Mechanical Vibrations
Chapter 1: Fundamental of Vibration
Dr. Azma Putra
Faculty of Mechanical Engineering
Department of Structure and Materials
Semester I/2014-2015
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
2
What you will get from this course
Nothing
Something
Lots of thing
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
3
Introduction
Most engineering structures vibrate:
machines, cars, aircraft, etc
Materials becoming lighter, more flexible, engines becoming faster
need to model, design, analyze, understand, treat
Vibration is usually (relatively) small, oscillatory motion about a static equilibrium position
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
4
Effect of vibration: Structural failure
Fatigue
*Courtesy of Mobius ILearn interactive
Machine failure
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
5
Effect of vibration: Structural failure
Fatigue in pipeline
Case study:
PETRONAS Malaysia LNG, Bintulu, Sarawak
Cracked pipe, one module had to be shut down
Loss RM25 million/day
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
6
Effect of vibration: Noise
Piling Railway
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
7
Effect of vibration: Instabilities
Flutter
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
8
Effect of vibration: Health
White finger: Hand-arm vibration syndrome
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
9
What causes vibration?
Imperfection in machine or structure
Design
Manufacturing defect
Installation
Assembly
Operations
Maintenance
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
10
How to quantify the vibration?
1. Measurement and/or Simulation
2. Analysis: frequency, amplitude and phase
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
11
Where to apply the knowledge?
1. Vibration isolation
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
12
2. Aircraft – Ground vibration test
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
13
3. Structural vibration
Noise, Vibration and Harshness (NVH)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
14
Which one requires more treatment to control vibration?
Comfort - Customer demand
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
15
4. Maintenance/Vibration monitoring
Rotating machinery Plant piping system: oil and gas
Most industrial problems !!
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
16
5. Building design
Millenium Bridge – London (2000)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
17
Most vibrations are undesirable, but there are many instances where vibrations are useful:
Tooth cleaning
Massage chair
Music instrument
Heartbeat
Vibration energy harvesting
VIBRATION: CAN IT BE FRIEND?
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
18
Terminologies and Definitions
Mass - store of kinetic energy
Stiffness - store of potential (strain) energy
Damping - dissipate energy
Force - provide energy
Ingredients in vibration:
Can you identify mass, stiffness and damping from a piece sheet of a panel?
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
19
Most contents of this course assume that the force and the resulting motion are time harmonic
i.e. a periodic motion that repeats at regular interval.
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
20
Three important results from vibration measurement:
1. Amplitude - HOW MUCH
2. Frequency - HOW FAST
3. Phase - HOW IT IS VIBRATING
*Courtesy of Mobius ILearn interactive
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
21
What is amplitude?
Amplitude is the level of vibration from the ‘equilibrium position’.
It can be displacement, velocity or acceleration.
maxy A
( )oy t t B
maximum amplitude:
instantaneous amplitude:
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
22
Am
pli
tud
e
Time Peak-Peak
Peak RMS Average
Amplitude: Time signal Desciptor
2
0
1( )
T
x t dtT
x(t)
RMS
0
1| ( ) |
T
x t dtT
Average
T
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
23
What is frequency?
Frequency is the number of ‘cycles’ repeated per second.
The unit is Heartz (Hz)
1f
T
T is the period,
i.e. the time required to complete one cycle
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
24
Low frequency vs. High frequency
Which light has the highest flashing frequency?
A
C
B
Which wave has the highest frequency?
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
25
What is phase?
Blue curve leads the green curve by π/2 radians
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
26
Example of a sinusoidal signal
General expression of a sinusoidal signal: ( ) sin( )x t A ωt φ
( ) 0.2sin(150 3.14)x t t
Amplitude (peak value) = 0.2 m
Frequency = 150 rad/s or 23.9 Hz
Peak-peak = 0.4 m
Phase = 3.14 rad or 180o
Period = 0.04 s
Average = 0.1 m
RMS = 0.14 m
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
27
Complex Exponential Notation
cos sinjωte ωt j ωt
Write time harmonic quantity as ( ) jωtx t Xe
amplitude (usually complex)
frequency
where
However in ‘real world’ we see Re x(t)
( ) cos( )x t X ωt φ ( ) jωtx t XeFor short, we just write
phase magnitude (absolute, real)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
28
Suppose X a jb
2 2X a b
Show that:
jφX X e
(1).
(2).
(3). ( ) cos( )x t X ωt φ
| |X = max. magnitude 1tanb
φa
= phase
HOW?
