25
AMME5510 VIBRATIONS AND ACOUSTICS LECTURE NOTES
AMME5510 VIBRATIONS AND ACOUSTICS LECTURE NOTES
1
Contents Introduction ............................................................................................................................................ 4
Vibration at work ................................................................................................................................ 4
Single Degree of freedom vibration ........................................................................................................ 5
Consider the pendulum: ..................................................................................................................... 5
Equation of motion: ........................................................................................................................ 5
Static experiment: ........................................................................................................................... 5
Single degree of freedom equation of motion ................................................................................... 5
Vibration: ............................................................................................................................................ 7
Newton approach ........................................................................................................................... 7
2nd order DE ......................................................................................................................................... 9
Equivalent solutions: ....................................................................................................................... 9
Lagrange energy method: ................................................................................................................. 10
Potential and dissipative energy: .................................................................................................. 10
Network anaylsis: .......................................................................................................................... 10
SDOF equivalent systems .................................................................................................................. 11
Bending- fundamental mode of a cantilever ................................................................................ 12
SDOF characteristic polynomials................................................................................................... 14
Logarithmic decrement ................................................................................................................. 16
Transients ...................................................................................................................................... 16
SDOF Forced vibration .................................................................................................................. 17
Numerical simulation ........................................................................................................................ 22
1st order ODE ................................................................................................................................. 22
2nd order ODE with damping ......................................................................................................... 22
Multiple Degrees of Freedom (MDOF) ............................................................................................. 23
MDOF equations of motion .......................................................................................................... 23
MDOF Forced vibrations ................................................................................................................... 26
Using Lagrange .............................................................................................................................. 27
Continuous systems .............................................................................................................................. 29
Rayleigh Rits expansion..................................................................................................................... 29
Flexibility index approach ................................................................................................................. 30
Finite Element approach ................................................................................................................... 32
Fourier Transform ................................................................................................................................. 33
Time and frequency domain ............................................................................................................. 34
Formally ........................................................................................................................................ 36
Sampling ........................................................................................................................................ 38
2
Aliasing .......................................................................................................................................... 41
Fast Fourier Transform...................................................................................................................... 42
Problems with FT .......................................................................................................................... 42
Short Time Fourier Transform .......................................................................................................... 43
Wavelets ....................................................................................................................................... 44
Hilbert transform: ............................................................................................................................. 45
Periodicity and windowing ................................................................................................................ 46
