professor mike brennan - unesp · impact as a source of vibration m v o local strain m m model...
TRANSCRIPT
Sources of Vibration
impacts rotating machines
gears
acoustic
excitation
vortex shedding
flutter
Self-excited vibration
Impact as a source of vibration
Impact force
Piston slap
pile driver
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• Impacts occur in many engineering situations. As they take place over
a very short time-period, the frequency content of the impacting force
tends to be rich. Some examples are illustrated below
Impact as a source of vibration
ovm
local strain
m
m model
transmitted force, f
k
• Consider an elastic sphere that strikes a rigid surface as shown below
Example
• Once the mass-spring system is in contact with the surface it will begin
to oscillate with a period
2m
Tk
• The transmitted force is the product of the displacement and the
contact stiffness. Thus there is a a half-sine force-time curve.
• The displacement will be sinusoidal during the half-period the body
stays in contact with the surface.
Impact as a source of vibration
Time-history
Energ
y s
pectr
um
of
the f
orc
e
frequency 1.5
cT
2.5
cT
4
1
(-12dB/octave)
Energy spectrum
• To increase the contact time the stiffness of the impacting body should be
decreased. The high frequency content is then reduced.
Contact time = 2T m k
time
( )f t ov km
cT
Vibrations generated by rotating machinery
Gears
• The driving wheel is assumed to rotate at constant angular velocity Ω1
and the driven wheel has an average angular velocity Ω2, and has
angular displacement
2 2t
where 12 1
2
r
r and is known as th e transmission error
Transmission error is defined as the difference between the actual
position of the output gear and the position it would take if it were perfect
Vibrations generated by rotating machinery
Gears
• transmission error occurs due to
• the fundamental meshing frequency of the meshing force is given by
fundamental meshing frequency = number of teeth
deflection of gear teeth under load manufacturing defects
Vibrations generated by rotating machinery
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local strain model
o
Gears Typical spectrum
frequency (Hz)
am
plit
ud
e (
dB
)
Vibrations generated by rotating machinery
Rotating unbalance
em
xtm
2k 2kc
In a rotating machine, unbalance exists if the mass centre of the rotor does
not coincide with the axis of rotation
The unbalance me is measured in terms of an equivalent point mass m with
an eccentricity e.
The equation of motion is given by
2 sintm x cx kx me t
Assuming a harmonic response
results in sinx X t
2
2
1
1 2n n
meX
k j
where and 2n t tk m c km
Vibrations generated by rotating machinery
Rotating unbalance
em
xtm
2k 2kc
2
2
1
1 2
t
n n n
Xm
me j
10-1
100
101
10-2
10-1
100
101
102
n
tm X
me
increasing
damping
• At low speeds is small and hence the vibration amplitude is also small n
• At resonance when damping controls the vibration amplitude 1n
• At high speeds when , which means that the
vibration is constant and is independent of speed
1, Xn tme m
Vibrations generated by rotating machinery
Reciprocating unbalance
m
xtm
2k 2kc
e
• The excitation force is equal to the inertia force
of the reciprocating mass, which is approximately
equal to
2dynamic force me si sin2n te
tL
Note the second harmonic
L
Vibrations generated by rotating machinery
Critical speed of a rotating shaft
The critical speed occurs when the speed of rotation of the shaft is
equal to the frequency of lateral vibration of the shaft
•Mass of the shaft is negligible
•Lateral stiffness of the shaft is k
•M is the mass of the rotating disk
•G is the mass centre of the disk
•S is the geometric centre of the disk
•The bearings are much stiffer than
the shaft
Vibrations generated by rotating machinery
Critical speed of a rotating shaft
em
tm
2k c
2k
c
x
y The equations of motion in the x and y directions are
2 sintm x cx kx me t
2 costm y cy ky me t
Assuming a harmonic response gives
2
2
1
1 2
t
n n n
Xm
me j
2j
Y Xe
and
Since the two harmonic motions x(t) and y(t) have the same amplitude, and are in
quadrature, their sum a circle. The radius r = X or Y
O
G
S
n O
G
S
n
O
GS
n
Vibrations generated by rotating machinery
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Critical speed of a rotating shaft – effects of bearing stiffness
• The stiffness of the bearings acts in series with the stiffness of the shaft. Thus
the system becomes more flexible lowering the critical speed
Vibrations generated by rotating machinery
Critical speed of a rotating shaft – effects of bearing stiffness
The equations of motion in the x and y directions are
2 sinxtm x cx x me tk
2 cosytm y cy y me tk
If the system has two natural frequencies and therefore two critical speeds. yxk k
Since the two harmonic motions x(t) and y(t) have different amplitudes, and are in
quadrature, their sum an ellipse.
