PART IA - MECHANICAL VIBRATIONS HANDOUT 4 CONTENTS 5. MULTI DEGREE-OF-FREEDOM SYSTEMS 5.1 Degrees of Freedom 5.2 Equations of Motion 5.3 Natural Frequencies and Mode Shapes 5.4 Eigen Approach 5.5 Undamped Free Vibration 5.6 Harmonic Excitation 5.7 The Dynamic Vibration Absorber J.P. Talbot, 2021 Based on the notes of R.S. Langley 1
5. MULTI DEGREE OF FREEDOM SYSTEMS 5.1 DEGREES OF FREEDOM Definition: The number of degrees of freedom N is the number of coordinates needed to describe the configuration of a system. For example: Assuming plane motion 9 N=1 T 00 N = 2 9. y ( x ) W - K N=3 N=2 N= 00 91 The continuous beam shown in the table above has an infinite number of degrees of freedom, i.e. fully to describe the position of the beam it would be necessary to describe the position of every one of the infinite number of points along the beam. The same situation would arise for a car structure, but in practice we approximate the deformation by using the displacement at a finite number of points (or nodes). This forms the basis of the finite element method, which is used extensively in industry. A typical engineering structure: 92 ys F.E.A. In this section we consider harmonic motion of undamped mdof systems. OS N = 6 2 a. N =24
5.2 EQUATIONS OF MOTION Consider a two degree of freedom system: N=2 TIT W MI 9, M2 Kg 92 f2 kg E k?y?. f, M, k2 (y2-9) y 2 F2 M2 ksyz 0 k2 (y2 I y )l kyp f1 my 2 f2 1 k2 (y2[ y ] k3y2 m2 2 These equations can be written in matrix form: .. 1 => [ M] ] ] 2) => [0 m2/52 + [kitkz -kg 75, 7 1 he katks lly? ) = 4 4" Lot y = [ 1] and = = = ] > [1]+ [k] = f cf. my+ ky = f for SDOF The same notation can be used regardless of the number of degrees of freedom - all that changes is the dimension of the matrices and the vectors. For an N degree of freedom system, the matrices are NuN and the vectors are Nu1. 3