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NORMAL-MODE FORCE APPROPRIATION—THEORY AND APPLICATION
Mechanical Systems and Signal Processing (1999) 13(2), 217—240 Article No. mssp.1998.1214 Available online at http://www.idealibrary.com on
NORMAL-MODE FORCE APPROPRIATION— THEORY AND APPLICATION J. R. WRIGHT, J. E. COOPER AND M. J. DESFORGES Dynamics and Control Research Group, School of Engineering, University of Manchester, Manchester M13 9PL, U.K. Normal-mode force appropriation is a method of physically exciting and measuring the undamped natural frequencies and normal-mode shapes of a structure which is distinct from the more common phase separation approaches. In this paper, the theory of normal modes and force appropriation is reviewed and a comparison made of a number of common force appropriation techniques. Both simulated and experimental data are used to highlight the relative merits of the different approaches, and comparisons to phase separation results are presented. Further advancements and applications of normal-mode testing—to non-proportionally damped structures, non-linear structures and consideration of the optimal exciter placement problem—are also discussed. 1999 Academic Press
1. INTRODUCTION
Normal-mode force appropriation (or phase resonance testing) [1] is a method of extracting the undamped natural frequencies and normal-mode shapes of a structure. By contrast to phase separation (curve-fitting) approaches which are largely mathematical, phase resonance testing is a physical technique in which the individual modes of the system are excited in turn and the mode shapes measured directly at each resonance condition. Existing force appropriation techniques may be divided into direct and iterative approaches. Iterative force appropriation methods [2, 3] were developed but are rarely used as they tend to be time-consuming and difficult to apply. The requirement of a suitable initial force vector is a limitation and convergence problems have been experienced. By comparison, direct methods have been found to be straightforward to apply, reliable and efficient. As a result, only direct force appropriation methods will be considered in this paper. The direct appropriation of the normal modes of a structure is essentially a three-stage procedure. Firstly, frequency response function (FRF) matrices are measured for multiple input and response positions, using random or stepped sine excitation. The undamped natural frequencies of the normal modes are then estimated using one of a number of matrix-based approaches [4] which are described in a later section. The force appropriation methods derive a monophase vector of excitation forces for each normal mode which will, in theory, generate a single-mode monophase response in the structure at the undamped natural frequency. The final stage of the force appropriation procedure is to apply the force vectors corresponding to each mode to the structure using a sinusoid of the relevant undamped natural frequency, and to directly measure the normal-mode shape. A normal mode is said to be ‘tuned’ when the response across the structure at the undamped natural frequency is in monophase, and in quadrature to the excitation, to within some specified level of accuracy. Normal-mode force appropriation has, traditionally, been used in the aerospace industry for ground vibration testing [5, 6], and in other areas where accurate normal-mode 0888—3270/99/020217#24 $30.00/0
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estimates are required such as when direct comparisons are to be made between appropriated normal-mode responses and the results of finite-element analyses. These techniques are of most benefit in situations of significant modal overlap where modes are close in frequency and/or coupled by damping forces. Phase separation techniques may have trouble identifying and separating these types of mode and other approaches are then needed to convert the estimated complex modes into normal modes. In this paper, an overview and comparison will be presented of a number of normal-mode force appropriation algorithms. The ability of such approaches to appropriate accurate normal modes in situations where phase separation techniques fail will be demonstrated. Finally, current and future developments and uses of force appropriation methods will be discussed.
2. COMPARISON OF FORCE APPROPRIATION METHODS
2.1. FORCE APPROPRIATION Consider a linear system subject to e monophase (0 or 180° phase) sinusoidal excitation forces at a frequency u, giving an (e;1) force vector + f ,. In the steady state, the complex displacement response at the r response measurement positions on the structure is +x,"[A(u)#iB(u)]+ f ,
(1)
where +x, is the (r;1) vector of responses, while [A] and [B] are the real and imaginary parts of the frequency response function (FRF) matrix respectively, relating responses at r positions to excitation at e positions. The jth undamped normal mode is excited at the corresponding undamped natural frequency (u ) when the response of the structure is in monophase, and in quadrature H (90° phase) with the excitation (for displacement or acceleration data) [7]. At this condition, the real part of the response will be zero while the imaginary part corresponds to the undamped normal-mode shape, + , . Thus, for undamped normal mode j, H Re+x,"[A]+ f , "0 H
(2)
Im+x,"[B]+ f , "+ , H H
(3)
where + f , is the appropriated force vector for mode j. This is known as the ‘phase H resonance condition’. It should be noted that a variety of normalisation methods may be applied to the mode shape. In principle, the force vector derived for a particular normal mode will excite only that mode. For proportionality damped systems, therefore, the force vector will be derived from a modal force input to the mode of interest and no contribution to the other modes. For non-proportionally damped systems, however, modal force contributions are also included for any coupled modes in order to cancel the unwanted modal responses due to the modal cross-damping terms. 2.2. FORCE APPROPRIATION METHODS A number of direct methods have been developed for the estimation of the normal-mode frequencies and appropriated force vectors of a system from measured FRF data. In all of these methods, an important parameter is the number of effective degrees of freedom (dof ) at any point in the frequency range of interest, n*. If the number of exciters used to excite
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a normal mode is less than the number of effective dof (e(n*) [8] at the undamped natural frequency, then the mode may not be accurately appropriated. Conversely, if the number of exciters is greater than the number of effective dof (e'n*), certain direct force appropriation methods will degrade and eventually fail due to rank deficiency of the FRF matrices [9]. In practice, the number of effective dof is an imprecise quantity which is generally unknown prior to testing. Direct force appropriation methods may be divided into three categories: methods that operate on square FRF matrices (r"e); methods that operate on rectangular FRF matrices (r'e); methods that operate on rectangular FRF matrices and utilise a rank-reduction technique [4]. The most common direct force appropriation techniques are discussed below. 2.2.1. Square FRF matrix methods For square FRF matrices (r"e), an exact non-trivial solution of equation (3) is possible. The Asher method [10] uses the determinant of A to obtain natural frequencies and then solves equation (3) directly using the adjoint of [A] or the Gauss—Seidel method. However, this process is not ideal as the force vector + f , may be shown [4] to be trivial if [A] is not H exactly singular. As a result, an eigenvalue solution is preferable. The Modified Asher method [4] thus solves [A]+ f ,"j+ f ,.
(4)
An alternative approach, the Traill—Nash method [11], uses the general eigenvalue of the form [A]+ f ,"j[B]+ f ,.
(5)
The undamped natural frequencies are then identified by zero crossings of the eigenvalues j. Note that the eigenvalues j behave differently for each of the methods discussed. The corresponding eigenvectors gives the appropriation force vectors for each mode. In general, the square FRF matrix methods have the inherent limitation that the phase resonance condition can be sought exactly for a limited number of response locations on the structure, fixed by the number of exciters. If the number of exciters is increased in order to allow more responses to be measured, the methods invariably degrade or fail due to the number of exciters exceeding the effective dof and subsequent rank deficiency of the FRF matrices. 2.2.2. Rectangular FRF matrix methods For rectangular FRF matrices, where m'e, an exact solution is not available. Instead, the real part of the response is minimised across all of the response measurements, with different specific cost functions being used in each of the methods. The Extended Asher method [12] minimises the sum of the squares of the real part of the response with respect to the force vector, leading to the eigenvalue problem [A]2[A]+ f ,"j+ f ,.
(6)
In the original formulation [12], it is stated that the eigenvalues j drop to zero at undamped natural frequencies. In fact, zero eigenvalues will only be generated if a quadrature response is realised on all r responses simultaneously [4]. Instead, minima in the Extended Asher eigenvalue trace should be sought. By comparison, the Multivariate Mode Indicator Function (MMIF) [13] minimises the ratio of the sum of the squares of the real part of the response to the sum of the squares of
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the moduli, leading to [A]2[A]+ f ,"j([A]2[A]#[B]2[B])+ f ,.