x
ymass centre
x
y
x
y
x y
Acoustic Excitation
txl
is the wavelength of the impinging sound field
is the trace wavelength and is given by t sint
The trace velocity is the speed at which the wavefront moves along the beam
and is given by where c is the speed of sound in air sintc c
The pressure distribution on the beam is given by
( , ) tj t k xp x t Pe
2where is the trace wavenumber given by t t
t t
k kc
Acoustic Excitation
The maximum response of the beam occurs at
A RESONANT FREQUENCY, when the length of the beam is equal to
an integer number of half-wavelengths
A COINCIDENT FREQUENCY, when the trace wavelength is equal to a
flexural wavelength
Acoustic Excitation
wavespeeds
frequency
Sound waves
Bending waves
Speed of sound
Critical
frequency
Acoustic Excitation
Critical frequency
The critical frequency fc is the
lowest frequency at which
coincidence can occur.
It is a frequency when the speed
of the impinging sound wave c,
equals the speed of the flexural
bending wave cb.
The critical frequency for a plate can
be determined using the equation
Product of plate thickness and
critical frequency in air (20°C)*
Material hfc (ms-1)
Steel
Aluminium
Brass
Copper
Glass
Perspex
Chipboard
Plywood
Light concrete
12.4
12.0
17.8
16.3
12.7
27.7
23
20
34
* For water multiply by 18.9 2
1.8c
l
cf
hc
where cl is the longitudinal wavespeed (≈5000 ms-1 for steel and aluminium)
and h is the plate thickness
Acoustic Excitation
Coincidence angle
The coincidence angle θco, is given by 1
12sinco c
c 1
co
incid
en
ce
an
gle
θco
2
Cri
tica
l fr
eq
ue
ncy
no coincidence
Self-excited vibrations
The force acting on a vibrating system is usually independent of the
motion of the system.
In a self-excited system the excitation force is a function of the
motion of the system.
Examples of such systems where this phenomenon occurs are
• Instabilities of rotating shafts
• The flutter of turbine blades
• The flow induced vibration of pipes
• Aerodynamically induced motion of bridges
• Brake squeal
Self-excited vibrations
Stable system
Vibrating system
Unstable system
Vibrating system +
feedback system
Self-excited vibrations
To see the circumstances that lead to instability, consider a SDOF system
k
m x
c
The equation of motion is
0mx cx kx
Substituting for 2
n k m and 2c mk
gives 22 0n nx x x
Assuming a solution of the form gives the characteristic equation stx Xe
2 22 0n ns s
which has a solution 2
1,2 1n ns
For oscillatory motion are complex. The system will be dynamically
Stable provided 1,2s
0
unstable stable
Self-excited vibrations
Locus of
real axis
imaginary axis
1 1
0
0
1
0 1
0 1
0
0
1 2 and S S
Self-excited vibrations
Simple example
The vibration of an aircraft wing can be crudely modelled as
mx cx kx Ax
where m, c and k are the mass, damping and stiffness values of the wing
modelled as a SDOF system and where is an approximate model
of the aerodynamic force on the wing.
