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MEC452 > Newtonian Dynamics > Flashcards

Newtonian Dynamics Flashcards

1
Q

Give the basic system equation

A

mx.. + kx + cx. = F

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2
Q

Natural frequnecy is defined as

A

frequency at which a system would osciallte if it were unforced and undamped

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3
Q

Unique natural frequency associated with

A

unique mode shapes

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4
Q

Why do we use simple models in vibration analysis

A

When modelling we take the approach that we would like the simplest possible model that provides the information that we need

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5
Q

What is a multi degree of freedom system

A

One that requires two or more coordinates to describe its motion

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6
Q

For each degree of freedom there is

A

an equation of motion

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7
Q

Why can we ignore deflection caused by mass weight

A

We take deflection from static equilibrium where mg = delta K, therefore mg and delta k cancel out

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8
Q

Derive the matrix equations of motions for a 3 DOF chain system

A

See powerpoint

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9
Q

For a linear system forced at a certain frequency how will it respond

A

Will respond at the same frequency

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10
Q

What type of equation is the force vibration equation of motion

A

Non homogenous differential equation

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11
Q

Why does it matter that the forced vibration equation is non homohenous differential equation

A

As it has two parts to the solution, a transient (complementary function) and a steady state response (particular integral)

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12
Q

What is the steady state response of a system

A

Solution of the system where frequency of the response is equal to the frequency of the forcing, only thing that can vary for each degree of freedom is the amplitude

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13
Q

What is the transient response

A

Homogenous solution, dies away quickly, solution for forcing equal to zero, all modeshapes are present

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14
Q

What is the solution to the nonhomogenous differential equation ax.. + bx. + cx = F

A

x(t) = xo(t) + x(t)
xo(t) homogenous solution (transient)
x(t) steady state response

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15
Q

For an unforced system what will be the reponse

A

Combination of all modeshapes. e.g. for 2 DOF
x1 = sin(w1t) + sin(w2t)
x2 = sin(w1t) + sin(w2t)

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16
Q

What do we assume when solving equations of motion

A

if forced that focing is harmonic F1 = Fsinvt, thus solution is harmonic x = Xsin(vt - phi)
if unforced that solution is harmonic

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17
Q

To determine the natural frequencies how would you go from the equations of motion to the eigenvalue problem

A 
m x.. + kx = 0
Assume harmonic response
- w^2 mX + kX = 0
kX = w^2 mX
m^-1 k X = w^2 X
Take determinant of dynamic matrix A (=m^-1 k) - lambda I and equate it to 0
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18
Q

How are modeshapes related

A

orthoganal, multiply them together and will get 0

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19
Q

What are modeshapes orthoganal with

A

other modeshapes, mass and stiffness matrix, and damping if proportional damping (i.e damping proportional to mass or stiffmess matrix)

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20
Q

What does the superscript and subscript refer to on X ^(1) sub 2

A

superscript is the natural frequency number ie first natural frequency, and the subscript is the mass

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21
Q

Why should you assume a solution including phase ie the form x1 = X1 cos (wt + phi)

A

Allows you to include damping, the phase cancels out so not an issue and assume damping but means we have a full solution with initial conditions

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22
Q

When doing ratio of modeshapes which one should always be on the bottom

A

Can write either way mathematically either way but as engineers should have first mass modeshape on bottom

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23
Q

When might you get a negative natural frequency and what do we do with it

A

-ve frequency doesnt make any sense so disregard at all times, occurs in fourier transform and when squarerooting the eigenvalues to give natural frequency

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24
Q

How would you write the full response of a system

A

first response
x1(t) = X1^(1) cos (w1t+phi1) +X1^(2) cos (w2t+phi2)
x2(t) = X2^(1) cos (w1t+phi1) +X2^(2) cos (w2t+phi2)

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25
Q

How would you work out full response of system

A

Eliminate some variable using modeshape ratios, then use initial conditions to determine X1 and X2 as well as phase values

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26
Q

Using vectors formulate FRF magnitude and phase

A

See powerpoint

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27
Q

what is the natural frequency of a SDOF system equal to

A

sqrt (k/m)

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28
Q

What is critical damping equal to in an SDOF system

A

2 m wn

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29
Q

What is damping factor equal to in an SDOF system

A

c/ critical damping or c / 2mwn

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30
Q

Nondemensionalise the FRF and phase for an SDOF system

A

See powerpoint

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31
Q

What happens to FRF magnitude as damping increases

A

The peak gets shorter and wider

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32
Q

Take the inverse of a 2x2 matrix [a b; c d]

A

1/dert * [d -b, -c a]

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33
Q

What is damping

A

energy dissipated in a structure

34
Q

What structures have damping

A

All structures dissipate energy to some extent, when the amount of energy dissipated is small we can attempt an analysis presuming the structure is undamped

35
Q

What are the type of damping

A

viscous damping
coloumb damping
hyteretic damping

36
Q

What is vsicour damping

A

Commonly used, viscous force proportional to velocity

critical damping given by sqrt 4mk, less than critical damping get exponential decay

37
Q

What is coloumb dampingq

A

steady friction force (dry friction) that occurs in many strucutres, forces are independent of amplitude and frequency, always oppose motion and magnitude may be approximated as constant

38
Q

What is hysteretic damping

A

Internal dissipation of energy (material damping) obseerved what a material is subjected to cyclic stress, independent of freqency of vibration, cant use vsicous, need to divide damping term by frequency of osciallation

39
Q

How is hysteretic damping simplified in equation of motion

A

As an increased stiffness value k* = k(1+i*eta) where eta is a loss factor for a material

