ABSTRACT
The dynamics of a linear structure subjected to periodic forces obeys the matrix differential equation
MX¨ + CX˙ + KX = F (t),
with initial conditions X(0) = D0 , X˙(0) = V0.
The solution vector X(t) has dimension n and M , C, and K are real square matrices of order n. The mass matrix, M , the damping matrix, C, and the stiffness matrix, K , are all real. The forcing function F (t), assumed to be real and having period L, can be approximated by a Þnite trigonometric series as
F (t) = N∑
k=−N cke
ıωkt where ωk = 2πk/L
and ı = √−1. The Fourier coefÞcients ck are vectors that can be computed using
the FFT. The fact that F (t) is real also implies that c−k = conj(ck) and, therefore,
F (t) = c0 + 2 real
( n∑
) .
