Nonlinear Dynamics Lec 5 (but actually 6) Flashcards
What is harmonic balance
simple and ffective means of approximating the FRFs of nonlinear systems
What is the simplest definition of an FRF
Based on harmonic/sinusoidal excitation
if sign X sin wt is input and response is Y sin wt + phi
FRF H(w) = magnitude (Y/X (w))* e^i*phi(w)
How is an FRF obtained experimentally
over a range of frequencies wmin -> wmax at fixed frequency increment delta w, sinusoids Xsin wt are inject sequentially into the system of interest
At each frequency the time histories of the input and respose signals are recorded after transients have died away
ratio of complex response spectrum to te input spectrum yields the FRF value
How will the response to a sinusoid input vary for a linear vs nonlinear system
linear respose always at the same frequency as input, does not depend on excitation ampltiude (Pure FRF)
Nonlinear response at frequency other than excitation frequencies and distribution of energy amongst these frequencies depends on the level of excitation (Composite FRF)
What is a composite FRF
FRF of a nolinear system resulting from experimentation carried out in the exact same fashion as linear
given the symbol LAMBDA
What is the harmonic balance
analytical analogue of the stepped sine test for an FRF
Used to approximate the response of nonlinear systems
What will the FRF of a nonlinear SDOF system with a cubic stiffness look like
From perturbation theory we get harmonics, so instead of getting a single peak at w, we get multiple at 3w, 5w, 7w etc
Whats the difference in harmonis for a cubic and qudratic stiffness
cubic get harmoincs at odd integers, quadratic get harmonics at even integers
Why is perturbation theory/plotting nonlinear FRFs important in real life
As many real life structures are nonlinear e.g. helicopter blades/cracks in structures etc and analysis shows we get harmonics thus we might get resonance at a frequency we werent expecting using just linear analysis
What is sin(a+-b) equal to
sin(a)cos(b) +- cos(a)sin(b)
What is sin^3 (a) equal to
3/4 sin(a) - 1/4 sin 3a
What form of equation is the harmonic balance usually carried out on
Duffing equation
my.. + cy. + ky + k3 y^3 = x(t)
In harmonic balance what do we usually assume the input and output are
input is X sin(wt - phi) and output Y sin(wt)
We put the phase on the input rather than output as it simplifies the maths
In harmonic balance once we’ve input our assumed solutions to the duffing equation what do we do next
equate the coeffiicents of sin wt and cos wt, we approximate and ignore harmonics e.g. sin 3wt terms
How do we find magnitude and phase of FRF after completing harmonic balance
Take two equations, square and add them together for magnitude, divide one from the other for phase
What do we find the effective stiffness is from harmonic balance
keq = k + 3/4 k3 Y^2
At a fixed level of excitation what is the effective FRF natural frequency
wn = sqrt ( (k + 3/4 k3 Y^2) / m)
What can mathematicall prove with the equivalent natural frequency from harmonic balance
wn = sqrt ( (k + 3/4 k3 Y^2) / m)
thus if k3 > 0 natural freq increases with X - hardening
if k3 < 0 natural freq decreases with X - softerning
Why is harmonic balance a linearisation of the FRF
As ignore the sin 3wt term
Why do we have multiple solutions for Y from harmoic balance
For given X and w displacement responce Y is obtained by solving cubic equation X^2 = Y^2 ((….Y^2)^2 +…), as complex roots occur in conjugate pairs either have one or three real solutions (complex solutions ignore as unphyiscal)
What is the implication for testing caused by multiple solutions for Y as a result of harmonic balance
Getting bifurcation point, small exciation FRF barely distored unique single real root for all w
As X increases FRF more distorted until hits point Xcirt where FRF has vertical tangent beyond which range of w values which there are three real solutions for the response
Why do you have to use a stepped sine or sine dwell test instead of white noise to accelerate a nonlinear system
If white noise excite all frequencies at once, cant see diffierent individual modes if nonlinear. Need small steps
Draw a diagram of what a theoretical FRF plot of a nonlinear system would like
See presentation
the void between B and C
As test or simulation passes point wlow two new responses become possible and persist until whigh is reached and they disappear
Three solutions Y1>Y2>Y3 Y2 is unstable and would never be observed in practice
What happens to nonlinear systems in real life as you do a stepped sine or sine dwell test
Depends which way you are going, if upward sweep in frequncy unique response upto wlow, stays on Y1 branch by continuity until whigh where Y1 ceases and it jumps to Y3 giving a disconinuity in the FRF reverse if going the other way
What are wlow and whigh
Points at which FRF discontinuity occur
Draw a diagram of stepped sine wave test on upward sweep
See presentation
Draw a diagram of stepped sine wave test on downward seep
See presentation
How does FRF discontinuity work for softening/harderning systems
if k3 > 0 resonance peak moves to higher frequencies and jumps occur on the right hand side of the peak
if k3 < 0 jump occurs on the left of the peak and the resonance shifts downward in frequency
Where would you see FRF discontinuity
in magnitude and phase plot
What is the issue with the simple harmonic balance
We equated fundamental components and ignore 1/4 k3 Y^2 sin 3wt, would require k3 or Y = 0 not true
Need to add sin 3wt to trial solution but this creates new system of equations with 5wt, 7wt and 9wt. Repeat and would get infinite series infinite harmonics
Why are harmonics odd in the simple harmonic balance
stiffness function ky + k3 y^3 is odd, if the function were even or generic all harmonics would be present
When might harmonics become a really big issue
In multiple degrees of freedom system where multiple harmonics coincide with another - get massive resonance as exciting two modes at once
What happens when excitation is not a pure tone
excitation through sum of multiple sine waves of different frequencies response ends up containing harmonics of all frequencies
if two sine waves freq w1 and w2 to lowest nonlinear order frequencies 3w1, 2w1 +- w2, w1+-2w2 and 3w2 would be present
