Nonlinear Dynamics Lec 4 Flashcards
What properties holds true for linear structures but not nonlinear structures
reciprocity H12 = H21 in a linear structure, measure at one point and force at another is the same either way round
What is the basic principle of superposition
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
Whats is the definining property of linear vs nonlinear structures
superposition
What is the mathematical definition of superposition
- 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)
Whats the issue with testing the mathematical definition of superposition and how is this overcome
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
What is the benefit of superposition for a linear system
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
For an encastre beam draw a diagram to show where superposition holds
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
Why does an encastre beam become nonlinear
Boundary conditions restrict the axial straining of the middle surface of the beam as the lateral amplitude is increased
Draw a diagram of an odd nonlinear characteristic
See powerpoint
odd function i.e. F(-y) = -F(y)
What would an even nonlinear characteristic look like
reflection about the y axis, this would create an unstable system as whichever direction you push in the force is in the same direction
Why is it important that easy to use procedures for detecting nonlinearity are avaliable
As impossible to fully implement the principle of superposition, need simple check to confirm if nonlnear so analysis can be carried as such
What are the most common procedures to check nonlinearity
harmonic distortion, homogenity, reciprocity and the coherence function
What is harmonic or wave distorting
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)
Why is harmonic distorting such a good test
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
Whats the difference between super harmonics and subharmonics
Subharmonics are integer multiples of the response 3w 5w etc, subharmonics are w/2 w/3
Need much stronger nonlinear system to get subharmonics
For harmonic distorting which is it important to check displacement, velocity and acceleration graphs
Acceleration will best show harmonic distortion, velocity and displacement may be difficult to see any distortion
Why does acceleration give the best of nolinearity in harmonic distortion
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
What is the homogenity test
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
What is the issue wih homogenity test
Some nonlinear system will show to pass homogenity i.e bilinear stiffness
How is homogenity test practicall carried out
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
When does reciporcity hold
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
How to check reciprocity
Overlay FRFs for impact A, response B and impact B, response A, if equal implies linearity
What is importatn to note when testing reciprocity
All response parameteres must be the same (e.g displacements or accelerations) and all inputs must be forces
What are some typical sources of nonlinearity
misalignment exciter problems looseness preloads cable rattle overloads/offset loads temperature effects impedance mismatching Poor transducer mounting