40
Q

How might excessive vibration in a machine or structure be resolved

A

Add or increase damping
Resiting or changing machinery that generate vibration if not an external source (wind or tuburlence causing)
Vibration isolation may be used to isolate machinery generation vibration from it surronding
Vibration absorbers may be attatched to machinery to alleviate excessive vibration at resonance

41
Q

What are vibration absorbers

A

Simple mass spring system added to a structure which is designed to greatly reduce the amplitude of vibration at a particular frequency

42
Q

When might you use a vibration absorber

A

If a particular resonance behaviour is an issue (particular frequency/mode)

43
Q

How does a vibration absorber work

A

If tuned correctly at wanted frequency, input force into main body will equal input force into main body from vibration absorber, meaning at that frequency amplitude of vibration will = 0

44
Q

What is often used when insufficient damping on a structure

A

viscoelastic material (polymers with such properties) widely used to provide additional damping

45
Q

Whats the issue/benefit of viscoelastic material

A

Both frequency and temperature have a large influence on the effectiveness of the material for damping

46
Q

why is constrained layer damping often used

A

polymers material that possess high damping properties often lack rigidity and cannot be used for structural purposes on their own, instead bonded to more rigid material (metals) to add damping, often in the form of sandwhich panels. Shear effects in damping layers dissipate energy

47
Q

What is free layer damping

A

Thermally sprayed coating such as ceramic deposited using thermal spray process, provide damping through complex microstructure, useful for gently curved surfaces

48
Q

What are particle dampers

A

Containers filled with particles attatched to the structure as discrete point damper, damping proportional to mass of particles, dissipate energy through inter particle friction

49
Q

What are the benefits and negatives of particles dampers

A

not temperature dependent, behaviour is amplitude and frequency dependent

50
Q

Where do the generalised mass and stiffness matricies come from

A

transpose (modeshape i) * mass matrix * modeshape i = Mii

transpose (modeshape i) * stiffness matrix * modeshape i = Kii

51
Q

What does the orthoganality of modeshapes mean

A

transpose (modeshape i) * mass matrix * modeshape j = 0

transpose (modeshape i) * stiffness matrix * modeshape j = 0

52
Q

When would you get orhoganality with damping matrix and a generalised damping quantity

A

When you have proportional damping

53
Q

What is the modal matrix psi

A

all normal modes assembled into one matrix

54
Q

If we use the modal matrix on the mass/stiffness matrix what can we get

A

Generalised mass and stiffness matrix

55
Q

What can we do with the generalised mass, stiffness and modal matrix

A

As they are all diagonal matricies we can solve each of the n equation of motion separately

56
Q

How do we create the othonormal modes

A

Divide the modeshapes by the square root of the generalised mass

57
Q

What can you do with the othonormal modes

A

othonormal when multiplied with the mass matrix give 1 and the stiffness matrix give the natural frequencies
With proportional damping matrix you would get 2damping ratiosqrt(natural frequency)

58
Q

Derive the orthognoality properties

A

See powerpoint

59
Q

What are the different types of FRFs

A

receptence when displacement is measured
mobility when velocity is measured
accelerance when acceleration is measured

60
Q

What is an FRF

A

Ratio of outputs/inputs

61
Q

What does a H12 FRF mean

A

Force applied at point 2 and response measured at point 1

62
Q

For a linear system what does Hij equal

A

Hji

63
Q

What other form of response can we assume for damped system

A

Xe^iwt

64
Q

Where are the natural frequencies on an FRF plot

A

Frequency at the peaks from any FRF as long as not measuring at a node

65
Q

What happens to the peak on an FRF is there is no damping

A

Goes to infinity

66
Q

What happens to the peak on an FRF is there is critical damping

A

Peak goes flat

67
Q

What happens to the peak of an FRF is the structure becomes less stiff

A

Peak moves to the left

68
Q

What can peaks moving be used to do an FRF

A

Use to detect damage, peak moving to the left means structure getting less stiff, peak moving to the right means structure getting more stiff. From this can determine damage

69
Q

How do you go from time to frequency domain

A

Using the fourier transform

70
Q

What is the FRF

A

Frequency response function ratio of output over input

71
Q

What is needed from an FRF to evulate modeshapes

A

complete row or coloumn, take the imaginary domain first peak in each FRF for first mode, second for second mode etc.

72
Q

Why does Hij equal Hji

A

As in linear systems we have reciprocity

73
Q

Why might younot see a peak in an FRF

A

If there is a node

74
Q

When might you see antiresonances

A

Will always see for drive point measurements i.e. Hii in magnitude plot see between the peaks, in phase plot will gain 180 degrees over a resonance and lose 180 degrees over antiresonance

75
Q

Whys it bad to take a measurement on a node

A

Will lose peak in FRF and so lose alot of information

76
Q

What are the difficulties with extracting FRFs from a bridge

A

Big structure dificult to excite, if background are things like traffic and wing strong enough to excite damage, difficult to work out damping, environmental facors like temperature impact stiffness and damping thus properties changes and therefore FRFs change

77
Q

In experimental modal analaysis what is often done to make the process easier

A

Single degree of freedom curve fit, look at each peak individually

78
Q

How can damping be estimated from an FRF

A

Using 3dB method, based on the width of the peaks

79
Q

What is the equation for damped natural frequency

A

wd = wn * sqrt(1 - zeta^2)

80
Q

What is the equation for resonant natural frequency

A

wr = wn * sqrt(1 - 2*zeta^2)

MEC452 (10 decks)

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