The University of New South Wales
School of Civil and Environmental Engineering
CVEN 4308 Structural Dynamics
Assignment
Due date: 5pm, Friday, 22/11/2019
Total Marks Available: 50
First Name:
Last Name:
zID:
Free Vibration Response of MDOFSystems
A 5storey building is idealized as a shear frame structure with rigid ﬂoors and consequently, 5
degrees of freedom as depicted in Fig. 1.
All storeys have the same height
h
, but the columns
have different stiffness coefﬁcients
k
i
. Experiments have found that damping is small and can
therefore be neglected. The stiffness of a column on the ith ﬂoor is given as
k
i
?
12
EI
C
i
h
3
;
@
i
?
1
;
2
:::;
5
:
(1)
Note that axial deformations of the columns are also neglected (
EA
Ñ 8
). The mass of each
storey is lumped at the ﬂoor level as indicated in Fig. 1.
The
set
of
parameters
describing
the
system
is
solely
based
on
the
zID
of
each
student.
The
zID
consists of a seven digit number as
z
C
1
C
2
C
3
C
4
C
5
C
6
C
7
;
where each constant
C
i
deﬁnes a parameter of the system.
Note that each digit in the
zID
that
is equal to
0
should be replaced by
C
i
?
5
. In the following, the system properties are listed:
Data:
I
C
i
?
C
i
?
1000
r
cm
4
s
m
?
C
6
?
1000
r
kg
s
h
?
C
7
r
m
s
E
?
200
r
kN/mm
2
s
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........
.
...........
...........





u
1
(
t
)
u
2
(
t
)
u
3
(
t
)
u
4
(
t
)
u
5
(
t
)
?
?
?
?
?
6
6
6
6
6
h
h
h
h
h
k
1
k
2
k
3
k
4
k
5
k
1
k
2
k
3
k
4
k
5
5
m
3
m
4
m
m
2
m
1
Figure 1: 5storey shear frame.
1
Questions
1.
Derive the equations of motion for the 5storey shear frame structure.
(10 Marks)
(a)
Provide the mass matrix
m
.
(5 Marks)
(b)
Provide the stiffness matrix
k
.
(5 Marks)
2.
Solve the general eigenvalue problem to determine the mode shapes and eigenfrequencies
of the frame structure.
(10 Marks)
(a)
Calculate the circular natural frequencies
!
i
and the natural periods
T
i
of each mode.
(5 Marks)
(b)
Determine
the
mode
shapes
'
i
and
normalize
each
mode
such
that
the
maximum
value is 1.0.
(5 Marks)
3.
Determine the analytical solution
u
p
t
q
for the free vibration response based on the modal
expansion technique.
(18 Marks)
(a)
Calculate the generalized mass matrix.
(5 Marks)
(b)
Determine the vector of the modal contribution factors and its temporal derivative at
t
?
0
s, i.e.
q
p
0
q
and
9
q
p
0
q
. The initial conditions of the frame structure are given as:
(5 Marks)
u
0
?
?
?
?
?
?
?
?
?
C
1
?
C
2
?
C
3
C
4
C
5
˝
˚
˚
˚
˚
˚
˛
;
9
u
0
?
0
:
(c)
Provide the expression for the free vibration response of the system.
(5 Marks)
(d)
Evaluate the analytical solution for the time interval
0
¤
t
¤
r
5
T
1
s
1
using the time
increment
?
t
1
and determine the maximum and minimum values for the displace
ment of ﬂoor 5, i.e
max
r
u
?
t
1
5
p
t
qs
and
min
r
u
?
t
1
5
p
t
qs
.
(3 Marks)
?
t
1
?
T
5
20
:
(2)
Round the value of the time increment to two signiﬁcant digits!
4.
Determine
the
numerical
solution
~
u
p
t
q
for
the
free
vibration
response
based
on
New
mark’s method (use the constant average acceleration method).
(12 Marks)
(a)
Compute the numerical solution with the time increment
?
t
1
in the same time in
terval
(
0
¤
t
¤
r
5
T
1
s
) and
determine
the
maximum
and
minimum
values
for
the
displacement of ﬂoor 5, i.e
max
r
~
u
?
t
1
5
p
t
qs
and
min
r
~
u
?
t
1
5
p
t
qs
.
(4 Marks)
(b)
Plot the solution for
u
5
p
t
q
and
~
u
5
p
t
q
in one diagram.
(2 Marks)
(c)
Discuss the differences in the results.
(2 Marks)
1
The
mathematical
operator
r
?
s
denotes
the
socalled
ceiling
function.
This
function
rounds
the
value
to
the
next greater integer number.
2
(d)
Determine
a
suitable
time
increment
?
t
2
for
the
numerical
solution
such
that
the
relative error
?
in the maximum displacement is less than 0.1%.
(4 Marks)
?
?

max
r
u
?
t
1
5
p
t
qs?
max
r
~
u
?
t
1
5
p
t
qs
max
r
u
?
t
1
5
p
t
qs
?
100%
(3)
Use the following approach to determine
?
t
2
:
?
t
2
?
T
5
20
n
(4)
where
n
P
N
2
. Round the value of the time increment to two signiﬁcant digits!
Note:
?
To
determine
the
solution
either
a
spreadsheet
developed
in
Excel
or
a
computer
pro
gram written in Matlab, Fortran, C++, etc.
has to be used.
This has to be explained and
documented in the submitted report.
?
Late submissions of the assignment will be penalised at a rate of 20% per day after the
due time and date have expired
3
.
?
Submit
1
pdf
ﬁle
through
the
Moodle
assignment
item
containing
all
relevant
data
(as
signment task, report, source code).
2
N
denotes the set of natural numbers.
3
Note that even if the submission is only seconds too late the penalty applies!
3