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MEC201 > Equations > Flashcards

Equations Flashcards

1
Q

Whats is Newtons Second Law

A

sum F = ma

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2
Q

Rotational equivalent of Newtons Second Law

A

sum MG= IG * alpha

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3
Q

general equation for sinusoidal motion

A

x = a sin(wt + phi)

where a is amplitude, w is frequency, t is time and phi phase shift

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4
Q

alternative form of sinusoidal motion

A

x = Acos(wt) + Bsin(wt)

where A = a sin(phi) and B = a cos (phi)

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5
Q

Velocity equation and how is it found

A

x. = -Awsin(wt) + Bwcos(wt)

by taking the time derivative

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6
Q

Acceleration equation

A

x.. = - Aw^2 * cos (wt) - Bw^2 * sin(wt) = -w^2 x

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7
Q

eulers equation of motion

A

x = a*e^jwt

x. = jwae^jwt = jw x
x. . = -w^2e^jwt = -w^2 x

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8
Q

undamped natural frequency

A

wn = sqrt (k/m)

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9
Q

Equation of free vibrations

A

x(t) = Acos(wnt) + Bsin(wnt)

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10
Q

natural frequency is

A

the preferred vibration frequency of an undamped system

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11
Q

equation of motion of damped system

A

mx.. + cx. + kx = 0

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12
Q

characteristic equation of motion for damped system

A

ms^2 + cs + k =0

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13
Q

critical damping equation

A

cc = 2 sqrt (km) = 2m wn

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14
Q

damping ration =

A

c/cc

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15
Q

characteristic equation of motion for undamped system

A

mx.. + kx = 0

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16
Q

Overdamped system equation

A

see book

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17
Q

damped natural frequency

A

wc = wn sqrt ( 1 - zeta^2)

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18
Q

underdamped system repsonse

A

see book

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19
Q

log decrement =

A

ln (xn-1/xn)

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20
Q

alternative form of underdamped free vibration equation

A

see book

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21
Q

log decrement simplified form (time period)

A

zeta * wn * taud

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22
Q

log decrement simplified form

A

2 PI() Zeta/ sqrt(1 - zeta^2)

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23
Q

logDec for several cycles

A

logdec = 1 /n * ln (xo/xn)

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24
Q

low damping zeta from logdec

A

zeta = ln(x0/xn) / wn * tdrop

where tdrop is the time taen to fall from x0 to xn

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25
Q

Force vibration equation X/F =

A

1/ (k -w^2 m + jwc)

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26
Q

what is the magnification factor equal for an undamped forced vibration system

A

1/ (1 - r^2)

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27
Q

Dynamic magnification =

A

X / Xstat where X stat = F/k

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28
Q

normalised frequency r =

frequency ratio

A

w / wn

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29
Q

equation of motion of forced damped system

A

mx.. + cx. + kx = f(t)

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30
Q

Force vibration equation using system terms X/Xtat =

A

1/ (1 - r^2 + j *2 * zeta * r)

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31
Q

Magnitude of damped frequency response

A

Xdyn / X stat = 1/sqrt((1-r^2)^2 + (2zetar)^2)

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32
Q

phase angle =

A

tan(phi) = 2zetar/(1-r^2)

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33
Q

When does the maximum response occur for damped forced vibrations

A

r = sqrt ( 1 - 2*zeta^2) when zeta is less than 1/root 2

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34
Q

What is the maximum response equal to at zeta less than 1/root 2

A

Xk/F = 1 / 2*zeta * sqrt ( 1 - zeta^2)

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35
Q

Q =

A

mag @max X dyn / X stat = 1 /2*zeta * sqrt (1-zeta^2)

36
Q

Q at low damping

A

Q = 1/2*zeta

37
Q

Force transmission equation =>

A

FT/F = (k + jwc)/(k - w^2m + jwc)

38
Q

Force transmission equation in system properties =>

A

FT/F = (1 + j2zetar)/(1-r^2 + j2zetar)

39
Q

equation of motion for base motion

A

mx.. + c(x.-y.) + k(x-y) = 0

40
Q

Z/Y =

A

w^2 m / k - w^2 m + jwc
r^2 / 1 - r^2 + j2zeta*r
where Z = X - Y

41
Q

Displacement response of rotating unbalance

A

X = Me/m * (r^2) / (1 - r^2 + j2zeta*r)

42
Q

equation of motion of system excited by rotating unbalance

A

mx.. + cx. + kx = Mew^2 *exp (jwt)
where e is eccentricity of the unbalance
and M the mass of the eccentricity

