Coupled Oscillators

Name:
Student number:

1. Coupled Oscillators

As seen in class, a set of equations for two coupled harmonic oscillators (e.g. two masses attached to each other and confining walls  with springs) can be written as,
m  (((d^2) (Subscript[x, 1]))/(d (t^2)))= -k (Subscript[x, 1]) + k ((Subscript[x, 2])-(Subscript[x, 1]))
m  (((d^2) (Subscript[x, 2]))/(d (t^2)))= -k (Subscript[x, 2]) - k ((Subscript[x, 2])-(Subscript[x, 1]))
a) For oscillators of mass m=1kg and a spring constant k=20 N/m, use DSolve to find (Subscript[x, 1])(t) and (Subscript[x, 2])(t) assuming the system is initially at rest with initial displacements (Subscript[x, 1])(0)=1 and (Subscript[x, 2])(0)=1. Use exact numbers for your parameter values, not floating point numbers.

For a single oscillator with the same mass and spring constant, what angular frequency would you expect the solution to have?

Now plot (Subscript[x, 1])(t) for the first 7 s.

b) Solve the same system of equations, but now use initial displacments (Subscript[x, 1])(0)=1 and (Subscript[x, 2])(0)=-1.

Now plot (Subscript[x, 1])(t) for the first 7 s.

The solution this time should have the same amplitutde, but a higher frequency.

c) Solve the same system of equations, but now use initial displacments (Subscript[x, 1])(0)=1 and (Subscript[x, 2])(0)=1/2.

Now plot (Subscript[x, 1])(t) for the first 7 s.

The graph of (Subscript[x, 1])(t) looks a little messy; it is a superposition of cosine functions with the two different frequencies seen earlier.  The initial conditions used in parts b) and c) were special in that they picked out the two independent normal modes present in this system.  In general, the solution for arbitrary initial conditions will be a linear combination of the normal modes, as illustrated in the analytic form of the solution in part c).

2. Matrix Approach

We saw in class that by beginning with the set of equations
m  (((d^2) (Subscript[x, 1]))/(d (t^2)))= -k (Subscript[x, 1]) + k ((Subscript[x, 2])-(Subscript[x, 1]))
m  (((d^2) (Subscript[x, 2]))/(d (t^2)))= -k (Subscript[x, 2]) - k ((Subscript[x, 2])-(Subscript[x, 1]))
we can define a new set of "normal coordinates" (Subscript[q, 1])=(Subscript[x, 2])-(Subscript[x, 1]) and (Subscript[q, 2])=(Subscript[x, 2])+(Subscript[x, 1]).  We then get a decoupled set of equations
m  (((d^2) (Subscript[q, 1]))/(d (t^2)))= -3 k (Subscript[q, 1]) 
m  (((d^2) (Subscript[q, 2]))/(d (t^2)))= -k (Subscript[q, 2]) 
describing an oscillator with coordinate  (Subscript[q, 1]) and angular frequency (Subscript[ω, 1])=(Sqrt[((3k)/m)]) and another, independent oscillator with coordinate (Subscript[q, 2]) and angular frequency (Subscript[ω, 2])=(Subscript[ω, 0])=(Sqrt[(k/m)]).  It seems a bit fortuitous that we picked these coordinates.  How do we do it in general?
To get to these normal mode coordinates and frequencies we begin by recasting the original set of equations in matrix form,

((d^2)/(d (t^2)))(({{(Subscript[x, 1])}, {(Subscript[x, 2])}}))=((Subscript[ω, 0])^2)(({{-2, 1}, {1, -2}}))(({{(Subscript[x, 1])}, {(Subscript[x, 2])}}))

Doing the same for the q equation we get

((d^2)/(d (t^2)))(({{(Subscript[q, 1])}, {(Subscript[q, 2])}}))=((Subscript[ω, 0])^2)(({{-3, 0}, {0, -1}}))(({{(Subscript[q, 1])}, {(Subscript[q, 2])}}))

The main difference between these equations is that the coefficient matrix A=(({{-2, 1}, {1, -2}})) is not diagonal, while the matrix d=(({{-3, 0}, {0, -1}})) IS diagonal.  By having a diagonal matrix, we get decoupled equations.  So how do we get the diagonal form of the matrix?  We introduce a matrix P that does this.

