Post on 13-Mar-2020
Vibration Isolation
Professor Mike Brennan
Vibration Isolation
• In vibration Isolation we modify the transmission path
Vibration
source Receiver Transmission path
• Simple approach
• Mobility approach – concept of isolator effectiveness
• Wave effects in the isolator
• Power flow approach
• Examples
Vibration Isolation by Resilient Connections
machines pipework
Low frequency models of vibration isolation
• Two problems
j t
tF e
Isolator
Receiver
Source
j t
eF e
j tXe
k c
m
(1)
Isolator
Host structure
equipment
k c
m
j t
tX e
j t
eX e
(2)
Low frequency models of vibration isolation
• The simplest model for vibration isolation is:
j t
tF e
Isolator
Receiver
Source
j t
eF e
j tXe
k c
m
The transmitted force, Fr is given by:
The equation of motion is:
( )mx cx kx f t For harmonic excitation:
2- em j c k X F
tF k j c X
(1)
(2)
Combining (1) and (2) gives the
force transmissibility
2
tF
e
F k j cT
F k m j c
Force transmissibility from SDOF model
(viscous damping)
• Transmissibility
can be written as:
2
1 2
1 2
nF
n n
j
T
j
where
,n
k
m
2 n
c
m
=0.01 =0.03 =0.1 =0.3
1 10 10
-3
10 -2
10 -1
10 0
10 1
10 2
Non-dimensional frequency n
Forc
e t
ran
sm
issib
ility
, T
F
2n
amplification isolation
Displacement transmissibility
Displacement Transmissibility
=
Force Transmissibility
t t
e e
X FT
X F
Isolator
Host structure
equipment
k c
m
j t
tX e
j t
eX e
Transmissibility- notes
•At low frequencies when n then 1T
•At resonance when n then 1
12
T
•At high frequencies when n then 2
1 2n
n
j
T
High damping T reduces at 20 dB/decade
Low damping T reduces at 40 dB/decade
Transmissibility – Low frequency
Below resonance At resonance Well-above resonance
Random excitation
Discussion of the SDOF model
• The simple model has a number of limitations for practical applications:
• Isolators often behave as hysteretically damped springs at
low frequencies. Replace (k+jc) by k(1+j) .
• The foundation is not rigid.
• At high frequencies internal resonances occur in the isolators.
• At high frequencies the source and receiver have modal behaviour
and can no longer be considered to be rigid.
• There are usually multiple mounts / mounting points.
• Usually more than one direction is important, including rotations.
Calculation of the fundamental natural frequency
from the static deflection
k
Undeformed
spring
k
m
System equilibrium
position
n
k
m
n
g
mg
k Static deflection
Isolator configurations- examples
Compression
l
A
stiffness, EA
kl
Shear
A
h
stiffness, GA
kh
2 1E G
Poisson's ratio
Young's modulusE
shear modulusG
Damping mechanism: complex Young’s modulus, * 1E E j
Loss factor
Isolator configurations- examples
Vibration isolation – The mobility approach
Source
sYA
Equipment to be mounted here F
Requirement • An item of flexible equipment is required to be
mounted on a flexible structure (source) that is
excited by unknown forces
• minimum vibration transmission
f
AVfree
• Before attaching the equipment the free velocity at A due to F
is
• The source mobility at point A is Ys. Now attach the equipment
(receiver) with mobility Yr to A through an isolator with mobility Yi
Vibration isolation – The mobility approach
Source Isolator Receiver
sYiY rY
fVrV
F A
• The problem is to find Vr in terms of and the system mobilities,
i.e., given , Ys,Yi and Yr, what is Vr?
f
AVf
AV
• Assumption: The isolator is massless, i.e., the inertia forces
in the isolator are negligible
Vibration isolation – The mobility approach
( ) (2)
( ) (
without isolator
with isolator 1)r
r
VE
VIsolator effectiveness
So
1 i
s r
YE
Y Y
Source Isolator Receiver
sYiY rY
fV rVFree velocity
. rr f
r s i
YV V
Y Y Y
(1)
• If no isolator is fitted, i.e., the receiver is rigidly connected directly
to the source, then Yi=0, and
. rr f
r s
YV V
Y Y
(2)
Vibration isolation - notes
1 i
s r
YE
Y Y
• E should be as large as possible for good isolation
• E depends on the source, isolator and receiver mobilities
• For good isolation i s rY Y Y
• We require high isolator mobility
• At high frequencies:
Ys and Yr can be large ~ resonance frequencies
Yi can be small ~ wave effects in the isolator
(the above theory is not strictly applicable because of inertia forces in
the isolator)
Some Measurements on a Container Ship
Some Measurements on a Container Ship
Some Measurements on a Container Ship
Some Measurements on a Container Ship
Some Measurements on a Container Ship
0 5 10 15 20 25 30 35 40
0
100
200
300
400
500
600
700
800
Engin
e s
peed(r
pm
)
Time(sec)
Some Measurements on a Container Ship
Horizontal transverse motion
Some Measurements on a Container Ship
Vertical motion
Some Measurements on a Container Ship
Vertical motion; 75% engine load and synchronized speed of 720 rpm
Some Measurements on a Container Ship
Transverse motion; 75% engine load and synchronized speed of 720 rpm
Some Measurements on a Container Ship
Vertical acceleration either side of an engine mount
