DETECTION OF NONLINEARITY IN VIBRATIONAL SYSTEMS USING THE SECOND TIME DERIVATIVE OF ABSOLUTE ACCELERATION

Transcription

1 DETECTION OF NONLINEARITY IN VIBRATIONAL SYSTEMS USING THE SECOND TIME DERIVATIVE OF ABSOLUTE ACCELERATION Masaki WAKUI 1 and Jun IYAMA and Tsuyoshi KOYAMA 3 ABSTRACT This paper shows a criteria to detect nonlinearity in the restoring force using the nd tie derivative of the absolute acceleration, which is often referred to as snap. In order to investigate the capability of snap in detecting changes in the stiffness of the syste, earthquake response analysis was conducted. The analysis results show that discontinuous jup like signals, which coincide with the yielding locations, can be observed in the snap tie histories. Thus one can say that in general, it is possible to detect daage due to change in stiffness through observation of the snap tie history. INTRODUCTION Much research has been conducted in the area of structural health onitoring and daage detection, in order to develop ethods which are capable of detecting aging deterioration and daage due to earthquakes as well as estiating the aintenance tie. When a steel structure experiences a strong earthquake, daage such as yielding and fracture occur in structural ebers. Such daage appears as degradation in stiffness and strength. Most developed ethods estiate this type of daage through the dynaic transfer function of the structure, which requires either easureent of ground otion or icrotreor input as well as the response of the structure. Such ethods are coplicated and costly, since the input and output ust be synchronized. Therefore it is advantageous to be able to detect daage of structures and their coponents only through the output. This paper investigates a ethod which is capable of detecting daage due to nonlinearity in the restoring force using the nd tie derivative of the absolute acceleration, which is often referred to as snap. Such a ethod is possible when the degradation of the stiffness occurs abruptly, resulting in discontinuities in the tie history of snap. The ethod is cost effective, since snap can be obtained nuerically as the nd tie derivative of the easured absolute acceleration, a quantity easily easured. The physical significance of snap is discussed in this paper as well as its applicability to the detection of daage and nonlinearity in the stiffness of structural ebers. DETECTING CHANGES IN THE STIFFNESS THROUGH SNAP In this paper, the tie derivative of the jerk is referred to as snap. 1 Graduate Student, The University of Tokyo, Tokyo Japan, wakui@stahl.arch.t.u-tokyo.ac.jp Assoc. Prof., The University of Tokyo, Tokyo Japan, iyaa@arch.t.u-tokyo.ac.jp 3 Asst. Prof., The University of Tokyo, Tokyo Japan, koyaa@arch.t.u-tokyo.ac.jp 1

2 A single degree of freedo dynaical syste with nonlinear restoring force Q( can be represented as, x cx Q( x. (1) Here, is the structural ass and c is the daping coefficient. The ters x, x, and x represent the relative acceleration, velocity, and relative displaceent; x is the ground acceleration. When there is no daping, c =, one obtains the following equation. ( x x ) Q(. () By defining the absolute acceleration a x x, the equation above can be rewritten as follows, A tie derivative of this equation yields, Q( a. (3) 1 dq( dx a, (4) dx dt where a is the jerk. By defining the tangent stiffness of the restoring force k( dq( dx, Eq.(4) becoes, 1 dx 1 a k( k( x. (5) dt Another tie derivative of this equation yields the snap a as, 1 dk( dx dx 1 a x k( k'( x k( x. (6) dx dt dt Here, k' ( is defined as the derivative of k( with respect to the relative displaceent. In the case that a syste behaves linear elastically with constant initial stiffness k, Eq.(6) reduces to, 1 a k x. (7) Therefore, in a non-daaged syste with constant initial stiffness, snap will haronically oscillate proportionally to the relative acceleration. On the other hand, in a syste with variable stiffness, such as an elasto-plastic syste, k' can becoe non-zero. Eq.(6) shows that a sudden change in the stiffness can result in an instantaneous large value of snap. If this change in snap can be properly observed, it can lead to a daage detection ethod for systes in which the stiffness is affected by the accuulated daage. However, it is not an easy task, since the first ter in Eq.(6)includes both k' and the square of the relative velocity; when the relative velocity is alost zero, the contribution of the first ter is sall copared to the second and difficult to detect. When a syste is vibrating linearly elastically, Q( is less than Q y, the yield force of the syste. In such a case, the following inequality can be derived fro Eq.(), x x Q y. (8)

