odeled by an equivalent viscous daping coponent. It can be shown that only the viscous coponent actually accounts for energy loss fro the syste and the reaining portion of the daping is due to nonlinearities that do not cause energy dissipation. Therefore, only the viscous ter needs to be easured to characterize the syste when using a linear odel. The equivalent viscous daping coefficient is obtained fro energy considerations as illustrated in the Figure 1.17 Definition of transfer function Transfer Function = G(s) = Output Input Y(s) X(s) hysteresis loop in Figure E is the energy dissipated per cycle of vibration, c eq is the equivalent viscous daping coefficient and X is the aplitude of vibration. Note that the criteria for equivalence are equal energy distribution per cycle and the sae relative aplitude. 11

12 Frequency Response Function and Transfer Function Relationship The transfer function is a atheatical odel defining the input-output relationship of a physical syste. Figure 1.16 shows a block diagra of a single input-output syste. Syste response (output) is caused by syste excitation (input). The casual relationship is loosely defined as shown in Figure Matheatically, the transfer function is defined as the Laplace transfor of the output divided by the Laplace transfor of the input. The frequency response function is defined in a siilar anner and is related to the transfer function. Matheatically, the frequency response function is defined as the Fourier transfor of the output divided by the Fourier transfor of the input. These ters are often used interchangeably and are occasionally a source of confusion. Figure 1.18 S-plane representation σ This relationship can be further explained by the odal test process. The easureents taken during a odal test are frequency response function easureents. The paraeter estiation routines are, in general, curve fits in the Laplace doain and result in the transfer functions. The curve fit siply infers the location of syste poles in the s-plane fro the frequency response functions as illustrated in Figure The frequency response is siply the transfer function easured along the jω axis as illustrated in Figure ω n iω ω d s-plane σ 12

13 Figure D Laplace representation Real Part σ jω Syste Assuptions The structural dynaics background theory and the odal paraeter estiation theory are based on two ajor assuptions: The syste is linear. The syste is stationary. Iaginary Part σ jω There are, of course, a nuber of other syste assuptions such as observability, stability, and physical realizability. However, these assuptions tend to be addressed in the inherent properties of echanical systes. As such, they do not present practical liitations when aking frequency response easureents as do the assuptions of linearity and stationarity. Magnitude jω σ Phase σ Transfer Function surface Frequency Response dashed 13

14 Chapter 2 Frequency Response Measureents Introduction This chapter investigates the current instruentation and techniques available for acquiring frequency response easureents. The discussion begins with the use of a dynaic signal analyzer and associated peripherals for aking these easureents. The type of odal testing known as the frequency response function ethod, which easures the input excitation and output response siultaneously, as shown in the block diagra in Figure 2.1, is exained. The focus is on the use of one input force, a technique coonly known as single-point excitation, illustrated in Figure 2.2. By understanding this technique, it is easy to expand to the ultiple input technique. Figure 2.1 Syste block diagra Figure 2.2 Structure under test Excitation X( ω) H( ω) Y( ω) Structure Response With a dynaic signal analyzer, which is a Fourier transfor-based instruent, any types of excitation sources can be ipleented to easure a structure s frequency response function. In fact, virtually any physically realizable signal can be input or easured. The selection and ipleentation of the ore coon and useful types of signals for odal testing are discussed. Transducer selection and ounting ethods for easuring these signals along with syste calibration ethods, are also included. Techniques for iproving the quality and accuracy of easureents are then explored. These include processes such as averaging, windowing and zooing, all of which reduce easureent errors. Finally, a section on easureent interpretation is included to aid in understanding the coplete easureent process. Force Transducer Shaker 14

15 General Test Syste Configurations The basic test setup required for aking frequency response easureents depends on a few ajor factors. These include the type of structure to be tested and the level of results desired. Other factors, including the support fixture and the excitation echanis, also affect the aount of hardware needed to perfor the test. Figure 2.3 shows a diagra of a basic test syste configuration. The heart of the test syste is the controller, or coputer, which is the operator s counication link to the analyzer. It can be configured with various levels of eory, displays and data storage. The odal analysis software usually resides here, as well as any additional analysis capabilities such as structural odification and forced response. The analyzer provides the data acquisition and signal processing operations. It can be configured with several input channels, for force and response easureents, and with one or ore excitation sources for driving shakers. Measureent functions such as windowing, averaging and Fast Fourier Transfors (FFT) coputation are usually processed within the analyzer. Figure 2.3 General test configuration Structure For aking easureents on siple structures, the exciter echanis can be as basic as an instruented haer. This echanis requires a iniu aount of hardware. An electrodynaic shaker ay be needed for exciting ore coplicated structures. This shaker syste requires a signal source, a power aplifier and an attachent device. The signal source, as entioned earlier, ay be a coponent of the analyzer. Transducers Exciter Controller Analyzer Transducers, along with a power supply for signal conditioning, are used to easure the desired force and responses. The piezoelectric types, which easure force and acceleration, are the ost widely used for odal testing. The power supply for signal conditioning ay be voltage or charge ode and is soeties provided as a coponent of the analyzer, so care should be taken in setting up and atching this part of the test syste. 15

16 Supporting The Structure The first step in setting up a structure for frequency response easureents is to consider the fixturing echanis necessary to obtain the desired constraints (boundary conditions). This is a key step in the process as it affects the overall structural characteristics, particularly for subsequent analyses such as structural odification, finite eleent correlation and substructure coupling. Figure 2.4a Exaple of free support situation Free Boundary Analytically, boundary conditions can be specified in a copletely free or copletely constrained sense. In testing practice, however, it is generally not possible to fully achieve these conditions. The free condition eans that the structure is, in effect, floating in space with no attachents to ground and exhibits rigid body behavior at zero frequency. The airplane shown in Figure 2.4a is an exaple of this free condition. Physically, this is not realizable, so the structure ust be supported in soe anner. The constrained condition iplies that the otion, (displaceents/rotations) is set to zero. However, in reality ost structures exhibit soe degree of flexibility at the grounded connections. The satellite dish in Figure 2.4b is an exaple of this condition. In order to approxiate the free syste, the structure can be suspended fro very soft elastic cords or placed on a very soft cushion. By doing this, the structure will be constrained to a degree and the rigid body odes will no longer have zero frequency. However, if a sufficiently soft support Figure 2.4b Exaple of constrained support situation syste is used, the rigid body frequencies will be uch lower than the frequencies of the flexible odes and thus have negligible effect. The rule of thub for free supports is that the highest rigid body ode frequency ust be less than one tenth that of the first flexible ode. If this criterion is et, rigid body odes will have negligible effect on flexible odes. Figure 2.5 shows a typical frequency response easureent of this type with nonzero rigid body odes. Constrained Boundary The ipleentation of a constrained syste is uch ore difficult to achieve in a test environent. To begin with, the base to which the structure is attached will tend to have soe otion of its own. Therefore, it is not going to be purely grounded. Also, the attachent points will have soe degree of flexibility due to the bolted, riveted or welded connections. One possible reedy for these probles is to easure the frequency 16

17 response of the base at the attachent points over the frequency range of interest. Then, verify that this response is significantly lower than the corresponding response of the structure, in which case it will have a negligible effect. However, the frequency response ay not be easurable, but can still influence the test results. Figure 2.5 Frequency response of freely suspended syste db /Div Rigid Body Mode 1st Flexible Mode There is not a best practical or appropriate ethod for supporting a structure for frequency response testing. Each situation has its own characteristics. Fro a practical standpoint, it would not be feasible to support a large factory achine weighing several tons in a free test state. On the other hand, there ay be no convenient way to ground a very sall, lightweight device for the constrained test state. A situation could occur, with a satellite for exaple, where the results of both tests are desired. The free test is required to analyze the satellite s operating environent in space. However, the constrained test is also needed to assess the launch environent attached to the boost vehicle. Another reason for choosing the appropriate boundary conditions is for finite eleent odel correlation or substructure coupling analyses. At any rate, it is certainly iportant during this phase of the test to ascertain all the conditions in which the results ay be used FxdXY 0 Hz k 17