Some Measurements on a Container Ship
Transverse acceleration either side of an engine mount
Vibration isolation at high frequencies
source
receiver
isolator
Wave effects in the isolator
• At high frequencies there are standing waves in
the isolator
• Simple theory based upon the massless element
is not appropriate
• Suitable model is a distributed parameter
element for the isolator (including mass and
stiffness)
Wave effects in isolators
j tVe
j tFe
Example: Cylindrical distributed parameter element
l
A Appropriate quantity is the transfer impedance
sin
F jkEA
V kl (No damping) from tables
kE
(longitudinal wavenumber)
Young's modulusE
density
F
V
10 - 1
10 0
10 1
10 - 1
10 0
10 1
Non-dimensional frequency
Wave effects in isolators
Wave effects in isolators
j tVe
j tFe
l
sin
F jkEA
V kl
As 0, then
skF jEA
V l j
where s
AEk
l static stiffness
(1)
Substituting for ks in (1) gives
sin
sj k mF
V kl
where mass of isolatorm Al
(2)
The modulus of equation (2) has a
minimum when sin 1, which iskl
min
s
Fk m
V
Wave effects in isolators - damping
We can include damping by using the complex Young’s modulus
* 1E E j
This results in a complex wavenumber * 1 2k k j
Substituting into (1) gives
* *
*
1 1 2
sinsin cos cos sin
2 2
jkEA j jF jk E A
jkl jklV k lkl kl
max
2 skF
V so
Assuming sin 0
and <<1,gives
kl
0 1 12
jkl
0 0
1 sk m To find the first natural frequency set , to give sin 0kl
Wave effects in isolators
isolator
mass m
stiffness ks
j tVe
j tFe
10 -1
10 0
10 1
10 -1
10 0
10 1
ks/
2ks/
(ksm)
F
V
1
• At low frequencies the transmitted force is controlled by the static stiffness
• At high frequencies, above the first isolator resonance, the isolator
impedance . Thus decreasing the isolator mass will improve the
situation.
• Decreasing the mass will also increase
• At isolator resonances, the transmission of vibration is controlled by the
isolator damping
sk m
1 sk m
Isolator effectiveness
Frequency (arbitrary units)
10 -1
10 0
10 1
10 2
-40
-20
0
20
40
60
80
100
Iso
lato
r e
ffe
ctive
ne
ss (
dB
)
Effectiveness of isolator with rigid
source and receiver and massless
isolator
Isolator resonance
frequencies
Fundamental resonance
frequency
Source and receiver
resonance frequencies
10 -1
10 0
10 1
10 2 10
-3
10 -2
10 -1
10 0
10 1
10 2
10 3
Wave Effects in Isolators
10 -1
10 0
10 1
10 2 10
-3
10 -2
10 -1
10 0
10 1
10 2
10 3
No mass in isolator
0.01
0.1T
ran
sm
issib
ility
, T
Non-dimensional frequency, n
j t
eF e
j t
tFe
mass of isolator
0.05mass of equipment
Transmissibility of a simple mounting system on the assumption that
the mount behaves as a rod with internal damping
Further considerations
1. Isolators should be placed symmetrically with respect to
the centre of gravity
2. The centre of gravity should be located as low as possible
to avoid rocking effects
machine
mass
rubber or cork etc
isolation
foundation
centre of gravity
Example “floating floor”
Further considerations
3. Greater isolation can be achieved if a compound system is
used
machine
added mass
j t
eF e
j t
tFe 10
-110
010
110
2-80
-60
-40
-20
0
20
40
compound system
Simple system
1
Tra
nsm
issib
ility
(dB
)
Miscellaneous - isolation from
earthquake motion
Miscellaneous - isolation from
earthquake motion
Miscellaneous - isolation from
earthquake motion
Miscellaneous – rotating shafts
Miscellaneous – Isolation of buildings
and railways
Building isolator
Power flow approach
machinery
(source)
pipework
pipe hangers
isolators
substructure (receiver) 6 dof – displacements and rotations
Consider the machinery installation
• Can measure pressures, accelerations and strain
• Cannot compare measurements, thus cannot determine which paths
are important
• Isolators, pipes, shafts, electrical connections, acoustics, etc
• Unifying concept is power transmission or “power flow”
Vibration power input to a structure
structure
force, f
velocity, v
Power input to the structure is given by
0
T
in
1 = f(t) v(t) dt
T (1)
For harmonic inputs ( ) Re j tf t Fe and
( ) Re j tv t Ve equation (1) simplifies to
*1 1
cos Re2 2
in F V FV
Phase angle between
force and velocity
Complex
conjugate
Force source 21Re
2in F Y Velocity source
21Re
2in V Z
mobility of structure impedance of structure
1
2
beams
Power input to infinite beams and plates is constant
1
2
• Moments are better at inputting
power at high frequencies
machine
plates
is constant
pow
er
frequency
Plate
force
Plate
moment Beam
force Beam
moment
Summary
• Vibration isolation – simple approach
• Vibration isolation – mobility approach
• Wave effects in isolators
• Vibration power
References
• C.M. Harris, 1987, Shock and Vibration Handbook, Third
Edition, McGraw Hill.
• R.G. White and J.G. Walker, 1982, Noise and Vibration, Ellis
Horwood Publishers.
• L.L. Beranek and I.L. Ver, 1992. Noise and Vibration Control
Engineering, John Wiley and Sons.
• S.S. Rao, 1990, Mechanical Vibrations, Second Edition, Adison
Wesley.