Solid Mechanics Fundamentals - Meshless Methods in Biomechanics - page 39

u ð t Þ ¼ X

m

/ i x i ð t Þ ð 2 : 124 Þ

i¼1

Therefore the response analysis requires, first, the solution of the eigenvalues
and eigenvectors of the problem, Eq. ( 2.113 ), then the solution of the decoupled
equilibrium equations in Eq. ( 2.122 ) and, finally, the superposition of the response
in each eigenvector as expressed in Eq. ( 2.124 ).

2.3.7 Forced Vibrations

In this topic when forced vibrations are imposed only three different time-
dependent loading conditions are considered, f ð t Þ ¼f g ð t Þ . A time constant
load—load case A,

g A ð t Þ ¼1 ð 2 : 125 Þ

A transient load—load case B,

g B ð t Þ ¼1 ft t i
g B ð t Þ ¼0 ft [ t i

ð 2 : 126 Þ

And a harmonic load—load case C,

g C ð t Þ ¼sin ð c t Þ ð 2 : 127 Þ

The solution of each equation in Eq. ( 2.123 ) can be calculated using the
Duhamel integral,

Z

t

x i ð t Þ ¼ 1
x i

f i ð s Þ sin x i ð t s Þ ð Þ ds þ a i sin x i ðÞþ b i cos x i ðÞ 2 : 128 Þ

0

where a i and b i are determined from the initial conditions: Eq. ( 2.123 ) and
f i ð t Þ ¼/ i f ð t Þ . For load case A and load case B the obtained solution is defined as,

Þ þ x t¼0

x i ð t Þ ¼ f i ð t Þ
x i

i
x i

sin x i ðÞþ x t¼0 ð 1 cos x i ðÞ cos x i ðÞ 2 : 129 Þ

i

For load case C the obtained solution is,

x i ð t Þ ¼ f i ð t Þ
x i c 2

sin c ðÞ c
x i

sin x i ðÞ ð 2 : 130 Þ