Sampling ........................................................................................................................................ 46
System identification ............................................................................................................................ 48
Carrying our experiment ................................................................................................................... 48
Signals ............................................................................................................................................... 48
Signal descriptors: ......................................................................................................................... 49
Gigo: Garbage in garbage out ....................................................................................................... 49
Time domain integration .............................................................................................................. 55
Integration in the frequency domain: ........................................................................................... 58
Differentiation of data .................................................................................................................. 59
Non linear Behaviours ........................................................................................................................... 60
Equation of motion ........................................................................................................................... 60
Examples: ...................................................................................................................................... 60
Effect of non linearities ................................................................................................................. 63
Detection of non linearities .......................................................................................................... 63
Stability ............................................................................................................................................. 65
Bifurcation plots ............................................................................................................................ 65
Phase plots .................................................................................................................................... 66
Types of limit cycles ...................................................................................................................... 68
Types of bifurcations ..................................................................................................................... 68
Numerical Simulation of Non linearities ........................................................................................... 68
Clustering: ..................................................................................................................................... 69
System identification ........................................................................................................................ 70
Rational Fraction Polynomials....................................................................................................... 70
Eigenvalue realisation algorithm .................................................................................................. 70
Direct identification ...................................................................................................................... 71
Time domain integration: ............................................................................................................. 72
Acoustics ............................................................................................................................................... 72
Human ear: ................................................................................................................................... 72
Sound parameters ............................................................................................................................. 72
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Sound power ................................................................................................................................. 72
Sound intensity ............................................................................................................................. 73
Sound intensity level (loudness) ................................................................................................... 73
Acoustics wave equation .................................................................................................................. 73
Acoustics ........................................................................................................................................... 73
Physics of sounds .......................................................................................................................... 73
Wave properties and characteristics ............................................................................................ 74
Human ear perception .................................................................................................................. 75
Velocity of sound in a medium ..................................................................................................... 80
Reverberation and absorption (and sabins) ................................................................................. 83
Definitions of things: ......................................................................................................................... 85
Sound pressure/acoustic pressure: .............................................................................................. 85
dB scale (decibel scale) ..................................................................................................................... 86
Sound power ................................................................................................................................. 87