(7)
A weighting matrix (possibly an estimated mass matrix) may also be included in the MMIF formulation. The undamped natural frequencies are thus identified by minima of the eigenvalues j. The rectangular FRF matrix methods allow the phase resonance condition to be sought approximately at many points on the test structure and are not limited to coincident exciter and response positions. However, when the number of exciters exceeds the number of effective dof, these methods will tend to degrade or fail due to rank deficiency of the FRF matrices [9]. Methods have therefore been formulated which incorporate rank reduction techniques and may be applied in cases where the number of exciters exceeds the number of effective dof. 2.2.3. Rectangular FRF matrix methods with rank reduction In the Modified Multivariate Mode Indicator Function approach [14], the eigenvalues of [B] are calculated as an indication of the rank (n*) of the FRF matrix. A rank-reduction technique is then employed to reduce the size of the eigenvalue problem and so generate principal force vectors. These may be transformed back to physical force vectors for the purpose of tuning. The Juang—Wright method [15] attempts to minimise the real part of the response, as for the Extended Asher method, whilst simultaneously maximising the imaginary response. The SVD of [A] is first carried out and the decomposition partitioned according to the effective rank of [A] at a given resonance. The singular values are also used to identify the undamped natural frequencies. A second SVD is then carried out which yields the appropriated force vectors. 2.3. MODAL PURITY INDICATOR (MPI) The quality of a tuned normal mode can be judged by calculating the normal-mode purity D [2], defined as Re+x,2+x, D"1! . +x,2+x,
(8)
This expression returns a value between zero and unity which relates the degree to which the responses deviate from being in quadrature with the applied appropriated force vector; a value of unity indicates a perfect undamped normal mode, while a value greater than 0.90 is usually deemed satisfactory. 2.4. USE OF FORCE APPROPRIATION METHODS Each of the methods discussed may be solved at each measured frequency point to give eigenvalues (j , j"1,2, e) or singular values, and corresponding eigenvectors or singular H vectors (+ f , , j"1,2, e). The eigenvalue (or singular value) traces are then used to indicate H the undamped natural frequencies of the system, either by maxima, minima or zero crossings depending upon the method employed, whilst the corresponding eigenvectors (or singular vectors) represent the appropriated force vectors required to tune the individual normal modes. The appropriated force vector can then be applied sinusoidally to the structure at the estimated undamped natural frequency and the frequency and/or force pattern adjusted
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until the undamped normal mode has been excited to some predetermined level of accuracy. This process is referred to as ‘hard tuning’. Alternatively, the appropriated force vector can be ‘applied’ to the FRF at the estimated undamped natural frequency [using equation (1)] to simulate the normal-mode tuning process and so estimate the resulting response and mode shape numerically. This process is referred to as ‘soft tuning’. 2.5. COMPARISON STUDIES A number of comparison studies have been carried out, including references [1, 16, 17], each using a limited number of the methods discussed. Wider-ranging comparisons have also been undertaken of both direct and iterative force appropriation methods using both simulated data [4], and experimental data from a benchmark structure [18]. In general, it has been found that the rectangular FRF matrix methods are superior to the square FRF matrix methods as the phase resonance criterion may be sought over the entire structure rather than just at a limited number of points. In cases where the number of exciters exceeds the number of effective dof, the rank-reduction procedures are of benefit as the rectangular matrix methods appear to be more sensitive to noise. The MMIF and Modified MMIF approaches are the easiest to interpret for the identification of the undamped natural frequencies because of the form of the eigenvalue signature, and also because they return an eigenvalue between 0 and 1 which can be used as an early indication of the modal purity that may be expected from the tuned mode. A brief comparison of the methods described above is now presented to illustrate the use of the mode indicator functions resulting from each approach. FRF data was simulated from a mathematical model of a proportionally damped free—free perspex plate with six rigid-body modes and 10 flexural modes in the frequency range 40—270 Hz. Four modes are present in the frequency range of analysis, 80—150 Hz, including two close modes around 110 Hz; 4% modal damping was included for each mode in the model. Excitation at the four corners has been used with 24 response positions across the plate (i.e. e"4, r"24). For the square FRF matrix techniques, only the corner responses have been included, although there is no necessity that the chosen excitation and response positions coincide for these methods. Figures 1—7 show the Asher determinant, the eigenvalues derived from the Modified Asher, Traill—Nash, Extended Asher, MMIF and Modified MMIF approaches and the singular values of the Juang—Wright approach. The undamped natural frequencies of the four modes in the chosen frequency range are indicated by vertical dashed lines, and given in Table 1, together with the other frequencies in the model. TABLE 1 Modal parameters of mathematical model of a perspex plate Mode
Frequency/(Hz)
Damping ratio/(%)
1 2 3 4 5 6 7 8 9 10
40.193 40.737 92.013 109.377 114.084 142.831 165.841 201.714 227.745 269.171
4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
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It is clear from Figs 1—3 that the square FRF matrix methods are not particularly successful in identifying the presence of normal modes. In all three cases, the mode 4 at 109 Hz has been missed. The reason that the traces derived from the square FRF matrix approaches do not cross zero at the undamped natural frequency of mode 4 is that the chosen exciter combination (four corners) is not capable of exactly appropriating this mode at the chosen responses, i.e. [A]+ f ,"0 is not satisfied at 109 Hz. It is certain that the use of fewer exciters would further degrade the results. Although it is possible that an alternative subset of the response positions may allow mode 4 to be identified, the need for an exact solution is a major shortfall of the square FRF matrix methods. The rectangular FRF matrix approaches overcome this problem by considering all of the responses simultaneously and thus identifying those frequencies at which the phase resonance condition is most closely satisfied on average over the structure. The Extended Asher and MMIF approaches both indicate normal modes by minima in the eigenvalues. It is clear from Figs 4 and 5 that the MMIF is easier to interpret as all of the minima occur in the primary MMIF; this will always be the case unless very close modes are present when the secondary and tertiary eigenvalues are seen to drop towards zero. The MMIF indicates the mode at 109 Hz by a clear minimum, although the fact that the eigenvalue does not drop to zero implies that the mode will not be accurately appropriated by excitation at the corner positions. The use of fewer exciters would be expected to cause the eigenvalue troughs corresponding to the other modes not to drop to as low a value, although a good indication of the undamped natural frequencies should still be obtained. By comparison, the mode at 109 Hz could easily be missed on the Extended Asher plot, particularly if the original FRF measurements were noisy. Both the the methods with rank reduction indicate the four modes present in the frequency range 80—150 Hz. The Juang—Wright approach has a very similar behaviour to the Extended Asher method, though the troughs are sharper because the singular values are essentially the square root of the eigenvalues. Again, it is arguable that the Modified MMIF
Figure 1. Asher determinant.
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Figure 2. Modified Asher eigenvalues.
approach is the easier to interpret and, in the authors’ experience, this is generally the case. Currently, the MMIF and the Modified MMIF are the direct force appropriation methods most commonly used in practice, although the Asher method has been implemented on some commercial systems. The MMIF methods will therefore be used for the remainder of this paper (Figs 6 and 7).
Figure 3. Trail—Nash eigenvalues.
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Figure 4. Extended Asher eigenvalues.
3. PHASE RESONANCE VS. PHASE SEPARATION
3.1. COMPLEX AND NORMAL MODES The key difference between force appropriation and the traditional curve-fitting techniques is in the form of the identified parameters. Phase separation methods, in general,
Figure 5. Multivariate Mode Indicator Function eigenvalues.
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Figure 6. Modified Multivariate Mode Indicator Function eigenvalues.
yield complex/damped mode shapes plus frequencies and damping ratios derived from the complex eigenvalues. In contrast, phase-resonance methods permit direct measurement of the undamped normal modes and the corresponding undamped natural frequencies; the modal damping ratio is generally estimated using a half-power points method from the Nyquist plot generated by a mini-sweep around the natural frequency.
Figure 7. Juang—Wright singular values.
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When a system exhibits proportional damping, the complex modes are related to the normal modes by a simple transformation. In theory, both approaches will yield the same undamped natural frequency and damping estimates, which will be correct. However, in systems where the damping is non-proportional, the modal damping matrix will be non-diagonal and some of the resulting modes are coupled by the damping forces. The complex/damped modes which result from the solution of the first-order eigenvalue problem differ from the undamped normal modes and there is no simple transformation between them. A variety of methods have been proposed although none are entirely satisfactory. Furthermore, the derivation of frequency and damping values from the complex eigenvalues is somewhat contrived as the underlying theory assumes a proportional damping distribution. As a result, the estimated frequency does not equal the undamped natural frequency and the concept of modal damping is inappropriate for non-proportionally damped systems. Results derived from such systems can thus be misleading. As the phase resonance approach has no limitations regarding the damping distribution of the system, force appropriation methods may equally be applied to nonproportionally damped systems without error. A force vector is again derived which allows a normal mode to be excited at its undamped natural frequency and no additional approximations are required. In practice, excitation must be provided at a sufficient number of correctly chosen positions if the normal modes of the non-proportionally damped systems are to be well appropriated. Such procedures have been implemented experimentally on significantly non-proportionally damped structures [19] with success. The importance of exciter placement to achieve well-appropriated modes was again highlighted by this work. 3.2. BENCHMARK PLATE STRUCTURES Some experimental force appropriation results are now presented from two real benchmark structures—a perspex plate [18], and an aluminium plate with a significant level of non-proportional damping [19, 20]. 3.2.1. Proportionally damped perspex plate Modal analyses were performed on an experimental perspex plate equivalent to that modelled mathematically above [18]. An LMS-DIFA SCADAS data acquisition system was employed throughout these analyses. In this case, it is reasonable to assume a proportional modal damping distribution. As for the simulated case, 24 response positions were used with excitation applied at the corners of the plate. Figure 8 shows one of the drive point FRFs measured during the modal test with multiple uncorrelated random excitation. These experimental results are of high quality, partly helped by the fairly high level of damping present. Figure 9 shows the corresponding MMIF, which correctly identifies the presence of eight modes in the frequency range 20—220 Hz. The true number of modes is far from clear if only the FRF amplitude is inspected; there would appear to be only six modes although the MMIF reveals two pairs of close modes around 39 and 110 Hz. Indeed, use of a mode indicator function such as the MMIF should be encouraged even if a phase separation analysis is to be used, to identify the correct number of modes in a given frequency range. The tuned normal-mode results and the results of a phase separation approach, the Least Squares Complex Exponential (LSCE) method, are given in Table 2. It can be seen from Table 2 that six of the modes were tuned to a high degree of accuracy whilst modes 4 and 8 could not be adequately tuned with the chosen exciter combination.
