線性預測模型外文翻譯--采用線性預測模型對鐵道車輛車體進行模態(tài)分析（英文）

1、Journal of System Design and DynamicsVol.3, No.6, 2009Modal Analysis of Railway Vehicle Carbodies Using a Linear Prediction Model?Takahiro TOMIOKA??, Tadao TAKIGAMI?? and Ken-Ichiro AIDA???? Vehicle Noise and Vibration L

2、ab., Railway Technical Research Institute,2–8–38 Hikari–cho, Kokubunji-shi, Tokyo, 185–8540, JapanE-mail: tomioka@rtri.or.jpAbstractThis paper reports on a study regarding modal property identification for railway vehi-c

3、les using a linear prediction model. The relationship between input (excitation forceor axlebox acceleration) and output (carbody acceleration) of an actual railway vehi-cle obtained by stationary or running tests is exp

4、ressed by means of an ARX (Auto-Regressive eXogenious) model, and the procedure for the extraction of modal proper-ties is described in detail. Determination of an appropriate model order (i.e., the orderof the predictio

5、n coeffi cients in the ARX model) is specifically discussed from theviewpoint of practical use. The implementation of average estimation errors for twodiff erent parts of the analyzed data is proposed, and their eff ecti

6、veness in determin-ing the model order is evaluated. Suitability for the MIMO (multiple-input multiple-output) problem using the ARX model is also described. It is shown that detailedmodal characteristics can be successf

7、ully identified using the proposed method frommeasured data for both stationary and running tests.Key words : Railway, Modal Analysis, Linear Prediction Model, Signal Processing,Bending Vibration1. IntroductionThe suppre

8、ssion of vertical bending vibration in carbodies is important in improvingthe ride quality of railway vehicles. The first step to take in vibration countermeasures is toidentify vibration characteristics such as the freq

9、uency and modal properties of the carbody.Running and stationary vibration tests are usually conducted for this purpose. Running tests onan actual commercial-service track are carried out to analyze frequency characteris

10、tics duringrunning and evaluate the ride quality of vehicles. Stationary vibration testing is suitable foruse in ascertaining the modal properties of a carbody because the relationship between inputforces and response ac

11、celeration is clear. Since the costs and eff ort involved in conductingrailway vehicle measurement tests are high, it is very eff ective to be able to evaluate bothride quality and modal properties from a single measurem

12、ent test. The authors have alreadyreported on a method of evaluating running quality from stationary testing(1)(2). This paperdeals with a technique to evaluate the modal properties of carbodies from running tests.The in

13、put/output relationships of a railway vehicle are complex and unsteady during run-ning; the vehicle is subjected to multiple-inputs, and the excitation conditions change overshort periods of time. It has therefore been d

14、iffi cult to ascertain the detailed frequency andmodal properties of carbodies from running tests. To address this challenge, the authors ex-amined the application of a linear prediction model (LPM) such as ARX (the Auto

15、-RegressiveeXogenious model) to analyze railway vehicle vibration(3)～(7), since LPMs are advantageousin treating short-time data and multiple-input multiple-output (MIMO) problems. However,determining the model order (i.

16、e., the order of the prediction coeffi cients in the ARX model)is problematic. This paper describes identification of the modal properties of railway vehiclecarbodies using the ARX model, and the determination of appropr

17、iate model order is specifi-?Received16Sep.,2009(No.T2-08-0862) Japanese Original: Trans. Jpn. Soc. Mech. Eng., Vol.75, No.753, C (2009), pp.1295–1303 (Received 24 Sep., 2008) [DOI: 10.1299/jsdd.3.918]918Journal of Syste

18、m Design and DynamicsVol.3, No.6, 2009x(n) =M ?m=1 A(M) m x(n ? m) + ε (M)(n) , (4)where x(n) = ? u T (n) y T (n) ?T is a combined vector consisting of input and output time seriesdata, ε (M)(n) = ? ε (M) uT (n) ε (M) yT

19、 (n) ?T expresses the combined error vector, and A(M) m denotesthe following block matrix containing prediction coeffi cients:A(M) m =? ? ? ? ? ? d(M) m c(M) m b(M) m a(M) m? ? ? ? ? ? .Equation (4) means that x(n) can b

20、e expressed as an AR (Auto-Regressive) model with P+ Qindependent variables, and A(M) m can be calculated by applying existing algorithms for anordinary AR model. In this study, we employ the Burg-method(11), which is co

21、nsidered ad-vantageous in spectrum estimation from short-time data and has several effi cient algorithmsfor numerical calculation of prediction coeffi cients.In addition to the forward ARX model expressed in Eqs. (2)-(4)

22、, we define anotherbackward ARX model in order to apply the Burg-method as follows:y(n ? M) =M ?m=1 e(M) m y(n ? M + m) +M ?m=1 f (M) m u(n ? M + m) + η (M) y (n) , (5)u(n ? M) =M ?m=1 g(M) m u(n ? M + m) +M ?m=1 h(M) m

23、y(n ? M + m) + η (M) u (n) , (6)x(n ? M) =M ?m=1 B(M) m x(n ? M + m) + η (M)(n) , (7)where the backward prediction error vector η (M)(n) and the backward prediction coeffi cientmatrix B(M) m (n) can be expressed asη (M)(

24、n) =? ? ? ? ? ? η (M) u η (M) y? ? ? ? ? ? , B(M) m =? ? ? ? ? ? h(M) m g(M) m f (M) m e(M) m? ? ? ? ? ? .The basic concept of the Burg-method is that prediction coeffi cients are calculated inorder to minimize the sum o

25、f prediction error variances in Eqs. (4) and (7) by solving thefollowing recurring formulas obtained by such a condition:? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?A(M) M = R(M) ? N ?n=M η (M?1)(n ? 1) η (M?1)T(n ? 1)??1 ,B(M) M =

26、 R(M)T ? N ?n=M ε (M?1)(n) ε (M?1) T(n)??1 ,(8)? ? ? ? ? ? ? ? ? ? ?A(M) m = A(M?1) m ? A(M) M B(M?1) M?m ,B(M) m = B(M?1) m ? B(M) M A(M?1) M?m ,(9)? ? ? ? ? ? ? ? ? ? ?ε (M)(n) = ε (M?1)(n) ? A(M) M η (M?1)(n ? 1) ,η (