Sound pressure level ..................................................................................................................... 87
Frequency composition of sound: .................................................................................................... 87
Complex sound patterns: .............................................................................................................. 88
Sound intensity: ............................................................................................................................ 89
Sound power ................................................................................................................................. 90
Acoustic variables ............................................................................................................................. 91
Pressure/velocity relationship: ..................................................................................................... 92
Wave equation in terms of pressure ............................................................................................ 93
Ducts: ................................................................................................................................................ 96
Changing cross sectional area: ...................................................................................................... 97
Small ducts: ................................................................................................................................... 97
Right angle bend in ducts .............................................................................................................. 98
Branches in duct ............................................................................................................................ 99
Silencers/Mufflers ............................................................................................................................. 99
Helmholtz resonator ....................................................................................................................... 100
Walls:............................................................................................................................................... 100
Mass law for wall transmission: .................................................................................................. 101
Cavity walls.................................................................................................................................. 101
Combination of walls: ................................................................................................................. 102
4
AMME5510 VIBRATIONS AND ACOUSTICS LECTURE NOTES
Lecture 1. Thursday, 28 July 2016
Introduction Gareth Vio.
Room N306
Assessments:
Hammer testing; 15%; week 9
Fluid structure interaction; 20%; week 12
Research project; 55%; week 13 (group)
Labs 10%
Week 1 Introduction, Equation of Motion - 1DOF discrete systems Introduction to Acoustics
Week 2 Equation of Motion - MDOF systems Basic physics of acoustics
Week 3 MDOF Forced Vibration – Modal Coordinates Hearing and sound measurement
Week 4 Vibration laboratory Sound level estimation and environmental noise
Week 5 Equation of Motion - MDOF continuous systems / wave theory Ducts and Silencers
Week 6 Damping – real / complex modes Walls and surfaces
Week 7 Vibration testing Noise from machinery
Week 8 Human - Structure and Fluid - Structure Interaction Room acoustics
Week 9 Rotating System Balancing - static and dynamic Assessment Due: Hammer Testing*
Week 10 Signal Processing
Week 11 Modal Updating Acoustic lab
Week 12 Non-Linear vibration Acoustics lab Assessment Due: Fluid Structure Interaction*
Week 13 Revision and guest lecture Acoustics guest lecture Assessment Due: Research Project*
Vibration at work - Shimmy
- Hunting
- Resonace
- Unwanted vibrwtion
- Acoustic modes
- Sound problems
Lecture 2. Friday, 29 July 2016
5
Single Degree of freedom vibration
Consider the pendulum:
- Sketch the structure or part of interest
- Freebody diagram
o Newton or Lagrange
∑ 𝑀 = 𝐽𝛼; 𝐽 = 𝑚𝑙2
Equation of motion:
𝐽0𝛼(𝑡) = −𝑚𝑔𝑙 sin 𝜃(𝑡) ⟹ 𝑚𝑙2�̈�(𝑡) + 𝑚𝑔𝑙 sin 𝜃(𝑡) = 0
(and 𝑚𝑔𝑙 sin 𝜃(𝑡) is our restoring force)
We can linearise:
�̈�(𝑡) +𝑔
𝑙𝜃(𝑡) = 0
And we need to know the initial conditions 𝜃0; �̇�0
Static experiment: From experiments we know the relationship between force and displacement 𝐹 = 𝑘𝑥
Single degree of freedom equation of motion Basic form of a SDOF system:
6
Where 𝑘 is stiffness (N/m), 𝑐 is viscous dampng constant (Ns/m), 𝑥(𝑡) is displacement (m) and 𝑓(𝑡)
is force (N)
- Viscous damping is the most common linear representation of energy dissipation
- Energy converted to heat/sound (via internal friction
(hsteresis)/friction/magnetic/aerodynamics forces)
- Viscous model assumes damping proportional to velocity
- Symbolically shown as a dashpot
Other types of damping are hysteric and coulomb
Equations of moion can be set up a varity of ways:
- Free body diagrmas
- D’almbert’s principle
- Conservation of energy
- Lagrange
- Rayleigh-Rits
�̇�
𝐹
𝐶𝑜𝑢𝑙𝑜𝑚𝑏 𝑑𝑎𝑚𝑝𝑖𝑛𝑔
7
Vibration: - Caused by interaction of 2 forced
- Stiffness 𝑓𝑘 = −𝑘𝑥(𝑡)
- Mass 𝑓𝑚 = 𝑚𝑎(𝑡)
Newton approach
𝑚�̈� + 𝑐�̇� + 𝑘𝑥 = 𝑓(𝑡)
�̈� +𝑔
𝑙sin 𝜙 = 0
8
Pendulum with spring:
Rotational springs exert a moment when twiseted by 𝜙 ; 𝑀 = 𝐾𝜙
�̇� + (𝑔
𝑙+
𝐾
𝑚𝑙2) 𝜙 = 0
Effect of gravity on springs:
Undeformed position: 𝑚�̈� + 𝐾𝑥 = 𝑚𝑔
Equilibrium: 𝑚�̈� + 𝐾𝑦 = 0
9
Spings in parallel and series
Springs extension are equation in parallel: 𝐾 = 𝐾! + 𝐾2
In series: 1
𝐾=
1
𝐾1+
1
𝐾2
2nd order DE
If system is vibrating something must have happened: moved a distance 𝑥0 and then released. For
each degree of freedom, you need 2 states; velocity and displacement for that DOF
Final solution to SHM is
𝑥(𝑡) =√𝜔𝑛
2𝑥02 + 𝑣0
2
𝜔𝑛sin (𝜔𝑛𝑡 + 𝑡𝑎𝑛−1 (
𝜔𝑛𝑥0
𝑣0))
( atan2 in matlab is 0→2𝜋)
Equivalent solutions: 𝑥𝑡 = 𝐴 sin(𝜔𝑡 + 𝜙)
𝑥(𝑡) = 𝐴1 sin 𝜔𝑛𝑡 + 𝐴2 cos 𝜔𝑛𝑡
𝑥(𝑡) = 𝑎1𝑒𝑖𝜔𝑛𝑡 + 𝑎2𝑒−𝑗𝜔𝑛𝑡
10
Lagrange energy method: In single DOF:
𝑑
𝑑𝑡(
𝜕𝑇
𝜕�̇�) −
𝜕𝑇
𝜕𝑞+
𝜕𝑈
𝜕𝑞+
𝜕𝐷
𝜕�̇�= 𝑄
Where 𝑞 is the generalised coordinate (eg, 𝜃 in a pendulum), 𝑇 is kinetic energy; 𝑈 is potential
energy, 𝐷 is dissipation energy and 𝑄 is generalised force.