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Figure 8. Experimental drive point FRF of perspex plate.
Figures 10—17 show the mode shapes and phase scatter plots of the tuned and LSCE estimates of the first two modes which are almost coincident in frequency. The identification and separation of close modes provided much of the impetus to the development of force appropriation techniques, an area in which phase separation approaches have sometimes
Figure 9. Experimental MMIF eigenvalues of perspex plate.
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TABLE 2 Results of experimental phase separation/phase resonance analysis of a perspex plate Phase separation (LSCE)
Phase resonance
Mode
Frequency/(Hz)
Frequency/(Hz)
MPI
1 2 3 4 5 6 7 8
38.97 39.01 90.97 100.29 114.77 141.26 165.66 206.72
38.93 38.99 90.87 100.76 114.53 141.02 165.43 206.78
0.979 0.966 0.974 0.690 0.949 0.959 0.949 0.490
been seen to fail or to give significantly complex modal results. The level of complexity of the phase separation estimates is apparent in Figs 10—17 whilst the tuned responses are very close to pure normal modes. The complexity could be due to some non-proportional damping or to the phenomenon mentioned in Section 3.3 below. Commonly, comparisons are made between the results of modal analyses and finiteelement (FE) predictions. Indeed, FE updating may even be the sole reason for performing an experimental modal test. Generally, FE analyses generate pure normal modes which may thus be directly compared to the appropriated modes resulting from force appropriation. The comparison to the results of a phase separation analysis is less straightforward, particularly for the case of non-proportionally damped structures. 3.2.2. Non-proportionally damped aluminium plate Further modal tests were performed using an experimental aluminium plate with large dash-pot oil dampers positioned at opposite corners in order to introduce a significant level of non-proportional damping into the structure. Again 24 response measurements and four corner exciters were used. The experimental rig is shown in Fig. 18. Figures 19 and 20 show a drive point FRF and the MMIF eigenvalue plot resulting from the modal test of the non-proportionally damped aluminium plate. It would be very difficult
Figure 10. First mode of perspex plate, tuned normal-mode shape.
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Figure 11. First mode of perspex plate, tuned phase scatter plot.
to judge the number and position of the modes from the FRF amplitude alone (arguably five modes), while they are clearly identified by the MMIF eigenvalues (actually seven modes). The separation of the primary and secondary MMIF eigenvalues between the first two modal frequencies is indicative of modal coupling due to a high level of non-proportional damping [20]. Even for the simple perspex plate, which was assumed to be proportionally damped, some non-proportionality is apparent between modes 4 and 5 in Fig. 9. This effect can be simulated on a simple two-dof system where it can easily be shown that the two MMIF eigenvalues may only coincide for the case of a proportionally damped system.
Figure 12. First mode of perspex plate, LSCE mode shape.
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Figure 13. First mode of perspex plate, LSCE phase scatter plot.
Figures 21 and 22 show the hard-tuned mode shape and phase scatter plot of the second mode, whilst Figs 23 and 24 show similar plots resulting from a phase separation approach, the Frequency Domain Direct Parameter Identification (FDPI) method. It can be seen that the complexity of the structure is now such that comparison of phase separation results with FE would prove very difficult. The tuned responses, however, are very close to the underlying normal modes of the structure. Arguably the best method for transforming from complex to real modes is to perform a soft-tuning force appropriation on the synthesised FRF data derived from the curve-fitted results. 3.3. SMOOTHNESS OF MODE SHAPES When a normal mode is physically excited using a monophase force vector, the structure acts as a spatial filter, so ensuring that the final mode shape has a smooth appearance with no abrupt changes or discontinuities. However, when a phase separation method is used for multiple input/output data, the resulting global frequency and damping estimates are
Figure 14. Second mode of perspex plate, tuned normal mode shape.
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Figure 15. Second mode of perspex plate, tuned phase scatter plot.
a compromise across all of the measurement positions. There is thus a possibility of some errors occurring in the derived mode shapes. In addition, modelling of complicated upper and lower residual effects by simple additional terms or extra modes may cause different levels of error at each measurement position. As a result, the final plotted mode shape may appear less smooth than the corresponding appropriated normal-mode shape. 3.4. TEST TIME The implementation of force appropriation techniques and the tuning of a set of modes of a structure is clearly a sizeable task; it would invariably be quicker to use a phase separation approach to curve-fit the multiple input/output data. However, given the time taken to set
Figure 16. Second mode of perspex plate, LSCE mode shape.
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Figure 17. Second mode of perspex plate, LSCE phase scatter plot.
up a modal test and perform pretest checks, the additional time requirements may seem less significant, particularly if improved parameter estimates are obtained. A compromise is to use phase separation techniques for the majority of modes whilst reserving phase resonance methods for particularly close or complex modes.
Figure 18. Aluminium plate with added non-proportional damping.
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Figure 19. Experimental drive point FRF of non-proportionally damped aluminium plate.
4. RECENT ADVANCES IN PHASE RESONANCE TESTING
4.1. OPTIMAL EXCITER LOCATION The number and positioning of exciters has been shown to be critical to the successful application of normal-mode force appropriation techniques. However, the choice of exciter
Figure 20. Experimental MMIF of non-proportionally damped aluminium plate.
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Figure 21. Second mode of non-proportionally damped aluminium plate, tuned normal mode shape.
configuration is traditionally left to the judgement of the test engineer, with alterations being made on a trial and error basis. Although a number of techniques are available for the positioning of exciters for phase separation testing, these methods are unsuitable for phase resonance testing as they all aim to maximise the energy input into the target modes simultaneously, rather than to excite a single mode whilst suppressing others. One approach to optimal exciter placement by Niedbal [21], applicable to phase resonance testing, makes use of the condition number of the modal matrix. It was proposed that the optimum subset of excitation positions for a given number of target modes should correspond to the modal matrix with the lowest condition number. However, there appears to be no theoretical reason that the exciter configuration chosen according to this criterion should necessarily excite the target normal modes in an optimum way. No account is taken of the damping levels in this technique, and proportional damping is assumed throughout.
Figure 22. Second mode of non-proportionally damped aluminium plate, tuned phase scatter plot.
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Figure 23. Second mode of non-proportionally damped aluminium plate, LSCE mode shape.
Recently, a method has been developed [22] which uses an a priori model of a test structure to predict the optimum exciter configuration for force appropriation of a set of normal modes. Two measures of combined normal mode purity (sum and product) may be defined which are suitable for different test situations. For the case of the proportionally damped perspex plate described above, the results of an a priori FE model have been validated by a normal mode test. Both exhaustive search and Genetic Algorithm approaches are considered. In cases when an a priori model of a structure is not available, it has been shown [23] that the mathematical model derived from a phase separation analysis may also be used for optical exciter placement prediction. This procedure particularly lends itself to mid-test
Figure 24. Second mode of non-proportionally damped aluminium plate, LSCE phase scatter plot.
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exciter optimisation in cases when some of the modes of a structure may not be well excited by the initial choice of exciter positions. In both cases, non-proportionally damped systems may be considered by estimating the positions and magnitudes of representative discrete dampers. The two methods of optimal exciter selection described above have been compared for excitation of the first four modes of the perspex plate model. All combinations of four exciters out of a possible 24 were considered. Figure 25 shows the condition number of the first approach [21] plotted against the average modal purity value (D ) of the second approach [22]. It is clear that the exciter configurations with the highest values of D are spread across the range of condition numbers. Thus, whilst a fairly good exciter configuration could be selected on the basis of the condition number, equally good alternatives would be discarded. The optimal configuration based on D corresponds to a condition number of only 0.077 and would thus not be considered under Niedbal’s method. Conversely, some configurations which lie in the top few per cent in terms of condition number (e.g. values above 0.3), have relatively low D values and are thus suboptimal in terms of normal-mode force appropriation. Figure 26 shows the exciter configuration [8, 12, 15, 16], chosen for optimal excitation of the first eight modes of the proportionally damped perspex plate. Using this configuration in practice, each of the first eight modes was hard-tuned to a modal purity above 0.92, with an average modal purity value of 0.941. These values represent a significant improvement over the results seen in Table 2 which were measured using excitation at the corner positions and gave an average modal purity of only 0.870.
Figure 25. Comparison of optimal excitation approaches for the perspex plate model.
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Figure 26. Optimal exciter configuration for the perspex plate model.
