9.10
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Insert isolator between vibrating mass and vibration source to reduce response
• Passive isolators: springs, cork, felt etc.
• E.g. Mounting of high-speed punch press
台灣師範大學機電科技學系
-75-9.10 Vibration Isolation
• Active isolator comprised of servomechanism with sensor, signal processor and actuator.
• Effectiveness given in terms of transmissibility Tr which is the ratio of amplitude of the transmitted force to that of the exciting force
• 2 types of isolation situations:
Protect base of vibrating machine against large unbalanced or impulsive forces
Protect system against motion of its foundation
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Protect base of vibrating machine against large unbalanced or impulsive forces
• Protect system against motion of its foundation
t kx
t cx
t Fi
t mx
t k
x
t y t
c
x
t y t
Fi
台灣師範大學機電科技學系
-77-9.10 Vibration Isolation
• Vibration Isolation System with Rigid Foundation
Resilient member placed between vibrating machine and rigid foundation
Member is modeled as a spring k and a dashpot c as shown:
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Vibration Isolation System with Rigid Foundation
Reduction of force transmitted to foundation:
Equation of motion:
Steady state solution:
where
Force Ft transmitted to the foundation:
Magnitude of total transmitted force FT:
t
台灣師範大學機電科技學系
-79-9.10 Vibration Isolation
• Vibration Isolation System with Rigid Foundation
Reduction of force transmitted to foundation:
Transmissibility
Following graphs shows the variation of Tr with r.
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Vibration Isolation System with Rigid Foundation
Reduction of force transmitted to foundation:
台灣師範大學機電科技學系
-81-9.10 Vibration Isolation
• Vibration Isolation System with Rigid Foundation
Reduction of force transmitted to mass:
Displacement transmissibility
Td is also the ratio of the maximum steady-state accelerations of the mass and the base.
y x
z y
m kz
z c z
m where
2
2
22
2 1
2 1
r r
r Y
T
dX
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Isolation of Source of Vibration from Surroundings
By defining
台灣師範大學機電科技學系
-83-9.10 Vibration Isolation
• Isolation of Source of Vibration from Surroundings
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Isolation of Source of Vibration from Surroundings
Reduction of force transmitted to foundation due to rotating unbalance
Excitation force:
Force transmissibility:
t F t me t
台灣師範大學機電科技學系
-85-9.10 Vibration Isolation
• Vibration Isolation System with Flexible Foundation
If the foundation moves, the system has 2 DOF Equations of motion:
Assuming solution of the form xj=Xjcosωt, j=1,2
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Vibration Isolation System with Flexible Foundation
Natural frequencies given by roots of
Amplitude of m1 at steady-state:
台灣師範大學機電科技學系
-87-9.10 Vibration Isolation
• Vibration Isolation System with Flexible Foundation
Amplitude of m2 at steady-state:
Force transmitted to structure:
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Vibration Isolation System with Flexible Foundation
Transmissibility of isolator:
where ω2 is the natural frequency of the system Ft decrease as ω2 decrease
台灣師範大學機電科技學系
-89-9.10 Vibration Isolation
Example 9.4
Spring Support for Exhaust Fan
An exhaust fan, rotating at 1000rpm, is to be supported by 4 springs, each having a stiffness of K. If only 10% of the unbalanced force of the fan is to be transmitted to the base, what should the value of K?
Assume the mass of the exhaust fan to be 40kg.
C. R. Yang, NTNU MT
9.10 Vibration Isolation
Example 9.4
Spring Support for Exhaust Fan Solution
Transmissibility = 0.1
Forcing frequency
2 2
台灣師範大學機電科技學系
-91-9.10 Vibration Isolation
Example 9.4
Spring Support for Exhaust Fan Solution
Natural frequency:
Assuming ζ=0,
To avoid imaginary values,
1623
C. R. Yang, NTNU MT
9.10 Vibration Isolation
Example 9.6
Isolator for Stereo Turntable
A stereo turntable, of mass 1kg, generates an excitation force at a
frequency of 3Hz. If it is supported on a base through a rubber mount, determine the stiffness of the rubber mount to reduce the vibration transmitted to the baes by 80%.
