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• W2034802257 abstract "In 1, the authors consider n-degree-of-freedom non-classically damped linear systems. They write the system representation in the modal coordinates as (1)X¨t+2CX˙t+Λ2Xt=0,X0=X0,X˙0=X˙0for all t⩾0, where the vector of displacements Xt∈Rn, the symmetric and non-negative matrix C∈Rn×n corresponds to the modal damping matrix, and the diagonal matrix (2)Λ2=diag[ω12,ω22,…,ωn2]∈Rn×nhas the square of the undamped natural frequencies of the system, ω12,ω22,…,ωn2, on its diagonal. The authors write that “The currently available methods for vibration analysis of the non-classically damped linear system include: direct numerical integration of the equations of motion;….” They further write that such a technique “only had limited applications. Several factors contribute to this including either huge computational effort involved, questionable accuracy,….” In order to circumvent “drawbacks” of the available techniques, the authors propose the following time-varying transformation (3)Xt=e−CtYtfor all t⩾0, where Yt∈Rn. Application of the transformation (3) to system (1) results in Y¨t+eCtΛ2−C2e−CtYt=0,(4)Y0=X0,Y˙0=CX0+X˙0for all t⩾0. The transformed system (4) is a set of 2nd-order differential equations without a damping term, i.e., a Y˙s˙ term. The authors propose a numerical technique to approximately solve system (4) for Ys˙, and consequently obtain Xs˙ via (3). Their approximate solution is not accurate as is evident from the example presented in 1. Regarding system (4), the authors, however, claim that “The new set of 2nd-order differential equations leads to computational advantage over the original system….” In this Discussion, it is shown that the transformed system (4) is not an easy system to analyze and is not suitable for computing the system responses: i) it requires large computational effort; ii) it results in less accurate solutions. The state-space representations of systems (1) and (4) are, respectively, (5a)ddt XtX˙t=AXtX˙t,X0X˙0=X0X˙0(5b)ddt YtY˙t=BtYtY˙t,Y0Y˙0=X0CX0+X˙0for all t⩾0, where A=0In−Λ2−2C∈R2n×2n,(6)Bt=0In−eCtΛ2−C2e−Ct0∈R2n×2nand In is the n×n identity matrix. In the following, it is argued why system (5a) [equivalently, (1)] is more suitable than system (5b) [equivalently, (4)] for analytical studies and numerical integration: 1) System (5a) is time-invariant, hence, its stability can be decided by the eigenvalues of the matrix A in (6). System (5b) is time-varying, the stability of which may not be easily established. It is well known that sets of eigenvalues of Bt for every fixed (frozen) t⩾0 cannot establish the stability of system (5b); see, e.g., 23. 2) It may happen that system (1) is stable, however, the transformation (3) results in an unstable transformed system (4). That is, it may happen that t↦Yt and t↦Y˙t are unbounded functions of time. In this case, solving system (4) for Ys˙ and then obtaining Xs˙ via (3) would be computationally demanding and can possibly lead to less accurate Xs˙.3) Solving system (5b) for Ys˙ requires the computation of matrices eCt and e−Ct. Once Ys˙ is obtained, Xs˙ is obtained via the transformation (3), which depends on e−Ct. In 1, the authors propose an approximate technique to compute the exponential of matrices at finitely many instances of time. It should be remarked that the computation of matrix exponentials is numerically a challenging problem: There could be reliability, stability, accuracy, efficiency, and storage problems 4. Such considerations attest that the transformed system (4) and the transformation (3) are not suitable for computing the solutions of system (1). In order to elucidate the points made above, the example given in 1 is revisited. In this example, the coefficient matrices of system (1) are (7)C=0.1−0.1−0.10.1,Λ2=1004For the matrices in (7), it can be easily verified that the matrix A in (6) is Hurwitz. Therefore, Xt and X˙t both converge to the zero vector in R2 as t→∞.By integrating system (5b) numerically, it is observed that this system is unstable. The time history of t↦y1t in the solution vector t↦Yt≔[y1t y2t]T is depicted in Fig. 1. It is evident that |y1t|→∞ as t→∞. By applying the transformation (3), the solution t↦Xt≔[x1t x2t]T can be obtained, where both x1t and x2t converge to zero as t→∞.For the matrices in (7), systems (5a) and (5b) were integrated numerically over the time interval of [0,100] using the Runge-Kutta 45 algorithm in MATLAB5. A fixed step size of 0.005 was used in the integration. Moreover, conditions of integration for both systems (5a) and (5b) were chosen identically. Each system was integrated 10 times and the spent CPU time was recorded for each integration using the command cputime in MATLAB. It was determined that, on average, system (5a) used 1.23 CPU time units, whereas system (5b) used 7.65 units. As it was explained earlier, the numerical integration of system (5b) is computationally demanding. Based on the preceding arguments, this Discussion draws the following conclusions: It is not advantageous to transform the time-invariant modal representation of non-classically damped systems via the transformation (3) to the time-varying system (4). The reason is that the transformed system (4) could be unstable. Moreover, since the transformed system depends on the exponentials of matrices: i) large computational effort is needed to compute its solution; ii) the solution could be less accurate." @default.
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• W2034802257 date "2005-02-01" @default.
• W2034802257 modified "2023-09-24" @default.
• W2034802257 title "Discussion on “Vibration Analysis of Non-Classically Damped Linear Systems”" @default.
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• W2034802257 doi "https://doi.org/10.1115/1.1857926" @default.
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