Finally, a method has been proposed [24] which seeks optimal exciter combinations for each normal mode of a system. The modal purity indicator, as defined in equation (8) is used to indicate the presence of the normal modes of the system, and is also used as the optimisation criterion. A Genetic Algorithm is used to perform the optimisation. Again, proportional damping is assumed, although the method may be extended to the situation where an a priori estimate of a non-proportional damping matrix is available. As a different set of exciter positions is found for each mode, this approach will prove impractical in real-test situations. 4.2. ESTIMATION OF MODAL DAMPING AND CROSS-DAMPING TERMS It has recently been demonstrated [25] that force appropriation may be used to estimate the direct modal damping and cross-damping terms for a general system. In this technique, a single normal mode is tuned and then the excitation is removed. For the case of a proportionally damped system, the modal damping may be found from a time-domain curve-fit to the resulting single-mode decays at each measurement position. For a nonproportionally damped system, the modal cross-damping terms cause other coupled modes to respond once the appropriated forces are removed, resulting in a multi-modal decay. A curve-fit to these responses allows both direct and cross-damping terms to be estimated. These methods have been validated on a benchmark aluminium plate structure. 4.3. APPLICATION TO NON-LINEAR STRUCTURES Recent work [26] has been directed at the application of force appropriation techniques to non-linear systems. The concept of normal modes is not directly applicable to non-linear systems, and it has been shown on simulated non-linear systems that the appropriated force vectors derived from the Modified MMIF do not significantly reduce the contribution of the coupled mode(s) for the non-linear case. In other words, classical force appropriation can yield a single-mode response at low excitation levels, but other modes respond at higher
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Figure 27. Modal Restoring force surfaces of a 2 DoF benchmark non-linear structure using FANS approach.
excitation levels where non-linear modal couplings are active. However, it has been demonstrated theoretically and experimentally that the linear mode shape of a single mode may be excited in the non-linear region for a multiple-dof non-linear system through the use of optimised force vectors with both fundamental and harmonic sinusoidal terms included in the multipoint force vector. The resulting responses are then suitable for analysis by single dof non-linear techniques such as the restoring force surface and the sensitivity approach which cannot be easily applied to coupled multiple-dof responses. An extension of the approach could yield important non-linear cross-coupling terms. Figure 27 shows the single-dof modal restoring force surfaces of the modes of a benchmark two-dof structure with a cubic stiffness non-linearity, appropriated using the force appropriation for non-linear systems (FANS) approach described in [26]. A comparison of this new approach with a classical restoring force method is presented in Table 3 . It can be seen that similar modal parameter estimates are obtained using the FANS technique as for the classical method. However, it is proposed that the FANS approach may be extended to real non-linear systems of arbitrary dimension, whereas the classical restoring force method begins to become impractical for systems with more than two dof. The non-linear identification could be restricted to those modes which demonstrate non-linear behaviour, and a classical modal model, with additional non-linear terms, may thus be developed. TABLE 3 Comparison of Restoring Force and FANS approaches on benchmark 2 DOF non-linear structure Restoring Force Method Modal parameter Mass, m/(kg) Damping, c/(Ns/m) Linear stiffness, k/(N/m) Cubic stiffness, b/(N/m)
Mode 1 2.60 10.11 4.87;10 3.83;10
FANS approach
Mode 2
Mode 1
3.06 9.49 6.49;10 1.73;10
3.01 8.73 4.97;10 6.54;10
Mode 2 2.84 10.90 6.54;10 2.29;10
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5. CONCLUSIONS
An overview of normal-mode force appropriation techniques has been presented. It has been shown that the rectangular FRF matrix methods are superior to the square FRF matrix methods in the identification of the normal modes of a structure. In situations where the number of exciters exceeds the number of effective dof, rank reduction techniques are of value. The MMIF is, arguably, the clearest of the methods to interpret, and is the most commonly used of the presented techniques. The advantages of phase resonance approaches over phase separation—accuracy, ease of comparison to FE results and smoothness of mode shapes—have been described along with the disadvantages—the requirement of extra analysis and testing time and the need for adequate excitation. The application of normal-mode force appropriation techniques to both proportionally and non-proportionally damped structures has been discussed. It has also been shown that phase resonance techniques may offer a way forward for non-linear identification with a modal model. Some approaches to optimal exciter placement have also been reviewed, and a demonstration of the need for good exciter positioning shown. It is the belief of the authors that phase resonance techniques should be used alongside, rather than instead of, phase separation analysis. In many cases, curve-fitted mode shapes are adequate and the time required for force appropriation may prove prohibitive. However, in cases of significant modal overlap, non-proportional damping or the need to correlate closely with FE analysis, normal-mode force appropriation in a valuable element in the modal analyst’s toolbox.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of LMS International and LMS-DIFA Measuring Systems in the provision of software and hardware.
REFERENCES 1. D. OTTE, H. VAN DER AUWERAER, J. DEBILLE and J. LEURIDAN 1992 Proceedings of the 17th International Seminar on Modal Analysis, 643—658. Enhanced force vector appropriation methods for normal mode testing. 2. E. BREITBACH 1973 Proceedings of the 3rd ESRO ¹esting Symposium, 519—528. A semi-automatic modal survey test technique for complex aircraft and spacecraft structures. 3. G. MOROSOW and R. S. AYRE 1978 Sound and »ibration Bulletin 48, 39—48. Force apportioning for modal vibration testing using incomplete excitation. 4. K. ALEXIOU and J. R. WRIGHT 1993 Modal Analysis 8(2), 119—136. Comparison of some multipoint force appropriation methods. 5. R. D. BRILLHART and D. L. HUNT 1992 Proceedings of the 10th International Modal Analysis Conference. Multiple input excitation methods for the C-17 transport aircraft GVT. 6. M. DEGENER 1995 Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, 70.1—70.12. Ground vibration testing for validation of large aircraft structural dynamics. 7. B. F. DE VEUBEKE 1956 AGARD Report 39. A variational approach to pure mode excitation based on characteristic phase lag theory. 8. G. W. ASHER 1967 American Institute of Aeronautics and Astronautics Journal 5(4), 822—824. A note on the effective degrees of freedom of a vibrating structure. 9. M. NASH 1988 Proceedings of the 6th International Modal Analysis Conference, 740—744. Use of the multivariate mode indicator function for normal mode determination. 10. G. W. ASHER 1958 Proceedings of the National Specialists Meeting on Dynamics and Aeroelasticity. Journal of Aeronautical Sciences 6, 56—96. A method of normal mode excitation using admittance measurements.
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11. R. W. TRAILL-NASH 1961 Structures and Materials Report 280, Dept. of Supply, Australian Defence Scientific Service, Aeronautical Research ¸aboratories. Some theoretical aspects of resonance testing and proposals for a technique combining experiment and computation. 12. P. IBANEZ 1976 SAE Paper No. 760873, SAE Aerospace Engineering and Manufacturing Meeting. Force appropriation by Extended Asher’s method. 13. R. WILLIAMS, J. CROWLEY and H. VOLD 1986 Proceedings of the 3rd International Modal Analysis Conference, 66—70. The multivariate mode indicator function in modal analysis. 14. M. NASH 1991 Proceedings of the 9th International Modal Analysis Conference, 688—693. A modification of the multivariate mode indicator function employing principal force vectors. 15. J.-N. JUANG and J. R. WRIGHT 1991 Journal of »ibration and Acoustics 113, 176—181. A multipoint force appropriation method based upon a singular value decomposition approach. 16. P. IBANEZ and K. D. BLAKELY 1984 Proceedings of the 2nd International Modal Analysis Conference, 2, 903—907. Automatic force appropriation—a review and suggested improvements. 17. M. RADES 1992 Proceedings of the 17th International Seminar on Modal Analysis, 563—581. A comparison of some mode indicator functions. 18. J. E. COOPER, M. J. HAMILTON and J. R. WRIGHT 1993 Modal Analysis 10(2), 118—130. Experimental evaluation of various normal mode force appropriation methods on a rectangular perspex plate using four exciters. 19. P. S. HOLMES, J. R. WRIGHT and J. E. COOPER 1995 Proceedings of the 15th ASME Conference on Mechanical »ibration and Noise, 753—759. Identification of a significantly non-proportionally damped structure using force appropriation. 20. P. S. HOLMES, J. E. COOPER and J. R. WRIGHT 1996 Proceedings of the 21st International Seminar on Modal Analysis, 1237—1250. Normal mode estimation from non-proportionally damped systems. 21. N. NIEDBAL and E. KLUSOWSKI 1990 Proceedings of the American Institute of Aeronautics and Astronautics Dynamics Specialists Conference, 130—141. Optimal exciter placement and force vector tuning required for experimental modal analysis. 22. P. S. HOLMES, J. R. WRIGHT and J. E. COOPER 1996 Proceedings of the 14th International Modal Analysis Conference, 1—7. Optimum exciter placement for normal mode force appropriation using an a priori model. 23. P. S. HOLMES, J. R. WRIGHT and J. E. COOPER 1996 Proceedings of the 2nd D¹A/NAFEMS International Conference. Mid-test optimum exciter placement for normal mode force appropriation. 24. A. BIFULCO and M. DEGENER 1996 International Conference on Identification in Engineering Systems, 377—387. Theoretical prediction and experimental verification of optimum exciter configurations for modal testing. 25. S. NAYLOR, J. R. WRIGHT and J. E. COOPER 1997 Proceedings of the 15th International Modal Analysis Conference, 137/1—137/7. On the estimation of modal matrices with non-proportional damping. 26. P. A. ATKINS and J. R. WRIGHT 1997 Proceedings of the 6th International Conference on Recent Advances in Structural Dynamics, 2, 1099—1112. Estimating the behaviour of a nonlinear experimental multi degree of freedom system using a force appropriation approach.
