Download Mid-Term Exam in MAE351 Mechanical Vibrations F(t)

Mid-Term Exam in MAE351 Mechanical Vibrations
Mechanical Engr. Dept.
1.
KAIST
April 22, 2015
Damping of a single-DOF under-damped system can be estimated by measuring consecutive peaks of
free vibration in an experiment.
1)
Explain how the logarithmic decrement is defined. (10)
2)
Derive a relation between the logarithmic decrement and the damping ratio of a system with viscous
damping. (10)
2.
Consider a base excitation problem with the
configuration shown in Fig. 1, where the base
motion is transmitted through a dashpot.
1)
Derive an expression for the amplitude of
force transmitted to the support as a function of
the excitation frequency. (5)
2)
Derive an expression for magnitude of the
base of negligible mass
force transmissibility, which is defined as the ratio
F(t)
of the force onto the support to the force required
to excite the base. (5)
3)
Fig. 1
For m = 100 kg, c = 50 N·s/m, Y = 0.03 m and ω = 3 rad/s, determine the stiffness k that
b
makes the force transmissibility equal to 0.5 and the amplitude of the force F(t) required on the
base of negligible mass to enable this excitation. (10)
3.
Fig. 2 is a schematic diagram for vibration
transducers, whose basic principle is to transduce an
electric signal in proportion to the relative displacement
z  x y .
The
equation
is
given
by
mz  cz  kz  my and the relative displacement
response z(t) to a harmonic displacement excitation
y  t   Y cos t is given by z  t   Z cos t    with
1
Fig. 2
Z  Y 2 
1)
1
1


2

(1  2 ) 2   2

n
 n 
2
n
2
.
Draw a figure of magnitude of the frequency response function for a seismometer which is
used to measure vibration displacement.
Make a note on the figure for a useful frequency range
and explain about the reasoning for your choice. (10)
2)
Draw a figure of magnitude of the frequency response function for an accelerometer which is
used to measure vibration acceleration.
Make a note on the figure for a useful frequency range and
explain about the reasoning for your choice.
Further, explain about the relationship between
sensitivity and useful frequency range for the accelerometers. (10)
4.
Response of a under-damped single-DOF system to an arbitrary input F(t) can be obtained using
ent
sin d t is the unit impulse response
convolution integral x(t )   F ( )h(t   )d , where h(t ) 
md
0
t
function. It can be also obtained using Laplace transforms in Table 1.
1) Choose your own method, i.e., either convolution integral or Laplace transform & Table 1 to show
that response of the system subject to a step function of magnitude of F0 at time t = 0 can be
obtained as follows:
x t  

F0 
1

ent cos d t    ,  tan -1
.
1 
2
k 
1 
1  2

and draw an approximate graph. (10)
2) Using the equation given in 1) for the step response, determine the first peak time tp:1 at which the
response will take its maximum value. (5)
3) Using the equation in 1), derive the n-th peak time tp:n.
at which magnitude of the n-th peak becomes
Defining the settling time ts as the time tp:n
F0
ln 
approximately for
1    , show that ts  
k
n
small values of  and  . For an example, ts 
60
n
for  ==. (5)

F ( s)   e st f (t )dt
f(t)
0
1
1
1
s
2
2
1
sa
e at
3
a
s  a2
sin at
s
s  a2
cos at
a
ebt sin at
2
4
2
5
 s  b
2
 a2
sb
6
 s  b
2
ebt cos at
 a2
Table 1
5.
Four explanations on a single-DOF system are given and a statement for question follows after each
explanation. Answer whether the statement for question is right or wrong.
the statement must be also provided.
positive points.
1)
Reasoning for your answer to
If your answer together with the reasoning is right, you will get
If it is wrong, however, you may get negative points.
For a mass-damper-spring system whose motion is governed by:
d 2x
dx
m 2  c  kx(t )  0
dt
dt
dynamic behavior is said to be critically damped when (a)  
c
 1 and over-damped when (b)
2 mk
  1 . The homogeneous solution in case of (a) is given by x(t )  a1e t  a2te t and the one in case
n
of (b) is given by x(t )  ent (a1e
 2 1nt
 a2e
 2 1nt
) , where n 
n
k
. It can be seen that both
m
responses decrease exponentially as time t goes to infinite.
Statement for question: By expressing responses subject to
an initial condition: x(0)  x0 , x(0)  0 , it can be shown
that the response in case of (a) becomes zero in a finite time
while the response in case of (b) does not become zero in a
finite time. (5)
2)
For a simple pendulum, the equation of motion is
given by a nonlinear differential equation as follows:
Fig. 3
3
d 2 g
 sin  (t )  0
dt 2 l
Two solutions for the above equation are shown in Fig. 3, where one is a solution of the nonlinear
differential equation and the other is a solution of the linearized
differential equation.
Statement for question: The solution of the linearized differential
equation is the dashed curve. (5)
3)
A pendulum consisting of a rigid body as shown in Fig. 4 is
r
called compound pendulum, for which center of percussion C is
O
y
G

defined as a distance qo such that a simple pendulum (a massless rod
pivoted at O with the same point mass as the rigid body at its tip
qo
C
point) of length qo has the same period.
Statement for question: When the compound pendulum is a
uniform rod of length l, the center of percussion is given by
q0 
l
. (5)
2
4)
Response of a under-damped system subject to an initial condition
x(t )  x0 2  (
Fig. 4
v0  n x0
d
 x0 , v0 
is given by
 x0d  
c

) 2 ent sin d t  tan 1 
.
  , where n  k / m and  
2 mk

 v0  n x0  
 x0 , v0 
and
obtained
as
Statement for question: The response of the system subject to both an initial condition
an
impulse
x(t ) 
of
magnitude
Fˆ  mv0

 1  2
ent sin d t  tan 1 
 
1  2


x0
at
 
  . (5)


4
time
t 0
can
be