MEEN 617 – HD 8 Modal Analysis with Proportional Damping. L. San Andrés © 2008 1Handout 8 Modal Analysis of MDOF Systems with Proportional Damping The governing equations of motion for a n-DOF linear mechanical system with viscous damping are: () ()ttMU+DU+KU =F  (1) where andU, U, Uare the vectors of generalized displacement, velocity and acceleration, respectively; and ()tF is the vector of generalized (external forces) acting on the system. M,D,K represent the matrices of inertia, viscous damping and stiffness coefficients, respectively1. The solution of Eq. (1) is uniquely determined once initial conditions are specified. That is, (0) (0)at 0 ,oot =→ = =UUUU (2) Consider the case in which the damping matrix D is of the form αβ=+DMK (3) where α, β are constants2, usually empirical. This type of damping is known as PROPORTIONAL, i.e proportional to either the mass M of the system, or the stiffness K of the system, or both. 1 The matrices are square with n-rows = n columns, while the vectors are n-rows. 2 These constants have physical units, α is given in [1/sec] and β in [sec]MEEN 617 – HD 8 Modal Analysis with Proportional Damping. L. San Andrés © 2008 2Proportional damping is rather unique, since only one or two parameters, α, β, appear to fully describe the complexity of damping, irrespective of the system number of DOFs, n. This is clearly not realistic. Hence, proportional damping is not a rule but rather the exception. Nonetheless the approximation of proportional damping is useful since, most times damping is quite an elusive phenomenon, i.e. difficult to model (predict) and hard to measure but for a few DOFs. Next, consider one already has found the natural frequencies and natural modes (eigenvectors) for the UNDAMPED case, i.e. given MU+KU=0, {}()1,2...,iiinω=φsatisfying 2() 1,...,iiinω=⎡⎤−⎣⎦M+Kφ =0. (4) with properties [][];TTMK==Φ M ΦΦK Φ (5) As in the undamped modal analysis, consider the modal transformation () ()tt=U Φ q (6) And with () () () ();tttt==U Φ qU Φ q, then EOM (1) becomes: ()t+M Φq+DΦqKΦq=F  (7) which offers no advantage in the analysis. However, premultiply the equation above by TΦ to obtain ()()()()TTTTt+Φ M Φ q+ Φ DΦ q Φ K Φ q=Φ F  (8) And using the modal properties, Eq. (5), andMEEN 617 – HD 8 Modal Analysis with Proportional Damping. L. San Andrés © 2008 3 ()TT T Tαβ α β=+= +Φ DΦΦ MKΦΦM ΦΦK Φ [][][]TMKDαβ=+→Φ DΦ (9) i.e. a diagonal matrix known as proportional modal damping. Then Eq. (7) becomes [][][]()TtMDK+=q+ q q=Q Φ F  (10) Thus, the equations of motion are uncoupled in modal space, since [M], [D], and [K] are diagonal matrices. Eq. (10) is just a set of n-uncoupled ODEs. That is, 11 11 11 122 22 22 2.....nn nn nn nMqDqKqQMqDqKqQMqDqKqQ++=++=++=    (11) Or 1,2...,jj jj jj j j nMq Dq Kq Q=++=  (12) Where jjjKMnω= and jjjDM Kαβ=+ . Modal damping ratios are also easily defined as 22jjjjjj jjDMKKM KMαβζ+== ; j=1,2,….n (13) For damping proportional to mass only, 0β=, and 22jjjnjjMKMααζω== (13a)MEEN 617 – HD 8 Modal Analysis with Proportional Damping. L. San Andrés © 2008 4i.e., the j-modal damping ratio decreases as the natural frequency increases. For damping proportional to stiffness only, 0α=, (structural damping) and 22jnjjjjKKMβωβζ== (13b) i.e., the j-modal damping ratio increases as the natural frequency increases. In other words, higher modes are more increasingly more damped than lower modes. The response for each modal coordinate satisfying the modal Eqn. 1,2...,jj jj jj j j nMq Dq Kq Q=++=  proceeds in the same way as for a single DOF system (See Handout 2). First, find initial values in modal space {},jjooqq. These follow from either 11;ooo o−−==q Φ UqΦ U (14) or [][]11,TooTooMM−−==q Φ MUq Φ MU (15a) ()()() ()11,kkTTokookokkqqMM==φ MU φ MU (15b) k=1,….nMEEN 617 – HD 8 Modal Analysis with Proportional Damping. L. San Andrés © 2008 5Free response in modal coordinates Without modal forces, Q=0, the modal EOM is 0jj jjH jH jH jMqDqKq Q++==  (16) with solution, for an elastic underdamped mode 1jζ< ()()()cos sinjdjjjjtHjdjdqe C tS tζωωω−=+ if 0jnω≠ (17a) where 21,jjjj jKMdn jnωω ζω=− =and ;jjjjjojnojo jdqqCq Sζωω+== (17b) See Handout (2a) for modal responses corresponding to overdamped and critically damped SDOF system. Forced response in modal coordinates For step-loads, SjQ , the modal equations are jj jj jj SjMqDqKqQ++=  (18) with solution, for an elastic underdamped mode 1jζ< ()()()cos sinjdjjjjjtjdjd Sqe C tS t qζωωω−=++ 0jnω≠ (19a)MEEN 617 – HD 8 Modal Analysis with Proportional Damping. L. San Andrés © 2008 6where 21,jjjj jKMdn jnωω ζω=− = and ();;jjjjjjjSojnjSjoSjjdQqCqCqqSKζωω+==−= (19b) See Handout (2a) for physical responses corresponding to overdamped and critically damped SDOF system. For periodic-loads, Consider the case of force excitation with frequency jnωΩ≠ and acting for very long times. The EOMs in physical space are ()cos t+ΩPMU+DU KU=F  The modal equations are cos( )jjj jj jj PMqDqKqQ t++= Ω  (20) with solutions for an elastic mode, 0jnω≠ () ()()() ()()cos sincos sinjnjjjjjj transient ss ttjdjdcsqq qeC tS tCtCtζωωω−=+=++Ω+ Ω (21) The steady state or periodic response is of importance, since the transient response will disappear because of damping dissipative effects. Hence, the j-mode response is:MEEN 617 – HD 8 Modal Analysis with Proportional Damping. L. San Andrés © 2008 7()cosjjPPSj jjQqAtKψ⎛⎞=Ω−⎜⎟⎜⎟⎝⎠ (22) Let jjnfωΩ=be a jth-mode excitation frequency ratio. Then, define ()()222112 jjjjAffζ=−+and ()()22tan1jjjjffζψ=− (23) Recall that jϕ is a phase angle and Aj is an amplitude ratio for the jth-mode. Note that depending on the magnitude of the excitation frequency Ω, the frequency ratio for a particular mode, say k, determines the regime of operation, i.e. below, above or around the natural frequency. Using the mode displacement method, the response in physical coordinates is ()1cosjmPjj jjjQAtKψ=⎛⎞≈Ω−⎜⎟⎜⎟⎝⎠∑U φ (24) And recall that 2() ()jTjnj j jKMω==φ K φ