18 Exciting the Structure The next step in the easureent process involves selecting an excitation function (e.g., rando noise) along with an excitation syste (e.g., a shaker) that best suits the application. The choice of excitation can ake the difference between a good easureent and a poor one. Excitation selection should be approached fro both the type of function desired and the type of excitation syste available because they are interrelated. The excitation function is the atheatical signal used for the input. The excitation syste is the physical echanis used to prove the signal. Generally, the choice of the excitation function dictates the choice of the excitation syste, a true rando or burst rando function requires a shaker syste for ipleentation. In general, the reverse is also true. Choosing a haer for the excitation syste dictates an ipulsive type excitation function. Excitation functions fall into four general categories: steady-state, rando, periodic and transient. There are several papers that go into great detail exaining the applications of the ost coon excitation functions. Table 2.1 suarizes the basic characteristics of the ones that are ost useful for odal testing. True rando, burst rando and ipulse types are considered in the context of this note since they are the ost widely ipleented. The best choice of excitation function depends on several factors: available signal processing equipent, characteristics of the structure, general easureent considerations and, of course, the excitation syste. A full function dynaic signal analyzer will have a signal source with a sufficient nuber of functions for exciting the structure. With lower quality analyzers, it ay be necessary to obtain a signal source as a separate part of the signal processing equipent. These sources often provide fixed sine and true rando functions as signals; however, these ay not be acceptable in applications where high levels of accuracy are desired. The types of functions available have a significant influence on easureent quality. Table 2.1 Excitation functions Periodic* Transient in analyzer window in analyzer window Sine True Pseudo Rando Fast Ipact Burst Burst steady rando rando sine sine rando state Minize leakage No No Yes Yes Yes Yes Yes Yes Signal to noise Very Fair Fair Fair High Low High Fair high RMS to peak ratio High Fair Fair Fair High Low High Fair Test easureent tie Very Good Very Fair Fair Very Very Very long good good good good Controlled frequency content Yes Yes* Yes* Yes* Yes* No Yes* Yes* Controlled aplitude content Yes No Yes* No Yes* No Yes* No Reoves distortion No Yes No Yes No No No Yes Characterize nonlinearity Yes No No No Yes No Yes No * Requires additional equipent or special hardware 18

19 The dynaics of the structure are also iportant in choosing the excitation function. The level of nonlinearities can be easured and characterized effectively with sine sweeps or chirps, but a rando function ay be needed to estiate the best linearized odel of a nonlinear syste. The aount of daping and the density of the odes within the structure can also dictate the use of specific excitation functions. If odes are closely coupled and/or lightly daped, an excitation function that can be ipleented in a leakagefree anner (burst rando for exaple) is usually the ost appropriate. Excitation echaniss fall into four categories: shaker, ipactor, step relaxation and self-operating. Step relaxation involves preloading the structure with a easured force through a cable then releasing the cable and easuring the transients. Self-operating involves exciting the structure through an actual operating load. This input cannot be easured in any cases, thus liiting its usefulness. Shakers and ipactors are the ost coon and are discussed in ore detail in the following sections. Another ethod of excitation echanis classification is to divide the into attached and nonattached devices. A shaker is an attached device, while an ipactor is not, (although it does ake contact for a short period of tie). Shaker Testing The ost useful shakers for odal testing are the electroagnetic shown in Fig. 2.6 (often called electrodynaic) and the electro hydraulic (or, hydraulic) types. With the electroagnetic shaker, (the ore coon of the two), force is generated by an alternating current that drives a agnetic coil. The axiu frequency liit varies fro approxiately 5 khz to 20 khz depending on the size; the saller shakers having the higher operating range. The axiu force rating is also a function of the size of the shaker and varies fro approxiately 2 lbf to 1000 lbf; the saller the shaker, the lower the force rating. With hydraulic shakers, force is generated through the use of hydraulics, which can provide uch higher force levels soe up to several thousand pounds. The axiu frequency range is uch lower though about 1 khz and below. An advantage of the hydraulic shaker is its ability to apply a large static preload to the structure. This is useful for assive structures such as grinding achines that operate under relatively high preloads which ay alter their structural characteristics. 19

20 There are several potential proble areas to consider when using a shaker syste for excitation. To begin with, the shaker is physically ounted to the structure via the force transducer, thus creating the possibility of altering the dynaics of the structure. With lightweight structures, the echanis used to ount the load cell ay add appreciable ass to the structure. This causes the force easured by the load cell to be greater than the force actually applied to the structure. Figure 2.7 describes how this ass loading alters the input force. Since the extra ass is between the load cell and the structure the load cell senses this extra ass as part of the structure. Figure 2.6 Electrodynaic shaker with power aplifier and signal source Power Aplifier Since the frequency response is a single input function, the shaker should transit only one coponent of force in line with the ain axis of the load cell. In practical situations, when a structure is displaced along a linear axis it also tends to rotate about the other two axes. To iniize the proble of forces being applied in other directions, the shaker should be connected to the load cell through a slender rod, called a stinger, to allow the structure to ove freely in the other directions. This rod, shown in Figure 2.8, has a strong axial stiffness, but weak bending and shear stiffnesses. In effect, it acts like a truss eber, carrying only axial loads but no oents or shear loads. Figure 2.7 Mass loading fro shaker setup Structure Ax F s Loading Mass Load Cell F F 1 Fs = F - MAx 20

21 The ethod of supporting the shaker is another factor that can affect the force iparted to the structure. The ain body of the shaker ust be isolated fro the structure to prevent any reaction forces fro being transitted through the base of the shaker back to the structure. This can be accoplished by ounting the shaker on a solid floor and suspending the structure fro above. The shaker could also be supported on a echanically isolated foundation. Another ethod is to suspend the shaker, in which case an inertial ass usually needs to be attached to the shaker body in order to generate a easurable force, particularly at lower frequencies. Figure 2.9 illustrates the different types of shaker setups. Another potential proble associated with electroagnetic shakers is the ipedance isatch that can exist between the structure and the shaker coil. The electrical ipedance of the shaker varies with the aplitude of otion of the coil. At a resonance with a sall effective ass, very little force is required to produce a response. This can result in a drop in the force spectru in the vicinity of the resonance, causing the force easureent to be susceptible to noise. Figure 2.10 illustrates an exaple of this phenoenon. The proble can usually be corrected by using shakers with different size coils or driving the shaker with a constantcurrent type aplifier. The shaker could also be oved to a point with a larger effective ass. Figure 2.8 Stinger attachent to structure Figure 2.9 Test support echaniss Figure 2.10 Shaker/structure ipedance isatch 30 Avg 50%Ovlp Hann 40.0 db Fxd Y 1.1k Hz 2.1k POWER SPEC1 30 Avg 50%Ovlp Hann FRF db rs V 2 Force Spectru -100 Fxd Y 1.1k Hz 2.1k 21

22 Ipact Testing Another coon excitation echanis in odal testing is an ipact device. Although it is a relatively siple technique to ipleent, it s difficult to obtain consistent results. The convenience of this technique is attractive because it requires very little hardware and provides shorter easureent ties. The ethod of applying the ipulse, shown in Figure 2.11, includes a haer, an electric gun or a suspended ass. The haer, the ost coon of these, is used in the following discussion. However, this inforation also applies to the other types of ipact devices. Since the force is an ipulse, the aplitude level of the energy applied to the structure is a function of the ass and the velocity of the haer. This is due to the concept of linear oentu, which is defined as ass ties velocity. The linear ipulse is equal to the increental change in the linear oentu. It is difficult though to control the velocity of the haer, so the force level is usually controlled by varying the ass. Ipact haers are available in weights varying fro a few ounces to several pounds. Also, ass can be added to or reoved fro ost haers, aking the useful for testing objects of varying sizes and weights. Figure 2.11 Ipact devices for testing Figure 2.12 Frequency content of various pulses FILT TIIME1 Real v /Div Fxd Y -46.9µ POWER SPEC /Div Hard Mediu Soft 0%Ovlp Pulse t Sec Avg 0%Ovlp Fr/Ex The frequency content of the energy applied to the structure is a function of the stiffness of the contacting surfaces and, to a lesser extent, the ass of the haer. The stiffness of the contacting surfaces affects the shape of the force pulse, which in turn deterines the frequency content. db rs v Fxd X 0 Soft Mediu Hard Hz 2.5k 22

