The Research Of Structural Reanalysis Algorithms And Application In Vibration System

Combined approximations are used for reanalysis and repeated analysis. The advantage is that the efficiency of local approximations and the improved quality of global approximations are combined to obtain effective solution procedures.The equations of motion for non-linear dynamic analysis areM^tr + C^tr + ^tF_s = ^tR (14)WhereWhen t +â–³t, The equations of motion isM ^t+â–³tr + C ^t+â–³tr + ^t+â–³tF_s =^t+â–³tR (16)ThenM^tâ–³r + C^tâ–³r +^tâ–³F_s = ^tâ–³R (17)Using the first-order Taylor series approximations of the resisting force we obtainM ^tâ–³r + C^tâ–³r +^tK ^tâ–³r = ^â–³R (18)Equation (14) must be solved repeatedly many times. Consider ing modal analysis,so we have Where ^tΦ_k are the non-linear mode shapes at time t.To obtain these modes and the corresponding eigenvalues ^tΦ_k, we solve the non-linear eigenproblem:^tK^kΦ_k=^tλ_k^tM^tΦ_k (20)Where ^tK is tangent stiffness matrix. Since the stiffness matrix is changed in time, the values of ^tΦ_k and ^tλ_k are changed accordingly. Evidently, repeated solutions of Eq. (18) involve more computational effort than solution of a linear eigenproblem. Eq.(18)consists of the p uncoupled equations:^tâ–³Z_k + 2^tω_kζ_k^tâ–³Z_k+^tω_k²â–³Z_k= ^tP_k ,k = 1,2,…,p (21)In this study eigenproblems analysis ,using the recently developed combined approximationsmethod and usually eigenmethods including Inverse Iteration(RII), Inverse Iteration with Shifts(RIIS), Subspace Interation(SI) and Refined Shift-and-Invert Arnoldi Algorithm(RSIA),is presented. And in this paper, a new refined CA method is presented. As follows:(a) Refined initial matrices K₀,M₀, and the corresponding eigenpairsΦ_0i,λ_0i, i = 1,…,p using RSIA and SI;(b) Calculate the modified matrices K, M;(c) Calculate the matrix of basis vectors r_Br_B = [r₁,r₂,…,r_s] (22)Where r₁, r₂,…, r_s are the basis vectors,and s is much smaller than the number of degrees of freedom n, i.e s << n. For anyΦ_i,λ_i, the basis vectors are determined separately;(d) Calculate the reduced matrices K_r, M_r byK_R = r_BKr_B^T , M_R = r_BMr_B^T (23)(e) Solve the reduced s x s eigenproblem for the first eigenpairλ₁,y₁,K_RY₁ =λ₁M_Ry₁ (24) Where y₁ is a vector of unknown coefficientsy₁^T = [y₁,y₂,…,y_s] (25)Solve (21) using the Refined eigenmethods(e.g.RII ,RIIS or RILCA).(f) Evaluate the requested mode shapeΦ_iΦ_i = y₁r₁+y₂r₂ +…+ y_sr_s = r_By₁ (26)The requested eigenvalue is already given from Eq. (24):λ_i =λ₁Next, Consider the equations (14) of motion at time t for a system subjected to dynamic forces, using Nemark method , finite differences or a new method reduced in this paper.