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
29
a. A time harmonic motion at 1 kHz has a peak acceleration of 100g (1 g = 9.8 ms-2). What is the peak displacement? (Ans. 0.0248 mm)
b. A time harmonic force f(t) = (4 + j3)ejωt produces a response x(t) = (2 - j)ejωt. Calculate the magnitude and the phase of the force. Calculate the magnitude and the phase of the displacement. Find the velocity per unit force magnitude, assuming ω = 5. (Ans. 5 N, 0.644 rad, 2.24 m, -0.464, 0.447 N/m, 2.24 m/s/N)
Problem examples
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
30
jeXX = |X|cos (ϕ) + j|X|sin (ϕ)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
31
Makes life easy but introduce complex numbers
Summary of Complex Exponential Notation
tωjXetx )(
Used widely in frequency response methods
Implicitly carries phase information
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
32
Vibration units: Definitions
Displacement:
The distance travel by the mass, how far up and down it is moving
Velocity:
Rate change of displacement. For example, how far it can cover in one second
Acceleration:
Rate change of velocity. How quickly the mass is speeding up or slowing down
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
33
Converting vibration units
( ) jωtx t Xe
Displacement:
Velocity:
( )( )
dx tv t x
dt
Acceleration: 2
2
( ) ( )( )
dv t d x ta t x
dt dt
*Note phase difference between them
Velocity leads displacement by 90o
Velocity lags acceleration by 90o
jωtjωXe
2 jωtω Xe
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
34
( ) jωtx t Xe
( ) jωtv t jωXe
2( ) ( ) jωta t j j ω Xe
Lead
Lag
Note on the imaginary sign
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
35
90o
270o
180o90
o
270o
180o90
o
270o
180o ωt
Displacement
Velocity
Acceleration
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
36
displacement probe
B
C
A
D
Maximum
displacement
If the sensor measures maximum displacement at point A, at which location is the maximum acceleration?
The maximum velocity?
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
37
Free vibration - no external forces act on the system
Forced vibration - forces act on the system
Damped/undamped system - damping does/doesn’t exist
What is the phase difference between the two systems?
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
38
FREE VIBRATION
( )x t
System vibrates at its natural frequency
Natural frequency
( ) sin( )nx t A ω t
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
39
FORCED VIBRATION
( )x t
System vibrates at its forcing frequency
Forcing frequency
( ) sin( )fx t A ω t
( )F t
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
40
We deal with only linear system.
same frequency as input
magnitude change
output proportional
to input
superposition holds
all components linear linear vibration (idealisation)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
41
Modeling
Degree of Freedom (DOF): number of independent coordinates to describe the motion.
Coordinates may be:
displacement of some points
rotation
other (modal amplitudes, waves, etc)
Number depends on:
how complex the system is
modeling simplifications + assumptions
(how we choose to model it)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
42
Example:
Modelling vibration of a motor cycle
suspension design (1 DOF)
Bounce +wheel hop (3 DOF)
Effect of motorcycle vibration to rider comfort (4 DOF) Simplification (2 DOF)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
43
Could you try to model this system?
Missile carrier: isolating the vibration from the ground
Mass of missile
Mass of truck
Suspension
Isolator
road input
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
44
Stiffness and Flexibility
Stiffness arises from any structural component that deforms (elastically)
under action of forces.
F kx
x : deformation (m) k : spring constant (N/m)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
45
Examples
vibration mounting
Engineering structures
?
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
46
Spring in series
Carry same force !!
1 ,F k y 2( )F k x y 21
Fk x F
k
1 2
1 1 1
eq
x
k F k k
Extension = sum of individual extension
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
47
Spring in parallel
Have same extension !!
Total force = sum of forces in each spring
1 1F k x
2 2F k x
1 2F F F
1 2eq
Fk k k
x
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
48
The equivalent stiffness can be found by considering:
force-deformation relation
potential energy-deformation relation (Chapter 2)
Fk
x
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
49
Cantilever beam
For cantilever with length L and a tip load F giving displacement x
3
3EIF x
L
EI : bending stiffness
Note that this is the stiffness only at the tip of the cantilever 3
3EIk
L
L F
x
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
50
Series or Parallel?
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
51
Shaft
Torque T produces rotation θ
GJT θ
L
G : Shear modulus
J : Polar moment of area of
shaft cross section
GJk
L
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
52
Helical spring
4
364
Gdk
nR
G : Shear modulus of spring material
d : Diameter of wire
2R : Diameter of turn
n : Number of turn
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
53
STIFFNESS TABLE
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
54
STIFFNESS TABLE
(cont.)
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
55
Linearisation
sinx r θ
If small displacement, i.e. θ is small:
3
sin ...3!
θθ θ θ
x rθThus
xθ
r
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
56
bx 1
ax 2
cx 3
1 ?x
2 ?x
3 ?x
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
57
POTENTIAL ENERGY
2120 0
( )x x
V Fdx kx dx kx
‘Linear’ spring equivalent stiffness ‘Torsional’ spring equivalent stiffness
= work done from unstreched position
212 eqV k x 21
2 eqV k θ
2
21 Rkkkeq 22
1 kR
kkeq
**note: we should relate x and θ
More discussion in Chapter 2
Lecture Note, Mechanical Vibrations, Semester I/2014-2015
58
References
B. R. Mace, ISVR Lecture Notes, Univ. of Southampton
D. J. Inman, Engineering Vibrations, 3rd Ed., Pearson
Animations courtesy of Dr. Dan Russell, Kettering University, USA
S. Rao, Mechanical Vibrations, 5th Ed., Pearson