B Linear Differential Equations
137
derivatives of x p (t) and the differential equation give
n
i=1
α
i (t) β
(n−1)
i
= f (t).
(B.10)
The conditions of Eqs. B.9 and B.10 define a linear system of equations for the
first derivatives of the unknown functions {α i (t)} of the form α
i (t) = g i (t), i =
1, . . . , n, where the functions g i (t) depend on the basis functions and the forcing
function.
Example B.1 Suppose x(t) is the displacement of the SDOF system defined by
Eq. 2.8 with damping ratio 0 < ζ < 1. The basis functions are β 1 (t) =
exp(−ζ ω t) cos(ω d t) and β 2 (t) = exp(−ζ ω t) sin(ω d t). The differential operator
of the defining equation of x(t) is D = d 2 /dt 2 + 2 ζ ω d/dt + ω 2 , and the order of
this equation is n = 2. From Eq. B.9, we have α
1 (t) β 1 (t) + α
2 (t) β 2 (t) = 0, so that
˙
x p (t) = α 1 (t) β
1 (t)+α 2 (t) β
2 (t), ¨
x p (t) = α
1 (t) β
1 (t)+α
2 (t) β
2 (t)+α 1 (t) β
1 (t)+
α 2 (t) β
2 (t), and
D[x p (t)] = α 1 (t) D[β 1 (t)] + α 2 (t) D[β 2 (t)] + α
1 (t) β
1 (t) + α
2 (t) β
2 (t)
= α
1 (t) β
1 (t) + α
2 (t) β
2 (t),
where the latter equality holds since {β i (t)} satisfies the homogeneous equation.
Since x p (t) must satisfy the inhomogeneous equation, we have α
1 (t) β
1 (t) +
α
2 (t) β
2 (t) = f (t). The resulting conditions for {α
i (t)} are (see also Eqs. B.9
and B.10)
α
1 (t) β 1 (t) + α
2 (t) β 2 (t) = 0
α
1 (t) β
1 (t) + α
2 (t) β
2 (t) = f (t)
whose solutions are α
i (t) = g i (t), i = 1, 2, where
g 1 (t) = −β 2 (t) f (t)/
β 1 (t) β
2 (t) − β
1 (t) β 2 (t)
and
g 2 (t) = β 2 1(t) f (t)/
β 1 (t) β
2 (t) − β
1 (t) β 2 (t)
.
The particular solution has the form x p (t) =
g 1 (t) dt
β 1 (t)+
g 2 (t) dt
β 2 (t).
We have not used the method of the variation of constants to construct particular
solutions for the forced vibration of SDOF systems for two reasons. First, the
method is rather abstract. Second, particular solutions can be constructed by direct
observations in simple cases, e.g., constant and harmonic forces, and, for arbitrary
forcing functions, from dynamical responses of SDOF systems to elementary
actions (see Duhamel’s integral in Sect. 2.4.4).