<s>
Structural	B-Algorithm
dynamics	I-Algorithm
is	O
a	O
type	O
of	O
structural	B-Algorithm
analysis	I-Algorithm
which	O
covers	O
the	O
behavior	O
of	O
a	O
structure	O
subjected	O
to	O
dynamic	O
(	O
actions	O
having	O
high	O
acceleration	O
)	O
loading	O
.	O
</s>
<s>
Structural	B-Algorithm
analysis	I-Algorithm
is	O
mainly	O
concerned	O
with	O
finding	O
out	O
the	O
behavior	O
of	O
a	O
physical	O
structure	O
when	O
subjected	O
to	O
force	O
.	O
</s>
<s>
Dynamic	O
analysis	O
for	O
simple	O
structures	O
can	O
be	O
carried	O
out	O
manually	O
,	O
but	O
for	O
complex	O
structures	O
finite	B-Application
element	I-Application
analysis	I-Application
can	O
be	O
used	O
to	O
calculate	O
the	O
mode	O
shapes	O
and	O
frequencies	O
.	O
</s>
<s>
A	O
simple	O
single	O
degree	O
of	O
freedom	O
system	O
(	O
a	O
mass	O
,	O
M	O
,	O
on	O
a	O
spring	O
of	O
stiffness	B-Algorithm
k	O
,	O
for	O
example	O
)	O
has	O
the	O
following	O
equation	O
of	O
motion	O
:	O
</s>
<s>
where	O
is	O
the	O
acceleration	O
(	O
the	O
double	O
derivative	B-Algorithm
of	O
the	O
displacement	O
)	O
and	O
x	O
is	O
the	O
displacement	O
.	O
</s>
<s>
As	O
the	O
number	O
of	O
degrees	O
of	O
freedom	O
of	O
a	O
structure	O
increases	O
it	O
very	O
quickly	O
becomes	O
too	O
difficult	O
to	O
calculate	O
the	O
time	O
history	O
manually	O
–	O
real	O
structures	O
are	O
analysed	O
using	O
non-linear	O
finite	B-Application
element	I-Application
analysis	I-Application
software	O
.	O
</s>
<s>
The	O
natural	O
frequency	O
of	O
a	O
system	O
is	O
dependent	O
only	O
on	O
the	O
stiffness	B-Algorithm
of	O
the	O
structure	O
and	O
the	O
mass	O
which	O
participates	O
with	O
the	O
structure	O
(	O
including	O
self-weight	O
)	O
.	O
</s>
<s>
It	O
is	O
useful	O
to	O
know	O
the	O
modal	O
frequencies	O
of	O
a	O
structure	O
as	O
it	O
allows	O
you	O
to	O
ensure	O
that	O
the	O
frequency	O
of	O
any	O
applied	O
periodic	O
loading	O
will	O
not	O
coincide	O
with	O
a	O
modal	O
frequency	O
and	O
hence	O
cause	O
resonance	B-Application
,	O
which	O
leads	O
to	O
large	O
oscillations	B-Algorithm
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
calculate	O
the	O
frequency	O
of	O
different	O
mode	O
shape	O
of	O
system	O
manually	O
by	O
the	O
energy	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
For	O
a	O
given	O
mode	O
shape	O
of	O
a	O
multiple	O
degree	O
of	O
freedom	O
system	O
you	O
can	O
find	O
an	O
"	O
equivalent	O
"	O
mass	O
,	O
stiffness	B-Algorithm
and	O
applied	O
force	O
for	O
a	O
single	O
degree	O
of	O
freedom	O
system	O
.	O
</s>
<s>
"	O
The	O
frequency	O
ω	O
of	O
an	O
arbitrary	O
mode	O
of	O
vibration	O
,	O
calculated	O
by	O
the	O
energy	B-Algorithm
method	I-Algorithm
,	O
is	O
always	O
greater	O
than	O
–	O
or	O
equal	O
to	O
–	O
the	O
fundamental	O
frequency	O
ωn.	O
"	O
</s>
<s>
For	O
an	O
assumed	O
mode	O
shape	O
,	O
of	O
a	O
structural	O
system	O
with	O
mass	O
M	O
;	O
bending	O
stiffness	B-Algorithm
,	O
EI	O
(	O
Young	O
's	O
modulus	O
,	O
E	O
,	O
multiplied	O
by	O
the	O
second	O
moment	O
of	O
area	O
,	O
I	O
)	O
;	O
and	O
applied	O
force	O
,	O
F(x )	O
:	O
</s>
<s>
To	O
superpose	O
the	O
individual	O
modal	O
responses	O
manually	O
,	O
having	O
calculated	O
them	O
by	O
the	O
energy	B-Algorithm
method	I-Algorithm
:	O
</s>
