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MEC452 - Advanced Dynamics > Advanced Dynamics Theory > Flashcards

Advanced Dynamics Theory Flashcards

(80 cards)

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1
Q

Why is Vibration Analysis Important

A

It allows us to find dynamic characteristics of systems and predict responses to conditions

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2
Q

What does homogeneity mean

A

FRFs don’t change regardless of excitation intensity

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3
Q

What does reciprocity mean

A

Resonant frequency peaks are the same for a constant system regardless of where you excite and measure

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4
Q

What is the principle of virtual work

A

A system in equilibrium that experiences virtual displacement due to forces has a net virtual work of zero

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5
Q

What does a conservative system mean

A

The sum of the kinetic and potential energies is constant so dE/dt = 0

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6
Q

What makes a system nonconservative

A

Energy losses, such as through damping

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7
Q

What is the Lagrangian

A

L = T - U

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8
Q

What is viscous dampng and where does it occur

A

Damping force is proportional to the velocity, used in vehicle shock absorbers and aircraft landing gear

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9
Q

What is coloumb damping and where does it occur

A

Damping force is approximately constant, independent of time-varying parameters, always opposes motion, used in brake systems and occurs as friction in door hinges

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10
Q

What is hysteretic damping

A

Internal dissipiation of energy when a material is subjected to cyclic stresses, found in building structures during earthquakes and railway tracks

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11
Q

What is complex stiffness

A

Most real structures difficult to distinguish stiffness and damping effects so considered together

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12
Q

When is damping used for troubleshooting

A

Increased damping where there is excessive vibration in structures
Alternatively change machinery but external vibration needs damping control
VIbration isolation
Vibration absorbers attached to alleviate resonance vibration

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13
Q

What makes up a simple vibration absorber

A

Simple mass-spring system

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14
Q

What are some examples of positive resonance effects

A

Playground swings
Guitars
Radios
Microwave ovens
MRI scanners
Lasers

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15
Q

What are some examples of negative resonance effects

A

Unstable oscillations in bridges
Shattering glass

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16
Q

What do orthogonal properties allow us to do with MDOF systems

A

MDOF equations of motion can be decoupled and each treated as a a single DOF system

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17
Q

What are the names given to FRFs for displacement, velocity and acceleration responses

A

Receptance, Mobility, Accelerance

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18
Q

What is point receptance

A

We excite and measure at the same place

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19
Q

What is transfer receptance

A

We excite at one place and measure at another

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20
Q

What effect do cracks have on FRFs

A

They cause a decrease in stiffness meaning natural frequency also decreases

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21
Q

Why is FRF analysis useful

A

Identify natural frequencies
Find mode shapes
Find damping ratio
Identify non-linearities and harmonics
Check system performance for damage
Update and validate models/FEA

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22
Q

How can unwanted vibrations be managed

A

Passive damping systems e.g. attach additional mass to aircraft
Active damping systems e.g. shock absorbers

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23
Q

What is SDOF curve fitting

A

Focus on frequency range around each peak separately and assume they can be treated as the response of an SDOF system to find equation of motion

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24
Q

Why can the Hamiltonian be useful as opposed to Lagrangian

A

It captures the whole energy of the system
It is only a first order equation
It simplifies things when there are many particles under consideration

