Simple Modification of Proper Orthogonal Coordinate Histories for Forced Response Simulation

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1 Simple Modification of Proper Orthogonal Coordinate Histories for Forced Response Simulation Timothy C. Allison, A. Keith Miller and Daniel J. Inman I. Review of Computation of the POD The POD can be computed by many methods. This paper explains how the POD is computed with the singular value decomposition (SVD) of a snapshot matrix. First, a system response is generated by either forcing the system or imposing an initial velocity or displacement profile. In this paper, we will assume that forcing function is applied to generate a response. Next, the displacement at n degrees of freedom is sampled m times and the data are arranged in a snapshot matrix W : w 1 (t 1 ) w 1 (t 2 ) w 1 (t m ) w W = 2 (t 1 ) w 2 (t 2 ).... (1) w n (t 1 ) w n (t m ) Next, the SVD of W is computed: W = UΣV T. (2) In equation (2), the columns of U R n n are the POMs, the columns of V R m n are the proper orthogonal coordinate (POC) time histories that correspond to each POM and, Σ R n n is a diagonal matrix whose diagonal elements σ i are the proper orthogonal values (POVs) corresponding to each POM. The POC histories describe the amplitude modulation of each POM, and the POVs describe the relative significance of each POM in the sampled displacement field. The percentage of energy in the signal captured by the i th POM is given by σ i ɛ i = n σ. (3) i Typically, only POMs that constitute a certain percentage of signal energy (e.g. 99% or 99.9%) are considered. II. Analytical Solution for Forced Response of a Linear Beam is The boundary value problem for the forced transverse vibration of a cantilever beam with no axial force ( ) m(x) 2 w 2 + d(x) w + 2 x 2 EI(x) 2 w x 2 = p(x, t) (4) Ph.D. Candidate, Center for Intelligent Material Systems and Structures, Virginia Polytechnic Institute and State University, Department of Mechanical Engineering, 31 Durham Hall, Mail Code 261, Blacksburg, VA 2461, Phone: , Fax: , talliso@vt.edu, AIAA Member. Ph.D. and Principal Member of Technical Staff, Sandia National Laboratories, Analytical Structural Dynamics Department, P.O. Box 58, Albuquerque, NM , Phone: , Fax: , akmille@sandia.gov, AIAA Member. G.R. Goodson Professor and Director, Center for Intelligent Material Systems and Structures, Virginia Polytechnic Institute and State University, Department of Mechanical Engineering, 31 Durham Hall, Mail Code 261, Blacksburg, VA 2461, Phone: , Fax: , dinman@vt.edu, AIAA Member.

2 with boundary conditions and initial conditions w = at x = (5) w = at x = (6) ( x ) EI 2 w x x 2 = at x = L (7) EI 2 w = at x = L (8) x2 w = f(x) at t = (9) w = g(x) at t =, (1) where w is the transverse displacement of a beam with length L, mass per unit length m(x), distributed damping d(x), and flexural rigidity EI(x) subjected to a transverse forcing function p(x, t). This problem may be solved by assuming a solution of the form w(x, t) = φ j (x)t j (t) (11) j=1 where φ j (x) are shape functions (not necessarily eigenmodes) for the system. Inserting equation (11) into equation (4), multipying both sides by φ i (x), and integrating over the interval [, L] gives a differential equation governing the generalized coordinate functions T j (t): m ij 2 T j 2 + T j d ij + k ij T j = q j (t), j = 1, 2, (12) where m ij, d ij, k ij, and q j (t) are defined as k ij = m ij = d ij = 2 x 2 q j (t) = Equation (12) can be written in matrix form as M 2 T(t) 2 m(x)φ i (x)φ j (x)dx. (13) d(x)φ i (x)φ j (x)dx. (14) ( EI(x)φ i (x) 2 φ j (x) x 2 + D T(t) ) dx. (15) φ j (x)p(x, t)dx. (16) + KT(t) = q(t) (17) If the shape functions φ j (x) are eigenmodes, then M and K are diagonal. In general, however, M and K are not diagonal and the solution to equation (17) is found by transforming into state form where s(t), A, and B are defined as s(t) s(t) = = As(t) + Bq(t) (18) [ T(t) T(t) ] (19)

3 A = [ I M 1 K M 1 D [ ] B = M 1 The solution to equation (17) can be written directly in the form ] (2) (21) s(t) = Ψ(t)s() + in which Ψ(t) is the state transition matrix, defined by the infinite series Ψ(t) = t i= Ψ(t τ)bq(τ)dτ (22) t i i! Ai (23) If the beam starts at rest, i.e. f(x) = g(x) =, then s() = and the solution for T(t) is T(t) = t C(t τ)q(τ)dτ (24) which is simply a convolution integral. The matrix C(t) is the upper half partition of Ψ(t)B and is only diagonal if the shape functions φ i (x) are the eigenmodes. Once the generalized coordinate functions T i (t) are found from equation (24), the beam response can be obtained from equation (11). It is important to note that the shape functions φ i (x) are unaffected by the forcing function. In other words, the same shape functions are used to form response solutions to all loading conditions. The generalized coordinate functions T i (t) depend on the forcing function because of the q(t) term in the convolution integral in equation (24). If the matrix product C(t) is known, the response to a variety of forcing functions may be obtained by changing the q(t) term in equation (24). III. Forced Response Calculation Using POD We now address the POD from the viewpoint of the methodology in section II. Suppose the snapshot matrix D is formed from the response to a forcing function F. We compare the shape functions φ i (x) to the POMs U i and the generalized coordinate functions T i (t) to the POC histories V i. The POVs σ i are simply scalars and we may simplify the problem by lumping them into the POC histories, i.e. if we modify V i as follows V i = σ i V i (25) then we may reconstruct a low-order approximation of the displacement snapshot matrix by W = U i V T i. (26) This expression for W is quite similar to the expression for w(x, t) in equation (11). The i th POC history V i is equal to the time-sample generalized coordinate function T i (t). We now wish to use the theoretical framework established in section II to modify each POC history in order to predict the response to a forcing function ˆF other than the one used to generate W. First, we calculate the modal force matrices by forming inner products of the POMs with the original measured force matrix F Q = U T F (27) and with the new forcing function ˆF ˆQ = U T ˆF. (28) We can now modify the POC histories by doing a numerical deconvolution to eliminate the effects of the modal forcing functions in Q. The deconvolution will give us the matrix C(t) in equation (24) without

