Low order continuous-time filters for approximation of the ISO human vibration sensitivity weightings

Transcription

1 Journal o Sound and Vibration 265 (23) JOURNAL OF SOUND AND VIBRATION Letter to the Editor Low order continuous-time ilters or approximation o the ISO 263- human vibration sensitivity weightings L. Zuo, S.A. Nayeh* Department o Mechanical Engineering, Massachusetts Institute o Technology, 77 Massachusetts Avenue, Cambridge, MA 239, USA Received 24 October 22; accepted 4 November 22. Introduction Human health and comort under whole-body vibration depend on the level, requency, direction, location, and duration o the acceleration. The international standard ISO 263- [] speciies a method o evaluation o the eect o exposure to vibration on humans by weighting the root-mean square (r.m.s.) acceleration with human vibration sensitivity curves. There are six requency-weighting curves given as magnitudes W i at the one-third octave requencies i : WðoÞj o¼2pi ¼ W i ; i ¼ ; 2; 3; y; N: ðþ In order to use these weightings in design, it is sometimes necessary to make a good approximation o these curves using stable rational transer unctions o the orm WðsÞ ¼ b ms m þ?b s þ b s n : ð2þ þ?a s þ a Unlike previous versions o the standard published in 974 and 985, the recent ISO 263- (997) standard gives high order s-plane equations to describe the weighting curves. However, low order ilter approximations are still preerred in practical applications, especially in controller design. Muller et al. [2] approximate the ISO 263- (974) weighting W k (or vertical acceleration) by a second order ilter given by 5s þ 5 WðsÞ ¼ s 2 þ 5s þ 2 : ð3þ This shape ilter has been cited by many researchers in passive and active vehicle suspension design (e.g., Res. [3 7]), even though the ISO weighting curves have been updated several times since 974. Other second order ilter approximations have also been used by Kim and Yoon [8] and Wang and Wilson [9]. As reported by Lewis and Griin [], the weighted r.m.s. accelerations *Corresponding author. Tel.: ; ax: address: nayeh@mit.edu (S.A. Nayeh) X/3/$ - see ront matter r 23 Elsevier Science Ltd. All rights reserved. doi:.6/s22-46x(2)567-5

2 46 L. Zuo, S.A. Nayeh / Journal o Sound and Vibration 265 (23) can dier by more than 2% among the ISO 263- (985) and ISO 263- (997) standards. It is thereore necessary to design good low order ilters to approximate the ISO 263- weighting curves. 2. Method and results A least-pth approximation in the requency domain is usually desired, where p ¼ 2 or a leastsquare approximation and p-n or a minimax (Chebyshev) approximation. The problem then is to determine the coeicients a k and b k such that X N i¼ jjwðs i Þj W i j p ; where s i ¼ 2pj i ð4þ is minimized. This is a highly non-linear problem, and no analytical solution can be obtained even or the irst order least-square approximation []. Fortunately, in the ield o digital signal processing, a great eort has been devoted to advanced ilter design in the z-domain, and many standard codes are available. Thereore, we use the method o bilinear transormation [2] to design the ISO 263- ilters. Step : Map the speciications o the weighting curves in continuous-time requency o i onto the unit circle (z-domain) via the bilinear transormation O i ¼ 2 T tan To i 2 ¼ 2 T tan ðtp i Þ; where O i is the discrete-time requency and T is a constant. Step 2: Design the optimal least-pth digital ilter W D ðzþ to minimize where z i ¼ e jo i and X N i¼ jjw D ðz i Þj W i j p : b W D ðzþ ¼ % n z n þ? %b z þ %b z n þ?%a z : ð7þ þ %a The Matlab Filter Design Toolbox [3] provides the unction iirlpnorm, which computes the IIR digital ilter that best approximates (in the least-pth sense) the desired requency response within single or multiple bands. Step 3: Obtain the s-domain ilter via the bilinear transormation WðsÞ ¼W D ðzþj z¼ðþts=2þ=ð Ts=2Þ : ð8þ This ilter is not an optimal least-pth analog ilter, but it is a very good approximation to the ideal continuous-time requency response. We call such ilters quasi-least-pth continuous-time ilters. Setting px6 [4] yields a good approximation to the minimax (Chebyshev) design. The ollowing are the second through ith order quasi-least-square ðp ¼ 2Þ ilters obtained by taking T ¼ :5 to approximate the ISO 263- (997) requency-weighting curve W k or ð5þ ð6þ