43
Q

Force due to spring

A

F = kx

44
Q

Work done/strain energy stored in a spring

A

W or U = 1/2 k x^2

45
Q

When mass elements are rigidly connected together equivalent mass for translational motion is

A

sum of the masses

meq = m1 + m2 + m3

46
Q

Rigid connection for inertia,

rigidly connected together for rotation about a point

A

J0 = (JG1 + m1r1^2) + (JG2 + m1r2^2) + (JG3 + m1*r3^2)

47
Q

Equivalent Rotation Inertia Jeq =

A

Jeq = mR^2 + J0

48
Q

Equivalent translational mass meq =

A

m + J0/R^2

49
Q

Equivalent kinetic energy equation =

A

T = sum 1/2 mx.^2 + sum 1/2 Jtheta.^2

50
Q

Energy dissipated in damper

A

delta W = pi() * c * w * X^2

51
Q

Method for equivalent damper

A

use delta W = pi() * c * w * X^2, and related X^2 for each damper using similar triangles

52
Q

simplest representation in hysteretic damping

A

DETLAW = pi() h X^2

53
Q

an equivelent viscous unit for hysteretic damping

A

ceq = h/w = eta * keq / w

54
Q

spring and hysteretic damper under the general harmonic excitation equation of motion

A 
f = h/w *x. + kx 
Fe^jwt = h/w * jwX*e^jwt + kX*e^jwt
55
Q

For hysteretic damper model what does F/X =

A

jh + k = k(1 + j*eta)

56
Q

What is the complex stiffness of hysteretic damping

A

k(1 + j*eta)

57
Q

SDOF system with hysteric damped equation

A

mx.. + ceq *x. + keq * x = 0

58
Q

Reaction force of coulomb damping

A

f = -sign(x.) mu N where the sign function takes the sign of the velocity ie if velocity is -ve f = mu N

59
Q

what is the drop in amplitude of coulomb damping per cycle

A

4 * mu * N / k

60
Q

What is the equation of the line coulomb damping will follow

A

xline = x0 - (4 * mu * N / k) t/T

61
Q

energy dissipated per cycle by coulomb damping

A

DELTAW = 4 * mu * N * X

62
Q

Equivalent damping for coulomb ceq =

A

4 * mu * N / pi() * w * X

63
Q

Relative Velocity

A

vB/A = w * rAB

64
Q

Vector dot product or scalar product =

A

a dot b = mag a * mag b * cos theta

65
Q

Component of force F in direction r

A

F dot r / mag of r

66
Q

vector cross product

A

mag a cross b = mag a * mag b * sin theta

67
Q

equation for tangential acc

A

(aB/A)t = alpha x rB/A

68
Q

equation for normal acc

A

(aB/A)n = w x ( w x rB/A)

69
Q

va =

A

vb + va/b = vb + w * ra/b

70
Q

aa =

A

ab + aa/b = ab + alpha x ra/b + w x (w x ra/b)

71
Q

Rotating frame of reference velocity equation

A

va = vb + w x r + vrel

72
Q

Rotating frame of reference acceleration equation

A

aa = ab + w. x r + w x (w x r) + 2 w x vrel + arel

73
Q

Equation for resultant moment

A

Sum Mg = Ig alpha

where Mg is the resultant moment Ig the mass moment of inertia alpha the angular acceleration

74
Q

Linear momentum =

A

L = mv

75
Q

L. =

A

d/dt (mv) = ma = sum F

76
Q

symbol of angular momentum =

A

H0 = r x mv

77
Q

Rate of change of angular momentum =

A

H0. = r x mv. = sum of moments

78
Q

sum of moments relationship with sum of forces

A

Sum of M0 = r x sum of Forces

79
Q

sum of moment about an arbitrary point

A

sum MP = sum rate of change of angular momentum + rhoG (dist from arbitrary point to centre of mass) x m ag

80
Q

Sum of moments about an arbitrary point in a planar system

A

sum MP = IG alpha + rhoG x maG

81
Q

what is the acceleration of single out of balance mass m located at radius r from axis of rotation

A

under constant velocity rotating

a = -w^2 r

82
Q

Force applied to rotor with single out of balance mass m located at radius r from axis of rotation

A

F = m w^2 r

needs to be equal and opposite

83
Q

moment vector of out of balance masses in multiple planes what is this system

A

M = d x F = d x mw^2 r = dk x mw^2 ri = dmw^2r j

dynamically unbalanced

84
Q

How do you obtain bearing loads from tabular method

A

mag F = mrw^2 angle will = theta

85
Q

Moment needed do gyroscopic motion

A

M = I * omega x w

where I is mass moment of inertia

86
Q

spin axis rotates in the same direction

A

as the gyroscopic moment

MEC201 (19 decks)

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