2.6 Numerical Methods
43
where the constants A and B result from the initial conditions x(0) = x 0 and ˙
x(0) =
˙
x 0 .
Suppose that the oscillator is subjected to a large family of forcing functions
{f k (t)}, k = 1, . . . , m, which are defined on the same time interval [0, τ ]. The
displacement functions {x k (t)}, k = 1, . . . , m, to these family of forces can be
obtained by standard numerical algorithms by calling these algorithms m times. A
more efficient approach is to calculate the displacements {x k (t)} to Fourier series
representations of the forcing functions {f k (t)}. The implementation of this method
involves the following three steps.
– Step 1: Represent the forcing functions {f k (t)} by Fourier series. Since the forces
{f k (t)} are active on the same time interval [0, τ ], they admit the representations
(see Eq. 2.55)
f k,n (t) =
a k,0
2
+
n
i=1
a k,i cos(ν i t) + b k,i sin(ν i t)
,
n = 1, 2, . . . ,
(2.73)
where the coefficients {a k,i } and {b k,i } are given by the formulas of {a i } and {b i }
in Eq. 2.53 with f k in place of f . The truncation level n has to be such that all
members of the family of forces {f k (t)} are represented accurately.
– Step 2: Calculate and store the integrals
I c,i (t) =
t
0
h(t − u) cos(ν i u) du andI s,i (t) =
t
0
h(t − u) sin(ν i u) du
(2.74)
for t in [0, τ ]. Note that the family of integrals
I k,n (t) =
t
0
h(t − u) f k,n (u) du =
n
i=1
a k,i I c,i (t) + b k,i I s,i (t)
, 0 ≤ t ≤ τ,
(2.75)
corresponding to the integral of Eq. 2.72 with {f k,n } in place of f result by
elementary calculations by using the integrals of Eq. 2.74 which have been
stored.
– Step 3: The displacements {x k (t)} have the form
x k (t) = e
−ζ ω t
A k cos(ω d t)+ B k sin(ω d t)
+ I k,n (t), 0 < t < τ,
(2.76)
where the constants A k and B k can be obtained from the initial conditions x 0 and
˙
x 0 . The integrals {I c,i (t)} and {I s,i (t)} in the expression of x k (t) are zero at the
initial time t = 0 and so are their time derivatives
43
where the constants A and B result from the initial conditions x(0) = x 0 and ˙
x(0) =
˙
x 0 .
Suppose that the oscillator is subjected to a large family of forcing functions
{f k (t)}, k = 1, . . . , m, which are defined on the same time interval [0, τ ]. The
displacement functions {x k (t)}, k = 1, . . . , m, to these family of forces can be
obtained by standard numerical algorithms by calling these algorithms m times. A
more efficient approach is to calculate the displacements {x k (t)} to Fourier series
representations of the forcing functions {f k (t)}. The implementation of this method
involves the following three steps.
– Step 1: Represent the forcing functions {f k (t)} by Fourier series. Since the forces
{f k (t)} are active on the same time interval [0, τ ], they admit the representations
(see Eq. 2.55)
f k,n (t) =
a k,0
2
+
n
i=1
a k,i cos(ν i t) + b k,i sin(ν i t)
,
n = 1, 2, . . . ,
(2.73)
where the coefficients {a k,i } and {b k,i } are given by the formulas of {a i } and {b i }
in Eq. 2.53 with f k in place of f . The truncation level n has to be such that all
members of the family of forces {f k (t)} are represented accurately.
– Step 2: Calculate and store the integrals
I c,i (t) =
t
0
h(t − u) cos(ν i u) du andI s,i (t) =
t
0
h(t − u) sin(ν i u) du
(2.74)
for t in [0, τ ]. Note that the family of integrals
I k,n (t) =
t
0
h(t − u) f k,n (u) du =
n
i=1
a k,i I c,i (t) + b k,i I s,i (t)
, 0 ≤ t ≤ τ,
(2.75)
corresponding to the integral of Eq. 2.72 with {f k,n } in place of f result by
elementary calculations by using the integrals of Eq. 2.74 which have been
stored.
– Step 3: The displacements {x k (t)} have the form
x k (t) = e
−ζ ω t
A k cos(ω d t)+ B k sin(ω d t)
+ I k,n (t), 0 < t < τ,
(2.76)
where the constants A k and B k can be obtained from the initial conditions x 0 and
˙
x 0 . The integrals {I c,i (t)} and {I s,i (t)} in the expression of x k (t) are zero at the
initial time t = 0 and so are their time derivatives