APPENDIX A: NOMENCLATURE [A] [B] e +f, j n* r +x, j D D ?T + , u
real part of the FRF matrix imaginary part of the FRF matrix number of exciters vector of forces integer number of effective modes number of response positions vector of responses eigenvalues normal-mode purity average normal-mode purity normal-mode shape frequency
NORMAL-MODE FORCE APPROPRIATION— THEORY AND APPLICATION J. R. WRIGHT, J. E. COOPER AND M. J. DESFORGES Dynamics and Control Research Group, School of Engineering, University of Manchester, Manchester M13 9PL, U.K. Normal-mode force appropriation is a method of physically exciting and measuring the undamped natural frequencies and normal-mode shapes of a structure which is distinct from the more common phase separation approaches. In this paper, the theory of normal modes and force appropriation is reviewed and a comparison made of a number of common force appropriation techniques. Both simulated and experimental data are used to highlight the relative merits of the different approaches, and comparisons to phase separation results are presented. Further advancements and applications of normal-mode testing—to non-proportionally damped structures, non-linear structures and consideration of the optimal exciter placement problem—are also discussed. 1999 Academic Press
1. INTRODUCTION
Normal-mode force appropriation (or phase resonance testing) [1] is a method of extracting the undamped natural frequencies and normal-mode shapes of a structure. By contrast to phase separation (curve-fitting) approaches which are largely mathematical, phase resonance testing is a physical technique in which the individual modes of the system are excited in turn and the mode shapes measured directly at each resonance condition. Existing force appropriation techniques may be divided into direct and iterative approaches. Iterative force appropriation methods [2, 3] were developed but are rarely used as they tend to be time-consuming and difficult to apply. The requirement of a suitable initial force vector is a limitation and convergence problems have been experienced. By comparison, direct methods have been found to be straightforward to apply, reliable and efficient. As a result, only direct force appropriation methods will be considered in this paper. The direct appropriation of the normal modes of a structure is essentially a three-stage procedure. Firstly, frequency response function (FRF) matrices are measured for multiple input and response positions, using random or stepped sine excitation. The undamped natural frequencies of the normal modes are then estimated using one of a number of matrix-based approaches [4] which are described in a later section. The force appropriation methods derive a monophase vector of excitation forces for each normal mode which will, in theory, generate a single-mode monophase response in the structure at the undamped natural frequency. The final stage of the force appropriation procedure is to apply the force vectors corresponding to each mode to the structure using a sinusoid of the relevant undamped natural frequency, and to directly measure the normal-mode shape. A normal mode is said to be ‘tuned’ when the response across the structure at the undamped natural frequency is in monophase, and in quadrature to the excitation, to within some specified level of accuracy. Normal-mode force appropriation has, traditionally, been used in the aerospace industry for ground vibration testing [5, 6], and in other areas where accurate normal-mode 0888—3270/99/020217#24 $30.00/0
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estimates are required such as when direct comparisons are to be made between appropriated normal-mode responses and the results of finite-element analyses. These techniques are of most benefit in situations of significant modal overlap where modes are close in frequency and/or coupled by damping forces. Phase separation techniques may have trouble identifying and separating these types of mode and other approaches are then needed to convert the estimated complex modes into normal modes. In this paper, an overview and comparison will be presented of a number of normal-mode force appropriation algorithms. The ability of such approaches to appropriate accurate normal modes in situations where phase separation techniques fail will be demonstrated. Finally, current and future developments and uses of force appropriation methods will be discussed.
2. COMPARISON OF FORCE APPROPRIATION METHODS
2.1. FORCE APPROPRIATION Consider a linear system subject to e monophase (0 or 180° phase) sinusoidal excitation forces at a frequency u, giving an (e;1) force vector + f ,. In the steady state, the complex displacement response at the r response measurement positions on the structure is +x,"[A(u)#iB(u)]+ f ,
(1)
where +x, is the (r;1) vector of responses, while [A] and [B] are the real and imaginary parts of the frequency response function (FRF) matrix respectively, relating responses at r positions to excitation at e positions. The jth undamped normal mode is excited at the corresponding undamped natural frequency (u ) when the response of the structure is in monophase, and in quadrature H (90° phase) with the excitation (for displacement or acceleration data) [7]. At this condition, the real part of the response will be zero while the imaginary part corresponds to the undamped normal-mode shape, + , . Thus, for undamped normal mode j, H Re+x,"[A]+ f , "0 H
(2)
Im+x,"[B]+ f , "+ , H H
(3)
where + f , is the appropriated force vector for mode j. This is known as the ‘phase H resonance condition’. It should be noted that a variety of normalisation methods may be applied to the mode shape. In principle, the force vector derived for a particular normal mode will excite only that mode. For proportionality damped systems, therefore, the force vector will be derived from a modal force input to the mode of interest and no contribution to the other modes. For non-proportionally damped systems, however, modal force contributions are also included for any coupled modes in order to cancel the unwanted modal responses due to the modal cross-damping terms. 2.2. FORCE APPROPRIATION METHODS A number of direct methods have been developed for the estimation of the normal-mode frequencies and appropriated force vectors of a system from measured FRF data. In all of these methods, an important parameter is the number of effective degrees of freedom (dof ) at any point in the frequency range of interest, n*. If the number of exciters used to excite
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a normal mode is less than the number of effective dof (e(n*) [8] at the undamped natural frequency, then the mode may not be accurately appropriated. Conversely, if the number of exciters is greater than the number of effective dof (e'n*), certain direct force appropriation methods will degrade and eventually fail due to rank deficiency of the FRF matrices [9]. In practice, the number of effective dof is an imprecise quantity which is generally unknown prior to testing. Direct force appropriation methods may be divided into three categories: methods that operate on square FRF matrices (r"e); methods that operate on rectangular FRF matrices (r'e); methods that operate on rectangular FRF matrices and utilise a rank-reduction technique [4]. The most common direct force appropriation techniques are discussed below. 2.2.1. Square FRF matrix methods For square FRF matrices (r"e), an exact non-trivial solution of equation (3) is possible. The Asher method [10] uses the determinant of A to obtain natural frequencies and then solves equation (3) directly using the adjoint of [A] or the Gauss—Seidel method. However, this process is not ideal as the force vector + f , may be shown [4] to be trivial if [A] is not H exactly singular. As a result, an eigenvalue solution is preferable. The Modified Asher method [4] thus solves [A]+ f ,"j+ f ,.
(4)
An alternative approach, the Traill—Nash method [11], uses the general eigenvalue of the form [A]+ f ,"j[B]+ f ,.
(5)
The undamped natural frequencies are then identified by zero crossings of the eigenvalues j. Note that the eigenvalues j behave differently for each of the methods discussed. The corresponding eigenvectors gives the appropriation force vectors for each mode. In general, the square FRF matrix methods have the inherent limitation that the phase resonance condition can be sought exactly for a limited number of response locations on the structure, fixed by the number of exciters. If the number of exciters is increased in order to allow more responses to be measured, the methods invariably degrade or fail due to the number of exciters exceeding the effective dof and subsequent rank deficiency of the FRF matrices. 2.2.2. Rectangular FRF matrix methods For rectangular FRF matrices, where m'e, an exact solution is not available. Instead, the real part of the response is minimised across all of the response measurements, with different specific cost functions being used in each of the methods. The Extended Asher method [12] minimises the sum of the squares of the real part of the response with respect to the force vector, leading to the eigenvalue problem [A]2[A]+ f ,"j+ f ,.
(6)
In the original formulation [12], it is stated that the eigenvalues j drop to zero at undamped natural frequencies. In fact, zero eigenvalues will only be generated if a quadrature response is realised on all r responses simultaneously [4]. Instead, minima in the Extended Asher eigenvalue trace should be sought. By comparison, the Multivariate Mode Indicator Function (MMIF) [13] minimises the ratio of the sum of the squares of the real part of the response to the sum of the squares of
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the moduli, leading to [A]2[A]+ f ,"j([A]2[A]#[B]2[B])+ f ,.
