basic phenomenology of simple nonlinear vibration … phenomenology of simple nonlinear vibration!...
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Basic phenomenology of simple nonlinear vibration
!
(free and forced)
Manoj Srinivasan (2016)
MassSpring
Damper
x(t)
x(t)
x(t)
e
mass m
length lgravity g
A
O
Hardening
Softening
Nonlinear spring-mass system
No damping
Nonlinear spring-mass systemNo damping
Frequency (time period) of free vibration oscillations
depends on oscillation amplitude-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
xDot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1oscillation amplitude
6.25
6.3
6.35
6.4
6.45
6.5
6.55
time
perio
d of
osc
illatio
n
softening cubic nonlinearity
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1oscillation amplitude
6.05
6.1
6.15
6.2
6.25
6.3
time
perio
d of
osc
illatio
n
hardening cubic nonlinearity
unlike for linear spring-mass system
Timeperiod
Oscillation amplitude Oscillation amplitude
HardeningSoftening Timeperiod
guitar “pitch glide” due to amplitude dependence
of frequency? (assume stiffening)
how would you expect the frequency to change as the oscillation amplitude
decreases?
0 20 40 60 80 100 120 140 160 180 200t
-10
-8
-6
-4
-2
0
2
4
6
8
10
x
Damped Duffingwith alpha>0
stiffening spring
x(t)
t
Damped free vibrations of Duffing equation
https://www.youtube.com/watch?v=VkkOFLouQDs
notice frequency change
MassSpring
Damper
Damping ratio Natural frequency
( Static deflection to
static force F0 )
External forcing
Normalized force amplitude =(units of length)
Linear spring-mass-damper system
Frequency response oflinear spring-mass-damper system
figure source: Wikipedia
MassSpring
Damper
External forcing
Duffing equation with forcing
x(t)
x(t)
x(t)
e
mass m
length lgravity g
A
O
Hardening
Softening
cubic nonlinearity
Cubic nonlinearity with or without quadratic nonlinearity
cubic
When we do Taylor series of an odd function about an equilibrium for with spring force = 0
When we do Taylor series of odd function about an equilibrium with spring force not = 0
quadratic
(Recall HW1 forillustrative example)
cubic
(or) just Taylor series of a not-odd function
Duffing’sequation
Frequency response ofDuffing equation
(cubic nonlinearity)
Primary resonance . Big response amplitude when forcing frequency ωo ~ ‘linear’ natural frequency ωn
Super-harmonic resonance: Big response amplitude when forcing frequency ωo ~ ωn / integer (example: ωo ~ ωn/3)!Sub-harmonic resonance: Big response amplitude when forcing frequency ωo ~ ωn x integer. (example: ωo ~ 3ωn)!!
Secondary resonance
Primary resonance and secondary resonances
Multiple peaks due to nonlinearity even though it is a single DOF system
( Recall that an N DOF linear system (N>1) will have multiple peaks due to there being N modes and corresponding natural frequencies)
Linear vs Hardening vs SofteningHow the primary resonance’s frequency response changes
Linearsystemresponse
Hardening springSoftening spring
forcing frequencyforcing frequency
Response amplitude (y axis) vs forcing frequency (x axis)for 3 cases
arrows indicate response obtained in forward and backward sweeps
Nayfeh and Mook
Hardening vs SofteningAmplitude response by simulation
until steady state
0 0.5 1 1.5 2 2.5 3 3.5Forcing frequency �
0
1
2
3
4
5
6
7
8
9
10
Stea
dy O
scilla
tion
Ampl
itude
Steady response amplitude vs frq. with the points connectedcubic nonlinearity alpha = 0.5
forward frequency sweepbackward frequency sweep
0 0.5 1 1.5 2 2.5 3 3.5Forcing frequency �
0
10
20
30
40
50
60
Stea
dy O
scilla
tion
Ampl
itude
Steady response amplitude vs frq. with the points connectedcubic nonlinearity alpha = -4
forward frequency sweepbackward frequency sweep
Hardening Softening
Forward sweep (magenta) and Backward sweep (blue) shown
see MATLAB program
Frequency response for forced Duffing with
hardening spring
0.5 1 1.5 2 2.5 3 3.5 4forcing frequency
0
0.5
1
1.5
2
2.5
3
3.5
4
resp
onse
am
plitu
de
Frequency response for forced Duffing with
softening spring
Amplitude response obtained by finding fixed points of Poincare maps
(so we can find both stable and unstable motions)
blue = stable periodic responsered = unstable periodic response
forcing frequency forcing frequency
resp
onse
am
plitu
de
resp
onse
am
plitu
desee MATLAB program
Ueda shows that the fully nonlinear forced Duffing(linear stiffness = 0) has many parameter regimes with
many different behavior
Dampingcoeff
c
Forcing amplitude A
See paper uploaded,Ueda 1991.
