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7/31/2019 Mechanical Vibration Forced Undamped
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BFF3103 Mechanical Vibration
Forced Vibration Undamped
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Lesson-03 Objectives
Students will be able to:
Obtain the harmonic response of systemshaving a SDOF
Obtain the transfer function from the EOM
Determine the total response frequency, peakresponse and bandwidth
Analyze the displacement and transmitted force
of the systems having base excitation
Solve the problem related to undamped anddamped harmonic response SDOF vibration
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Todays Objectives
Students will be able to: Identify the harmonic response of SDOF
Obtained the response of an undamped
system under harmonic force
Solve the problem related to damped SDOF
force vibration
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Forced vibrations occurs when external
energy is supplied to the system during
vibration
The external force can be supplied
through either an applied force or an
imposed displacement excitation, which
may be harmonic, nonharmonic but
periodic, nonperiodic, or random in
nature.
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Harmonic response results when thesystem responses to a harmonicexcitation
Transient response is defined as theresponse of a dynamic system tosuddenly applied nonperiodic
excitations
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There are 4 categories of F(t):
(1) Harmonic (sin, cos) (3) Transient
(2) Periodic (4) Random
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Equation of Motion
The equation of Motion Using Newtons Second
Law of Motion:
)(tFkxxcxm
The homogeneous
solution of the equation:
0 kxxcxm
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Response of an Undamped System Under
Harmonic Force
tFtF cos)( 0
tFkxxm cos0
tnDtnCthx sincos)(
The homogeneous solution is
where is the natural frequency.2/1)/( mkn
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where denotes the static deflection
where X is the maximum amplitude ofxp(t)
tXtxp cos)(
220
1
n
st
mkFX
kFst
/0
Thus, t
mk
FtnDtnCtx
cos
2
0sincos)(
Because the exciting force and particular solution is
harmonic and has same frequency, we can assume a
solution in the form:
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Using initial conditions
Hence,
00
)0(and)0( xtxxtx
n
xD
mk
FxC
0,2
00
tmk
F
tx
tmk
Fxtx
n
n
n
cos
sincos)(
2
0
0
2
0
0
The max amplitude: 2
1
1
n
st
X
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where the quantity is
called the magnificationfactor, amplification factor, or
amplitude ratio. The
response of the system can
be identified to be of three
types from the figure.
stX /
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Case 1. When 0 < < 1, the denominator is positive
and the response is given without change. The
harmonic response of the system is in phase with
external force.
n/
2
1
1
n
st
X
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where the amplitude,
Case 2. When > 1, the denominator is negative
and the steady-state solution can be expressed asn/
tXtxp cos)(
1
2
n
stX
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Case 3. When = 1, the amplitude X given
becomes infinite. This condition, for which the forcing
frequency is equal to the natural frequency of the
system, is called resonance. Hence, the total responseif the system at resonance becomes
tt
tx
txtx
n
nst
n
n
n
sin2
sincos)( 00
n/
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Total Response
1for;cos
1
)cos()( 2
n
n
stn ttAtx
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1for;cos
1
)cos()(2
n
n
stn ttAtx
and
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Beating Phenomenon
If the forcing frequency is close to, but not exactly equal
to, the natural frequency of the system, beating may
occur. The phenomenon of beating can be expressed as:
ttmF
tx
sinsin
2
/)( 0
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The time between the points of zero amplitude or the
points of maximum amplitude is called theperiod of
beating and is given by
n
b
2
2
2
The frequency of beating defined as
nb 2