An Estimation of Error-Free Frequency Response Function from Impact Hammer Testing

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1 85 An Estimation of Error-Free Frequency Response Function from Impact Hammer Testing Se Jin AHN, Weui Bong JEONG and Wan Suk YOO The spectrum of impulse response signal from the impact hammer testing is widely used to obtain the frequency response function (FRF). However the FRFs obtained from the impact hammer testing have not only leakage errors but also finite record length errors when the record length for the signal processing is not sufficiently long. The errors cannot be removed by the conventional signal analyzer which treats the signals as if they are always steady and periodic. Since the response signals generated by the impact hammer testing are transient and damped, they are undoubtedly non-periodic. It is inevitable that the signals acquired for limited recording time should cause several errors. This paper makes clear the relation between the errors in the FRF and the length of the recording time. A new method is suggested to reduce the errors in the FRF in this paper. Several numerical examples involving 1-dof models are considered to elucidate the properties of the errors and the validity of the proposed method. Key Words: Impact Hammer Testing, Impulse Response Function, Frequency Response Function, Fourier Transform, Discrete Signal, Record Length, Leakage Error, Finite Record Length Error 1. Introduction The impact hammer testing is used widely to obtain the frequency response function (FRF) because of its simplicity (1), (). Although the signals acquired during the impact hammer testing are neither periodic nor stationary, they are dealt with as periodic and stationary signals by conventional analyzers. The FRFs from the impact hammer testing should be influenced by the record length, while the FRFs from the exciter testing are not. It means that the FRFs from the impact hammer testing inevitably have a finite record length error as well as a leakage error when the record length is not long enough. There have been various works aimed at reducing the leakage error from the windowed data (3). It is known that a window function distorts the phase of the FRFs, and the method for correcting the phase was studied (4).However,mostof these works assumed the signals to be periodic. A method for reducing the errors without any window function was introduced recently (5), in which a finite record length error was not considered. In the impact hammer testing, it is in- Received 4th September, 003 (No ) School of Mechanical Engineering, Pusan National University, 30 Jangjeon-dong, Kumjung-ku, Pusan , Korea. wbjeong@pusan.ac.kr Series C, Vol. 47, No. 3, 004 evitable that the signals of impulse input and its responses be acquired for limited record length, which causes both a finite record length error and a leakage error. This paper shows that the FRFs of 1-dof model obtained from the impact hammer testing depend on the record length, and elucidates the characteristics of the errors due to insufficient record length. Furthermore, this paper proposes a new method to remove these errors and to estimate the error-free FRFs.. Finite Record Length Error. 1 Theoretical background The equation of motion of a viscously damped 1-dof system in Fig. 1 is described as follows. mẍ(t)+c ẋ(t)+kx(t) = f (t) (1) If the force in the right hand side of Eq. (1) is an impulse, it can be expressed by Dirac delta function: f (t) = F 0 δ(t) () The response of displacement under this impulse can be described as e ζω n t x(t) = F 0 h(t) = F 0 sinω d t, t 0 (3) m ω d where h(t) means unit impulse response (6). Since the signal is acquired for T RL, which is called the record length, JSME International Journal

