Lecture 6 Forced Vibrations Flashcards
what are the two types of sources of excitation
steady state
transient
what is an example of a steady state source
rotating/reciprocating machinery
example of transient source
one off events, explosion, impact random or unstabke
steady state response to harmonic excitation is also
harmonic
if forcing f is harmonic what does f =
e^jwt
resulting motion of forced vibration x=
Xa *e^(jwt - phi)
equation of motion of forced damped system
mx.. + cx. + kx = f(t)
normalised frequency r =
w / wn
Dynamic magnification =
X / Xstat where X stat = F/k
what is the magnification factor equal for an undamped forced vibration system
1/ (1 - r^2)
What happens to dynamic magnification as you approach normalised frequency r = 1
tends to + inf from the below and - inf from above
what is the theoretical magnitude of response of a undamped system at r = 1
infinity
what happens to phase angle as your pass through r =1 for undamped system
it switches suddenly by 180 degrees
phase angle at r = 1 ie phi =
90 degrees
Force vibration equation X/F =
1/ (k -w^2 m + jwc)
Force vibration equation using system terms X/Xtat =
1/ (1 - r^2 + j *2 * zeta * r)
Graph of damped frequency response
real part FRF and normalised freq
imaginary part FRF and normalised freq
see book
Magnitude of damped frequency response
Xdyn / X stat = 1/sqrt((1-r^2)^2 + (2zetar)^2)
phase angle =
tan(phi) = 2zetar/(1-r^2)
Damped frequency response graph
see book
at w = wn what happnes
spring forces balances inertia force
damping force equals excitation force
maximum response occurs
near resonance frequency
magnitude controlled by damping
what happnes to phase
180 degree change as we go through resonance
damping makes the change more gradual
What effect does damping have far away from r = 1
damping does not change the response
as forcing freq approaches zero what happens to dynamic magnification
goes to unity
When does the maximum response occur
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)