4 computing it directly using the modal mass or stiffness matrices. After we have found C(t) in this manner, we may employ numerical convolution to impose the effects of the new modal forcing functions in ˆQ. C(t) is generally not diagonal and each generalized coordinate function depends on a convolution with every generalized force, i.e. T i (t) = C ij (t) q j (t). (29) j=1 In this paper, however, we assume that the POMs are similar to the eigenmodes and that the modal mass and stiffness matrices are diagonally dominant. This assumption is reasonable in many cases. In fact, if the system is linear and lightly damped with a diagonal (finite element) mass matrix, the POMs are equal to the eigenmodes. 1 We also assume that the damping is either light or modal so that D is insignificant compared to K or diagonally dominant. If we make these assumptions then C(t) is also diagonally dominant and the convolutions in equation (29) are uncoupled: T i (t) = C ii (t) q i (t) (3) In this paper, the deconvolution and convolution are performed in the time domain. First, we will discuss the convolution of two discrete signals and then the deconvolution of them. Suppose that we desire to form v(t) as the convolution of signals q(t) and s(t), i.e. v(t) = c(t) q(t). (31) In discrete form, we have v, s, and q as vectors of length m. The value of v at time t i can be formed as v(t i ) = t i c(t j )q(i 1 j). (32) Performing the operation in equation (32) for every time step and writing in matrix form yields j=1 v = tqc (33) where Q = q(t 1 ) q(t 2 ) q(t 1 ) q(t 3 ) q(t 2 ) q(t 1 ) q(t m ) q(t m 1 ) q(t 2 ) q(t 1 ) is a convolution matrix. We may now deconvolve q out of v by performing the multiplication (34) c = 1 t Q 1 v. (35) We now let q(t) equal the i th modal force from equation (27) and ˆq(t) the i th new modal force from equation (28). Q and ˆQ are convolution matrices formed from q(t) and ˆq(t), respectively. As long as Q is full rank we may calculate each modified POC history as follows: Finally, the displacement response to the new force ˆF can be formed as ˆ V i = ˆQ i Q 1 i V i (36) Ŵ = U i ˆ V T i (37) This method will be most accurate for forcing functions that generate shapes in the response similar to those found in W. When forming W, it is desirable to choose forcing functions that excite the widest variety of shapes in the response in order to generate POMs that can represent responses to a large selection of forcing functions.

5 IV. Examples This section applies the methods described in section III to two models: an undamped beam and a jointed beam with a dashpot at the end (see Figure 1). Finite element (FE) models were created for each beam, and a four-parameter Iwan model 2 was used to model the joint. These models were used to simulate the exact response of each beam to several forcing functions applied vertically at the tip of each beam, shown in Figure 2. The responses of each model to the impulsive tip force F 1 were simulated for.5 seconds and vertical displacements at 24 points (linear beam) and 25 points (nonlinear beam) were captured every.1 seconds to form W. Thus the dimensions of W were (24 x 5) for the linear model and (25 x 5) for the nonlinear model. The nonlinear beam displacement was captured next to both sides of the joint to allow for representation of the joint slip. Figure 1. (A) Linear Beam Model and (B) Nonlinear Beam Model (a) Impulsive F 1 (t) (b) Random F 2 (t) (c) Constant F 3 (t) Figure 2. Forcing functions Applied Vertically to Each Beam Tip Next, the POD was computed for each beam, and the methods explained in section III were applied to simulate the responses of both systems to the random forcing function F 1 and constant tip force F 2 using 6 POMs. The tip displacements at each time step for the linear beam responses are shown in Figures 3-4. The tip displacements for the nonlinear beam responses are shown in Figures 5 and 6. The joint slip in the nonlinear model is shown in Figures 7 and 8. References 1 Feeny, B. F. and Kappagantu, R., On the Physical Interpretation of Proper Orthogonal Modes in Vibrations, Journal of Sound and Vibration, Vol. 211, 1998, pp Segalman, D. J., An Initial Overview of Iwan Modeling for Mechanical Joints, SAND21-811, Sandia National Laboratories, Albuquerque, NM, March 21.

6 Figure 3. Tip Displacement of Linear Beam Model s Response to F 2 Figure 4. Tip Displacement of Linear Beam Model s Response to F 3 Figure 5. Tip Displacement of Nonlinear Beam Figure 6. Tip Displacement of Nonlinear Beam Model s Response to F 2 Model s Response to F 3 Figure 7. Joint Slip in Nonlinear Beam Model s Response to F 2 Figure 8. Joint Slip in Nonlinear Beam Model s Response to F 3