3 L. Zuo, S.A. Nayeh / Journal o Sound and Vibration 265 (23) vertical acceleration: W ð2þ 86:5s þ 546: k ðsþ ¼ s 2 þ 82:7s þ 892 ; W ð3þ k ðsþ ¼ 8:3s2 þ 989:s þ :28 s 3 þ 78:92s 2 þ 242s þ 564 ; W ð4þ k ðsþ ¼ 8:89s3 þ 796:6s 2 þ 937s þ :446 s 4 þ 8:s 3 þ 2264s 2 þ 772s þ 296 ; W ð5þ k ðsþ ¼ 87:72s 4 þ 38s 3 þ 336s 2 þ 5453s þ 559 s 5 þ 92:6854s 4 þ 2549:83s 3 þ 25969s 2 þ 857s þ : The magnitude requency responses are shown in Fig.. The second through ourth order ilters similarly obtained or the weighting curve W d horizontal acceleration are W ð2þ d ðsþ ¼ 3:55s s 2 þ 2:9s þ 47:6 ; or W ð3þ d ðsþ ¼ 4:55s2 þ 6:26s þ 7:725 s 3 þ 5:2s 2 þ 5:63s þ 47:6 ; Fig.. ISO263- requency-weighting curve W k (circles) and quasi-least-square ilter approximations: second order (dot), third order (dash), ourth order (solid), ith order (dash dot).

4 462 L. Zuo, S.A. Nayeh / Journal o Sound and Vibration 265 (23) W ð4þ d ðsþ ¼ 2:66s3 þ 63:7s 2 þ 6:64s þ 2:79 s 4 þ 23:77s 3 þ 236:s 2 þ 692:8s þ 983:4 ; and their magnitude requency responses are shown in Fig. 2. The third principal requency-weighting curve W is used or motion sickness. The second through ith order quasi-least-square ilter approximations are W ð2þ :8892s ðsþ ¼ s 2 þ :8263s þ :63 ; W ð4þ W ð3þ :5726s 3 þ 3:876s ðsþ ¼ s 3 þ 4:263s 2 þ 4:777s þ 4:396 ; ðsþ ¼ :2633s4 þ :238s 3 þ 2:286s 2 þ :2335s þ :292 s 4 þ 2:527s 3 þ 4:584s 2 ; þ 2:993s þ :373 W ð5þ ðsþ ¼ :457s4 þ :233s 3 þ 3:75s 2 þ :75s þ :3596 s 5 þ 7:757s 4 þ 9:6s 3 þ 28:37s 2 þ 8:52s þ 7:23 ; and their magnitude requency responses are shown in Fig. 3. Low order quasi-least-square ilter approximations or the three additional requency weightings can be similarly obtained Fig. 2. ISO263- requency-weighting curve W d (circles) and quasi-least-square ilter approximations: second order (dot), third order (dash), ourth order (solid).

5 L. Zuo, S.A. Nayeh / Journal o Sound and Vibration 265 (23) Fig. 3. ISO263- requency weighting curve W (circles) and quasi-least-square ilter approximations: second order (dot), third order (dash), ourth order (solid), ith order (dash dot) Fig. 4. Second order ilter approximations: Muller s design [2] (dot), Hankel model reduction (dash), balance model reduction (dash dot), quasi-least-square (solid) ISO263- weighting curve W k (circles).