(7)
A weighting matrix (possibly an estimated mass matrix) may also be included in the MMIF formulation. The undamped natural frequencies are thus identified by minima of the eigenvalues j. The rectangular FRF matrix methods allow the phase resonance condition to be sought approximately at many points on the test structure and are not limited to coincident exciter and response positions. However, when the number of exciters exceeds the number of effective dof, these methods will tend to degrade or fail due to rank deficiency of the FRF matrices [9]. Methods have therefore been formulated which incorporate rank reduction techniques and may be applied in cases where the number of exciters exceeds the number of effective dof. 2.2.3. Rectangular FRF matrix methods with rank reduction In the Modified Multivariate Mode Indicator Function approach [14], the eigenvalues of [B] are calculated as an indication of the rank (n*) of the FRF matrix. A rank-reduction technique is then employed to reduce the size of the eigenvalue problem and so generate principal force vectors. These may be transformed back to physical force vectors for the purpose of tuning. The Juang—Wright method [15] attempts to minimise the real part of the response, as for the Extended Asher method, whilst simultaneously maximising the imaginary response. The SVD of [A] is first carried out and the decomposition partitioned according to the effective rank of [A] at a given resonance. The singular values are also used to identify the undamped natural frequencies. A second SVD is then carried out which yields the appropriated force vectors. 2.3. MODAL PURITY INDICATOR (MPI) The quality of a tuned normal mode can be judged by calculating the normal-mode purity D [2], defined as Re+x,2+x, D"1! . +x,2+x,
(8)
This expression returns a value between zero and unity which relates the degree to which the responses deviate from being in quadrature with the applied appropriated force vector; a value of unity indicates a perfect undamped normal mode, while a value greater than 0.90 is usually deemed satisfactory. 2.4. USE OF FORCE APPROPRIATION METHODS Each of the methods discussed may be solved at each measured frequency point to give eigenvalues (j , j"1,2, e) or singular values, and corresponding eigenvectors or singular H vectors (+ f , , j"1,2, e). The eigenvalue (or singular value) traces are then used to indicate H the undamped natural frequencies of the system, either by maxima, minima or zero crossings depending upon the method employed, whilst the corresponding eigenvectors (or singular vectors) represent the appropriated force vectors required to tune the individual normal modes. The appropriated force vector can then be applied sinusoidally to the structure at the estimated undamped natural frequency and the frequency and/or force pattern adjusted
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until the undamped normal mode has been excited to some predetermined level of accuracy. This process is referred to as ‘hard tuning’. Alternatively, the appropriated force vector can be ‘applied’ to the FRF at the estimated undamped natural frequency [using equation (1)] to simulate the normal-mode tuning process and so estimate the resulting response and mode shape numerically. This process is referred to as ‘soft tuning’. 2.5. COMPARISON STUDIES A number of comparison studies have been carried out, including references [1, 16, 17], each using a limited number of the methods discussed. Wider-ranging comparisons have also been undertaken of both direct and iterative force appropriation methods using both simulated data [4], and experimental data from a benchmark structure [18]. In general, it has been found that the rectangular FRF matrix methods are superior to the square FRF matrix methods as the phase resonance criterion may be sought over the entire structure rather than just at a limited number of points. In cases where the number of exciters exceeds the number of effective dof, the rank-reduction procedures are of benefit as the rectangular matrix methods appear to be more sensitive to noise. The MMIF and Modified MMIF approaches are the easiest to interpret for the identification of the undamped natural frequencies because of the form of the eigenvalue signature, and also because they return an eigenvalue between 0 and 1 which can be used as an early indication of the modal purity that may be expected from the tuned mode. A brief comparison of the methods described above is now presented to illustrate the use of the mode indicator functions resulting from each approach. FRF data was simulated from a mathematical model of a proportionally damped free—free perspex plate with six rigid-body modes and 10 flexural modes in the frequency range 40—270 Hz. Four modes are present in the frequency range of analysis, 80—150 Hz, including two close modes around 110 Hz; 4% modal damping was included for each mode in the model. Excitation at the four corners has been used with 24 response positions across the plate (i.e. e"4, r"24). For the square FRF matrix techniques, only the corner responses have been included, although there is no necessity that the chosen excitation and response positions coincide for these methods. Figures 1—7 show the Asher determinant, the eigenvalues derived from the Modified Asher, Traill—Nash, Extended Asher, MMIF and Modified MMIF approaches and the singular values of the Juang—Wright approach. The undamped natural frequencies of the four modes in the chosen frequency range are indicated by vertical dashed lines, and given in Table 1, together with the other frequencies in the model. TABLE 1 Modal parameters of mathematical model of a perspex plate Mode
Frequency/(Hz)
Damping ratio/(%)
1 2 3 4 5 6 7 8 9 10
40.193 40.737 92.013 109.377 114.084 142.831 165.841 201.714 227.745 269.171
4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
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It is clear from Figs 1—3 that the square FRF matrix methods are not particularly successful in identifying the presence of normal modes. In all three cases, the mode 4 at 109 Hz has been missed. The reason that the traces derived from the square FRF matrix approaches do not cross zero at the undamped natural frequency of mode 4 is that the chosen exciter combination (four corners) is not capable of exactly appropriating this mode at the chosen responses, i.e. [A]+ f ,"0 is not satisfied at 109 Hz. It is certain that the use of fewer exciters would further degrade the results. Although it is possible that an alternative subset of the response positions may allow mode 4 to be identified, the need for an exact solution is a major shortfall of the square FRF matrix methods. The rectangular FRF matrix approaches overcome this problem by considering all of the responses simultaneously and thus identifying those frequencies at which the phase resonance condition is most closely satisfied on average over the structure. The Extended Asher and MMIF approaches both indicate normal modes by minima in the eigenvalues. It is clear from Figs 4 and 5 that the MMIF is easier to interpret as all of the minima occur in the primary MMIF; this will always be the case unless very close modes are present when the secondary and tertiary eigenvalues are seen to drop towards zero. The MMIF indicates the mode at 109 Hz by a clear minimum, although the fact that the eigenvalue does not drop to zero implies that the mode will not be accurately appropriated by excitation at the corner positions. The use of fewer exciters would be expected to cause the eigenvalue troughs corresponding to the other modes not to drop to as low a value, although a good indication of the undamped natural frequencies should still be obtained. By comparison, the mode at 109 Hz could easily be missed on the Extended Asher plot, particularly if the original FRF measurements were noisy. Both the the methods with rank reduction indicate the four modes present in the frequency range 80—150 Hz. The Juang—Wright approach has a very similar behaviour to the Extended Asher method, though the troughs are sharper because the singular values are essentially the square root of the eigenvalues. Again, it is arguable that the Modified MMIF
Figure 1. Asher determinant.
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Figure 2. Modified Asher eigenvalues.
approach is the easier to interpret and, in the authors’ experience, this is generally the case. Currently, the MMIF and the Modified MMIF are the direct force appropriation methods most commonly used in practice, although the Asher method has been implemented on some commercial systems. The MMIF methods will therefore be used for the remainder of this paper (Figs 6 and 7).
Figure 3. Trail—Nash eigenvalues.
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Figure 4. Extended Asher eigenvalues.
3. PHASE RESONANCE VS. PHASE SEPARATION
3.1. COMPLEX AND NORMAL MODES The key difference between force appropriation and the traditional curve-fitting techniques is in the form of the identified parameters. Phase separation methods, in general,
Figure 5. Multivariate Mode Indicator Function eigenvalues.
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Figure 6. Modified Multivariate Mode Indicator Function eigenvalues.
yield complex/damped mode shapes plus frequencies and damping ratios derived from the complex eigenvalues. In contrast, phase-resonance methods permit direct measurement of the undamped normal modes and the corresponding undamped natural frequencies; the modal damping ratio is generally estimated using a half-power points method from the Nyquist plot generated by a mini-sweep around the natural frequency.
Figure 7. Juang—Wright singular values.
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When a system exhibits proportional damping, the complex modes are related to the normal modes by a simple transformation. In theory, both approaches will yield the same undamped natural frequency and damping estimates, which will be correct. However, in systems where the damping is non-proportional, the modal damping matrix will be non-diagonal and some of the resulting modes are coupled by the damping forces. The complex/damped modes which result from the solution of the first-order eigenvalue problem differ from the undamped normal modes and there is no simple transformation between them. A variety of methods have been proposed although none are entirely satisfactory. Furthermore, the derivation of frequency and damping values from the complex eigenvalues is somewhat contrived as the underlying theory assumes a proportional damping distribution. As a result, the estimated frequency does not equal the undamped natural frequency and the concept of modal damping is inappropriate for non-proportionally damped systems. Results derived from such systems can thus be misleading. As the phase resonance approach has no limitations regarding the damping distribution of the system, force appropriation methods may equally be applied to nonproportionally damped systems without error. A force vector is again derived which allows a normal mode to be excited at its undamped natural frequency and no additional approximations are required. In practice, excitation must be provided at a sufficient number of correctly chosen positions if the normal modes of the non-proportionally damped systems are to be well appropriated. Such procedures have been implemented experimentally on significantly non-proportionally damped structures [19] with success. The importance of exciter placement to achieve well-appropriated modes was again highlighted by this work. 3.2. BENCHMARK PLATE STRUCTURES Some experimental force appropriation results are now presented from two real benchmark structures—a perspex plate [18], and an aluminium plate with a significant level of non-proportional damping [19, 20]. 3.2.1. Proportionally damped perspex plate Modal analyses were performed on an experimental perspex plate equivalent to that modelled mathematically above [18]. An LMS-DIFA SCADAS data acquisition system was employed throughout these analyses. In this case, it is reasonable to assume a proportional modal damping distribution. As for the simulated case, 24 response positions were used with excitation applied at the corners of the plate. Figure 8 shows one of the drive point FRFs measured during the modal test with multiple uncorrelated random excitation. These experimental results are of high quality, partly helped by the fairly high level of damping present. Figure 9 shows the corresponding MMIF, which correctly identifies the presence of eight modes in the frequency range 20—220 Hz. The true number of modes is far from clear if only the FRF amplitude is inspected; there would appear to be only six modes although the MMIF reveals two pairs of close modes around 39 and 110 Hz. Indeed, use of a mode indicator function such as the MMIF should be encouraged even if a phase separation analysis is to be used, to identify the correct number of modes in a given frequency range. The tuned normal-mode results and the results of a phase separation approach, the Least Squares Complex Exponential (LSCE) method, are given in Table 2. It can be seen from Table 2 that six of the modes were tuned to a high degree of accuracy whilst modes 4 and 8 could not be adequately tuned with the chosen exciter combination.