For an 𝑛 DOF there are a set of 𝑁 equations and derivatives
𝑑
𝑑𝑡(
𝜕𝑇
𝜕𝑞�̇�) −
𝜕𝑇
𝜕𝑞𝑖+
𝜕𝑈
𝜕𝑞𝑖+
𝜕𝐷
𝜕𝑞�̇�= 𝑄𝑖; 𝑖 = 1, … , 𝑛
Potential and dissipative energy: Potential:
𝑈 =1
2𝐾𝑥2
Potential in rotational spring
𝑈 =1
2𝐾𝜙2
Height 𝑈 = 𝑚𝑔ℎ
Energy dissipated dampler 𝐷 =1
2𝑐�̇�2
Network anaylsis:
11
Lagrange for linear spring
Applying lagrange:
𝑑
𝑑𝑡(
𝜕𝑇
𝜕�̇�) =
𝑑
𝑑𝑡(𝑚�̇�) = 𝑚�̈�
𝜕𝑈
𝜕𝑥= 𝑘𝑥
𝜕𝐷
𝜕�̇�= 𝑐�̇�
∴ 𝑚�̈� + 𝑐�̇� + 𝑘𝑥 = 0
Generalised force: for SDOF is
𝑄 =𝜕(𝛿𝑊)
𝜕(𝛿𝑞)
Lagrange for pendulum
𝑇 =1
2𝑚(𝑙�̇�)
2
𝑈 = 𝑚𝑔ℎ
𝑑
𝑑𝑡(
𝑑𝑇
𝑑�̇�) = 𝑚𝑙2�̈�;
𝑑𝑈
𝑑𝜙= 𝑚𝑔𝑙 sin 𝜙
𝑚𝑙�̇�2 + 𝑚𝑔𝑙 sin 𝜙 = 0
SDOF equivalent systems Continuous MDOF systems idealised as SDOF systems to allow behaviour of fundamental modes
- Fundamental mode is how structure vibrates at lowest natural frequency
- Equivalent energy argument is used:
o Equivalent mass has same 𝐾𝐸 as distributed mass/lumped mass
o Equivalent stif- apply point load 𝑃 to continuous structure and determine deflection
𝛿 the equivalent stiffness will be 𝑘𝑒 =𝑃
𝛿
Lecture 3. Thursday, 4 August 2016 (Guest lecture)
Vibration project ideas
- Railway carriage model
o How do multiple rail carts interact with eacherother/loose connections
- Complete car and look at roughness over bump/loose road
o Driver/passenger comfort over bumpy road
- Assumed modal model
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o Approximations vs full model
- Mimo response experimental/computational
- Bispectrum/trispectrum for non linear systems
o Which frequencies interact and where energy foes
- Bridge dynamics
- Energy harvesting
- Earthquake response
- Stadium response
o Fans jumping up and down in stands
- Ship/helicopter model
o How does ship motion effect motion of a helicopter on deck
- Broken?
o Can you tell something is broken by vibration response
Bending- fundamental mode of a cantilever
- Equivalent SDOF system
(𝑀 + 0.23𝑚)�̈� + (3𝐸𝐼
𝐿3) 𝑣 = 0
Table of equivalent system:
Rod in axial deformation
Shaft in torsion
Tantiliver beam
13
- Non uniform axial deformation:
Rod approximated as series spring
𝐾1 =𝐸𝐴1
𝐿1; 𝐾2 =
𝐸𝐴2
𝐿2
14
∴1
𝐾𝑒𝑞=
1
𝐾1+
1
𝐾2
Equivalent mass table
MDOF equivalent systems:
- Car can be thought of as a collection of masses and springs for preliminary vibration
calculations
SDOF characteristic polynomials
15
𝑚�̈� + 𝑐�̇� + 𝑘𝑥 = 0
𝜔𝑛 = √𝑘
𝑚;
𝑐
2𝑚= 𝜁𝜔𝑛
→ 𝜆 = 𝜁𝜔𝑛 ± 𝜔𝑛√1 − 𝜁2
Lecture 4. Friday, 5 August 2016
Undamped case:
𝜁 = 0 → 𝜆1,2 = ±𝑖𝜔𝑛
So we just get
𝑥(𝑡) = 𝑐1 sin 𝜔𝑛𝑡 + 𝑐2 cos(𝜔𝑛𝑡)
Overdamped:
𝜁 > 1
𝑥(𝑡) = 𝑐1𝑒 (−𝜁+√𝜁2−1)𝜔𝑛𝑡
+ 𝑐2𝑒(−𝜁−√𝜁2−1)𝜔𝑛𝑡
Underdamped:
𝜁 < 1
𝑥(𝑡) = 𝑒−𝜔𝑛 𝜁𝑡(𝑐1 sin 𝜔𝑑𝑡 + 𝑐2 cos 𝜔𝑑𝑡)
Typical values of damping ratio
built up metallic structures 0.01-0.03 (1-3%) steel / concrete structures 0.02-0.1 (2-10%) plastic
0.03-0.05 (3-5%) machined metal components 0.001-0.005 (0.1-0.5%) rubber 0.1-0.5 (10-50%)
16
Logarithmic decrement Method to determine natural frequency and damping ratio from underdamped decay
- Damping related to loss of energy between peaks
Let 𝑥1,..𝑛 be ths successive peak amplitureds for decay, assuming envelope amplityde at peak; the
rakio of adjacent peaks is then
𝑥𝑛−1
𝑥𝑛=
𝐴𝑒−𝜁𝜔𝑛𝑡
𝐴𝑒−𝜁𝜔𝑛(𝑡+𝜏)= 𝑒
2𝜋𝜁
√1−𝜁2
Definition:
𝛿 ≔ ln (𝑥𝑛−1
𝑥𝑛) =
2𝜋𝜁
√1 − 𝜁2
Transients - Steady state solution prevails for large 𝑡
- Often we ignore the transient term (how large is 𝜁, how long is 𝑡)
- Always check to make sure transient is not significant
o For example, earthquake, starting a car, starting a power station, explosions have
much larger load at begibeginning so transient are important
o However, many machine applications transient may be ignored
17
To earthquake proof buildings; damped spring mass system added at the top, with natural frequency
the same as the building; shifting the response out of the frequency range of the earthquake.