23 It is not feasible to change the stiffness of the test object, therefore the frequency content is controlled by varying the stiffness of the haer tip. The harder the tip, the shorter the pulse duration and thus the higher the frequency content. Figure 2.12 illustrates this effect on the force spectru. The rule of thub is to choose a tip so that the aplitude of the force spectru is no ore than 10 db to 20 db down at the axiu frequency of interest as shown in Figure A disadvantage to note here is that the force spectru of an ipact excitation cannot be band-liited at lower frequencies when aking zoo easureents, so the lower out-of-band odes will still be excited. Figure 2.13 Useful frequency range of pulse spectru POWER SPEC /Div db rs v Fxd Y 0-72 db 1Avg 0%Ovlp Fr/Ex -88 db Hz 2.5k Ipact testing has two potential signal processing probles associated with it. The first noise can be present in either the force or response signal as a result of a long tie record. The second leakage can be present in the response signal as a result of a short tie record. Copensation for both these probles can be accoplished with windowing techniques. Figure 2.14 Force pulse with force window applied FILT TIIME /Div Real v 120 0%Ovlp Since the force pulse is usually very short relative to the length of the tie record, the portion of the signal after the pulse is noise and can be eliinated without affecting the pulse itself. The window designed to accoplish this, called a force window, is shown in Figure The sall aount of oscillation that occurs at the end of the pulse is actually part of the pulse. It is a result of signal processing and should not be truncated Fxd X 250µ Apl 1.0 τ Sec Force Pulse 100 Force Window Tie 23

24 The response signal is an exponential decaying function and ay decay out before or after the end of the easureent. If the structure is heavily daped, the response ay decay out before the end of the tie record. In this case, the response window can be used to eliinate the reaining noise in the tie record. If the structure is lightly daped, the response ay continue beyond the end of the tie record. In this case, it ust be artificially forced to decay out to iniize leakage. The window designed to accoplish either result, called the exponential window, is shown in Figure The rule of thub for setting the tie constant, (the tie required for the aplitude to be reduced by a factor of 1/e), is about one-fourth the tie record length, T. The result of this is shown in Figure Figure 2.15 Decaying response with exponential window applied FILT TIIME2 Real v /Div -250 Fxd Y Apl 0.0 No Data Response Sec Exponential Decaying Response Exponential Window 512 Unlike the force window, the exponential window can alter the resulting frequency response because it has the effect of adding artificial daping to the syste. The added daping coefficient can usually be backed out of the easureent after signal processing, but nuerical probles ay arise with lightly daped structures. This can happen when the added daping fro the exponential window is significantly ore than the true daping in the structure. A better easureent procedure in this case would be to zoo, thus utilizing a longer tie record in order to capture the entire response, instead of relying on the exponential window. WIND TIIME /Div Real v 250 0% Ovlp Windowed Response Tie -250 Fxd Y 0.0 Sec

25 Transduction Now that an excitation syste has been set up to force the structure into otion, the transducers for sensing force and otion need to be selected. Although there are various types of transducers, the piezoelectric type is the ost widely used for odal testing. It has wide frequency and dynaic ranges, good linearity and is relatively durable. The piezoelectric transducer is an electroechanical sensor that generates an electrical output when subjected to vibration. This is accoplished with a crystal eleent that creates an electrical charge when echanically strained. Figure 2.16 Frequency response of transducer Response Transducer (Acceleroeter) The echanis of the force transducer, called a load cell, functions in a fairly siple anner. When the crystal eleent is strained, either by tension or copression, it generates a charge proportional to the applied force. In this case, the applied force is fro the shaker. However, due to ounting ethods discussed earlier, this is not necessarily the force transitted to the structure. db Mag Useable Frequency Range The echanis of the response transducer, called an acceleroeter, functions in a siilar anner. When the acceleroeter vibrates, an internal ass in the assebly applies a force to the crystal eleent which is proportional to the acceleration. This relationship is siply Newton s Law: force equals ass ties acceleration. The properties to consider in selecting a load cell include both the type of force sensor and its perforance characteristics. The type of force sensing for which load cells are designed include copression, tension, ipact or soe cobination Figure 2.17 Stud ounted load cell Frequency 25

26 thereof. Most shaker tests require at least a copression and tension type. A haer test, for exaple, would require an ipact transducer. Figure 2.18 Frequency response of transducer 20k-25k Soe of the operating specifications to consider are sensitivity, resonant frequency, teperature range and shock rating. Sensitivity is easured in ters of voltage/force with units of V/1b or V/N. Analyzers have a range of input voltage settings; therefore, sensitivity should be chosen along with a power supply aplification level to generate a easurable voltage. The resonant frequency of a load cell is siply a function of its physical ass and stiffness characteristics. The frequency range of the test should fall within the linear range below the resonant peak of the frequency response of the load cell, as shown in Figure The rule of thub for shock rating is that the axiu vibration level expected during the test should not exceed one third the shock rating. Useable Frequency Range Useable Frequency Range Stud 15k-20k The load cell should be ounted to the structure with a threaded stud for best results as shown in Figure If this is not feasible, then an alternative ethod of first fixing a spacer to the structure with soe type of adhesive (such as dental ceent) and then stud ounting the load cell to this spacer will usually suffice for low force levels. Ceent, Wax 5k-7k The properties to consider in selecting an acceleroeter are very siilar to those of the load cell, although they are related to acceleration rather than force. The type of response is liited to acceleration as the ter iplies, since displaceent and velocity transducers are not available Useable Frequency Range Magnet 26

27 Figure 2.19 Mass loading fro acceleroeter /Div Frequency 3Avg 0%Ovlp FR/Ex Aplitude db Fxd Y 0 Hz 2.5k in the piezoelectric type. However, if displaceent or velocity responses are desired, the acceleration response can be artificially integrated once or twice to give velocity and displaceent responses, respectively. In general, the optiu acceleroeter has high sensitivity, wide frequency range and sall ass. Trade-offs are usually ade since high sensitivity usually dictates a larger ass for all but the ost expensive acceleroeters. The sensitivity, easured in V/G, and the shock rating should be selected in the sae anner as with the load cell. Although the resonant frequency of the acceleroeter (freely suspended) is a function of its ass and stiffness characteristics, the actual natural frequency (when ounted) is generally dictated by the stiffness of the ounting ethod used. The effect of various ounting ethods is shown in Figure The rule of thub is to set the axiu frequency of the test at no ore than one-tenth the ounted natural frequency of the acceleroeter. This is within the linear range of the ounted frequency response of the acceleroeter. Another iportant consideration is the effect of ass loading fro the acceleroeter. This occurs as a result of the ass of the acceleroeter being a significant fraction of the effective ass of a particular ode. A siple procedure to deterine if this loading is significant can be done as follows: Measure a typical frequency response function of the test object using the desired acceleroeter. Mount another acceleroeter (in addition to the first) with the sae ass at the sae point and repeat the easureent. Copare the two easureents and look for frequency shifts and aplitude changes. If the two easureents differ significantly, as illustrated in Figure 2.19, then ass loading is a proble and an acceleroeter with less ass should be used. On very sall structures, it ay be necessary to easure the response with a non-contacting transducer, such as an acoustical or optical sensor, in order to eliinate any ass loading. 27

28 Figure 2.20 Exaple of the input half ranging INST TIME /Div COHERENCE 1.1 Mag 50 Avg 50% Ovlp Hann Real v 0.0 Fxd Y 1.1k Hz 2.1k 50 Avg 50% Ovlp Hann 40.0 db Fxd XY 0.0 Sec Fxd XY 1.1k Hz 2.1k Figure 2.21 Exaple of the input under ranging INST TIME /Div COHERENCE 1.1 Mag 50 Avg 50% Ovlp Hann Real v 0.0 Fxd Y 1.1k Hz 2.1k 50 Avg 50% Ovlp Hann 40.0 db -130 Fxd XY 0.0 Sec Fxd XY 1.1k Hz 2.1k Figure 2.22 Exaple of the input over ranging INST TIME /Div Ov2 COHERENCE 1.1 Mag 50 Avg 50% Ovlp Hann Ov2 Real v 0.0 Fxd Y 1.1k Hz 2.1k 50 Avg 50% Ovlp Hann Ov db Fxd X 0.0 Sec Fxd XY 1.1k Hz 2.1k 28

