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Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
*
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Some Counterintuitive Problems in Vibration
Hugh Hunt
Cambridge University Engineering Department
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Something is counter-intuitive if:
• it requires advanced/specialist knowledge
• it is obscure or difficult to observe
• it doesn’t fit with our experience
• we’ve never noticed it before
• we believed what our teachers said
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
VIBRATION
“Common sense will carry one a long waybut no ordinary mortal is endowed with an inborn
instinct for vibrations”.
“Vibrations are too rapid for our sense of sight … common sense applied to these phenomena is too common to be other than a source of danger”.
Professor Charles Inglis, FRS from his “James Forrest” Lecture, Inst Civil Engineers, 1944
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
“design process”
concept
iteration
product
vibration problemsVibrationConsultant
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Important concepts
• StiffnessFrequency =
• Massmk
m
k c
The mkc model
• Nodal points• Vibration modes• Non-linearity• Damping
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Helmholtz ResonatorNeck plug of mass m
Contained air of stiffness k
mk
m
k
V1
Smallervolumeof air:
stiffnessincreased
V2
Wallsmade
flexible: stiffness
decreased
V2 V2
Water recreates rigid enclosure:
stiffnessincreased
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
The tips of the tuning fork move on thearcs of circles and centrifugal inertia forces are generated, twice per cycle.
Suppose tip amplitude is 0.2mm, oscillating frequency is 440Hz, moving mass is 20% of the fork mass, thenthe 880Hz component of tip force Fis about 10% of the weight of the fork.
F
P
Tuning Fork:“P” is a nodal point, so why do we get more sound when “P” is put on a table?
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
mode 1 mode 2 mode 3 mode 4
AXIALVIBRATION
cLn
fn 2
E
c where
1n 2n 3n 4n
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
mode 1 mode 2 mode 3 mode 4
EULERBENDINGVIBRATION
cAI
L
af n
n 2 2
E
c where
4.221 a 7.612 a 1213 a 2004 a
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
A vibrating beam marked out with the nodal points is very useful. The location of the nodal points are:
Position of nodal points for a beam of L=1000mm (measured in mm from one end)
mode 1: 224 776 mode 2: 132 500 868 mode 3: 94 356 644 906 mode 4: 73 277 500 723 927 mode 5: 60 226 409 591 774 940 mode 6: 51 192 346 500 654 808 949 mode 7: 44 166 300 433 567 700 834 956 mode 8: 39 147 265 382 500 618 735 853 961
See the Appendix for details of how to derive these
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Axisymmetric bodies
Turbocharger blade vibration
Questions: 1. Do the blades fatigue less
rapidly if they are perfectly tuned, or is it better to mistune them?
2. Can vibration measurements made on a rotor be used to estimate its fatigue life?
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Constrained-layer damping 1. Works by introducing damping
material in places where shear strain is large
2. Material selection is important(i) not too rubbery(ii) not too glassy
- just right!
3. Temperature dependent
4. Effective over wide range of frequencies
5. Compromises strengthfrequency
ampl
itude
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Tuned absorber 1. Works by attaching a resonant
element, with just the right amount of damping
2. Works at one frequency only
3. Material selection again is important owing to temperature dependence of damping
frequency
ampl
itude
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
m
k c
The mkc model has great virtues:
- simple
- huge range of application
- “intuitive” … with a bit of thought
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
Appendix
Nodes of a Vibrating Beam
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
zy
Equation of motion:
For vibration, assume y(x,t)=Y(x)cos(t), so
This has general solution
Boundary condition for a fee end at z=0:
02
2
4
4
t
ym
z
yEI
mass per unit length m
flexural rigidity EI, length L
EI
mY
dz
Yd 244
4
4
with ,0
Free vibration of a beam
zDzCzBzAzY sinhcoshsincos)(
0and00
3
3
0
2
2
zz
dz
Yd
dz
Yd
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
so
i.e. C=A and D=B
Boundary condition for a free end at z=L:
so
and
or, in matrix form,
0and0 DBCA
0and03
3
2
2
LzLz
dz
Yd
dz
Yd
0sinhcoshsincos LBLALBLA
0coshsinhcossin LBLALBLA
0coshcossinhsin
sinhsincoshcos
B
A
LLLL
LLLL
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
For a non-trivial solution, the determinant must be zero, so
0coshcossinhsin
sinhsincoshcos
LLLL
LLLL
0)sinh)(sinsinh(sin)cosh(cos 2 LLLLLL
0coshcos22 LL L
L
cosh
1cos
L0 2
32
25
27
29
1
Exact solutions for L: 4.730 7.853 10.996 14.137
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
From aL the frequencies of free vibration are found using
aj= 22.37, 61.67, 120.90, 199.86, ...
or aj
The corresponding mode shapes are obtained by substituting j intothe matrix equation to find the ratio between A and B
so that
The location of nodal points is then found by looking for where Y(z)=0
EI
m 24
4
2
mL
EIa jj
...,,,, 2
292
272
252
23
0)sinh(sin)cosh(cos BLLALL
)sinh)(sincosh(cos
)cosh)(cossinh(sin)(
zzLL
zzLLzY
jjjj
jjjj
Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk
The location of the nodal points needs to be computed numerically, and the values are:
Position of nodal points for a beam of L=1000mm (measured in mm from one end)
mode 1: 224 776 mode 2: 132 500 868 mode 3: 94 356 644 906 mode 4: 73 277 500 723 927 mode 5: 60 226 409 591 774 940 mode 6: 51 192 346 500 654 808 949 mode 7: 44 166 300 433 567 700 834 956 mode 8: 39 147 265 382 500 618 735 853 961