Lecture 5.
SDOF Forced vibration - Force added to system
𝑚�̈� + 𝑐�̇� + 𝑘𝑥 = 𝑓(𝑡)
Assuming 𝑓(𝑡) = 𝐹 sin 𝜔𝑡; we get 𝑥(𝑡) = 𝑋 sin(𝜔𝑡 − 𝜙)
Solving to get:
𝑋 =
𝐹𝑘
√(1 − 𝑟2)2 + (2𝜁𝑟)2; 𝜙 = tan−1 (
2𝜁𝑟
1 − 𝑟2)
With 𝑟 =𝜔
𝜔𝑛 frequency ratiol and damping ratio 𝜁 =
𝑐
2𝑚𝜔𝑛
Laplace representation: (frequency response function)
𝑋(𝑠) = ℒ(𝑥(𝑡)) = ∫ 𝑥(𝑡)𝑒−𝑠𝑡𝑑𝑡∞
0
Giving:
The frequency response function
𝑋(𝑠) =𝐹0𝑠
(𝑚𝑠2 + 𝑐𝑠 + 𝑘)(𝑠2 + 𝜔2)
∴𝑋(𝑠)
𝐹(𝑠)= 𝐻(𝑠) =
1
𝑚𝑠2 + 𝑐𝑠 + 𝑘
𝐻(𝑗𝜔) =1
𝑘 − 𝑚𝜔2 + 𝑐𝜔𝑗
Dynamic magnification factor
Static deflection under 𝐹 is 𝛿𝑠𝑠𝑡 =𝐹
𝑘; so the dynamic magnification factor is
18
|𝑀| =𝑋
𝛿𝑠𝑡=
1
√(1 − 𝑟2)2 + (2𝜁𝑟)2
Resonant peaks can be used to elimate damping value by identifying the half power points 1 and 2
It may be shown that 𝑟1,2 = √1 ∓ 2𝜁 and therefore that 𝜁 =𝑟2−𝑟2
2𝑟max=
𝜔2−𝜔2
2𝜔𝑛
Total response including transient
Is sum of steady state and homogenous (forced vibration) solutions
𝑥(𝑡) = 𝑋 sin(𝜔𝑡 − 𝜙) + 𝑍𝑒−𝜁𝜔𝑛𝑡 sin(𝜔𝑑𝑡 − 𝜓)
𝑋, 𝜙 defined for a given 𝜔 by analysis in this section and 𝐴, 𝜓 defined by initial conditions occurring
at the start of the excitation, namely that 𝑥0 and �̇�0
19
Wave phenomenon during forced vibrations
beats
When the drive frequency and natural frequency are close, a beating phenomena occurs
𝑥(𝑡) =2𝑓0
𝜔𝑛2 − 𝜔2
sin (𝜔𝑛 − 𝜔
𝑛𝑡) sin (
𝜔𝑛 + 𝜔
2𝑡)
- Have the smaller frequency waves inside the larger enveloper
Resonance
- When the drive frequency and natural frequency are the same the amplitude of the
vibration grows without bounds. This is known as a resonance condition
𝑥𝑝(𝑡) = 𝑡𝑋 sin(𝜔𝑡)
Which we substitution in and solve for 𝑋 =𝑓0
2𝜔
∴ 𝑥(𝑡) = 𝐴1 sin 𝜔𝑡 + 𝐴2 cos 𝜔𝑡 +𝑓0
2𝜔𝑡 sin 𝜔𝑡
20
(the 𝑓0
2𝜔𝑡 sin 𝜔𝑡 term means we grow without bound)
SDOF forced vibration with out of balance excitation
D’Almbert’s principle
Consider the system with pair of contra rotating out of balance masses; (mass 𝑚𝑟 and eccentricity
𝑒); such that the force is only applied in one direction. The out of balance force, 𝑓𝑜𝑜𝑏 will be