Following slidesshow some MATLABillustrations of this
complexity
-1 -0.5 0 0.5 1x
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
xdot
Five different co-existing periodic motionsFor the same parameter values: k = 0, A = 0.2, c = 0.08, omega = 1
0 10 20 30 40 50 60 70t
-1.5
-1
-0.5
0
0.5
1
1.5
x(t)x(t)
t
xDot
x
Chaotic motion (unique)For the same parameter values: k = 0, A = 10, c = 0.1, omega = 1
0 20 40 60 80 100 120 140 160 180 200t
-4
-3
-2
-1
0
1
2
3
4
x(t)
-8 -6 -4 -2 0 2 4 6 8x
-6
-4
-2
0
2
4
6
xdot
Chaotic
0 10 20 30 40 50 60 70 80 90 100t
-4
-3
-2
-1
0
1
2
3
4
x(t)
0 10 20 30 40 50 60 70 80 90 100t
-4
-3
-2
-1
0
1
2
3
4
x(t)
Co-existing periodic motion and chaotic motionFor the same parameter values: k = 0, A = 12.5, c = 0.1, omega = 1
-10 -8 -6 -4 -2 0 2 4 6 8 10x
-8
-6
-4
-2
0
2
4
6
8
xdot
periodic motion chaotic motion
0 1 2 3 4 5 6 7 8 9 10Omega (Hz) -- in angular frequency
0
1
2
3
4
Fourier Spectrum of External Forcing function
0 1 2 3 4 5 6 7 8 9 10in angular frequency (Hz)
0
0.5
1
1.5
2Fourier Spectrum of the oscillation Response
Frequency content of response
c = 0.1m = 1, k = 1epsilon = 0.4omega = 1
A = 5
Even when the responsehas the same frequency as the
forcing, there can be other harmonics in response due to
nonlinearity(unlike linear systems)
Primary resonance (forcing ω = 1)
Input
Response
Four
ier
ampl
itude
spe
ctru
m
frequency
0 2 4 6 8 10 12 14 16 18 20Omega (Hz) -- in angular frequency
0
1
2
3
4
Fourier Spectrum of External Forcing function
0 2 4 6 8 10 12 14 16 18 20in angular frequency (Hz)
0
0.5
1
1.5
2
2.5
3
3.5Fourier Spectrum of the oscillation Response
c = 0.1m = 1, k = 1epsilon = 0.4omega = 2
A = 5
Frequency content of responseForcing freq ω = 2
Input
Response
Four
ier
ampl
itude
spe
ctru
m
frequency
far fromresonance
we don’t see much higher harmonics in
response
0 1 2 3 4 5 6 7 8 9 10Omega (Hz) -- in angular frequency
0
2
4
6
8
Fourier Spectrum of External Forcing function
0 1 2 3 4 5 6 7 8 9 10in angular frequency (Hz)
0
0.2
0.4
0.6
0.8
1
1.2Fourier Spectrum of the oscillation Response
Chaos = broad band frequency content
40 60 80 100 120 140 160t
-4
-3
-2
-1
0
1
2
3
4
x
Phase space in 3D
x(t)
Four
ier
ampl
itude
spe
ctru
m
frequency
chaotic motion
0 1 2 3 4 5 6 7 8 9 10Omega (Hz) -- in angular frequency
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14Fourier Spectrum of External Forcing function
0 1 2 3 4 5 6 7 8 9 10in angular frequency (Hz)
00.05
0.10.15
0.20.25
0.30.35
Fourier Spectrum of the oscillation Response
Ueda(Manoj
notation)!
c = 0.08k = 0
epsilon = 1; omega = 1A = 0.2;
0 1 2 3 4 5 6 7 8 9 10Omega (Hz) -- in angular frequency
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14Fourier Spectrum of External Forcing function
0 1 2 3 4 5 6 7 8 9 10in angular frequency (Hz)
0
0.05
0.1
0.15
0.2
0.25
0.3Fourier Spectrum of the oscillation Response
period 2 motion
period 3 motion-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
x
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
xdot
x
xDot
-0.4 -0.2 0 0.2 0.4 0.6x
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
xdotxDot
x
Beware: !softening cubic nonlinearityand purely quadratic nonlinearity!have regimes where the stiffness is ‘negative’ for somelarge amplitudes => system can go to infinity if the forcing is not small enough!Fix:!A quadratic nonlinearity could be accompanied by a stiffeningcubic nonlinearity which keeps the motion bounded!A softening cubic nonlinearity could be accompanied by a stiffening x^5 nonlinearity, which keeps the motion bounded!