Ax
(1)
Rearranging (1) gives
0mx c A x kx
If C > A the system is stable If C < A the system is unstable
This is an example of flutter instability
Flow induced vibrations
There are three mechanisms for inducing vibration by
fluid flow
• Turbulence
• Vortex shedding
• Instabilities
Vibrations induced by turbulence
• Most common cause of flow induced vibration
• swaying of trees by gusts of wind
• vibration of a car aerial due to turbulent flow
• anchored ship bobbing in a tubulent sea
• heat exchanger vibration due to turbulent flow
• vibration of aircraft fuselage due to turbulent boundary layer
• random vibration problem
Vibrating system
2
gg ffS H j S
ffS ggS
Vortex induced vibration
• Note vortices being shed alternatively from each side of
the structure
Dynamic
forces
Vortex induced vibration
• Any structure with sufficiently bluff trailing edge sheds vortices in a
subsonic flow
Dynamic
forces
D U
• The separation of the flow is a function of the Reynolds number (Re)
inertial forceRe
viscous force
uD
v
u is the flow velocity
D is width normal to the free stream
v is the kinematic viscocity of the fluid which is the viscocity divided by the fluid
density
Vortex induced vibration regimes of flow
across a circular cylinder [after Blevins]
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Vortex induced vibration regimes of flow
across a circular cylinder [after Blevins]
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local strain model
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Frequency of vortex shedding
• The frequency of vortex shedding fs is given by
s
Suf
D
u is the flow velocity
D is width normal to the free stream
Vortex induced vibration – other effects
Self-Limiting Effect
If the amplitude of vibration increases beyond approximately 0.5D, the
symmetric pattern of alternatively spaced vortices begins to break up.
This means that the vortex induced forces are self-limiting at vibrations
of the order of D
Lock-In Effect
When the vortex shedding frequency
is close to the cylinder natural
frequency, the frequency of vortex
shedding can shift to the cylinder
natural frequency. This is known as
the lock-in effect
Vortex induced vibration - solutions
• Increase internal damping
• Avoid resonance nf Du
S
where fn is the natural frequency of the structure
• Take measures to eliminate vortex shedding
Galloping vibrations and stall flutter
If a structure vibrates in a steady flow, the flow in turn oscillates relative to the
moving structure, and acts as an oscillating aerodynamic force on the structure.
If this force tends to diminish the vibrations of the structure, then the structure is
said to be aerodynamically stable.
If this force tends to increase the vibrations of the structure, then the structure is
said to be aerodynamically unstable.
k c
m u
x
Wing stiffness
and damping
Flutter – aircraft wings
Examples
Galloping – ice coated power lines
k c
m u
x
Cable stiffness
and damping
cable
ice
Galloping vibrations and stall flutter
k c
m u
x
Wing stiffness
and damping
Example
x
u
relu
Causes a static
displacement
Causes instability
The vertical force acting on the aerofoil due to the fluid flow is given by
[Blevins]
21 1
2 2L
L D
cf u Dc uD c x
ρ is the density of the fluid
cL is the lift coefficient
cD is the drag coefficient
u is the free stream velocity
Galloping vibrations and stall flutter
The equation of motion is
mx cx kx f
Substituting for the force component related to velocity gives
1
2L
D
cuDmx cx k c xx
or
Unstable when this < 0
10
2L
D
cmx c x kD c xu
2
LD
cu
cD c
Thus the minimum flow velocity to cause instability is
The system is unstable if the lift force increases when the model is slightly
rotated in a steady wind such that 0L Du c c
section xc Re
Galloping vibrations and stall flutter
xc
note negative gradient
x LD
c cc
vertical force coefficientxc
Example – bridge vibrations
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local strain model
transmitted force ÃÃFÄÄ
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stable
unstable
ae
rod
yn
am
ic d
am
pin
g
structural damping
Noise from Vortices
U
Vortices impinge on
structure generating sound
Vortices propagate at
speed of the flow
Sound propagates at
speed of sound
Summary
• Impacts
• Rotating machinery
– Gears
– Mass unbalance
• Acoustic excitation
• Self-excited vibration
• Flow-induced vibration
The following sources of vibration have been discussed
References
• R.H. Lyon 1987. Machinery Noise and Diagnostics, Butterworth
Publishers.
• F.S. Tse, I.E. Morse and R.T. Hinkle 1978. Mechanical
Vibrations, Theory and Applications, Second Edition, Allyn and
Bacon, Inc.
• F.J. Fahy 1985. Sound and Structural Vibration; Radiation,
Transmission and Response, Academic Press.
• R.D. Blevins 1977. Flow Induced Vibration. Van Nostrand
Reinhold Company.