Equations Flashcards by Jamie Rayson

Whats is Newtons Second Law

sum F = ma

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Rotational equivalent of Newtons Second Law

sum MG= IG * alpha

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general equation for sinusoidal motion

x = a sin(wt + phi)

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

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alternative form of sinusoidal motion

x = Acos(wt) + Bsin(wt)

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

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Velocity equation and how is it found

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

by taking the time derivative

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Acceleration equation

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

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eulers equation of motion

x = a*e^jwt

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

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undamped natural frequency

wn = sqrt (k/m)

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Equation of free vibrations

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

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natural frequency is

the preferred vibration frequency of an undamped system

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equation of motion of damped system

mx.. + cx. + kx = 0

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characteristic equation of motion for damped system

ms^2 + cs + k =0

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critical damping equation

cc = 2 sqrt (km) = 2m wn

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damping ration =

c/cc

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characteristic equation of motion for undamped system

mx.. + kx = 0

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Overdamped system equation

see book

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damped natural frequency

wc = wn sqrt ( 1 - zeta^2)

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underdamped system repsonse

see book

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log decrement =

ln (xn-1/xn)

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alternative form of underdamped free vibration equation

see book

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log decrement simplified form (time period)

zeta * wn * taud

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log decrement simplified form

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

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logDec for several cycles

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

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low damping zeta from logdec

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

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

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Force vibration equation X/F =

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

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

1/ (1 - r^2)

Dynamic magnification =

X / Xstat where X stat = F/k

normalised frequency r = | frequency ratio

w / wn

equation of motion of forced damped system

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

Force vibration equation using system terms X/Xtat =

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

Magnitude of damped frequency response

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

phase angle =

tan(phi) = 2*zeta*r/(1-r^2)

When does the maximum response occur for damped forced vibrations

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

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

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

Q =

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

Q at low damping

Q = 1/2*zeta

Force transmission equation =>

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

Force transmission equation in system properties =>

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

equation of motion for base motion

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

Z/Y =

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

Displacement response of rotating unbalance

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

equation of motion of system excited by rotating unbalance

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

Force due to spring

F = kx

Work done/strain energy stored in a spring

W or U = 1/2 k x^2

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

sum of the masses | meq = m1 + m2 + m3

Rigid connection for inertia, | rigidly connected together for rotation about a point

J0 = (JG1 + m1*r1^2) + (JG2 + m1*r2^2) + (JG3 + m1*r3^2)

Equivalent Rotation Inertia Jeq =

Jeq = mR^2 + J0

Equivalent translational mass meq =

m + J0/R^2

Equivalent kinetic energy equation =

T = sum 1/2 *m*x.^2 + sum 1/2 *J*theta.^2

Energy dissipated in damper

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

Method for equivalent damper

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

simplest representation in hysteretic damping

DETLAW = pi() h X^2

an equivelent viscous unit for hysteretic damping

ceq = h/w = eta * keq / w

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

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

For hysteretic damper model what does F/X =

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

What is the complex stiffness of hysteretic damping

k(1 + j*eta)

SDOF system with hysteric damped equation

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

Reaction force of coulomb damping

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

what is the drop in amplitude of coulomb damping per cycle

4 * mu * N / k

What is the equation of the line coulomb damping will follow

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

energy dissipated per cycle by coulomb damping

DELTAW = 4 * mu * N * X

Equivalent damping for coulomb ceq =

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

Relative Velocity

vB/A = w * rAB

Vector dot product or scalar product =

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

Component of force F in direction r

F dot r / mag of r

vector cross product

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

equation for tangential acc

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

equation for normal acc

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

va =

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

aa =

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

Rotating frame of reference velocity equation

va = vb + w x r + vrel

Rotating frame of reference acceleration equation

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

Equation for resultant moment

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

Linear momentum =

L = mv

L. =

d/dt (mv) = ma = sum F

symbol of angular momentum =

H0 = r x mv

Rate of change of angular momentum =

H0. = r x mv. = sum of moments

sum of moments relationship with sum of forces

Sum of M0 = r x sum of Forces

sum of moment about an arbitrary point

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

Sum of moments about an arbitrary point in a planar system

sum MP = IG alpha + rhoG x maG

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

under constant velocity rotating | a = -w^2 r

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

F = m w^2 r | needs to be equal and opposite

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

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

How do you obtain bearing loads from tabular method

mag F = mrw^2 angle will = theta

Moment needed do gyroscopic motion

M = I * omega x w | where I is mass moment of inertia

spin axis rotates in the same direction

as the gyroscopic moment