2 853 the Fourier transform of the impulse signal can be written as F( jω) TRL > 0 = f (t) e j ω t dt TRL = F 0 δ(t) e j ω t dt = F 0 (4) 0 It shows that the spectrum of impulsive force has constant values, and has no relation with the record length. On the other hand, Fourier transform of response given in Eq. (3) becomes X( jω) TRL > 0 = x(t) e j ω t dt where = F 0 TRL = E( jω) TRL = e ( ζω n jω)t RL 0 e ζω n t m ω d j ω t sinω d te dt F 0 k ω m+ jω c [1 E( jω) T RL ] ζ 1 ζ sinω d T RL +cosω d T RL + j ω sinω d T RL ω d (5.a) The E( jω) TRL in Eq. (5.a) indicates the error of FRF due to the finite record length T RL. As record length T RL becomes infinite, the FRF in Eq. (5) will be X(ω) TRL 1 H(ω) = lim = (6) T RL F(ω) TRL k ω m+ jω c which is the same as the theoretical FRF.. Discrete signals In the previous paragraph, it is shown that the FRFs obtained by the impact hammer testing become different from theoretical ones due to the finite record length. In this paragraph, the errors caused by finite record length in discrete signal processing are discussed. The impulsive force signal in the impact hammer testing is assumed to be a half-sine function as follows. F 0 sin π t, if 0 t t 0 f (t) = t 0 (7) 0, else (5) Here, the time duration of signal t 0 is infinitesimal. And the response signal of 1-dof damped system can be described as x(t) = Ae σ t sinω t, t 0 (8) After digitalizing with sampling time interval t for the record length T RL, the impulse signal in Eq. (7) can be transformed by the Fourier transform as follows (7) ; F(k f ) = t N 1 N t0 = tf 0 jπ (k f ) i t f (i t) e ( ) π sin i t t 0 jπ ki e N (9) where N t0 is the number of data acquired for time duration t 0. Equation (9) shows that the Fourier transformed data of the impulse signal is not dependent on the record length, since the record length T RL is definitely longer than duration time t 0. Similarly, the discrete Fourier transformation of the response signal given in Eq. (8) is as follows. X(k f ) = ta N 1 e σ (i t) jπ ki sin[ω( i t)] e N (10) Equation (10) shows that the spectrum will converge to the true value if the record length is sufficiently long, but it will contain some errors in case the response signal does not decay fully during the record length.. 3 Finite record length error In the actual impact hammer testing, the errors in Eq. (10) will be inevitable since the signal is acquired in limited recording time. They will be referred to as the finite record length error in this paper. To see the tendency of the errors with changes in record length, selected values, F 0 = 10 3, t 0 = 10 3, A = 10 3, σ = 5, ω = π 100, t = 10 4 are used in Eqs. (7) and (8) respectively. The impulse and its response signal with these values are represented in Figs. and 3. Figure 4 shows the Fourier transformed spectrum of the impulse given in Fig.. As shown in Fig. 4, the spectrum of impulsive force signal is not influenced by the record length except for its frequency resolution. Figure 5 shows the Fourier transformed spectrum of response signal given in Fig. 3. As shown in Fig. 5, the magnitude of spectrum of response signal shows different Fig. 1 A model of 1-dof system Fig. Impulse force signal JSME International Journal Series C, Vol. 47, No. 3, 004

3 854 Fig. 3 Impulse response signal Fig. 6 Variation of peak values of impulse response spectrum to signal acquisition time Fig. 4 Spectrum of impulse force signal 3. A Method for Reducing the Finite Record Length Error In the previous section, it was shown that the exact FRF cannot be obtained from the impact hammer testing if the record length for signal processing is not sufficiently long. This chapter introduces a method which can eliminate the finite record length error and the leakage error simultaneously from the spectrum contaminated by these errors FRF with a finite record length error The unit impulse response function of a single degree-of-freedom model with mass m, stiffness k and damping c is described by (8) h(t) = Ae σ t sinπ f d t (11) where 1 A = π mf n 1 ζ (11.a) σ = πζ f n (11.b) Fig. 5 Variation of impulse response spectrum to signal acquisition time values according to changing the record length. In particular, the shorter the record length is, the smaller the peak value of spectrum will be. The changes of the peak value in Fig. 5 with respect to the record length are illustrated in Fig. 6. It makes sure that the peak value approaches to the true value as the record length is longer. Observing the detailed variation of the peak value with the variation of the record length shown in Fig. 6, it fluctuates due to the leakage error which occurs due to the disagreement between the period of signal and the record length. The peak value is strongly dependent on the leakage error in short record length ((a) in Fig. 6), while it is weakly dependent on the leakage error in long record length ((b) in Fig. 6). Series C, Vol. 47, No. 3, 004 f d = f n 1 ζ (11.c) Equation (11) can be expressed by the following equation with sampling time t H k = T RL N N 1 Ae σ i T RL N jπ ki sin[π fd i t] e N, k = N,, N 1 (1) Using sum s formula of geometric progression, Eq. (1) can be rewritten as H k = A T [ RL 1 e [σt RL jπ(m k)] j N [1 e [σt RL jπ(m k)]/n ] 1 e [σ T RL+ jπ(m+k)] ], N [1 e [σ T RL+ jπ(m+k)]/n ] k = N,, 1,0,1,, N 1 (13) where m = f d f = f d T RL (13.a) JSME International Journal

