Method of obtaining natural frequencies in between first and last one (Sweeping Technique) For understanding it is required to to clearly understand Orthogonality principle of modal vectors. Orthogonality principle of modal vectors x2 b 1 b  b2 a 1 a  a2 x1 Fig.11 Vector representation graphically a bare Consider two vectors in Fig.11. Vectors and orthogonal to each other if and only if aT b 0  b1 a 0  b2 1 0 b 0 0 1b2 1 Ib aT 0 (12)  I where, is Identity matrix. a From Eqn.(12), Vectors b and are orthogonal to each with respect to identity matrix. Application of orthogonality principle in vibration analysis Eqns. of motion of a vibratory system (having n DOF) in matrix form can be written as: M&x& Kx 0   x Asinωt   φ MA1 K A1 0   ω2
Method of obtaining natural frequencies in between first and last one (Sweeping Technique) For understanding it is required to to clearly understand Orthogonality principle of modal vectors. Orthogonality principle of modal vectors x2 b b  1 b2 a a  1 a2 x1 Fig.11 Vector representation graphically Consider two vectors in Fig.11. Vectors a and bare orthogonal to each other if and only if aT b  0 b1  0 b2 a 1 0 b 0 1 0 1b2 aT Ib  0 (12) where, I is Identity matrix. From Eqn.(12), Vectors a and b are orthogonal to each with respect to identity matrix. Application of orthogonality principle in vibration analysis Eqns. of motion of a vibratory system (having n DOF) in matrix form can be written as: M&x&  Kx  0 x  Asinωt  φ ω2 MA1  K A1  0
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