3 Using the triangle inequality, the equation above can be rewritten as follows, Qy x x. (9) If the predoinant period of the ground acceleration is nearly identical to the natural period of the syste, x will be uch saller than x. In such a case, Q y / in Eq.(9) can be considered an approxiate upper bound of x, and an estiate of the axiu absolute acceleration of the undaaged syste in Eq.(7) can be given as, k * Qy a ax. (1) The nuerical approxiation to snap can be coputed fro the nd order backward difference of the absolute acceleration using the following forula. a( t) a( t t) a( t t) a ( t). (11) t It is possible to detect nonlinearity in vibrational systes by coparing the threshold value calculated fro Eq.(1) with the snap tie history calculated fro the easured absolute acceleration tie history through Eq.(11). VALIDATION OF SNAP WITH EARTHQUAKE RESPONSE ANALYSIS In order to investigate the actual capability of snap in detecting changes in the stiffness of the syste, earthquake response analysis was conducted. The odel is a single degree of freedo elasto-plastic vibrational syste which has a predoinant period of 1. seconds and a daping coefficient of %. The three types of restoring force characteristics shown in Fig.1 are selected for the single degree of freedo syste. The horizontal axis represents the displaceent and the vertical axis force. δ y, Q y, and Q u represents the yield displaceent, the yield force, and the axiu force. As shown in Fig., these three types differ in the way the stiffness decreases to zero after yielding, where Case.1 is discontinuous, Case. is linearly varying, and Case.3 is parabolically varying. In Fig., the horizontal axis represents the restoring force and the vertical axis the stiffness. Figure 1. Restoring force characteristics 3

4 Figure. The relation between the restoring force and the stiffness Case.1(discontinuous) : The stiffness of the syste is initially k, and discontinuously drops to zero when the restoring force reaches the yield force. Thus k' will be infinite at this point and so will the value of snap. Case.(linearly varying) : The stiffness of the syste is initially k, and linearly decreases after the restoring force reaches the yield force. At the axiu restoring force, the stiffness will becoe zero. The axiu restoring force is defined as Q u, and is related to the yield force through the equation Q. k' can be represented as, y Q u k k'. (1) (1 ) Q u k' is a constant which depends on k, Q u, and. Because k' is discontinuous, the snap is also discontinuous in this restoring force odel. Case.3(parabolically varying) : The stiffness of the syste is initially k, and parabolically decreases after the restoring force reaches the yield force. In this case, k' linearly decreases, and k'' is a constant. At the axiu restoring force, the stiffness will becoe zero. The axiu restoring force is defined as Q u, and is related to the yield force through the equation Q Q. k'' can be represented as, y u 3 ) Q u 8k k''. (13) 9(1 k'' is a constant which depends on k, Q u, and. Since k' is continuous, the snap is also continuous in this restoring force odel. The axiu force Q u is defined as, Q. 4. (14) u Q e Here, Q e is the axiu elastic force response of the syste under the given ground otion., which deterines the upper bound for the rate at which the stiffness decreases after yielding for Case. and Case.3, is fixed as.9. A value close to 1 iplies that the rate is extreely large. The Newark Beta ethod is used to integrate the equations of otion with a fixed tie step of t.4seconds. INPUT EARTHQUAKE MOTION Since the ethod presented above is developed in the case where the predoinant period of the ground acceleration is nearly identical to the natural period of the syste, an artificial ground otion 4