29 Measureent Interpretation Having discussed the echanics of setting up a odal test, it is appropriate at this point to ake soe trial easureents and exaine their trends before proceeding with data collection. Taking the tie to investigate preliinaries of the test, such as exciter or response locations, various types of excitation functions and different signal processing paraeters will lead to higher quality easureents. This section includes preliinary checks such as adequate signal levels, iniu leakage easureents and linearity and reciprocity checks. The concept and trends of the driving point easureent and the cobinations of easureents that constitute a coplete odal survey are discussed. After the structure has been supported and instruented for the test, the tie doain signals should be exained before aking easureents. The input range settings on the analyzer should be set at no ore than two ties the axiu signal level as shown in Figure Often called half-ranging, this takes advantage of the dynaic range of the analog-to-digital converter without underranging or overranging the signals. The effect resulting fro underranging a signal, where the response input level is severely low relative to the analyzer setting, is illustrated in Figure Notice the apparent noise between the peaks in the frequency response and the resulting poor coherence function. In Figure 2.22, the response is severely overloading the analyzer input section and is being clipped. This results in poor frequency response and, consequently, poor coherence since the actual response is not being easured correctly. Figure 2.23 Rando test signals Figure 2.24 Transient test signals FILT TIME Real It is also advisable to verify that the signals are indeed the type expected, (e.g., rando noise). With a rando signal, it is advisable to easure the histogra to verify that it is not containated with other signal coponents, i.e., it has a Gaussian distribution as shown in Figure This can be visually checked as illustrated with the transient signals in Figure v Rando Noise 50%Ovlp Fxd Y 0.0 Sec 799 HIST 1 90 Avg 50%Ovlp 10.0 k Histogra Real 0.0 Fxd XY V 10.0 FILT TIME Real v 0%Ovlp -400 Fxd Y 0.0 Sec 512 FILT TIME 2 0%Ovlp 700 Real Exponential Decaying v Typical Force Response -700 Fxd Y 0.0 Sec

30 Chapter 3 Iproving Measureent Accuracy Measureent Averaging In order to reduce the statistical variance of a easureent with a rando excitation function (such as rando noise) and also reduce the effects of nonlinearities, it is necessary to eploy an averaging process. By averaging several tie records together, statistical reliability can be increased and rando noise associated with nonlinearities can be reduced. One ethod to gain insight into the variance of a easureent is to observe the Nyquist display of the frequency response. The circle appears very distorted for a easureent with few averages, but begins to sooth out with ore and ore averages. This process can be seen in Figure 3.1. With each data record acquired, the frequency spectru has a different agnitude and phase distribution. As these spectra are averaged, the nonlinear ters tend to cancel, thus resulting in the best linear estiate. Figure 3.1 Measureent averaging frequency response COHERENCE Avg 50% Ovlp Hann Avg 50% Ovlp Hann Mag 0.0 Fxd Y 1.1k Hz 2.1k 5 Avg 50% Ovlp Hann 40.0 Iag db Fxd Y 1.1k Hz 2.1k -100 Fxd Y Averages Real 128 COHERENCE Avg 50% Ovlp Hann Avg 50% Ovlp Hann Mag 0.0 Fxd Y 1.1k Hz 2.1k Avg 50% Ovlp Hann Iag db Fxd Y 1.1k Hz 2.1k -100 Fxd XY Averages Real

31 Windowing Tie Data There is a property of the Fast Fourier transfor (FFT) that affects the energy distribution in the frequency spectru. It is the result of the physical liitation of easuring a finite length tie record along with the periodicity assuption required of the tie record by the FFT. This does not present a proble when the signal is exactly periodic in the tie record or when a transient signal is copletely captured within the tie record. However, in the case of true rando excitation or in the transient case when the entire response is not captured, a phenoenon called leakage results. This has the effect of searing or leaking energy into adjacent frequency lines of the spectru, thus distorting it. Figure 3.2 illustrates an exaple of the effects of severe leakage probles with true rando excitation. The effect is to underestiate the aplitude and overestiate the daping factor. One of the ost coon techniques for reducing the effects of leakage with a non-periodic signal is to artificially force the signal to 0 at the beginning and end of the tie record to ake it appear periodic to the analyzer. This is accoplished by ultiplying the tie record by a atheatical curve, known as a window function, before processing the FFT. Another easureent is taken with a Hann window applied to the true rando excitation signal, shown in Figure 3.3. This easureent is ore accurate, but notice that the coherence is still less than unity at the resonance. The window does not Figure 3.2 Frequency response with true rando signal and no windows Figure 3.3 Frequency response with burst rando signal FILT TIME /Div Real v Fxd Y 0.0 COHERENCE 1.1 Mag Sec 799 Fxd Y 1.1k Hz 2.1k 50 Avg 50% Ovlp Unif 40.0 db %Ovlp True Rando Noise 50 Avg 50% Ovlp Unif Fxd Y 1.1k Hz 2.1k COHERENCE 1.1 Mag Fxd Y 1.1k Hz 2.1k 50 Avg 50% Ovlp Hann 40.0 db Avg 50% Ovlp Hann Fxd Y 1.1k Hz 2.1k 31

32 eliinate leakage copletely and it also distorts the easureent as a result of eliinating soe data. A better easureent technique is to use an excitation that is periodic within the tie record such as burst rando, in order to eliinate the leakage proble as illustrated in Figure 3.4. Increasing Measureent Resolution Another easureent capability that is often needed, particularly for lightly daped structures, is to obtain ore frequency resolution in the vicinity of resonance peaks. It ay not be possible in a baseband easureent to extract valid odal paraeters with inadequate inforation. Norally, the Fourier transfor is calculated over a frequency range fro 0 to soe axiu frequency. Zoo processing is a technique in which the lower and upper frequency liits are independently selectable over fixed ranges within the analyzer. The capability to zoo allows closely spaced odes to be ore accurately identified by concentrating the easureent points over a narrower band. The result of this increased easureent accuracy is shown in Figure 3.5. Another result is that distortion due to leakage is reduced, because the searing of energy is now within a narrower bandwidth, but not eliinated. Another related process associated with zooing is the ability to band-liit the excitation to concentrate the available energy within the given frequency range of the test. Figure 3.4 Frequency response with true rando signal and Hann window FILT TIME /Div Real v Fxd Y 0.0 COHERENCE 1.1 Mag Sec 799 Fxd Y 1.1k Hz 2.1k 15 Avg 0% Ovlp Unif 40.0 db 0.0 0%Ovlp Burst Rando Signal 15 Avg 0% Ovlp Unif Fxd Y 1.1k Hz 2.1k 32

33 Figure 3.5 Effects of increasing frequency resolution COHERENCE Avg 0% Ovlp Hann Avg 0% Ovlp Hann Mag 0.0 Fxd Y 0 Hz 4k 50 Avg 0% Ovlp Hann 50.0 Iag db Fxd Y 0 Hz 4k -200 Fxd Y -257 Baseband Measureent Real 257 COHERENCE Avg 5% Ovlp Hann Avg 5% Ovlp Hann Mag 0.0 Fxd Y Hz 50 Avg 5% Ovlp Hann 2.345k Iag db Fxd Y 745 Hz 2.345k -100 Fxd XY -128 Real 128 1st Zoo Span COHERENCE Avg 50% Ovlp Hann Avg 50% Ovlp Hann Mag 0.0 Fxd Y 1.2k 40.0 Hz 50 Avg 50% Ovlp Hann 1.6k Iag db Fxd Y 1.2k Hz 1.6k -100 Fxd Y -128 Real 128 2nd Zoo Span 33