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Figure 8. Experimental drive point FRF of perspex plate.
Figures 10—17 show the mode shapes and phase scatter plots of the tuned and LSCE estimates of the first two modes which are almost coincident in frequency. The identification and separation of close modes provided much of the impetus to the development of force appropriation techniques, an area in which phase separation approaches have sometimes
Figure 9. Experimental MMIF eigenvalues of perspex plate.
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TABLE 2 Results of experimental phase separation/phase resonance analysis of a perspex plate Phase separation (LSCE)
Phase resonance
Mode
Frequency/(Hz)
Frequency/(Hz)
MPI
1 2 3 4 5 6 7 8
38.97 39.01 90.97 100.29 114.77 141.26 165.66 206.72
38.93 38.99 90.87 100.76 114.53 141.02 165.43 206.78
0.979 0.966 0.974 0.690 0.949 0.959 0.949 0.490
been seen to fail or to give significantly complex modal results. The level of complexity of the phase separation estimates is apparent in Figs 10—17 whilst the tuned responses are very close to pure normal modes. The complexity could be due to some non-proportional damping or to the phenomenon mentioned in Section 3.3 below. Commonly, comparisons are made between the results of modal analyses and finiteelement (FE) predictions. Indeed, FE updating may even be the sole reason for performing an experimental modal test. Generally, FE analyses generate pure normal modes which may thus be directly compared to the appropriated modes resulting from force appropriation. The comparison to the results of a phase separation analysis is less straightforward, particularly for the case of non-proportionally damped structures. 3.2.2. Non-proportionally damped aluminium plate Further modal tests were performed using an experimental aluminium plate with large dash-pot oil dampers positioned at opposite corners in order to introduce a significant level of non-proportional damping into the structure. Again 24 response measurements and four corner exciters were used. The experimental rig is shown in Fig. 18. Figures 19 and 20 show a drive point FRF and the MMIF eigenvalue plot resulting from the modal test of the non-proportionally damped aluminium plate. It would be very difficult
Figure 10. First mode of perspex plate, tuned normal-mode shape.
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Figure 11. First mode of perspex plate, tuned phase scatter plot.
to judge the number and position of the modes from the FRF amplitude alone (arguably five modes), while they are clearly identified by the MMIF eigenvalues (actually seven modes). The separation of the primary and secondary MMIF eigenvalues between the first two modal frequencies is indicative of modal coupling due to a high level of non-proportional damping [20]. Even for the simple perspex plate, which was assumed to be proportionally damped, some non-proportionality is apparent between modes 4 and 5 in Fig. 9. This effect can be simulated on a simple two-dof system where it can easily be shown that the two MMIF eigenvalues may only coincide for the case of a proportionally damped system.
Figure 12. First mode of perspex plate, LSCE mode shape.
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Figure 13. First mode of perspex plate, LSCE phase scatter plot.
Figures 21 and 22 show the hard-tuned mode shape and phase scatter plot of the second mode, whilst Figs 23 and 24 show similar plots resulting from a phase separation approach, the Frequency Domain Direct Parameter Identification (FDPI) method. It can be seen that the complexity of the structure is now such that comparison of phase separation results with FE would prove very difficult. The tuned responses, however, are very close to the underlying normal modes of the structure. Arguably the best method for transforming from complex to real modes is to perform a soft-tuning force appropriation on the synthesised FRF data derived from the curve-fitted results. 3.3. SMOOTHNESS OF MODE SHAPES When a normal mode is physically excited using a monophase force vector, the structure acts as a spatial filter, so ensuring that the final mode shape has a smooth appearance with no abrupt changes or discontinuities. However, when a phase separation method is used for multiple input/output data, the resulting global frequency and damping estimates are
Figure 14. Second mode of perspex plate, tuned normal mode shape.
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Figure 15. Second mode of perspex plate, tuned phase scatter plot.
a compromise across all of the measurement positions. There is thus a possibility of some errors occurring in the derived mode shapes. In addition, modelling of complicated upper and lower residual effects by simple additional terms or extra modes may cause different levels of error at each measurement position. As a result, the final plotted mode shape may appear less smooth than the corresponding appropriated normal-mode shape. 3.4. TEST TIME The implementation of force appropriation techniques and the tuning of a set of modes of a structure is clearly a sizeable task; it would invariably be quicker to use a phase separation approach to curve-fit the multiple input/output data. However, given the time taken to set
Figure 16. Second mode of perspex plate, LSCE mode shape.
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Figure 17. Second mode of perspex plate, LSCE phase scatter plot.
up a modal test and perform pretest checks, the additional time requirements may seem less significant, particularly if improved parameter estimates are obtained. A compromise is to use phase separation techniques for the majority of modes whilst reserving phase resonance methods for particularly close or complex modes.
Figure 18. Aluminium plate with added non-proportional damping.
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Figure 19. Experimental drive point FRF of non-proportionally damped aluminium plate.
4. RECENT ADVANCES IN PHASE RESONANCE TESTING
4.1. OPTIMAL EXCITER LOCATION The number and positioning of exciters has been shown to be critical to the successful application of normal-mode force appropriation techniques. However, the choice of exciter
Figure 20. Experimental MMIF of non-proportionally damped aluminium plate.
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Figure 21. Second mode of non-proportionally damped aluminium plate, tuned normal mode shape.
configuration is traditionally left to the judgement of the test engineer, with alterations being made on a trial and error basis. Although a number of techniques are available for the positioning of exciters for phase separation testing, these methods are unsuitable for phase resonance testing as they all aim to maximise the energy input into the target modes simultaneously, rather than to excite a single mode whilst suppressing others. One approach to optimal exciter placement by Niedbal [21], applicable to phase resonance testing, makes use of the condition number of the modal matrix. It was proposed that the optimum subset of excitation positions for a given number of target modes should correspond to the modal matrix with the lowest condition number. However, there appears to be no theoretical reason that the exciter configuration chosen according to this criterion should necessarily excite the target normal modes in an optimum way. No account is taken of the damping levels in this technique, and proportional damping is assumed throughout.
Figure 22. Second mode of non-proportionally damped aluminium plate, tuned phase scatter plot.
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Figure 23. Second mode of non-proportionally damped aluminium plate, LSCE mode shape.
Recently, a method has been developed [22] which uses an a priori model of a test structure to predict the optimum exciter configuration for force appropriation of a set of normal modes. Two measures of combined normal mode purity (sum and product) may be defined which are suitable for different test situations. For the case of the proportionally damped perspex plate described above, the results of an a priori FE model have been validated by a normal mode test. Both exhaustive search and Genetic Algorithm approaches are considered. In cases when an a priori model of a structure is not available, it has been shown [23] that the mathematical model derived from a phase separation analysis may also be used for optical exciter placement prediction. This procedure particularly lends itself to mid-test
Figure 24. Second mode of non-proportionally damped aluminium plate, LSCE phase scatter plot.
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exciter optimisation in cases when some of the modes of a structure may not be well excited by the initial choice of exciter positions. In both cases, non-proportionally damped systems may be considered by estimating the positions and magnitudes of representative discrete dampers. The two methods of optimal exciter selection described above have been compared for excitation of the first four modes of the perspex plate model. All combinations of four exciters out of a possible 24 were considered. Figure 25 shows the condition number of the first approach [21] plotted against the average modal purity value (D ) of the second approach [22]. It is clear that the exciter configurations with the highest values of D are spread across the range of condition numbers. Thus, whilst a fairly good exciter configuration could be selected on the basis of the condition number, equally good alternatives would be discarded. The optimal configuration based on D corresponds to a condition number of only 0.077 and would thus not be considered under Niedbal’s method. Conversely, some configurations which lie in the top few per cent in terms of condition number (e.g. values above 0.3), have relatively low D values and are thus suboptimal in terms of normal-mode force appropriation. Figure 26 shows the exciter configuration [8, 12, 15, 16], chosen for optimal excitation of the first eight modes of the proportionally damped perspex plate. Using this configuration in practice, each of the first eight modes was hard-tuned to a modal purity above 0.92, with an average modal purity value of 0.941. These values represent a significant improvement over the results seen in Table 2 which were measured using excitation at the corner positions and gave an average modal purity of only 0.870.
Figure 25. Comparison of optimal excitation approaches for the perspex plate model.
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Figure 26. Optimal exciter configuration for the perspex plate model.
