duffing
DESCRIPTION
Duffing. Two Springs. A mass is held between two springs. Spring constant k Natural length l Springs are on a horizontal surface. Frictionless No gravity. l. k. m. k. l. Transverse Displacement. The force for a displacement is due to both springs. Only transverse component - PowerPoint PPT PresentationTRANSCRIPT
DuffingDuffing
Two SpringsTwo Springs
A mass is held between two A mass is held between two springs.springs.• Spring constant Spring constant kk
• Natural length Natural length ll
Springs are on a horizontal Springs are on a horizontal surface.surface.• FrictionlessFrictionless
• No gravityNo gravity
k l
m
k l
Transverse DisplacementTransverse Displacement
The force for a displacement The force for a displacement is due to both springs.is due to both springs.• Only transverse componentOnly transverse component
• Looks like its harmonicLooks like its harmonic
s
x
s
22 xls
22
222xl
xlxlkF
221
112
lxkxF
sin2 22 lxlkF
Purely NonlinearPurely Nonlinear
The force can be expanded The force can be expanded as a power series near as a power series near equilibrium.equilibrium.• Expand in Expand in xx//ll
The lowest order term is The lowest order term is non-linear.non-linear.• FF(0) = (0) = FF’(0) = ’(0) = FF’’(0) = 0’’(0) = 0
• FF’’’(0) = 3’’’(0) = 3
Quartic potentialQuartic potential• Not just a perturbationNot just a perturbation
221
112
lxl
xklF
3
l
xklF
424
xl
kV
Mixed PotentialMixed Potential
Typical springs are not at Typical springs are not at natural length.natural length.• Approximation includes a Approximation includes a
linear termlinear terms
x
s
l+d
l+d
3
3
2x
l
dlkx
l
kdF
4
32
4x
l
dlkx
l
kdV
Quartic PotentialsQuartic Potentials
The sign of the forces influence the shape of the The sign of the forces influence the shape of the potential.potential.
42
42x
kx
kV
42
42x
kx
kV
42
42x
kx
kV
hard softdouble well
Driven SystemDriven System
Assume a more complete, Assume a more complete, realistic system.realistic system.• Damping termDamping term
• Driving forceDriving force
Rescale the problem:Rescale the problem:• Set Set tt such that such that 00
22 = k = k//m = m = 11
• Set Set xx such that such that = k = k//m = m = 11
This is the Duffing equationThis is the Duffing equation
tfxkkxxbxm cos2 3
tfxxxm
bx cos2 32
0
tfxxxx cos2 3
Steady State SolutionSteady State Solution
Try a solution, match termsTry a solution, match terms )](cos[)()( tAtx
tftAtAtA cos)(cos)sin(2)cos()1( 332
tfxxxx cos2 3
trig identities )(3cos)cos()(cos 4
1433 ttt
)sin(sin)cos(coscos tftftf
0
)(3cos
)sin(]sin2[
)cos(]cos)1([
341
2432
tA
ttfA
ttfAA
0)(3cos
2sin
)1(cos
341
2432
tA
Atf
AAtf
Amplitude DependenceAmplitude Dependence
Find the amplitude-Find the amplitude-frequency relationship.frequency relationship.• Reduces to forced harmonic Reduces to forced harmonic
oscillator for A oscillator for A 0 0
Find the case for minimal Find the case for minimal damping and driving force.damping and driving force.• f, f, both near zero both near zero
• Defines resonance conditionDefines resonance condition
]4)1[(
4sin
)1(cos
222243222
22222
22432222
AAf
Atf
AAtf
])2()1[( 22222 Af
)1()(
10
]0)1[(0
234
2432
224322
A
A
AA
Resonant FrequencyResonant Frequency
The resonant frequency of a The resonant frequency of a linear oscillator is linear oscillator is independent of amplitude.independent of amplitude.
The resonant frequency of a The resonant frequency of a Duffing oscillator increases Duffing oscillator increases with amplitude.with amplitude.
A
Duffing oscillator
Linear oscillator
)1( 234 A
HysteresisHysteresis
A Duffing oscillator behaves A Duffing oscillator behaves differently for increasing and differently for increasing and decreasing frequencies.decreasing frequencies.• Increasing frequency has a Increasing frequency has a
jump in amplitude at jump in amplitude at 22
• Decreasing frequency has a Decreasing frequency has a jump in amplitude at jump in amplitude at 11
This is hysteresis.This is hysteresis.
A
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