台灣師範大學機電科技學系
-93-9.10 Vibration Isolation
Example 9.6
Isolator for Stereo Turntable Solution
Using N=3x60=180 cpm and R=0.8,
N/m 2179
. 59 or
81 . 9 1657 1
.
0
k k k
mg
st
1 00.80.80
or 0.1657m9092 2 .
29
180
st
st
C. R. Yang, NTNU MT
9.10 Vibration Isolation
Example 9.8
Isolation from Vibrating Base
A vibrating system is to be isolated from its supporting base. Find the required damping ratio that must be achieve by the isolator to limit the transmissibility at resonance to Tr=4. Assume the system to have a single degree of freedom.
台灣師範大學機電科技學系
-95-9.10 Vibration Isolation
Example 9.8
Isolation from Vibrating Base Solution
Setting ω=ωn,
1291 .
15 0 2
1 1
2 or 1
2 2 1
2 2
r
r T
T
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Vibration Isolation System with Partially Flexible Foundation
Base of isolator is partially flexible is as shown.
Mechanical impedance of base:
Displacementfrequency of
force
Applied
Z
台灣師範大學機電科技學系
-97-9.10 Vibration Isolation
• Vibration Isolation System with Partially Flexible Foundation
Equations of motion:
Harmonic solution:
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Vibration Isolation System with Partially Flexible Foundation
Amplitude of transmitted force:
Transmissibility of isolator
Z(ω) can be found experimentally by measuring the displacement produced by a vibrator.
台灣師範大學機電科技學系
-99-9.10 Vibration Isolation
• Shock Isolation
Shock load is a force load applied for less than one natural time period of the system
Impulse:
Velocity imparted to the mass,
i.e. application of shock load is equivalent to giving an initial velocity to the system
Initial conditions:
T F t dt F 0
m v F
x x x v
x 0
0 0 , 0
0
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Shock Isolation
Free vibration solution:
Force transmitted to the foundation due to spring and damper:
台灣師範大學機電科技學系
-101-9.10 Vibration Isolation
Example 9.11
Isolation Under Shock
An electronic instrument of mass 20kg is subjected to a shock in the form of a step velocity of 2m/s. If the maximum allowable values of deflection (due to clearance limit) and acceleration are specified as 20mm and 25g respectively, determine the spring constant of an undamped shock isolator.
C. R. Yang, NTNU MT
9.10 Vibration Isolation
Example 9.11
Isolation Under Shock Solution
Magnitude of velocity of mass:
Magnitude of acceleration of mass: where X is the displacement amplitude
X n
台灣師範大學機電科技學系
-103-9.10 Vibration Isolation
Example 9.11
Isolation Under Shock Solution
Selecting the value of ωn as 105.3681,
105.3681
2.2205 10 N/m20 2 5
2
m n
k
C. R. Yang, NTNU MT
9.10 Vibration Isolation
Example 9.12
Isolation Under Step Load
A sensitive electronic instrument of mass 100 kg is supported on springs and packaged for shipment. During shipping, the package is dropped from a height that effectively applied a shock load of intensity F0 to the instrument, as shown below.
台灣師範大學機電科技學系
-105-9.10 Vibration Isolation
Example 9.12
Isolation Under Step Load
Determine the stiffness of the springs used in the package if the maximum deflection of the instrument is required to be less than 2 mm. the response spectrum of the shock load is shown below with F0
=1000 N and t0 = 0.1s.
C. R. Yang, NTNU MT
9.10 Vibration Isolation
Example 9.12
Isolation Under Step Load Solution
Response spectrum
Making use of the known data,
The root can be found by MATLAB.
台灣師範大學機電科技學系
-107-9.10 Vibration Isolation
• Active Vibration Control
An active vibration isolation system is shown below.
C. R. Yang, NTNU MT
9.10 Vibration Isolation
• Active Vibration Control
System maintains a constant distant between vibrating mass and referee
Depending on the types of sensor, signal processor and actuator used, the system can be electromechanical, electrofluidic,
electromagnetic, piezoelectric or fluidic.
台灣師範大學機電科技學系