34 Coplete Survey As frequency response functions are being acquired and stored for subsequent odal paraeter estiation, an adequate set of easureents ust be collected in order to arrive at a coplete set of odal paraeters. This section describes the nuber and type of easureents that constitute coplete odal survey. Definitions and concepts, such as driving point easureent and a row or colun of the frequency response atrix are discussed. Optial shaker and acceleroeter locations are also included. Figure 3.6 Frequency response atrix H 11 H 12 H 1N H 21 H 22 H N1 H NN A coplete, although redundant, set of frequency response easureents would for a square atrix of size N, where the row corresponds to response points and the coluns correspond to excitation points, as illustrated in Figure 3.6. It can be shown, however, that any particular row or colun contains sufficient inforation to copute the coplete set of frequencies, daping, and ode shapes. In other words, if the excitation is at point 3, and the response is easured at all the points, including point 3, then colun of the frequency response atrix will be easured. This situation would be the result of a shaker test. On the other hand, if an acceleroeter is attached to point 7, and a haer is used to excite the structure at all points, including point 7, then row 7 on the atrix will be easured. This would be the result of an ipact test. The easureent where the response point and direction are the sae as the excitation point and direction is called a driving point easureent. The bea in Figure 3.7 illustrates a easureent of this type. Driving point easureents Figure 3.7 Driving point easureent setup for the diagonal of the frequency response atrix shown above. They also exhibit unique characteristics that are not only useful for checking easureent quality, but necessary for accoplishing a coprehensive odal analysis, which includes not only frequencies, daping factors and scaled ode shapes, but odal ass and stiffness as well. It is not necessary to ake a driving point easureent to obtain only frequencies, daping factors and unscaled ode shapes. However, a set of scaled ode shapes and consequently, odal ass and stiffness cannot be extracted fro a set of easureents that does not contain a driving point. 34

35 Recall fro Chapter 1, Structural Dynaics Background, that the response of a MDOF syste is siply the weighted su of a nuber of SDOF systes. The characteristics of the driving point easureent can be easily explained and presented as a consequence of this property. Figure 3.8 shows a typical driving point easureent displayed in rectangular and polar coordinates. As seen in the iaginary part of the rectangular coordinates, all of the resonant peaks lie in the sae direction. In other words, they are in phase with each other. This characteristic becoes ore intuitive when illustrated with the bea odes in Figure 3.9. The response point oves in the sae direction as the excitation point at all the odes, since it is easured at the sae physical location as the excitation. By observing the trends of this easureent in polar coordinates in Figure 3.9, a further understanding of its characteristics can be gained. When the agnitude is displayed in log forat (db), anti-resonances occur between every resonance throughout the frequency range. The individual SDOF systes su to 0 at the frequencies where the ass and stiffness lines of adjacent odes intersect since all the odes are in phase with each other. This results in the near 0 agnitude of an antiresonance. Also notice the phase lead as the agnitude passes through an anti-resonance and the opposite phase lag as the agnitude passes through a resonance. These trends of the driving point easureent should be observed and onitored throughout the easureent process as a check for aintaining a consistent set of data. Figure 3.8 Driving point frequency response Figure 3.9 Typical free bea ode shapes 56.0 Real Fxd Y 0 Hz 1.25k 3 Avg 0% Ovlp Fr/Ex 100 Iag 3 Avg 0% Ovlp Fr/Ex -100 Fxd Y 0 Hz 1.25k 40.0 db Fxd Y 0 Hz 1.25k Avg 0% Ovlp Fr/Ex Phase 3 Avg 0% Ovlp Fr/Ex -180 Fxd Y 0 Hz 1.25k 35

36 The reaining easureents, where the response coordinates are different fro the excitation coordinates, are called cross-point easureents. Figure 3.10 illustrates a typical crosspoint easureent. All the odes are not necessarily in phase with each other, as seen in the iaginary display. Since the response points are not at the sae location as the excitation point, the response can ove either in phase or out of phase with the excitation. This otion, which defines the ode shape, is a function of the easureent location, and will vary fro easureent to easureent. In the db display, if any two adjacent odes are in phase at a particular point, then an anti-resonance will exist between the. If any two adjacent odes are out of phase, then their ass and stiffness lines will not cancel at the intersection and a sooth curve will appear instead, as seen in Figure In order to excite all the odes within the frequency range of interest, several shaker or acceleroeter locations should be exained. A point or line on the structure that reains stationary is called a node point or node line. The node points of a cantilever bea are illustrated in Figure The nuber and location of these nodes are a function of the particular ode of vibration and increase as ode nuber increases. Figure 3.10 Cross-point frequency response Figure 3.11 Typical cantilever bea ode shapes 56.0 Real Fxd Y 0 Hz 1.25k 3 Avg 0% Ovlp Fr/Ex 100 Iag 3 Avg 0% Ovlp Fr/Ex -100 Fxd Y 0 Hz 1.25k 40.0 db Fxd Y 0 Hz 1.25k Avg 0% Ovlp Fr/Ex Phase 3 Avg 0% Ovlp Fr/Ex -180 Fxd Y 0 Hz 1.25k

37 If the response is easured at the end of the bea at point 1 and excitation is applied at point 3, all odes will be excited and the resulting frequency response will contain all the odes as shown in Figure However, if the response is at point 1 but the excitation is oved to point 2, the second ode will not be excited and the resulting frequency response will appear as shown in Figure Referring to Figure 3.11, note that point 2 is a node point for ode 2 and very near a node point for ode 3. Mode 2 does not appear in the iaginary display and ode 3 is barely discernible. In db coordinates, the ode 3 appears to exist but it is still difficult to observe ode 2. It ay be advisable or necessary at ties to gather ore than one set of data at different excitation locations in order to easure all the odes. It should also be noted that the sae observations can be ade in an ipact test where the response point would also be oved to various locations. Another concept associated with a linear structure concerns a property of the frequency response atrix. The frequency response atrix for a linear syste can be shown to be syetric due to Maxwell s Reciprocity Theore. Siply stated, a easureent with the excitation at point i and the response at point j is equal to the easureent with the excitation at point j and the response at point i. This is illustrated in Figure A check can be ade on the easureent process by coparing these two reciprocal easureents at various pairs of points and observing any differences between the. This can be helpful for noting nonlinearities when applying different force levels. Figure 3.12 Frequency response with excitation at node of vibration Figure 3.13 Syetric frequency response atrix 100 Iag Fxd Y 0 Hz 1.25k 3 Avg 0% Ovlp Fr/Ex 40.0 db Avg 0% Ovlp Fr/Ex Fxd Y 0 Hz 1.25k 56.0 Iag Fxd Y 0 Hz 1.25k Avg 0% Ovlp Fr/Ex db 3 Avg 0% Ovlp Fr/Ex Fxd Y 0 Hz 1.25k H 11 H 12 H 1N H 21 H 22 H N1 H ji H ij H NN H ij = H ji 37

38 Chapter 4 Modal Paraeter Estiation Introduction The previous chapter presented several techniques for aking frequency response easureents for odal analysis. Having acquired this data, the next ajor step of the process is the use of paraeter estiation techniques curve fitting to identify the odal paraeters. A vast aount of literature exists on the subject of curve fitting easured data to estiate the odal properties of a structure. However, this inforation tends to be atheatically vigorous and is generally biased toward a particular type of algorith. It is the intent of this chapter to categorize, in a conceptual anner, the different types of curve fttters and discuss the applications and probles associated with those ost coonly ipleented. It was discussed earlier that a iniu of one row or colun of the frequency response atrix, or its equivalent, ust be easured in order to identify a coplete set of odal paraeters. Although additional data is, in principle, redundant inforation, it can be used to verify and increase the confidence level of the estiated paraeters. The frequency and daping for each ode can be estiated fro any cobination of these easureents. The residues and, consequently, the odal coefficients are then coputed for each easureent point. The ode shapes are then scaled and sorted for each resonant frequency. Finally, the odal ass and stiffness can be deterined fro these scaled paraeters as illustrated in Figure 4.1. Figure 4.1 Typical flow of odal test Measure coplete set of frequency responses Identify Modal Frequency Daping Mode Shape Scale Mode Shape Modal Mass Modal Stiffness 38

