Page 70 - Dynamic Loading and Design of Structures

Page 50

(2.60)

where m and n correspond to two different eigenvectors. This fundamental property allows for
uncoupling the original N coupled equations of motion into N modal equations, each of which
is a dynamic equation of equilibrium for an SDOF oscillator whose natural frequency ω i
comes from the discrete spectrum ω 1, ω 2,…, ω N
Specifically, the nth such equation assumes the following form:

(2.61)

The subscript ∆in an eigenvector denotes the difference between two consecutive
components (i.e. . A comparison of the above equation with eqn (2.1) for
the SDOF system reveals that the equivalent mass, stiffness and loading coefficients for the
nth modal equation are

(2.62)

respectively. Thus, eqn (2.61) for the nth modal displacement A (t) can be rewritten as
n

(2.63)

Following the solution procedure outlined for the SDOF system in the previous section, the
modal static displacement A nst for the nth equation is given by

(2.64)

since we have that . For instance, Figure 2.19 plots the eigenvectors of
a two DOF oscillator.
The displacement amplitude given by the nth modal equation can be computed as

or as where the dynamic load factors DLF
depend on the particular form of the load’s time function f(t) and on natural frequency ωn We
note that DLFs for various load cases were presented in the previous section on SDOF
systems. The final displacement response of the rth DOF of the MDOF system is found by
superimposing all the modal displacement