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