39 Modal Paraeters One of the ost fundaental assuptions of odal testing is that a ode of vibration can be excited at any point on the structure, except at nodes of vibration where it has no otion. This is why a single row or colun of the frequency response atrix provides sufficient inforation to estiate odal paraeters. As a result, the frequency and daping of any ode in a structure are constants that can be estiated fro any one of the easureents as shown in Figure 4.2. In other words, the frequency and daping of any ode are global properties of the structure. In practical applications, it is iportant to include sufficient points in the test to copletely describe all the odes of interest. If the excitation point has not been chosen carefully or if enough response points are not easured, then a particular ode ay not be adequately represented. At ties it ay becoe necessary to include ore than one excitation location in order to adequately describe all of the odes of interest. Frequency responses can be easured independently with single-point excitation or siultanously with ultiple-point excitations. The ode shapes as a whole are also global properties of the structure, but have relative values depending on the point of excitation and scaling and sorting factors. On the other hand, each individual odal coefficient that akes up the ode shape is a local property in the sense that it is estiated fro the particular easureent associated with that point as shown in Figure 4.3. Figure 4.2 Concepts of odal paraeters Figure 4.3 Modal paraeters Vibrating Bea Measureent Points Daping Frequency Mode = 1 Mode = 2 Mode = 3 Daping, frequency sae at each easureent point Mode shape obtained at sae frequency fro all easureent points ω ζ Frequency Daping Global { φ} Mode Shape Local 39

40 Curve Fitting Methods Due to the large aount of literature and algoriths currently available for curve fitting structural data, it has becoe difficult to deterine the exact need for each ethod and which ethod is best. There is no ideal solution and the coon ethods are only approxiations. Also, any of the ethods are very siilar to each other and, in soe cases, siply extensions of a few basic techniques. Users generally fall into one of three ajor groups. The first group is priarily concerned with troubleshooting existing echanical equipent. They are usually concerned with tie and require a fast, ediu quality curve fitter. The second group is ore serious about quantitative paraeter estiates for use in a odal odel. For exaple, they require ore accuracy and are willing to spend ore tie obtaining results. The final group is pushing the state of the art and is involved with developent work. Accuracy, rather than tie, is of paraount iportance. Although there are several ways in which curve fitting ethods can be categorized, the ost straightforward is single-ode versus ultipleode classification. Besides the intuitive reasoning for single- and ultiple-ode approxiations, there are soe practical reasons for this classification. The ajor difference in the level of sophistication, or l evel of accuracy, aong curve fitters is between a single-ode and a ultiple-ode ethod. Also, the coputing resources needed (coputation speed, eory size and I/0 capability) for ultiple-ode ethods can increase treendously. Other sub-catagories and extensions that fall ostly within ultiple-ode ethods are shown in Figure 4.4. Figure 4.4 Increasing accuracy in curve fit ethods Single-Mode Methods Easy, fast Liited Multi-Mode Methods More Accurate Slower Multi-Measureent Global Accuracy Large Aount of Data Processing 40

41 Single-Mode Methods As stated earlier, the general procedure for estiating odal paraeters is to estiate frequencies and daping factors, then estiate odal coefficients. For ost singleode paraeter estiation techniques, however, this is not always the case. In fact, it is not absolutely necessary to estiate daping in order to obtain odal coefficients. This is typical in a troubleshooting environent where frequencies and ode shapes are of priary concern. Figure 4.5 Frequency response Magnitude Phase The basic assuption for single-ode approxiations is that in the vicinity of a resonance, the response is due priarily to that single ode. The resonant frequency can be estiated fro the frequency response data (illustrated in Figure 4.5) by observing the frequency at which any of the following trends occur: Real ω n Frequency The agnitude of the frequency response is a axiu. The iaginary part of the frequency response is a axiu or iniu. The real part of the frequency response is zero. The response lags the input by 90 phase. It was discussed earlier that the height of the resonant peak is a function of daping. The daping factor can be estiated by the halfpower ethod or other related atheatical or graphical ethod. In the half-power ethod, the daping is estiated by deterining the sharpness of the resonant peak. It can be shown fro Figure 4.6 that daping can be related to the width of the Iaginary ω n Frequency 41

42 peak between the half-power points: points below and above the resonant peak at which the response agnitude is.7071 ties the resonant agnitude. One of the siplest single-ode odal coefficient estiation techniques is the quadrature ethod, often called peak picking. Modal coefficients are estiated fro the iaginary (quadrature) part of the frequency response, so the ethod is not a curve fit in the strict sense of the ter. As entioned earlier, the iaginary part reaches a axiu at the resonant frequency and is 90 out of phase with respect to the input. The agnitude of the odal coefficient is siply taken as the value of the iaginary part at resonance as illustrated in Figure 4.7. The sign (phase) is taken fro the direction that the peak lies along the iaginary axis, either positive or negative. This iplies that the phase angle is either 0 or 180. Figure 4.6 Daping factor fro half power Figure 4.7 Quadrature peak pick /Div Mag Hz /Div Iag Daping The quadrature response ethod is one of the ore popular techniques for estiating odal paraeters because it is easy to use, very fast and requires iniu coputing resources. It is, however, sensitive to noise on the easureent and effects fro adjacent odes. This ethod is best suited for structures with light daping and well separated odes where odal coefficients are essentially real valued. It is ost useful for troubleshooting probles, however, where it is not necessary to create a odal odel and tie is liited. Figure 4.8 Circle fit ethod -640 Iaginary Modal Coefficient Real Modal Coefficient 42

43 Another single-ode technique, called the circle fit, was originally developed for structural daping but can be extended to the viscous daping case. Recall fro Chapter 1 that the frequency response of a ode traces out a circle in the iaginary plane. The ethod fits a circle to the real and iaginary part of the frequency response data by iniizing the error between the radius of the fitted circle and the easured data. The odal coefficient is then deterined fro the diaeter of the circle as illustrated in Figure 4.8. The phase is deterined fro the positive or negative half of the iaginary axis in which the circle lies. Frequency and daping can be estiated by one of the ethods discussed earlier or by soe of the MDOF ethods to be discussed later. Daping can also be estiated fro the spacing of points along the Nyquist plot fro the circle. The circle fit ethod is fairly fast and requires iniu coputer resources. It usually results in better paraeter estiates than obtained by the quadrature ethod because it uses ore of the easureent inforation and is not as sensitive to effects fro adjacent odes as illustrated in Figure 4.9. It is also less sensitive to noise and distortion on the easureent. However, it requires uch ore user interaction than the quadrature ethod; consequently, it is prone to errors, particularly when fitting closely spaced odes. A SDOF ethod related to the circle fit is a frequency doain curve fit to a single-ode analytical expression of the frequency response. This expression is generally forulated as a second order polynoial with residual ters to take into account the effects of out of band odes. Because of its siilarities to the circle fit, it possesses the sae basic advantages and disadvantages. Figure 4.9 Coparison of quadrature and circle fit ethods Figure 4.10 Analytical expression of frequency response φ Peak Pick H( ω) = n Σ r = 1 Concept of Residual Ters Before proceeding to ultiple-ode ethods, it is appropriate to discuss the residual effects that out-of-band odes have on estiated paraeters. In general, structures possess an infinite nuber of odes. However, there are only a liited nuber that are usually of concern. Figure 4.10 illustrates the analytical expression Iaginary φ Circle Fit φ Circle Fit φφ i j ( ω 2 n - ω 2 ) + j(2ζωω n ) Real φ Peak Pick 43

44 for the frequency response of a structure taking into account the total nuber of realizable odes. Unfortunately, the easured frequency response is liited to soe frequency range of interest depending on the capabilities of the analyzer and the frequency resolution desired. This range ay not necessarily include several lower frequency odes and ost certainly will not include soe higher frequency odes. However, the residual effects of these out-of-band odes will be present in the easureent and, consequently, affect the accuracy of paraeter estiation. Figure 4.11 Single ode suation of frequency response db Magnitude 0.0 Mass Line Stiffness Line ω 1 ω 2 ω 3 Frequency Although paraeters of the out-of-band odes cannot be identified, their effects can be represented by two relatively siple ters. It can be seen fro Figure 4.11 that the effects of the lower odes tend to have ass-like behavior and the effects of the higher odes tend to have stiffness-like behavior. The analytical expression for the residual ters can then be written as shown in Figure Notice that the residual ters are equivalent to the asyptotic behavior of the ass and stiffness of a SDOF syste discussed in the chapter on structural dynaics. Useful inforation can often be gained fro the residual ters that has soe physical significance. First, if the structure is freely supported during the test, then the low frequency residual ter can be a direct easure of rigid body ass properties of the structure. The high frequency ter, on the other hand, can be a easure of the local flexibility of the driving point. Figure 4.12 Residual ters of frequency response db Magnitude 0.0 H( ω) = H( ω) = ω 1 ω 2 ω 3 Frequency φφ i j -w Lower Modes φφ i j k Higher Modes 44

