Last Minute Panic Flashcards
When is nonlinear useful
Mathematical epidemiology Interactions between particles Accurately calculating large beam displacements Showing harmonics/predicting resonance Modelling Cracks
Why does nonlinear dynamics arise
Due to large deflections or from non hookian material behaviour
Common types of non linearity
Cubic stiffness Bilinear stiffness Nonlinear damping (quadratic) Coulomb friction Piecewise linear stiffness
What physical things cause nonlinearity
Polynomial stiffness and damping Clearances Impacts Friction Saturation effects Actuators Bearings Linkages Elastomeric materials
Where would you see a positive cubic stiffness
Clamped plates and beams
Where would you see a negative cubic stiffness
Buckling beams
Where would you see a bilinear damping
In shock absorbers, low damping for driver comfort, high damping for road contact
Where would you see cubic damping
Fluid flow through an oriface
Where would you see coulomb damping
Friction
Where would you see piecewise stiffness
Saturation or backlash
What’s wrong with the secular term in the cubic perturbation method
Expect a periodic solution, but t*sint term increases forever, due to to truncated series, would usually cancel out
What is the definition of superposition
The total response of linear structure to set of simultaneous inputs can be broken down into several experiments where each input is applied individually, and the output of each input can be summed to give the total response
What is the mathematical definition of superposition
A system with initial conditions S1(a1, b1) responds to input x1(t), with output y1(t), in a separate test with initial conditions S2(a2, b2) responds to input x2(t) with output y2(t). Superposition holds if with initial conditions S3(alphaa1+betaa2, alphab1+betab2) with all pairs of inputs x1(t) and x2(t) responds in outputs alphay1(t) + betay2(t)
What’s the issue with superposition as a test of linearity
Would require infinite tests for inputs of alpha, beta, x1(t) and x2(t), but need to violate just once to show nonlinear
Draw diagram showing no linearity of an encastre beam
See presentations
What are the checks for nonlinearity
Harmonic distortion
Homogeneity
Reciprocity
What is the harmonic distortion test
For a linear system monoharmonic input gives mono harmonic output, for nonlinear this does not happen as get harmonics in the response
Draw a diagram of harmonic distortion
Display displacement and velocity aren’t always good enough need to look at acceleration
How is harmonic distortion monitored
Using an oscilloscope
What is the homogeneity test
Idea that for linear systems x(t) -> y(t), alphax(t) -> alphay(t), thus FRF should be the same for all inputs as it is a ratio of Y(w)/X(w), overlay FRFs see if they are the same
What is the reciprocity test for nonlinearity
For linear system output yB at B due to an input xA at A, gives ratio numerically equal to reversing input and output i.e. yB/xA = yA/xB, all response parameters (velocity, acceleration) must be the same to hold
What are some sources of nonlinearity
Misalignment Exciter Problems Looseness Preloads Cable rattle Overloads/offset loads Temperature effects Impedence mismatching Poor transducer mounting
Describe the sum and difference theory
Input x(t) = X1 sinw1t + X sinw2t, trial solution of same form, sub in and equate harmonics, need another trial solution, this would need to contain all harmonics +- pw1 +- qw2, If try for duff in oscillator (k2=0) get the same result but p and q are only allowed to sum to add values, lowest nonlinear order means frequencies of 3w1, 2w1+-w2, w1+-w2, and 3w2 Thus FRF cannot encode info about sum and difference frequencies
What is an RV
Variable with uncertain outcome but can define probability of a single outcome
What are CRVs
Uncertain outcome but infinite possibilities, therefore a single value has no real chance of occurring
What does mutually exclusive mean
Occurrence of one precludes occurrence of the other
Statistically independent
Probability of one in no way effects the other
What is the mathematical definition of expectation
E(X) = sum for xi of P(X=xi)*xi
What is the central limit theorem
If Xi, i = 1,…,N where Xi are random variables, the sum of X = X1 + X2 + X3 + … has a Gaussian distribution
What is an independent random variable mean
Where the random variable Xt could depend on values of previous t but doesn’t, independent of previous times
What does identically distributed mean
For all values of t the probability P(Xt) = P(X), same for all values of time
What is an independent and identically distributed random variable
Has only one probability density function, as independent of previous state and it is the same for all time values
What is ensemble averaging
Trying to generate underlying physics for P(x) like the mean E(X), generate several example e relations of the process from same initial conditions
E(Xt) = 1/Np sum for i=1 to Np of x(i)(t)
What is time averaging
Similar to ensemble averaging but can compute but integrating with respect to time E[Xt] = 1/T * integral from 0 to T of x*p(x) dt
What is an ergodic signal
One for which ensemble averaging and time averaging are the same
What does a stationary mean for a random signal
Mean and standard deviation are constant
Draw the random signal classification tree
See presentation
What is the autocorrelation
Measure of how much a signal looks like itself when shifted by a value
If x(t) is zero mean what is the auto correlation of 0
The standard deviation squared, if non zero mean = mean square
Why is Hamilton dynamics useful
Really good for untraceable large number of particles (statistical mechanics) or system with no particles at all (quantum mechanics), astrony mechanics, and all first order terms therefore can be solved by computers quickly
