Y2, C8 - Modelling with Differential Equations Flashcards
Write v and a in terms of x (displacement)
v = dx/dt
a = d2x/dt2
What does simple harmonic motion mean
The acceleration is proportional to the displacement (x) of the particle from O
What is O in SHM
The centre of oscillation
What is the equation for SHM
a = -ω^2 * x
a = dv / dt = dv/dx * dx/dt = v * dv/dx
Where is the acceleration of SHM always towards
The centre O
What is ω
The angular velocity of the particle
(number of oscillations per 2pi seconds)
By the chain rule, what is v * dv/dx equal to
Acceleration a
In SHM, when x is maximum, what are a and v equal to
a = x
v = 0
Using the harmonic identity x = a * sin(ωt + α), what is a equal to
Amplitude
Using the harmonic identity x = a * sin(ωt + α), what is the period equal to
(2 * pi) / ω
What methods are there for finding maximum displacement
Using the harmonic identity (find R)
v = 0
x = Rsin(2t + α)
What is the period
One oscillation = 2pi / ω
ω = 2
Therefore:
One oscillation = pi
What is the equation for a particle moving with damped harmonic motion
d2x/dt2 = -k * dx/dt - ω^2 * x
Thus
d2x/dt2 + k * dx/dt + ω^2 * x = 0
What type of damping is caused by x = Ae^-αt + Be^-βt
(distinct roots of AE)
Heavy damping (no oscillations)
What type of damping is caused by x = (A + Bt)e^-αt (equal roots of AE)
Critical damping (the limit for which there are no oscillations)
What type of damping is caused by x = Ae^-αt sin(bt)
(no roots of AE) (imaginary)
Light damping (oscillates)
What is the period of oscillations of the G.S. e^-kt * (Pcos(kt) + Qsin(kt))
period = 2pi/ω
ω = k
Therefore:
period = 2pi/k seconds
What is the definition of coupled first-order linear differential equations
dx/dt = ax + by + f(t)
dy/dt = cx +dy +g(t)
When are coupled first-order linear differential equations homogenous
If f(t) = g(t) = 0 for all t
When given simultaneous differential equations, how do you find an equation in terms of x
Rearrange to find y
Differentiate to find dy/dt
Sub into equation to have everything in terms of x
What is a restoring force
A force acting against displacement in order to try to bring the system back to equilibrium
