forced vibration - chulalongkorn university: faculties and...
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Forced vibration
A system is said to undergo forced vibration whenever external energy is supplied to the system during vibration. External energy can be supplied through either an applied force or an imposed displacement excitation.
Applied force Displacement excitation
Types of forced vibrationClassified by input1. Harmonic (sinusoidal) input 2. Arbitrary periodic input
t
F
t
F
3. Impact 4. Arbitrary nonperiodic input
t
F
t
F
Harmonic excitation (undamped)
k mF(t)=F0cosωt
x
F(t)kx
mg
N
tFkxxm ω=+ cos0&&
FBD
EOM:or tfxx n ω=ω+ cos0
2&&
The response x(t) (solution) can be separated into 2 part;
1. Homogeneous solution xh(t)2. Particular solution xp(t)
0=+ kxxm &&
tFkxxm ω=+ cos0&&
)()()( txtxtx ph +=
Response of harmonic excitation (1)
The homogeneous solution xh(t) is the solution of 0=+ kxxm &&
tAtAtx nnh ω+ω= cossin)( 21
The homogeneous solution is given by:
)sin()( φ+ω= tAtx nhor
wheremk
n =ω is the natural frequency
The homogeneous solution has the same form as free vibration response
Response of harmonic excitation (2)
The particular solution xp(t) is the solution of tFkxxm ω=+ cos0&&
From physical observation, harmonic excitation causes vibration at its “driving frequency”,ω.
tXtxp ω= cos)(
The particular solution xp(t) can be written in the form:
Solve for X by substituting xp(t) into EOM
tFtkXtmX ω=ω+ωω− coscoscos 02
20
ω−=
mkFX t
mkFtxp ω
ω−= cos)( 2
0
Response of harmonic excitation (3)
tmkFtAtAtxtxtx nnph ω
ω−+ω+ω=+= coscossin)()()( 2
021
A1 and A2 can be solved by using initial condition
0)0( xtx == 00)0( vxtx === &&and
xh(t) xp(t)
n
vAω
= 01 2
002 ω−
−=mkFxA,
tmkFt
mkFxtvtx nn
n
ωω−
+ω⎟⎠⎞
⎜⎝⎛
ω−−+ω
ω= coscossin)( 2
02
00
0
Response of harmonic excitation (4)
For 0)0( xtx == 00)0( vxtx === &&and
tmkFt
mkFxtvtx nn
n
ωω−
+ω⎟⎠⎞
⎜⎝⎛
ω−−+ω
ω= coscossin)( 2
02
00
0
tftfxtvtxn
nn
nn
ωω−ω
+ω⎟⎟⎠
⎞⎜⎜⎝
⎛ω−ω
−+ωω
= coscossin)( 220
220
00or
Example of a total response
Amplitude of the response
tftfxtvtxn
nn
nn
ωω−ω
+ω⎟⎟⎠
⎞⎜⎜⎝
⎛ω−ω
−+ωω
= coscossin)( 220
220
00From
km F(t)=F0cosωt
Ex Natural freq. fn = 1 HzExcited force/mass f0 = 1 N/kgExcited freq. f = 0.4, 1.01, 1.6 Hzx0 = 0.01 mv0 = 1 m/s
xh
xp
x
0.4 Hz 1.01 Hz 1.6 Hz
+ + +
Phase of the response (1)Natural freq. fn = 0.1 HzExcited force/mass f0 = 40 N/kgExcited freq. f = 0.02 Hzx0 = 0.1 mv0 = 0 m/s
ω/ωn = 0.2
f0
xp
x 5f0
Phase of the response (2)Natural freq. fn = 0.1 HzExcited force/mass f0 = 40 N/kgExcited freq. f = 1 Hzx0 = 0.1 mv0 = 0 m/s
ω/ωn = 10
f0
xp
10x f0
ResonanceWhen forcing frequency is equal to the natural frequency of the system, the response grows without bound. This phenomenon is called resonance.
ttXtxp ω= cos)(In this case
The total response is
ttf
txtvtx
ωω
+
ω+ωω
=
sin2
cossin)(
0
00
ω = ωn
Example 1The engine is mounted on a foundation block which is spring-supported. Describe the steady-state vibration of the system if the block and engine have a total weight of 7500 N (≈ 750 kg) and the engine, when running, creates an impressed force F = 250sin(2t) N, where t is in seconds. Assume that the system vibrates only in the vertical direction, with the positive displacement measured downward, and that the total stiffness of the springs can be represented as k = 30 kN/m. Also determine the rotational speed ω of the engine which will cause resonance. [R. C. Hibbeler 22-55,56]
Example 2A 220-kg engine is mounted on a test stand with spring mounts at A andB, each with a stiffness of 105 kN/m. The radius of gyration of the engine about its mass center G is 115 mm. With the motor not running, calculate the natural frequency (fn)y of vertical vibration and (fn)θ of rotation about G. [J. L. Meriam & L. G. Kraige 8/113]
Example 3Determine the amplitude of vertical vibration of the spring-mounted trailer as it travels at a velocity of 25 km/h over the road whose contour may be expressed by a sinusoid. The mass of the trailer is 500 kg and that of the wheels alone may be neglected. During loading, each 75 kg added to the load caused the trailer to sag 3 mm on its springs. Assume that the wheels are in contact with the road at all times and neglect damping. At what critical speed vc is the vibration of the trailer greatest? [J. L. Meriam & L. G. Kraige 8/71]
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