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25
What makes a system nonlinear
The principle of superposition no longer applies
26
What kind of conditions can cause nonlinearity
Very high excitations and specific frequencies Clearances, impacts, friction, saturation effects
27
List the common types of nonlinearity
Cubic stiffness Bilinear stifness Nonlinear quadratic damping Coloumb friction damping Piecewise linear stiffness
28
What are characteristics of cubic stiffness
Linear near the origin but increasingly nonlinear as the force and displacement increase Can be softening or hardening Occur in buckling beams and clamped plates
29
What are characteristics of bilinear stiffness
Value changes depending on displacement Used in car shock absorbers No FRF distortion if threshold is at origin
30
What are characteristics of nonlinear damping
Most common Usually associated with drag and found in automotive dampers and hydromounts Increased excitation means increased damping
31
What are characteristics of Coloumb damping
Occurs when there is relative motion of interfaces such as in grandstands Most evident at low levels of excitation
32
What are characteristics of piecewise linear stiffness
Saturation is linear near origin and zero beyond thresholds Clearance is zero near the origin and linear at threshold points
33
What is the effect of a cubic stiffness term in Duffings oscillator
At low levels of excitation there is low FRF distortion At high levels of excitation there is high FRF distortion
34
What is a bifurcation point
The threshold of an FRF plot where there is only one response Between the low and high frequencies there are three responses
35
What is the principle of superposition
Total response for a linear structure to simultaneous inputs can be broken down to several experiments with each input individually and then summed
36
What are the tests for non-linearity
Harmonic distortion Homogeneity Reciprocity
37
What is harmonic distortion
If excitation to a linear system in a monoharmonic signal then response will be harmonic at same frequency (after transients decay) Distortion is greater in velocity and acceleration as weighted harmonic occur
38
What are some sources of nonlinearity
Misalignment Rattle Temperature
39
What does it mean to be a random signal
You cannot specify the value of it at some time, only that it wil take some value with some probability
40
What are the first levels of classification
IID and Non-IID Gaussian IID and non-Gaussian IID - Stationary Non-IID and non-Stationary IID
41
What does iid mean
Indepedent identically distributed Independence means it is impossible to predict the value of a signall at time 2 from a measurement at time 1 p(x1, x2) = p(x1)*p(x2) Identically distributed means each random variable has same probability
42
Why do random signals arise and what implications do they have
Forces on structures tend to be random causing random responses The behaviour of a system will be unpredictable
43
What are some examples of random responses
Marine vessels and offshore structures forced by sea waves Aeroplanes experiencing turbulent airflow Cars driving on rough roads Buildings responding to earthquakes
44
Why is it common to excite systems with random forces in labs
Allows engineers to sample response at many frequencies at the same time
45
What does the Fourier transform do
Constructs an angular frequency function from a time function which is reconstructible
46
What is the power spectrum of a signal equal to
The integral of the fast Fourier transform across infinite limits
47
What is the Nyquist frequency
The maximum frequency that can be resolved in a signal sampled at a certain timescale Sample frequency = 2*Nyquist frequency
48
How can we vary the frequency interbals of a sample
Collect more data points (favourable) Increase the sampling rate (affects Nyquist frequency)
49
What is the way to create a signal in time domain if the signal of interest is simple
Create white noise by sampling a Gaussian distribution Filter out signal energy of frequencies outside the target range
50
Describe the steps needed to synthesise a signal from the power spectral density
Create spectrum directly in discrete frequency domain from given formula Use inverse discrete Fourier transform to get time domain realisation Assume signal with constant magnitude in target range and 0 outside Long signal and number of sample is power of 2 Choose sampling frequency ensuring highest frequency of interest is less than Nyquist frequency Set magnitude to unity - scale signal after for desired variance Create array of spectral lines in spectrum
51
What is the Pierson-Moskowitz spectrum
Spectrum for fully-developed seas at equilbrium steady state Proportional to inverse of frequency^5 and exponential of inverse frequency^4
52
What is JONSWAP spectrum
Applies a correction to Pierson-Moskowitz as seas don't get to steady-state
53
Summarise the steps to find the responses of an MDOF system with initial conditions
Find equations of motion Assume harmonic solution Find the matrix of physical properties Set the determinant = 0 and solve for the natural frequencies Find the ratios and mode shapes Set up the responses as a sum of the two parts Set equal to initial conditions Eliminate and solve for the phase and magnitudes
54
Summarise the steps to find the magnitude and phase of an FRF using complex solutions
Assume a complex harmonic solution Subsitute into the equation of motion and cancel common terms Rearrange to find FRF Take the magnitude and phase as standard
55
What is the Lagrange equation
d/dt(dL/xdot) = dL/dx
56
What is the kinetic energy of a system
The sum of 1/2 * mass * velocity squared Or 1/2 * inertia * angular velocity squared for rotational
57
What is the potential energy of a system
The sum of elastic potential (1/2 * mass * stretch squared) and gravitational potential (mgh)
58
When is the Lagrange equation non-conservative
There is an additional non-conservative generalised force component
59
What is the equation for natural frequency
w = sqrt(k/m)
60
What is the equation for critical damping
Cc = 2wm = 2*sqrt(mk)
61
What is the equation for damping factor
Zeta = c/Cc
62
What is the equation for damped natural frequency
wd = w*sqrt(1 - zeta squared)
63
What happens to the FRF when a vibration absorber is added
There are two resonance peaks offset from the primary response with a lower magnitude
64
How are orthonormal modes constructed
Start with K matrix * eigenvector i = eigenvalue * M matrix * eigenvector i Premultiply with eigenvector j tranposed Do the same but switch i and j eigenvectors Substitute from one another and the stiffness term disappears due to symmetry Shows the mass term must also be 0 as long as eigenvalues aren't the same
65
What is the eigenvalue equal to
Natural frequency squared
66
What should the final Hamiltonian never include
Velocity term
67
Summarise the steps to find the Hamilton equations
Set up the Lagrangian Find conjugate momenta: p = dL/dx_dot Form Hamiltonian: H = px_dot - L Replace velcocity terms with conjugate momenta Find Hamilton equations: 1. x_dot = dH/dp 2. p_dot = - dH/dx
68
Describe what effects cubic stiffness have on Nyquist and Bode plots
Bode plots - natural frequency shifts to the left (softening) or right (hardening) Nyquist plots - there is a jump on the left (hardening) or right (softening)
69
Describe the effect of quadratic damping on Nyquist and Bode plots
Both become flatter
70
Describe the effect of Coloumb damping on Nyquist and Bode plots
Both become longer
71
Describe how to find the magnitude and phase of Duffing oscillator using harmonic balance
Assume a harmonic solution with Ysin(wt) and Xsin(wt - phi) Differentiate and substitute Use trig identities to expand out terms Ignore the higher harmonics and balance sin(wt) and cost(wt) coefficients Solve for the phase and magnitude
72
What is the sine cubed relation
sine cubed(wt) = 3/4*sine(wt) - 1/4*sine(3wt)
73
What is the sine summed angle relation
sine(a - b) = sin(a)cos(b) - sin(b)cos(a)
74
What is the Nyquist frequecy relation
fN = fs/2 = 1/(2*delta t)
75
What is the frequency size relation
Delta f = 1/(N * delta t)
76
What is the normal distribution
x ~ N(mean, standard deviation)
77
What are some ways that a signal could be non-stationary
The mean of the signal changes The variance of the signal changes
78
What is the Fourier transform equation
X(w) = integral across infinity (x(t)*e^-iwt) dt
79
What is the inverse Fourier transform equation
x(t) = 1/(2*pi) * integral across infinity (X(w)*e^iwt) dw
80
How are spectral lines synthesised
DFT has N lines, line at N = 0 represents mean value (usually 0) Lines from r = 1 to r = N/2 correspond to frequency intervals up to Nyquist frequency Lines after this not considered and interpreted as negative frequencies Find first line i, such that i*delta f equal to f low Find last line j, such that j*delta f equal to f high Magnitude between i and j equals unity Spectral lines have complex values so set each phase value as a random number from uniform distribution between -pi and pi Combine magnitudes and phases to get complex spectrum

MEC452 - Advanced Dynamics (1 decks)

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