𝑓𝑜𝑜𝑏 = 𝑚𝑒𝜔2
The equation of motion is thus (with 𝑚 including 𝑚𝑟)
𝑚�̈� + 𝑐�̇� + 𝑘𝑥 = 𝑓𝑜𝑜𝑏(𝑡) = 𝑚𝑟𝑒𝜔2 sin 𝜔𝑡
Base excitation – transmissibility
- Consider spring mass system in series below
21
∑ 𝐹 = −𝑘(𝑥 − 𝑦) − 𝑐(�̇� − �̇�) = 𝑚�̈�
→ 𝑚�̈� + 𝑐�̇� + 𝑘𝑥 = 𝑐�̇� + 𝑘𝑦
Lecture 6.
22
Numerical simulation - Solving differenential equations by numerical integration
- Euler, Runge-Kutta methods ect are in Matlab
- We will use these to examine the non linear vibration problems which do not have analytical
solutions
1st order ODE Soliving
�̇� = 𝑎𝑥(𝑡); 𝑥(0) = 𝑥0
Euler integration:
𝑥𝑖+1 − 𝑥𝑖
Δ𝑡= 𝑎𝑥𝑖
𝑥𝑖+1 = 𝑥𝑖(1 + 𝑎Δ𝑡)
- Note the final solution changes based on time step
Effect of the time step
- Large time steps can create large errors
- Small time steps are more computationally intensive
o In example below; look at difference between Δ𝑡 = 0.5 ; and 0.05 seconds
2nd order ODE with damping
(�̇�1
�̇�2) = (
0 1
−𝑘
𝑚−
𝑐
𝑚
) (𝑥1
𝑥2)
With state vector 𝑥 and state matrix 𝐴:
Using Euler integration, we then get
𝑥(𝑡𝑖+1) = 𝑥(𝑡𝑖) + Δt Ax(ti)
23
𝑥𝑖+1 = [𝐼 + Δ𝑡𝐴]𝑥𝑖
- We use numerical solutions to solve when we cannot find analytical ones; but there can still
be many problems
Multiple Degrees of Freedom (MDOF)
MDOF equations of motion Up to now we have only considered SDOF
MDOF examples can be:
- Many masses with one degree of freedom
- Single mass with more than 1 DOF
- Combinations of the above
Set up is the same:
If we divide this up along each mass:
24
We get:
𝑚1�̈�1 + (𝑐1 + 𝑐2)�̇�1 − 𝑐2𝑥2̇ + (𝑘1 + 𝑘2)𝑥1 − 𝑘2𝑥2 = 0
𝑚2�̈�2 + 𝑐2𝑥2̇ − 𝑐2�̇�1 − 𝑘2�̇�1 + 𝑘2𝑥2 = 0
Giving us:
(𝑚1 00 𝑚2
) (�̈�1
�̈�2) + (
𝑐1 + 𝑐2 −𝑐2
−𝑐2 𝑐2) (
�̇�1
�̇�2) + (
𝑘1 + 𝑘2 −𝑘2
−𝑘2 𝑘2) (
𝑥1
𝑥2) = (
00
)
Ie
𝑀�̇� + 𝐶�̇� + 𝐾𝑥 = 0
Undamped systems
For a previous example, taking this system as undamped
(𝑚1 00 𝑚2
) (�̈�1
�̈�2) + (
𝑐1 + 𝑐2 −𝑐2
−𝑐2 𝑐2 + 𝑐3) (
�̇�1
�̇�2) + (
𝑘1 + 𝑘2 −𝑘2
−𝑘2 𝑘2 + 𝑘3) (
𝑥1
𝑥2) = (
00
)