5 which has the desired tie-frequency characteristic is generated and used. The Fourier aplitude spectru is the given as the Kanai-Tajii spectru, where the predoinant period is set to 1. seconds and the coefficient hg deterining the spectral shape to.3. The standard deviation of the Fourier phase difference distribution is.15, which corresponds to the Fourier phase spectru of an interediate-field earthquake. The artificial earthquake otion is generated by an inverse Fourier transfor of the Fourier phase and Fourier aplitude spectru. The earthquake data is sapled at 5Hz, and the nuber of data points is 496. This results in a ground otion with a duration of 81.9 seconds. Fig.3 shows the Fourier aplitude spectru and the artificial ground otion. The earthquake data is linearly interpolated to atch the tie step in the earthquake response analysis. Figure 3. Artificial ground otion NUMERICAL RESULTS Fig.4 shows the snap tie history calculated fro Eq.(11) using quantities obtained fro the tie history response analysis of the three cases. The horizontal axis represents the tie and the vertical axis the snap. Moreover, the red line shows the threshold calculated fro Eq.(1), and the yellow circles denote yielding, i.e., change in the value of the stiffness. An enlarged figure of the portion of the tie history enclosed by the blue ellipse is shown beside the tie histories. The alphabet (A~E) shows the correspondence between the tie history of snap and the load deforation curve. Fig.4(a), which corresponds to Case.1(discontinuous), shows discontinuous jup like signals. These coincide with the yielding locations, such as point B. As previously explained, k is discontinuous at the onset of yielding, and the snap instantaneously increases or decreases. At the yield force, the stiffness and its derivative k' are zero, so the value of snap also becoes zero fro Eq.(6) at point D. The value of snap will reain zero fro point D until unloading at point E. At the onset of unloading, the stiffness k and relative acceleration are both nonzero, resulting in a nonzero value of snap, as can be seen fro Eq.(6) In Fig.4(b), which corresponds to Case.(linearly varying), discontinuous jup like signals, which coincide with the yielding locations, such as the point B, can be observed in the snap tie histories. At the axiu force, the stiffness and its derivative k' are zero, so the value of snap also becoes zero fro Eq.(6) at point D. The value of snap will reain zero fro point D until unloading at point E. This is siilar to Case.1. The difference between Case.1, is the gradual decrease of snap fro point C to point D. This is attributed to the fact that the decreent of the relative velocity is larger than the increent of the relative acceleration. In Fig.4(c), which corresponds to Case.3(parabolically varying), k' is continuous, and thus so is snap. k' decreases after yielding at the point B, and the value of snap increases fro point B to point C. After reaching the axiu force, the value of snap becoes zero, siilar to the two previous cases. The value of snap reains zero fro point D until unloading at point E. For all three cases, the value of snap exceeds the threshold value calculated fro Eq.(1) whenever yielding occurs, i.e., the stiffness changes. This change in stiffness or nonlinearity in the restoring force can be detected through the snap tie history not only when k and k' are discontinuous, such as Case.1 and Case., but also in the case where they are continuous, such as Case. 3. This clai 5

6 holds true as long as the change in stiffness is rapid. When the change in stiffness as well as the relative velocity is sall, the spiky behavior of snap also becoes sall and the nonlinearity becoes difficult to detect. (a)case.1(discontinuous) (b)case.(linearly varying) (c)case.3(parabolically varying) Figure 4. The nuerical results CONCLUSIONS This paper presents a ethod to detect daage due to change in the stiffness of structures using snap, the second tie derivative of the absolute acceleration. Since the ethod requires easureent of only the absolute acceleration, it can be easily and cost effectively ipleented. In the case of a single degree of freedo elasto-plastic syste without daping, the expression for calculating snap includes an additive ter involving the product of the derivative of the tangent stiffness and the square of the relative velocity. At the onset of yielding, if the change in stiffness of the syste is abrupt or nearly discontinuous, the value of snap will instantly increase and result in sharp peaks. By specifying an appropriate threshold, these peaks can be detected as the points at which yielding supposedly occur. In this paper, the proposed ethod is validated for single degree of freedo systes with 3 types of restoring force characteristics, which differ in the way the stiffness decreases to zero after yielding, where Case.1 is discontinuous, Case. is linearly varying, and Case.3 6

7 is parabolically varying. Case.1 and Case. show discontinuous jup like signals in the snap tie history since k and k' are discontinuous at the onset of yielding. In Case.3, k and k' are continuous at the onset of yielding, and the snap tie history is also continuous. For all three cases, the value of snap exceeds the threshold value whenever yielding occurs, so in the particular case presented in this paper, it is possible to detect nonlinearity in the vibrational systes. This is because the ability of snap in detecting nonlinearity does not depend solely on the discontinuity of the stiffness but on its rapid variation. In future work, the effect of noise arising in actual easured accelerations on the proposed ethod will be investigated. Such noise can be severely troublesoe in coputing snap through nuerical differentiation and a rational ethod ust be developed to eliinate the. Another issue that ust be addressed is the dependence of the ability of snap in detecting nonlinearity on the speed of variation of k with respect to displaceent. In this paper β is fixed as.9, but the appropriate value for β in real structures is an open issue. REFERENCES Julien C S (1997) Soe siple chaotic jerk functions, A. J. Phys. 65(6), June: Hiroshi T (196) A statistical ethod of deterining the axiu response of a building structure during an earthquake, Proceedings of the nd World Conference on Earthquake Engineering, Tokyo, Japan: Yoshihiro I and Hitoshi K (3) Earthquake input otions in perforance evaluation of steel building structures (Part Spectral Characteristics of Typical Recorded Earthquake Motions), Proc., 73th Archival Research Meeting, Chapter AIJ, 1: (in Japanese) 7