6 464 L. Zuo, S.A. Nayeh / Journal o Sound and Vibration 265 (23) Discussion Unlike previous versions o the standard, that published in 997 provides a high order s-plane description. So a low order approximation could also be obtained by model reduction. The most commonly used methods are the balanced and Hankel model reductions. Standard codes to perorm these reductions are also available, such as the unctions balmr and ohkapp in Matlab Robust Control Toolbox and the unction modred in Matlab Control System Toolbox [3]. Figs. 4 and 5 show the magnitude requency responses o second and ourth order shape ilters obtained by balanced and Hankel model reductions o the ISO 263- (997) weighting W k ðsþ; in comparison with the quasi-least-square ilter approximations and Muller s ilter (3). The r.m.s. errors at the one-third octave requencies are compared in Table. We can see that the quasi-least-square shape ilters are much better than Muller s ilter and those obtained by either model reduction technique Fig. 5. Fourth order ilter approximations: Hankel model reduction (dashed), balance model reduction (dash dot), quasi-least-square (solid), ISO263- weighting curve W k (circles). Table Comparison o r.m.s. errors at the one-third octave requencies Quasi-least-square Balanced model reduction Hankel model reduction Muller s design [2] Second order Fourth order

7 L. Zuo, S.A. Nayeh / Journal o Sound and Vibration 265 (23) Conclusion In this letter, we propose a simple procedure or design o low order quasi-least-pth continuoustime ilters that approximate the ISO 263- requency-weighting curves. We use the bilinear transormation to map the magnitude-requency speciications into the z-domain, design the optimal digital ilters using standard codes, and thence obtain the corresponding quasi-least-pth s-plane ilters. The method is very easy to implement, and the obtained ilters are a better match to the ISO 263- (997) requency weightings than those obtained by model reduction techniques and those widely cited in the literature. The low order ilters presented herein can be used in optimization o vehicle suspensions and other applications. Reerences [] International Organization or Standardization, Mechanical vibration and shock evaluation o human exposure to whole body vibration part : general requirements, ISO 263-:997, 997. [2] P. Muller, K. Popp, W. Schiehlen, Covariance analysis o nonlinear stochastic guideway-vehicle-systems, Proceedings o the Sixth IAVSD Symposium, 979, pp [3] M.M. Elmadany, A procedure or optimization o truck suspensions, Vehicle System Dynamics 6 (987) [4] W. Schiehlen, Modeling, analysis and estimation o vehicle systems, in: W. Schiehlen, W. Wedig (Eds.), Analysis and Estimation o Mechanical Systems, Springer, New York, 988, pp [5] E.M. Elbeheiry, D.C. Karnopp, M.E. Elaraby, A.M. Abdelraaou, Suboptimal control design o active and passive suspensions based on a ull car model, Vehicle System Dynamics 26 (996) [6] P. Eberhard, W. Schiehlen, D. Bestle, Some advantages o stochastic methods in multicriteria optimization o multibody systems, Archive o Applied Mechanicas 69 (999) [7] L. Zuo, S. Nayeh, Structured H 2 optimization o vehicle suspensions based on multi-wheel models, Vehicle System Dynamics, submitted or publication. [8] H. Kim, Y. Yoon, Semi-active suspension with preview using a requency-shape perormance index, Vehicle System Dynamics 24 (995) [9] J. Wang, D. Wilson, Mixed GL 2 =H 2 =GH 2 control with pole placement and its application to vehicle suspension systems, International Journal o Control 74 (2) [] C. Lewis, M. Griin, A comparison o evaluations and assessments obtained using alternative standards or predicting the hazards o whole-body vibration and repeated shocks, Journal o Sound and Vibration 25 (998) [] J. Taylor, Q. Huang, Handbook o Electronic Filters, CRC Press, New York, 996. [2] A. Oppenheim, R. Schaer, J. Buck, Discrete-Time Signal Processing, Prentice-Hall, Englewood Clis, NJ, 998. [3] The MathWorks, Inc., Filter Design Toolbox, Control System Toolbox, Robust Control Toolbox, 2, [4] T. Nishi, F. Lu, On the relation between the maximum errors o the least pth approximation and those o the minimax approximation by rational unction, Electronics and Communications in Japan, Part 3 84 (2)