Numerical methods (i) Matrix iteration method Using this method one can obtain natural frequencies and modal vectors of a vibratory system having multi-degree freedom. It is required to have Eqns. of motion of a vibratory system (having n DOF) in matrix form can be written as: M&x& Kx 0   where, x Asinωt   (8) φ substitute Eqn.(8) in (9) MA KA 0   (9) ω2 For principal modes of oscillations, for rth mode, MAr K Ar 0   ωr2 1 MA A  r r ω2 r 1    r r ω2 r D where, is referred as Dynamic matrix. Eqn.(10) converges to first natural frequency and first modal vector. The Equation, KA A  ω2 r r r  ωr2 Ar D1  is referred as inverse dynamic matrix. Eqn.(11) converges to last natural frequency and last modal vector. In above Eqns (10) and (11) by assuming trial modal vector and iterating till the Eqn is satisfied, one can estimate natural frequency of a system. Examples: 1 Find first natural frequency and modal vector of the system shown in the Fig.10 using matrix iteration method. Use flexibility influence co-efficients. Find influence coefficients. 1 2K      α11 α21 α12 α31 α13 3   α22 α32 
Numerical methods (i) Matrix iteration method Using this method one can obtain natural frequencies and modal vectors of a vibratory system having multi-degree freedom. It is required to have Eqns. of motion of a vibratory system (having n DOF) in matrix form can be written as: M&x&  Kx  0 where, x  Asinωt  φ (8) substitute Eqn.(8) in (9) ω2 MA  KA  0 (9) For principal modes of oscillations, for rth mode, ωr2 MAr  K Ar  0 MA  1 A ω2 r r r  1 r ω2 r r where, D is referred as Dynamic matrix. Eqn.(10) converges to first natural frequency and first modal vector. The Equation, KA  ω2 A r r r  ωr2 Ar D1  is referred as inverse dynamic matrix. Eqn.(11) converges to last natural frequency and last modal vector. In above Eqns (10) and (11) by assuming trial modal vector and iterating till the Eqn is satisfied, one can estimate natural frequency of a system. Examples: 1 Find first natural frequency and modal vector of the system shown in the Fig.10 using matrix iteration method. Use flexibility influence co-efficients. Find influence coefficients. 1 α11  α21  α12  α31  α13  2K α22  α32   3  
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