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MEC452 > Last Minute Panic > Flashcards

Last Minute Panic Flashcards

1
Q

When is nonlinear useful

A 
Mathematical epidemiology
Interactions between particles
Accurately calculating large beam displacements
Showing harmonics/predicting resonance
Modelling Cracks
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2
Q

Why does nonlinear dynamics arise

A

Due to large deflections or from non hookian material behaviour

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3
Q

Common types of non linearity

A 
Cubic stiffness
Bilinear stiffness
Nonlinear damping (quadratic)
Coulomb friction
Piecewise linear stiffness
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4
Q

What physical things cause nonlinearity

A 
Polynomial stiffness and damping
Clearances 
Impacts
Friction
Saturation effects
Actuators
Bearings
Linkages
Elastomeric materials
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5
Q

Where would you see a positive cubic stiffness

A

Clamped plates and beams

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6
Q

Where would you see a negative cubic stiffness

A

Buckling beams

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7
Q

Where would you see a bilinear damping

A

In shock absorbers, low damping for driver comfort, high damping for road contact

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8
Q

Where would you see cubic damping

A

Fluid flow through an oriface

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9
Q

Where would you see coulomb damping

A

Friction

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10
Q

Where would you see piecewise stiffness

A

Saturation or backlash

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11
Q

What’s wrong with the secular term in the cubic perturbation method

A

Expect a periodic solution, but t*sint term increases forever, due to to truncated series, would usually cancel out

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12
Q

What is the definition of superposition

A

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

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13
Q

What is the mathematical definition of superposition

A

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)

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14
Q

What’s the issue with superposition as a test of linearity

A

Would require infinite tests for inputs of alpha, beta, x1(t) and x2(t), but need to violate just once to show nonlinear

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15
Q

Draw diagram showing no linearity of an encastre beam

A

See presentations

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16
Q

What are the checks for nonlinearity

A

Harmonic distortion
Homogeneity
Reciprocity

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17
Q

What is the harmonic distortion test

A

For a linear system monoharmonic input gives mono harmonic output, for nonlinear this does not happen as get harmonics in the response

18
Q

Draw a diagram of harmonic distortion

A

Display displacement and velocity aren’t always good enough need to look at acceleration

19
Q

How is harmonic distortion monitored

A

Using an oscilloscope

20
Q

What is the homogeneity test

A

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

21
Q

What is the reciprocity test for nonlinearity

A

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

22
Q

What are some sources of nonlinearity

A 
Misalignment
Exciter Problems
Looseness
Preloads
Cable rattle
Overloads/offset loads
Temperature effects
Impedence mismatching
Poor transducer mounting
23
Q

Describe the sum and difference theory

A 
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
24
Q

What is an RV

A

Variable with uncertain outcome but can define probability of a single outcome

25
Q

What are CRVs

A

Uncertain outcome but infinite possibilities, therefore a single value has no real chance of occurring

26
Q

What does mutually exclusive mean

A

Occurrence of one precludes occurrence of the other

27
Q

Statistically independent

A

Probability of one in no way effects the other

28
Q

What is the mathematical definition of expectation

A

E(X) = sum for xi of P(X=xi)*xi

29
Q

What is the central limit theorem

A

If Xi, i = 1,…,N where Xi are random variables, the sum of X = X1 + X2 + X3 + … has a Gaussian distribution

30
Q

What is an independent random variable mean

A

Where the random variable Xt could depend on values of previous t but doesn’t, independent of previous times

31
Q

What does identically distributed mean

A

For all values of t the probability P(Xt) = P(X), same for all values of time

32
Q

What is an independent and identically distributed random variable

A

Has only one probability density function, as independent of previous state and it is the same for all time values

33
Q

What is ensemble averaging

A

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)

34
Q

What is time averaging

A

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

35
Q

What is an ergodic signal

A

One for which ensemble averaging and time averaging are the same

36
Q

What does a stationary mean for a random signal

A

Mean and standard deviation are constant

37
Q

Draw the random signal classification tree

A

See presentation

38
Q

What is the autocorrelation

A

Measure of how much a signal looks like itself when shifted by a value

39
Q

If x(t) is zero mean what is the auto correlation of 0

A

The standard deviation squared, if non zero mean = mean square

40
Q

Why is Hamilton dynamics useful

A

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

MEC452 (10 decks)

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