Finally, a method has been proposed [24] which seeks optimal exciter combinations for each normal mode of a system. The modal purity indicator, as defined in equation (8) is used to indicate the presence of the normal modes of the system, and is also used as the optimisation criterion. A Genetic Algorithm is used to perform the optimisation. Again, proportional damping is assumed, although the method may be extended to the situation where an a priori estimate of a non-proportional damping matrix is available. As a different set of exciter positions is found for each mode, this approach will prove impractical in real-test situations. 4.2. ESTIMATION OF MODAL DAMPING AND CROSS-DAMPING TERMS It has recently been demonstrated [25] that force appropriation may be used to estimate the direct modal damping and cross-damping terms for a general system. In this technique, a single normal mode is tuned and then the excitation is removed. For the case of a proportionally damped system, the modal damping may be found from a time-domain curve-fit to the resulting single-mode decays at each measurement position. For a nonproportionally damped system, the modal cross-damping terms cause other coupled modes to respond once the appropriated forces are removed, resulting in a multi-modal decay. A curve-fit to these responses allows both direct and cross-damping terms to be estimated. These methods have been validated on a benchmark aluminium plate structure. 4.3. APPLICATION TO NON-LINEAR STRUCTURES Recent work [26] has been directed at the application of force appropriation techniques to non-linear systems. The concept of normal modes is not directly applicable to non-linear systems, and it has been shown on simulated non-linear systems that the appropriated force vectors derived from the Modified MMIF do not significantly reduce the contribution of the coupled mode(s) for the non-linear case. In other words, classical force appropriation can yield a single-mode response at low excitation levels, but other modes respond at higher
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Figure 27. Modal Restoring force surfaces of a 2 DoF benchmark non-linear structure using FANS approach.
excitation levels where non-linear modal couplings are active. However, it has been demonstrated theoretically and experimentally that the linear mode shape of a single mode may be excited in the non-linear region for a multiple-dof non-linear system through the use of optimised force vectors with both fundamental and harmonic sinusoidal terms included in the multipoint force vector. The resulting responses are then suitable for analysis by single dof non-linear techniques such as the restoring force surface and the sensitivity approach which cannot be easily applied to coupled multiple-dof responses. An extension of the approach could yield important non-linear cross-coupling terms. Figure 27 shows the single-dof modal restoring force surfaces of the modes of a benchmark two-dof structure with a cubic stiffness non-linearity, appropriated using the force appropriation for non-linear systems (FANS) approach described in [26]. A comparison of this new approach with a classical restoring force method is presented in Table 3 . It can be seen that similar modal parameter estimates are obtained using the FANS technique as for the classical method. However, it is proposed that the FANS approach may be extended to real non-linear systems of arbitrary dimension, whereas the classical restoring force method begins to become impractical for systems with more than two dof. The non-linear identification could be restricted to those modes which demonstrate non-linear behaviour, and a classical modal model, with additional non-linear terms, may thus be developed. TABLE 3 Comparison of Restoring Force and FANS approaches on benchmark 2 DOF non-linear structure Restoring Force Method Modal parameter Mass, m/(kg) Damping, c/(Ns/m) Linear stiffness, k/(N/m) Cubic stiffness, b/(N/m)
Mode 1 2.60 10.11 4.87;10 3.83;10
FANS approach
Mode 2
Mode 1
3.06 9.49 6.49;10 1.73;10
3.01 8.73 4.97;10 6.54;10
Mode 2 2.84 10.90 6.54;10 2.29;10
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5. CONCLUSIONS
An overview of normal-mode force appropriation techniques has been presented. It has been shown that the rectangular FRF matrix methods are superior to the square FRF matrix methods in the identification of the normal modes of a structure. In situations where the number of exciters exceeds the number of effective dof, rank reduction techniques are of value. The MMIF is, arguably, the clearest of the methods to interpret, and is the most commonly used of the presented techniques. The advantages of phase resonance approaches over phase separation—accuracy, ease of comparison to FE results and smoothness of mode shapes—have been described along with the disadvantages—the requirement of extra analysis and testing time and the need for adequate excitation. The application of normal-mode force appropriation techniques to both proportionally and non-proportionally damped structures has been discussed. It has also been shown that phase resonance techniques may offer a way forward for non-linear identification with a modal model. Some approaches to optimal exciter placement have also been reviewed, and a demonstration of the need for good exciter positioning shown. It is the belief of the authors that phase resonance techniques should be used alongside, rather than instead of, phase separation analysis. In many cases, curve-fitted mode shapes are adequate and the time required for force appropriation may prove prohibitive. However, in cases of significant modal overlap, non-proportional damping or the need to correlate closely with FE analysis, normal-mode force appropriation in a valuable element in the modal analyst’s toolbox.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of LMS International and LMS-DIFA Measuring Systems in the provision of software and hardware.
REFERENCES 1. D. OTTE, H. VAN DER AUWERAER, J. DEBILLE and J. LEURIDAN 1992 Proceedings of the 17th International Seminar on Modal Analysis, 643—658. Enhanced force vector appropriation methods for normal mode testing. 2. E. BREITBACH 1973 Proceedings of the 3rd ESRO ¹esting Symposium, 519—528. A semi-automatic modal survey test technique for complex aircraft and spacecraft structures. 3. G. MOROSOW and R. S. AYRE 1978 Sound and »ibration Bulletin 48, 39—48. Force apportioning for modal vibration testing using incomplete excitation. 4. K. ALEXIOU and J. R. WRIGHT 1993 Modal Analysis 8(2), 119—136. Comparison of some multipoint force appropriation methods. 5. R. D. BRILLHART and D. L. HUNT 1992 Proceedings of the 10th International Modal Analysis Conference. Multiple input excitation methods for the C-17 transport aircraft GVT. 6. M. DEGENER 1995 Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, 70.1—70.12. Ground vibration testing for validation of large aircraft structural dynamics. 7. B. F. DE VEUBEKE 1956 AGARD Report 39. A variational approach to pure mode excitation based on characteristic phase lag theory. 8. G. W. ASHER 1967 American Institute of Aeronautics and Astronautics Journal 5(4), 822—824. A note on the effective degrees of freedom of a vibrating structure. 9. M. NASH 1988 Proceedings of the 6th International Modal Analysis Conference, 740—744. Use of the multivariate mode indicator function for normal mode determination. 10. G. W. ASHER 1958 Proceedings of the National Specialists Meeting on Dynamics and Aeroelasticity. Journal of Aeronautical Sciences 6, 56—96. A method of normal mode excitation using admittance measurements.
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11. R. W. TRAILL-NASH 1961 Structures and Materials Report 280, Dept. of Supply, Australian Defence Scientific Service, Aeronautical Research ¸aboratories. Some theoretical aspects of resonance testing and proposals for a technique combining experiment and computation. 12. P. IBANEZ 1976 SAE Paper No. 760873, SAE Aerospace Engineering and Manufacturing Meeting. Force appropriation by Extended Asher’s method. 13. R. WILLIAMS, J. CROWLEY and H. VOLD 1986 Proceedings of the 3rd International Modal Analysis Conference, 66—70. The multivariate mode indicator function in modal analysis. 14. M. NASH 1991 Proceedings of the 9th International Modal Analysis Conference, 688—693. A modification of the multivariate mode indicator function employing principal force vectors. 15. J.-N. JUANG and J. R. WRIGHT 1991 Journal of »ibration and Acoustics 113, 176—181. A multipoint force appropriation method based upon a singular value decomposition approach. 16. P. IBANEZ and K. D. BLAKELY 1984 Proceedings of the 2nd International Modal Analysis Conference, 2, 903—907. Automatic force appropriation—a review and suggested improvements. 17. M. RADES 1992 Proceedings of the 17th International Seminar on Modal Analysis, 563—581. A comparison of some mode indicator functions. 18. J. E. COOPER, M. J. HAMILTON and J. R. WRIGHT 1993 Modal Analysis 10(2), 118—130. Experimental evaluation of various normal mode force appropriation methods on a rectangular perspex plate using four exciters. 19. P. S. HOLMES, J. R. WRIGHT and J. E. COOPER 1995 Proceedings of the 15th ASME Conference on Mechanical »ibration and Noise, 753—759. Identification of a significantly non-proportionally damped structure using force appropriation. 20. P. S. HOLMES, J. E. COOPER and J. R. WRIGHT 1996 Proceedings of the 21st International Seminar on Modal Analysis, 1237—1250. Normal mode estimation from non-proportionally damped systems. 21. N. NIEDBAL and E. KLUSOWSKI 1990 Proceedings of the American Institute of Aeronautics and Astronautics Dynamics Specialists Conference, 130—141. Optimal exciter placement and force vector tuning required for experimental modal analysis. 22. P. S. HOLMES, J. R. WRIGHT and J. E. COOPER 1996 Proceedings of the 14th International Modal Analysis Conference, 1—7. Optimum exciter placement for normal mode force appropriation using an a priori model. 23. P. S. HOLMES, J. R. WRIGHT and J. E. COOPER 1996 Proceedings of the 2nd D¹A/NAFEMS International Conference. Mid-test optimum exciter placement for normal mode force appropriation. 24. A. BIFULCO and M. DEGENER 1996 International Conference on Identification in Engineering Systems, 377—387. Theoretical prediction and experimental verification of optimum exciter configurations for modal testing. 25. S. NAYLOR, J. R. WRIGHT and J. E. COOPER 1997 Proceedings of the 15th International Modal Analysis Conference, 137/1—137/7. On the estimation of modal matrices with non-proportional damping. 26. P. A. ATKINS and J. R. WRIGHT 1997 Proceedings of the 6th International Conference on Recent Advances in Structural Dynamics, 2, 1099—1112. Estimating the behaviour of a nonlinear experimental multi degree of freedom system using a force appropriation approach.
APPENDIX A: NOMENCLATURE [A] [B] e +f, j n* r +x, j D D ?T + , u
real part of the FRF matrix imaginary part of the FRF matrix number of exciters vector of forces integer number of effective modes number of response positions vector of responses eigenvalues normal-mode purity average normal-mode purity normal-mode shape frequency