Page 71 - Dynamic Loading and Design of Structures
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Figure 2.19 Eigenvectors of a simple two DOF system.
amplitudes as
(2.65)
In sum, there are a number of methods for computing eigenvalues and their associated
eigenvectors, which can be grouped into three basic categories as follows: (i) direct methods,
which essentially follow the procedure previously described, (ii) iterative methods such as
Jacobi’s method and (iii) approximate methods (e.g. Rayleigh’s method).
Using matrix notation, the equations of motion for an MDOF system assume the form
shown below
(2.66)
where square and curly brackets respectively denote a matrix and a vector. The orthogonality
property of the eigenvectors previously mentioned assumes the following form:
(2.67)
where overbars denote a diagonal matrix and superscript T denotes matrix transposition. If
T
eqn (2.66) is premultiplied by [Φ] and if modal co-ordinates are introduced as
, then we recover the following uncoupled form for the equations of motion:
(2.68)
