Modal Testing and Analysis Undamped MDOF systems Saeed

Modal Testing and Analysis Undamped MDOF systems Saeed Ziaei-Rad 1

Undamped MDOF systems [M] and [K] are N*N mass and stiffness matrices. {f(t)} is N*1 force vector x 1 k 1 f 1 m 1 k 2 f 2 x 2 k 3 m 2 2

Undamped 2 DOF system or 3

Undamped 2 DOF system 4

MDOF- Free Vibration 5

MDOF- Free Vibration Natural Frequencies: Mode shapes: Or in matrix form: Modal Model 6

2 DOF System- Free Vibration Solving the equation: Numerically: 7

Orthogonality Properties of MDOF The modal model possesses some very important Properties, stated as: Modal mass matrix Modal stiffness matrix Exercise: Prove the orthogonality property of MDOF 8

Mass-normalisation The mass normalized eigenvectores are written as And have the following property: The relationship between mass normalised mode shape and its more general form is: 9

Mass-normalisation of 2 DOF Clearly: 10

Multiple modes n n n The situation where two (or more) modes have the same natural frequency. It occurs in structures with a degree of symmetry, such as discs, rings, cylinders. Free vibration at such frequency may occur not only in each of the two modes but also in a linear combination of them. c a a Vertical mode b Horizontal mode b c Oblique mode 11

Forced Response of MDOF Or by rearranging: Which may be written as: Response model 12

Forced Response of MDOF The values of matrix [H] can be computed easily at each frequency point. However, this has several advantages: n It becomes costly for large N. n It is inefficient if only a few FRF expression is required. n It provides no insight into the form of various FRF properties. Therefore, we make use of modal properties for deriving the FRF parameters instead of spatial properties. 13

Forced Response of MDOF Premultiply both sides by and postmultiply by Inverse both sides Equation 1 Note that: Diagonal matrix 14

Forced Response of MDOF As H is a symmetric matrix then: or Principle of reciprocity Using equation 1: or Modal constant 15

Forced Response of 2 DOF Which gives: Numerically: 16

Forced Response of 2 DOF Or numerically: Receptance FRF ( ) for 2 dof system 17