45 Multiple-Mode Methods The single-ode ethods discussed earlier perfor reasonably well for structures with lightly daped and well separated odes. These ethods are also satisfactory in situations where accuracy is of secondary concern. However, for structures with closely spaced odes, particularly when heavily daped, (as shown in Figure 4.13) the effects of adjacent odes can cause significant approxiations. In general, it will be necessary to ipleent a ultiple-ode ethod to ore accurately identify the odal paraeters of these types of structures. The basic task of all ultiple-ode ethods is to estiate the coefficients in a ultiple- ode analytical expression for the frequency response function. This is done by curve fitting a ultiple-ode for of the frequency response function to frequency doain easureents. An equivalent ethod is to curve fit a ultiple response for of the ipulse response function to tie doain data. In either process, all the odal paraeters (frequency, daping and odal coefficient) for all the odes are estiated siultaneously. Figure 4.13 Daping and odal coupling /Div Mag /Div Mag Hz Light Daping and Coupling Hz Heavy Daping and Coupling k k There are a nuber of ultiple-ode ethods currently available for curve fitting easured data to estiate odal paraeters. However, there are essentially two different fors of the frequency response function which are used for curve fitting. These are the partial fraction for and the polynoial for which are shown in Figure They are equivalent analytical fors and can Figure 4.14 Frequency response representation H( ω) = H( ω) = n Σ k = 1 n h(t) = Σ k = 1 r k jω - pk as + a, s bs n + b, s n r k * + Partial Fraction jω - pk* r k e -σ k t + r k *e - σ* k t Polynoial Coplex Exponential 45

46 be shown to be related to the structural frequency response developed earlier in Chapter 1. The ipulse response function, obtained by inverse Fourier transforing the frequency response function into the tie doain, is also shown in Figure When the partial fraction for of the frequency response is used, the odal paraeters can be estiated directly fro the curve fitting process. A least squares error approach yields a set of linear equations that ust be solved for the odal coefficients and a set of nonlinear equations that ust be solved for frequency and daping. Because an iterative solution is required to solve these equations, there is potential for convergence probles and long coputation ties. If the polynoial for of the frequency response is used, the coefficients of the polynoials are identified during the curve fitting process. A root finding solution ust then be used to deterine the odal paraeters. The advantage of the polynoial for is that the equations are linear and the coefficients can be solved by a noniterative process. Therefore, convergence probles are inial and coputing tie is ore reasonable. The coplex exponential ethod is a tie doain ethod that fits decaying exponentials to ipulse response data. The equations are nonlinear, so an iterative procedure is necessary to obtain a solution. The ethod is relatively insensitive to noise on the data, but suffers fro sensitivity to tie doain aliasing, as a result of truncation in the frequency doain fro inverse Fourier transforing the frequency responses. In principle, it should not atter whether frequency doain data or tie doain data is used for curve fitting since the sae inforation is contained in both doains. However, there are soe practical reasons, based on frequency doain and tie doain operations, that see to favor the frequency doain. One, the easureent data can be restricted to soe desired frequency range and any noise or distortion outside this range can effectively be ignored. Another, the cross spectra and autospectra needed to copute frequency responses can be fored faster than the corresponding tie doain correlation functions. It is true that the tie doain can be used to select odes having different daping values, but this is usually not as iportant as the ability to select a frequency range of interest. Each ethod has its advantages and disadvantages, but the fundaental probles of noise, distortion and interference fro adjacent odes reain. As a result, none of the ethods work well in all situations. It is also unlikely that soe agic ethod will be discovered that eliinates all of these probles. All of the ethods work well with ideal data, but cannot be evaluated by analytical eans alone. The iportant factor is how well they work, or gracefully fail, with real experiental data coplete with noise and distortion. 46

47 Concept of Real and Coplex Modes The structural odel discussed so far is based on the concept of proportional viscous daping which iplies the existence of real, or noral, odes. Matheatically, this iplies that the physical daping atrix can be defined as linear cobination of the physical ass and stiffness atrices as shown in Figure The ode shapes, are, in effect real valued, eaning the phase angles differ by 0 or 180. Physically, all the points reach their axiu excursion at the sae tie as in a standing wave pattern. One of the consequences of this assuption, discussed earlier, is that the iaginary part of the frequency response reaches a axiu at resonance and the real part is 0 valued as illustrated in Figure Note also that the Nyquist circle lies along the iaginary axis. However, physical structures exhibit a ore coplicated for of daping which results in non-proportional daping. The ode shapes are, generally, coplex valued, eaning the phase angles can have values other than 0 or 180. Physically, the points reach their axiu excursions at various ties as in a traveling wave pattern. With nonproportional daping, the iaginary part of the frequency response no longer reaches a axiu at resonance nor is the real part nonzero valued as illustrated in Figure Note also that the Nyquist circle is rotated at an angle in the coplex plane. When daping is light, as is the case in ost echanical structures, the proportional daping assuption is generally an accurate approxiation. Although daping is not proportional to the ass and stiffness, the nonproportional coupling effects ay be sall enough not to cause serious errors. Physically, this eans that the daping is sufficiently sall so that Figure 4.15 Proportional daping representation Figure 4.16 Figure 4.17 [C] = a [M] + β [K] r1 k +r2 k 2σ k 2 2 ω = ω k coupling is a second-order effect. It should be noted that closely-spaced odes often appear coplex as a result of the effects fro adjacent odes as illustrated in Figure In reality, they ay actually be ore real than they appear. Iaginary a k ω = ω k Real 2σ k Iaginary Figure 4.18 Two closely spaced real odes Iaginary Real 47

48 Chapter 5 Structural Analysis Methods Introduction The basic techniques for perforing a odal test to identify the dynaic properties of a structure have been described in the previous chapters. An introduction to the applications of the resulting frequency responses and odal paraeters is the focus of this chapter. The discussion is specifically concerned with the uses of a response odel or a odal odel with structural analysis ethods as shown in Figure 5.1. The intent is to bring together the experiental and analytical tools for solving noise, vibration and failure probles. Figure 5.1 Alternative types of structural odels Physical Structure Model Response Model (easureents) FFT A response odel is siply the set of frequency response easureents acquired during the odal test. These easureents contain all the dynaics of the structure needed for subsequent analyses. A odal odel is derived fro the response odel and is a function of the paraeter estiation technique used. It not only includes frequencies, daping factors, and ode shapes, but also odal ass and odal stiffness. These asses and stiffnesses depend on the ethod that was used to scale the ode shapes. A subset of the odal odel consisting of only the frequencies and unscaled ode shapes can be useful for soe troubleshooting applications where frequencies and ode shapes are the priary concern. However, for applications involving analysis ethods, such as structural odification and Modal Model (paraeters) substructure coupling, a coplete odal odel is required. This definition of a coplete odal odel should not be confused with the concept of a truncated ode set in which all the odes are not included. ω Frequency ζ Daping { φ} Mode Shape Curve Fit 48

49 Structural Modification When troubleshooting a vibration proble or investigating siple design changes, an analysis ethod known as structural odification, illustrated in Figure 5.2, can be very useful. Basically, the ethod deterines the effects of ass, stiffness and daping changes on the dynaic characteristics of the structure. It is a straightforward technique and gives reasonable solutions for siple design studies. Soe of the benefits of using structural odification are reduced tie and cost for ipleenting design changes and eliination of the trial-and-error approach to solving existing vibration probles. The technique can be extended to an iterative process, often called sensitivity analysis, in order to categorize the sensitivity of specific aounts of ass, stiffness or daping changes. Figure 5.2 Structural odification siulation Engine Mount Troublesoe Mode SYNTHESIS /Div db Stiffness, Daping SM Pole Residue In general, structural odification involves two interrelated design investigations. In the first, a physical ass, stiffness or daping change can be specified with the analysis deterining the odified set of odal paraeters. The second involves specifying a frequency and having the analysis deterine the aount of ass, stiffness or daping needed to shift a resonance to this new frequency. For the user s convenience, specific applications of these basic ethods, such as tuned absorber, design are usually included. -20 FxdXY 1.375k Hz 2.975k Structural odification can be ipleented with either a response odel or a odal odel. The advantages of response odel are that it contains the effects of odes outside the analysis range and the easureents can be liited to the selected set of odified points. It is sensitive, though, to the quality of frequency response easureents and the effects of rigid body odes fro the support syste. A odal odel requires only one driving point easureent whereas a response odel requires a driving point easureent at every location to be odified. The odal odel ay be ore intuitive since it contains direct ass, stiffness and daping inforation directly. However, it does suffer fro being tedious and tie consuing to derive and is sensitive to the nuber and type of odes extracted. 49