4 855 and m is the number of sinusoidal waves within record length. Since m is real number generally, it is described as m = p (integer part) + q (decimal fraction part) (14) while p is a serial number which represents the peak position in the spectrum obtained by the impact hammer testing, q is a frequency error which can t be expressed on the spectrum. Provided the number of data in record length, N, issufficiently large, substituting Eq. (14) into Eq. (13) gives lim H k = A T [ RL 1 e σ T RL e jπ{(p+q) k} N π{(p+q) k}+ jσt RL + 1 e σt RL e jπ{(p+q)+k} ] (15) π{(p+q)+k} jσt RL Equation (15) is the frequency response spectrum in which the finite record length errors are taken into consideration. 3. Elimination of errors from unit impulse response function Thepeakvalue(k = p in Eq. (15)) and fore and after values of the peak (k = p 1, p+1 in Eq. (15)) will be used to eliminate the errors from the impulse response function, and they can be taken easily from the spectrum obtained by the impact hammer testing. The ratio of the peak values can be written as follows (5). R = H p R + = H p 1 = [p(q+1)+(σt RL/π) ] +[(σt RL /π)] [q p+(σt RL /π) ] +[(σt RL /π) p ] H p H p+1 (16) = [p(q 1)+(σT RL/π) ] +[(σt RL /π)] [q p+(σt RL /π) ] +[(σt RL /π) p] (17) X F (ω) = 1 m [(ω n ω )+ j ζω n ω ] 4. Numerical Example (19) 4. 1 FRF by conventional method To show that frequency response spectrum by DFT of impulse and its response signal has the errors when the record length is not sufficiently long, a proportionally damped 1-dof model is simulated numerically. Substituting m = 3.0 kg, k = N/m andc = N s/m into Eq. (1) and letting driving force f (t) the unit impulse, its displacement response is calculated numerically by Runge-Kutta method. The unit impulse force and its displacement response are shown in Fig. 7 and the theoretical FRF given in Eq. (6) is shown in Fig. 8. In order to compare the trend of the errors, the frequency response spectrum is calculated numerically for four cases and their Nyquist circles are plotted. The first case, in which record length is relatively long and is a multiple of the signal s period, is displayed in Fig. 9 (a). The spectrum shows a good agreement with the theoretical FRF. However, the second case in Fig. 9 (b), where record length is not sufficiently long and is a multiple of the signal s period, shows the finite record length error in the spectrum. Both the third case in Fig. 10 (a) and the fourth case in Fig. 10 (b) Using the fact that p 1andp q generally, these values can be approximated to p ± 1 p and p + q ± 1 p. Since the left hand side of Eqs. (16) and (17), R ± can be read easily from the spectrum obtained experimentally, the decaying ratio, σ, can be calculated as follows. σ = π ( R + R ) 8 + (18) T RL R + +R R + +R Substituting σ into Eq. (16) or Eq. (17), frequency error q can be calculated. After calculating q, damped natural frequency, f d, can be obtained from Eqs. (13.a) and (14). Finally, the amplitude A can be obtained by substituting decaying ratio σ and frequency error q into Eq. (13). Substituting damping ratio σ, damped natural frequency f d and amplitude A into Eqs. (11.a), (11.b) and (11.c) gives the estimation of mass m, damping ratio ζ and undamped natural frequency ω n. Using these parameters, FRF without finite record length error and leakage error can be estimated by the following equation. Fig. 7 Impulse force and displacement for 1-DOF damped model (by Runge-Kutta method) Fig. 8 Magnitude of theoretical F.R.F. JSME International Journal Series C, Vol. 47, No. 3, 004

