Nonlinear Dynamics Lec 5 (but actually 6)

Question 1

What is harmonic balance

Answer 1

simple and ffective means of approximating the FRFs of nonlinear systems

Question 2

What is the simplest definition of an FRF

Answer 2

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)

Question 3

How is an FRF obtained experimentally

Answer 3

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

Question 4

How will the response to a sinusoid input vary for a linear vs nonlinear system

Answer 4

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)

Question 5

What is a composite FRF

Answer 5

FRF of a nolinear system resulting from experimentation carried out in the exact same fashion as linear
given the symbol LAMBDA

Question 6

What is the harmonic balance

Answer 6

analytical analogue of the stepped sine test for an FRF

Used to approximate the response of nonlinear systems

Question 7

What will the FRF of a nonlinear SDOF system with a cubic stiffness look like

Answer 7

From perturbation theory we get harmonics, so instead of getting a single peak at w, we get multiple at 3w, 5w, 7w etc

Question 8

Whats the difference in harmonis for a cubic and qudratic stiffness

Answer 8

cubic get harmoincs at odd integers, quadratic get harmonics at even integers

Question 9

Why is perturbation theory/plotting nonlinear FRFs important in real life

Answer 9

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

Question 10

What is sin(a+-b) equal to

Answer 10

sin(a)cos(b) +- cos(a)sin(b)

Question 11

What is sin^3 (a) equal to

Answer 11

3/4 sin(a) - 1/4 sin 3a

Question 12

What form of equation is the harmonic balance usually carried out on

Answer 12

Duffing equation

my.. + cy. + ky + k3 y^3 = x(t)

Question 13

In harmonic balance what do we usually assume the input and output are

Answer 13

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

Question 14

In harmonic balance once we’ve input our assumed solutions to the duffing equation what do we do next

Answer 14

equate the coeffiicents of sin wt and cos wt, we approximate and ignore harmonics e.g. sin 3wt terms

Question 15

How do we find magnitude and phase of FRF after completing harmonic balance

Answer 15

Take two equations, square and add them together for magnitude, divide one from the other for phase

Question 16

What do we find the effective stiffness is from harmonic balance

Answer 16

keq = k + 3/4 k3 Y^2

Question 17

At a fixed level of excitation what is the effective FRF natural frequency

Answer 17

wn = sqrt ( (k + 3/4 k3 Y^2) / m)

Question 18

What can mathematicall prove with the equivalent natural frequency from harmonic balance

Answer 18

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

Question 19

Why is harmonic balance a linearisation of the FRF

Answer 19

As ignore the sin 3wt term

Question 20

Why do we have multiple solutions for Y from harmoic balance

Answer 20

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)

Question 21

What is the implication for testing caused by multiple solutions for Y as a result of harmonic balance

Answer 21

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

Question 22

Why do you have to use a stepped sine or sine dwell test instead of white noise to accelerate a nonlinear system

Answer 22

If white noise excite all frequencies at once, cant see diffierent individual modes if nonlinear. Need small steps

Question 23

Draw a diagram of what a theoretical FRF plot of a nonlinear system would like

Answer 23

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

Question 24

What happens to nonlinear systems in real life as you do a stepped sine or sine dwell test

Answer 24

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

Question 25

What are wlow and whigh

Answer 25

Points at which FRF discontinuity occur

Question 26

Draw a diagram of stepped sine wave test on upward sweep

Answer 26

See presentation

Question 27

Draw a diagram of stepped sine wave test on downward seep

Answer 27

See presentation

Question 28

How does FRF discontinuity work for softening/harderning systems

Answer 28

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

Question 29

Where would you see FRF discontinuity

Answer 29

in magnitude and phase plot

Question 30

What is the issue with the simple harmonic balance

Answer 30

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

Question 31

Why are harmonics odd in the simple harmonic balance

Answer 31

stiffness function ky + k3 y^3 is odd, if the function were even or generic all harmonics would be present

Question 32

When might harmonics become a really big issue

Answer 32

In multiple degrees of freedom system where multiple harmonics coincide with another - get massive resonance as exciting two modes at once

Question 33

What happens when excitation is not a pure tone

Answer 33

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