3. Mathematical Properties of MDOF Systems

Transcription

1 3. Mathematical Properties of MDOF Systems 3.1 The Generalized Eigenvalue Problem Recall that the natural frequencies ω and modes a are found from [ - ω 2 M + K ] a = 0 or K a = ω 2 M a Where M and K are the mass and stiffness matrices of the MDOF system NB : M & K are symmetric matrices, M = M T and K = K T

2 This is actually a more general version of the eigenvalue problem A x = λ x where A is a square matrix and the unknowns x and λ are called the eigenvector and eigenvalue. The eigenvalues are obtained by solving a characteristic equation det [ A - λ I ] = 0 And for each eigenvalue you can find the eigenvectors by solving [ A - λ I ] x = 0 Question : Where s the analogy?

3 3.2 Orthogonal property of natural modes (eigenvectors) Orthogonal Property: The natural modes are orthogonal with respect to both the mass and stiffness matrices Proof : Consider two modes i & j of the system Premultiply each equation with a mode vector

4 Since M and K are symmetric Hence subtracting the two equation yields Since the natural frequencies are distinct

5 Hence the natural modes are orthogonal with respect to the mass matrix. Similarly for the stiffnesss matrix. 0 = a j T K ai Homework : Prove this

6 Example: Verify the orthogonal property for the 2 DOF system k k k m 2m x 1 x 2 +ve Recall the mass and stiffness matrices are : m m 2 k k k 2 k

7 and the natural frequencies and normal modes are : ω 1 = (k/m) ω 2 = (k/m) a 1 = a 2 = a 1 T M a2 =

8 Similarly a 1 T K a2 = Homework : Verify the orthogonal property for other examples

9 Question : Can you suggest another way to normalize modes? Answer : Modes normalized this way are called orthonormal modes Question : So what s the big deal about orthogonality? The big deal :

10 3.3 Decoupling a MDOF System Let a 1,..., a N be the modes of an N DOF system : M x + K x = F with initial conditions x(0) = x o and x (0) = v o The modal matrix P is obtained by placing these mode vectors together column wise P = [ a 1... a N ]

11 Define a change of coordinates x = P y Substitute in the EOM : and initial conditions :

12 Pre multiply EOM by P T Pre multiply initial conditions by P T M

13 By the orthogonality of the modes, P T M P and P T K P are diagonal matrices. How so? P T M P = a T 1 [ M ] [ a 1... a N ]... a T N

14 The system decouples into N SDOF equations! m i y i + k i y i = a i T F y i (0) = ( a T i M xo ) / m i y i (0) = ( a T i M vo ) / m i i = 1,..., N where m i = a T i M ai k i = a T i K ai Are we done?

15 Question : How do we get the actual response x? Answer :

16 Example : Decoupling a MDOF system x 1 x 2 k m m k k M = m 0 K = 2k -k 0 m -k 2k ω 1 = (k/m) ω 2 = ( 3k/m) a 1 = 1 1 a 2 = 1 1

17 Find the response for initial conditions x(0) = {1,0} T and x (0) = {0,0} T Form the modal matrix P = The modal mass matrix P T M P

18 The modal stiffness matrix P T K P = Hence the decoupled EOM for the modal coordinates are : What else do we need?

19 Get the initial conditions for y 1 and y 2. ( P T M P ) y(0) = P T M x o

20 So we need to solve? 2 SDOF equations : y 1 + k/m y 1 = 0 y 1 (0) = 1/2 y 1 (0) = 0 y 2 + 3k/m y 2 = 0 y 2 (0) = -1/2 y 2 (0) = 0 The solution is :

21 Finally transform back to real coordinates x 1 = = x 2 = =

22 Notes 1. Decoupling may not work if a damping matrix is present. Eg : M x'' + C x' + K x = F An exceptional case is Rayleigh damping where C = a M + b K

23 2. The transformation x = P y is basically a summation of natural modes x = [ a 1... a N ] y 1... y N = y 1 a y N a N