Page 69 - Dynamic Loading and Design of Structures

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where α 1, α 2,…is the amplitude of vibration of each DOF. Substituting this result in the
equations of motion yields

(2.56)

From all the above equations we recover the relation and thus

(2.57)

where γs a phase angle. This implies that each and every one of the n DOF undergoes
i
harmonic vibration. Substituting this result in the equations of motion yields the following:

(2.58)

In order for the above system of equations to have a solution, its determinant must be set
equal to zero, i.e.

(2.59)

Upon solution, we recover N values, ω, ω,…, ω , for the eigenfrequencies of the system.
1
2
N
For each value ω which is inserted in eqns (2.58), a vector of coefficients
i
results which is the eigenvector corresponding to that particular
eigenfrequency. We normalize each eigenvector by setting , since they cannot be
completely determined from eqn (2.59), and proceed to solve for the remaining components
relative to the first one. In this case, the notation used for the ith normalized
eigenvector is

2.3.2 Eigenvalue Analysis
A basic property of the eigenvectors is orthogonality with respect to the mass coefficients, i.e.