Nonlinear Dynamics Lec 4

Question 1

What properties holds true for linear structures but not nonlinear structures

Answer 1

reciprocity H12 = H21 in a linear structure, measure at one point and force at another is the same either way round

Question 2

What is the basic principle of superposition

Answer 2

total response of linear structure to a set of simultaneous inputs can be proken down into serval experiments where each input is applied individually and the output to each of these separate inputs can be summed to give the total response

Question 3

Whats is the definining property of linear vs nonlinear structures

Answer 3

superposition

Question 4

What is the mathematical definition of superposition

Answer 4

• System in intial condition S1 = y1(0), y1.(0) responds to input x1(t) with an output y1(t) and in separate test an input x2(t) to the system initinally in state S2 = y2(0), y2.(0) produces an output y2(t)
• Superposition holds if input alphax1(t) + betax2(t) to system in initial state S3 = alhpay1(0) + betay2(0), alphay1.(0) + betay2.(0) results in output alphay1(t) + betay2(t) for all constants alpha and beta and all pairs of inputs x1(t) and x2(t)

Question 5

Whats the issue with testing the mathematical definition of superposition and how is this overcome

Answer 5

Need an infinite numbers of tests spanning all of alpha, beta, x1(t) and x2(t) which is nearly impossible. However to show nonlinearty without doubt only one set of alpha beta x1 x2 which violate superposition are needed

Question 6

What is the benefit of superposition for a linear system

Answer 6

Only need to know response to sine wave and then can use taylor series to work out response to any input, without this i.e. nonlinearity very difficult to predict response

Question 7

For an encastre beam draw a diagram to show where superposition holds

Answer 7

At low forces/deflections graph is linear but at higher forces resulting in deflections bigger than beam thickness start to see increased stiffness and odd nonlinear stiffness characteristic

Question 8

Why does an encastre beam become nonlinear

Answer 8

Boundary conditions restrict the axial straining of the middle surface of the beam as the lateral amplitude is increased

Question 9

Draw a diagram of an odd nonlinear characteristic

Answer 9

See powerpoint

odd function i.e. F(-y) = -F(y)

Question 10

What would an even nonlinear characteristic look like

Answer 10

reflection about the y axis, this would create an unstable system as whichever direction you push in the force is in the same direction

Question 11

Why is it important that easy to use procedures for detecting nonlinearity are avaliable

Answer 11

As impossible to fully implement the principle of superposition, need simple check to confirm if nonlnear so analysis can be carried as such

Question 12

What are the most common procedures to check nonlinearity

Answer 12

harmonic distortion, homogenity, reciprocity and the coherence function

Question 13

What is harmonic or wave distorting

Answer 13

if the excitation to a linear system is monoharmonic signal, sine or cosine wave of frequency w, the response will be monoharmonic at the same frequency (after any transients have decayed away)

Question 14

Why is harmonic distorting such a good test

Answer 14

sine waves are simple signals to generate in practice and distortion can easily be detected on oscilloscope
nonlinear systems will not output a sine wave for a sine wave input due to higher harmonics (3w and 5w) appear in the respone

Question 15

Whats the difference between super harmonics and subharmonics

Answer 15

Subharmonics are integer multiples of the response 3w 5w etc, subharmonics are w/2 w/3
Need much stronger nonlinear system to get subharmonics

Question 16

For harmonic distorting which is it important to check displacement, velocity and acceleration graphs

Answer 16

Acceleration will best show harmonic distortion, velocity and displacement may be difficult to see any distortion

Question 17

Why does acceleration give the best of nolinearity in harmonic distortion

Answer 17

if input x = sin wt out is y = A1 sin wt + A2 sin 2wt + A3 sin 3wt +…
differentiate twice y.. = - A1 w^2 sin wt - 4A3w^2 sin 2wt - 9A3w^2 sin 3wt
nth output accleartion term is weighted by factor n^2 compared to fundamental

Question 18

What is the homogenity test

Answer 18

Most common
Restriected form of the superposition principle
Homogenity x(t) - > y(t) , alpha x(t) -> alpha y(t)
If holds ratio (can transpose to freq and thus FRF) of output to input is indepedent of constant alpha

Question 19

What is the issue wih homogenity test

Answer 19

Some nonlinear system will show to pass homogenity i.e bilinear stiffness

Question 20

How is homogenity test practicall carried out

Answer 20

Dynamic testing to FRF where input levels are usually mapped over a range covering operating levels, if FRFs for different levels overlay, linearity is inferred

Question 21

When does reciporcity hold

Answer 21

if ouput yb at a point B due to input xa at point gives a ratio yb/xa numerically to that when the input and output points are revered giving ya/xb

Question 22

How to check reciprocity

Answer 22

Overlay FRFs for impact A, response B and impact B, response A, if equal implies linearity

Question 23

What is importatn to note when testing reciprocity

Answer 23

All response parameteres must be the same (e.g displacements or accelerations) and all inputs must be forces

Question 24

What are some typical sources of nonlinearity

Answer 24

misalignment
exciter problems
looseness
preloads
cable rattle
overloads/offset loads
temperature effects
impedance mismatching
Poor transducer mounting