5 856 (a) Long acquisition time (T RL : 4.0 sec, N : 4 096) (a) Long acquisition time (T RL : 4.05 sec, N : 4 096) (b) Short acquisition time (T RL : 1.0 sec, N : 1 04) Fig. 9 Nyquist plot of FRF obtained by conventional Fourier transform (no leakage error) show the Nyquist plot in case that the record length is not a multiple of the signal s period and the leakage errors occur. The spectrum in Fig. 10 (a) shows a good agreement with the theoretical FRF. It means that the spectrum is not contaminated by leakage error if record length is sufficiently long. The modal circle by the fourth case is greater than that of theoretical FRF due to leakage error. In conclusion, it is certain that if a spectrum is calculated for FRF without sufficiently long record length, it always has the finite record length error and is different from the theoretical modal circle. 4. FRF by the proposed method Four cases of spectra contaminated by the errors as shown in Fig. 9 of the previous paragraph are used to calculate the true FRFs. The parameters of unit impulse response function are estimated by the method in this paper for each case, and they are displayed at Table 1. The parameters for all cases agree well with exact values. Of Series C, Vol. 47, No. 3, 004 (b) Short acquisition time (T RL : 1.05 sec, N : 1 04) Fig. 10 Nyquist plot of FRF obtained by conventional Fourier transform (with leakage error) Table 1 The parameters of unit impulse response function calculated by the method proposed in this paper these cases, the case 4, in which the parameters are estimated worse than any other case here, is used for calculating new FRF which is shown in Fig. 11. When the new FRF in Fig. 11 is compared with the original FRF in Figs. 9 and 10, the new one shows a good agreement JSME International Journal

6 857 the true FRF from the impulse response spectrum contaminated by the errors. 4 ) Several numerical examples showed that the method in this paper is valid and useful to reduce not only the finite record length error but also the leakage error of 1-dof model. 5 ) The case of the FRFs for multi-dof systems will be undertaken in the near future. Acknowledgements The authors would like to thank the Ministry of Science and Technology of Korea for the financial support by a grant (M ) under the NRL (National Research Laboratory). Fig. 11 Nyquist plot of FRF obtained by the suggested method (for case 4 in Table 1 or Fig. 10(b)). with the theoretical FRF. This means that the errors are eliminated from the spectrum, and error-free FRF can be calculated. This method can also be applied for the FRFs with multi-dof systems if the peaks are sufficiently separate so that each peak value is not affected by the residuals of other peaks. However, it is difficult to use this method if the peaks of FRF are strongly coupled. Further work is needed to eliminate the errors in the multi-dof systems in which appreciable coupling between the modes exists. 5. Conclusion 1 ) This paper showed clearly the relations between the errors of impulse response spectrum and the record length of 1-dof model. ) The record length should be long enough to obtain the true impulse response spectrum, otherwise the spectrum is contaminated by the finite record length error and the leakage error. 3 ) A new method was proposed to remove the finite record error as well as the leakage error and to estimate References ( 1 ) Park, H.S. and Park, Y.S., Impact Force Reconstruction and Impact Model Identification Using Inverse Dynamics of an Impacted Beam, The Korea Society of Mechanical Engineering, Vol.19, No.3 (1995), pp ( ) Maia, N.M.M. and Silva, J.M.M., Theoretical and Experimental Modal Analysis, (1998), Research Studies Press Ltd. ( 3 ) Burgess, J.C., On Digital Spectrum Analysis of Periodic Signals, The Journal of the Acoustical Society of America, Vol.58, No.3 (1975), pp ( 4 ) Dishan, H., Phase Error in Fast Fourier Transform Analysis, Mechanical System and Signal Processing, Vol.9, No. (1995), pp ( 5 ) Jeong, W.B., Ahn, S.J., Chang, H.Y. and Chang, C.H., The Improvement of Leakage Error in Digital Fourier Transform, The Korean Society for Noise and Vibration Engineering, Vol.11, No.3 (001), pp ( 6 ) Rao, S.S., Mechanical Vibrations, (1990), Addison- Wesley Publishing Company, Inc. ( 7 ) Bendat, J.S. and Piersol, A.G., Random Data: Analysis and Measurement Procedures, (1986), John Wiley & Sons, Inc. ( 8 ) Nagamatsu, A., Modal Analysis, (1985), Baifukan. JSME International Journal Series C, Vol. 47, No. 3, 004