Chapter 1 - Free Undamped Vibration of 1 DOF Systems Flashcards
What is static deflection?
When the spring is slightly extended due to weight
What is dynamic deflection?
Displacement during movement/oscillation/vibration of the system
Can you draw the FBD of a mass spring system in static equilibrium?
YES OR NO
Resolving the forces from the static equilibrium FBD gives?
mg = kδst
Can you draw the FBD of the mass spring system during vibration?
YES OR NO
Resolving the forces from the dynamic FBD gives?
mg - k(x+δst) = mx(double dot)
=> kδst - kx - kδst = mx(double dot)
=> mx(doubledot) + kx = 0
What does the result from the dynamic system mean?
We can ignore gravity and draw our FBD’s with just dynamic forces. Gravity isn’t effecting the system: a mass on a spring will oscillate at the same frequency in a room as on the moon.
Can you therefore draw the new FBD?
YES OR NO
What graph could we use to represent the movement of a mass spring system?
A sine shape.
How can we get a sine shape without using sin or cos wave?
HARMONIC ADDITION THEOREM: adding a sin and cos wave will get a wave that is not precisely sine or cosine but will be one with the same shape but with a shift (due to different amplitudes).
What will the equations be if the behaviour is considered quasi-sinusoidal?
x= Asin(ωt)+Bcos(ωt)
x(dot)= A ωcos(ωt) - B ωsin(ωt)
x(double dot)= -A ω^2sin(ωt) - B ω^2cos(ωt)
If you have a higher A what will the wave look like?
More like a sine wave
What do you get if you substitute the quasi-sinusoidal expressions for x into mx(double dot) + kx =0?
ωn = sqrt(k/m)
Assuming that the solution looks like x=Ae^st, what are the expressions for position, velocity and acceleration?
x=Ae^st
x(dot)=sAe^st
x=(double dot)=s^2Ae^st
Plugging the Ae^st expressions into mx(double dot) + kx=0 gives?
s=(+/-) i ωn = A1 e^(i ωnt) or A2 e(-i ωnt)
