Oscillations Flashcards
What are the equations for w^2 for a massless pendulum and a massless spring?
- w^2 = g/L
- w^2 = k/m
What is are the relationships for resonant frequency and period of oscillation ?
- w(o) = 2 Pi/T
- T = 1 / f(o) = 2 Pi/ w(o)
What is the definition of simple harmonic motion ?
When can we say a system is simple harmonic
When the displacement of an object is a sinusoidal function of time
This is a characteristic of any system where the restoring force is proportional to the displacement
How do you set up the second order differential equations for oscillating systems ?
Take all your restoring forces and set them equal to f=ma and then switch you acceleration and velocity to be derivatives of displacement
What is the differential equation for an undampened simple harmonic oscillation ?
d^2 x/ d t^2 = - w^2 x(t)
What form does the equation for elastic potential energy , and kinetic energy of an elastic object take ?
Elastic potential - U = 1/2 k x^2
Elastic Kinetic - K = 1/2 m v^2
What is the total mechanical energy of a system ?
It is the elastic potential + the kinetic energy of a system
E = 1/2 k x^2 + 1/2 m v^2
What is the equation for the angular velocity, and period in terms of moment of inertia ?
- w(o) = sqrt( mgh / I )
- T = 2Pi sqrt( I / mgh )
What are the small angle approximations for sin, cos and tan ?
Sin x => x
Tan x => x
Cos x => 1
(This for a pendulum would mean that x = s / L)
What theorem describes an oscillating system about a pivot point that is not the centre of mass?
The parallel axis theorem
This is given by
I = I(com) + m d^2
What are the four types of damping ?
Un-damped
Under-damped
Critically-damped
Over-damped
What is the equation for the force of damping ?
F(damp) = -b v
What is special about critical damping in comparison to under or over damping ?
Critical damping is where the system returns to its equilibrium point in the shortest amount of time (w(o) = gamma)
Over damped takes longer (w(o) < gamma) and under damped will slowly decay as it loses energy (w(o) > gamma)
What is the solution to the differential equation in the under damped case ?
x(t) = A sin(wt+phi) exp( -gamma t)
In what direction does F(damp) apply?
F(damp) is always applied so that it opposed the velocity of the system
What is gamma equal too ?
gamma = b/2m
How do you find the damped angular frequency?
w(1) = sqrt(w(o)^2 - gamma^2)
What is the Q factor for a system ?
The Q factor or quality factor describes the amount of damping in a system
The higher the Q value, the weaker the damping in the system
This will be given by
Q= 2Pi (E(stored)/E(lost per cycle))
What is the solution to the differential equation in the critically damped case?
x(t) = (A+Bt) exp( -gamma t)
What can the Q factor be reduced to for under damped systems?
Q= w(o) m / b
How do you find the roots to second order differential equations for oscillating systems with some form damping ?
r = -gamma +/- sqrt( gamma^2 - w(o)^2 )
What is Tao for oscillating systems and what is it given by ?
Tao is the time constant of a system
it is given by
Tao = m / b
Describe what is meant by a forced oscillator
A forced oscillator is an oscillator subjected to a periodic external force
This is mainly be by a force that is sinusoidal
F(ext) = F(o) cos( w(dr) t )
What is w(dr) ?
It is the angular frequency for a driven oscillation in the steady state
What is the angular resonant frequency defined as ?
w(res) = sqrt( w(o)^2 - 2.gamma^2 )
When damping is small, what equations of Q can we equate in terms of decay rate and resonance?
Q(decay) = w(1) m / b Q(res) = w(res) / E(FWHM)
What can we say about the size of the value of Q if we see a sharp resonance curve and a large resonance amplitude ?
Q is large
True or False:
One normal mode can be excited by multiple resonant frequencies
False
one normal mode can only be excited by one resonant frequency
True or False:
One mode will never decay into another mode
True
How many modes will there be if three oscillators are coupled together?
What is the general formula for number of modes to number of oscillators?
What can we say about a system that has more modes than coupled oscillators?
- There will be 3 modes
- The number of modes equals the number of coupled oscillators
- The other modes that are greater than the number of coupled oscillators are just superposition of the initial modes
What can we say about the dependence of normal modes?
Normal modes are independent of one another
What is beating ?
Beating is an interference effect
In beating there is an exchange of energy between individual oscillators but there is no exchange of energy between normal modes
What can we say about the first normal mode in a coupled oscillating system ?
It is the same as the resonant frequency for the system as if it were one mass at its centre
What is the definition of a normal mode?
A collective motion at a single frequency that does not exchange energy with other modes