50 Finite Eleent Correlation Finite eleent analysis is a nuerical procedure useful for solving structural echanics probles. More specifically, it is an analytical ethod for deterining the odal properties of a structure. It is often necessary to validate the results fro this theoretical prediction with easured data fro a odal test. This correlation ethod is generally an iterative process and involves two ajor steps. First, the odal paraeters, both frequencies and ode shapes, are copared and the differences quantified. Second, adjustents and odifications are ade, usually to the finite eleent odel, to achieve ore coparable results. The finite eleent odel can then be used to siulate responses to actual operating environents. Table 5.1 Tabular coparisons of frequency Figure 5.3 Graphical coparison of frequencies FE (Hz) Test (Hz) Fe The correlation task is usually begun by coparing the easured and predicted frequencies. This is often done by aking a table to copare each ode frequency by frequency as shown in Table 5.1. It is ore useful, however, to graphically copare the entire set of frequencies by plotting easured versus the predicted results as shown in Figure 5.3. This shows not only the relative differences between the frequencies, but also the global trends and suggests possible causes of these differences. If there is a direct correlation the points will lie on a straight line with a slope of 1.0. If a rando scatter arises, then the finite eleent odel ay not be an accurate representation of the structure. This could result fro an inappropriate eleent type or a poor eleent esh in the finite eleent odel. It could also result fro incorrect boundary conditions in either the test or the analysis. If the points lie on a straight line, but with a slope other than 1, then the proble ay be a ass loading proble in the odal test or an incorrect aterial property, such as elastic odulus or aterial density, in the finite eleent odel. The paraeter coparison is not actually this siple, nor is it coplete, because the ode shapes ust also be copared at the sae tie to Test ensure a one-to-one correspondence between the frequency and the ode shape. Reeber that a distinct ode shape is associated with each distinct frequency. One technique for perforing this coparison is to siply overlay the plotted ode shapes fro the test and analysis and observe their general trends. This can becoe rather difficult, though, for structures with coplicated geoetry because the plots tend to get cluttered. 50

51 Nuerical techniques have been developed to perfor statistical coparisons between any two ode shapes, illustrated in Figure 5.4. The first results in the odal scale factor (MSF) a proportionality constant between the two shapes. If the constant is equal to 1.0, this eans the shapes were scaled in the sae anner such as unity odal ass. If the constant is any value other than 1.0, then the shapes were scaled differently. The second, and ore iportant ethod, results in the odal assurance criterion (MAC), a correlation coefficient between the two ode shapes. If the coefficient is equal to 1.0, then the two shapes are perfectly correlated. In practice, any value between 0.9 and 1.0 is considered good correlation. If the coefficient is any value less than 1.0, then there is soe degree of inconsistency, proportional to the value of the factor, between the shapes. This can be caused by an inaccurate finite eleent odel, as described earlier, or the presence of noise and nonlinearities in the easured data. It should be noted that in order for these coparisons to have a reasonable degree of accuracy, it is very iportant that coordinate locations in the odal test coincide with coordinates in the finite eleent esh. Figure 5.4 Nuerical coparison of ode shapes MSF Proportionality Constant How were the odes scaled? MAC Correlation Coefficient Mode 1 Mode 2 There are other nuerical ethods for coparing the easured and predicted odal paraeters of a structure. One such technique, called direct syste paraeter identification, is the derivation of a physical odel of a structure fro easured force and response data. However, techniques such as this are beyond the scope of this text and can be found in technical articles about odal analysis. [ M ] [ I ] Unit Modal Mass Are the odes the sae ode? 51

52 Substructure Coupling Analysis In analyses involving large structures or structures with any coponents it ay not be feasible to asseble a finite eleent odel of the entire structure. The tie involved in building the odel ay be unacceptable and the odel ay contain ore degrees of freedo than the coputer can handle. As a result, it ay be necessary to eploy a odeling reduction ethod known as substructure coupling or coponent ode synthesis illustrated in Figure 5.5. Figure 5.5 Substructure coupling analysis Airplane A odal odel of a coponent for substructure coupling ust contain the odal ass, stiffness and daping factors along with the odal atrix. The odal atrix of a structure is siply a atrix whose coluns are coprised of the respective odes of the structure. In the special case where the ode shapes have been scaled to unity odal ass, the odal odel reduces to the frequencies, daping and ode shapes. Substructure coupling involves the division of the structure into various coponents, odeling these coponents for their individual dynaics and then cobining these individual results into one odel to analyze the dynaics of the coplete structure. These coponent odels can take on several different atheatical fors each of which has a particular usefulness. Coon odels include odal odels and physical odels fro a finite eleent analysis, odal odels fro a odal test, rigid body odels and physical springs and dapers. The coponent odels are cobined through a transforation that relates their dynaics at the interfaces. The results fro the analysis of the coplete structure can then be correlated with equivalent odal test results in the sae anner as described earlier. Coponent Analysis Fuselage Wing Engine Wing Fuselage Engine 52

53 Forced Response Siulation One of the ajor design goals for ost engineering analyses is to be able to predict syste responses to actual operating forces. This can enable engineers to ultiately find optial solutions to troublesoe noise or vibration probles. This technique, illustrated in Figure 5.6, is coonly called forced response siulation or forced response prediction. Forces can be specified for any degree of freedo in the odal odel and displaceents, velocities or accelerations can be predicted for any degree of freedo. Figure 5.6 Forced response siulation Force Inputs Syste Model Engine RPM Road Surface ω, ζ, φ Syste Response Ride Motion X 53

54 Bibliography Robert D.Cook Concepts and Applications of Finite Eleent Analysis John Wiley & Sons 1981 Roy R. Craig, Jr. Structural Dynaics: An Introduction to Coputer Methods John Wiley & Sons 1981 Clare D. McGille, George R. Cooper Continuous and Discrete Signal and Syste Analysis Holt, Rinehart, and Winston, Inc Francis S. Tse, Ivan E. Morse, Rolland T. Hinkle Mechanical Vibrations: Theory and Applications Allyn and Bacon, Inc D. Brown, G.Carbon, K. Rasey Survey of Excitation Techniques Applicable to Autootive Structures Society of Autootive Engineers, # R. J. Alleang, D. L. Brown A Correlation Coefficient for Modal Vector Analysis Proceedings of the 1st International Modal Analysis Conference 1982 R. J. Alleang, D. L. Brown, R. W. Rost Multiple Input Estiation of Frequency Response Functions Proceedings of the 2nd International Modal Analysis Conference 1984 Dave Corelli, David L. Brown Ipact Testing Considerations Proceedings of the 2nd International Modal Analysis Conference 1984 Nor Olsen Excitation Functions for Structural Frequency Response Measureents Proceedings of the 2nd International Modal Analysis Conference 1984 John R. Crowley, G. Thoas Rocklin, Albert L. Klosteran, Havard Vold Direct Structural Modification Using Frequency Response Functions Proceedings of the 2nd International Modal Analysis Conference 1984 D. L. Brown, R. J. Alleang, Ray Zieran, M. Mergeay Paraeter Estiation Techniques for Modal Analysis Society of Autootive Engineers, # Richard Jones, Yuji Kobayashi Global Paraeter Estiation Using Rational Fraction Polynoials Proceedings of the 4th International Modal Analysis Conference 1986 Richard Jones, Kathleen Iberle Structural Modification: A Coparison of Techniques Proceedings of the 4th International Modal Analysis Conference

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