a) Enter the matrices A=(({{-2, 1}, {1, -2}})),  d=(({{-3, 0}, {0, -1}})) and P=(1/(Sqrt[2]))(({{-1, 1}, {1, 1}})) as well as the column vector x=(({{x1}, {x2}})).

b) The matrix A is symmetric, and always is for these coupled oscillator problems.  P is also orthogonal, i.e. the inverse of P is equal to its transpose.  Check this by multplying P by its transpose (it happens that P is symmetric here as well, but that is a coincidence), i.e. by entering Transpose[P].P//MatrixForm

c) The matrix P is special in that it transforms the set of equations for (Subscript[x, 1])and (Subscript[x, 2]) into the set of equations for (Subscript[q, 1]) and (Subscript[q, 2]).  Enter the following:
Transpose[P].x//MatrixForm    and
Transpose[P].A.P//Simplify//MatrixForm

d)  You will notice that the top row of Transpose[P].x is just (1/(Sqrt[2]))(Subscript[q, 1]) and the second row is (1/(Sqrt[2]))(Subscript[q, 2]), while Transpose[P].A.P is just the diagonal matrix "d".  Here is the sequence again (keep in mind that (P^(-1))=(P^T), and so P (P^T)=I).
((d^2)/(d (t^2))) x = ((Subscript[ω, 0])^2)A x
((d^2)/(d (t^2))) (P^T)x = ((Subscript[ω, 0])^2)(P^T)A P (P^T) x
((d^2)/(d (t^2))) ((P^T)x) = ((Subscript[ω, 0])^2)((P^T)A P)( (P^T) x)
(1/(Sqrt[2]))((d^2)/(d (t^2))) q =(1/(Sqrt[2])) ((Subscript[ω, 0])^2)d q
((d^2)/(d (t^2))) q = ((Subscript[ω, 0])^2)d q 	multiply both sides by (P^T) and insert (P^( ))(P^T)=I

We have used the matrix P to go from the original coupled equation for (Subscript[x, 1])and (Subscript[x, 2]) to the uncoupled equation for (Subscript[q, 1]) and (Subscript[q, 2]) (we have used the notation q=(({{(Subscript[q, 1])}, {(Subscript[q, 2])}}))).  It transforms the original equation involving the matrix A into one involving the diagonal matrix d. But how do we get P?
Enter
eigenvec=Eigenvectors[A]
and
eigenval=Eigenvalues[A]

e) Thus we see that the matrix P is related to the eigenvectors of A, while the frequencies of the normal modes are given by the square roots of the (negative of the) eigenvalues of A. The columns of P are (normalized) eigenvectors of A.  An eigenvector v and eigenvalue λ of a matrix A satisfy Av=λv.  Check this for one of the eigenvectors for our problem by typing
v=eigenvec[[1]]
λ=eigenval[[1]]
A.v
λ v

You will doubtless see eigenvalues and eigenvectors in your studies again elsewhere.  We have explored here one example of their use.

3. Resonance

You have already seen that in the case of a single, lightly damped, driven oscillator, if the driving frequency is near the natural frequency of the oscillator, resonance, that is a large reponse to the driving force, will occur.  Consider once again our system, this time under the influence of drag and a driving force on the first mass, rewritten in our standard form.

 ((d^2)/(d (t^2)))(Subscript[x, 1])+ ((Subscript[ω, 0])^2)(Subscript[x, 1])-((Subscript[ω, 0])^2) ((Subscript[x, 2])-(Subscript[x, 1]))+γ (d/(d t))(Subscript[x, 1])= cos(ω t)
 ((d^2)/(d (t^2)))(Subscript[x, 2])+ ((Subscript[ω, 0])^2)(Subscript[x, 2])+((Subscript[ω, 0])^2) ((Subscript[x, 2])-(Subscript[x, 1]))+γ (d/(d t))(Subscript[x, 2])=0

Here, as in 1a),  ((Subscript[ω, 0])^2)=20,  (Subscript[x, 1])(0)=1,  (Subscript[x, 2])(0)=1, and the oscillators start from rest.  Additionally, we have light damping with γ=0.1.  If this were a single oscillator with the same (Subscript[ω, 0])=(Sqrt[20]), we would obtain a resonant response to a driving force near ω=(Subscript[ω, 0]).  However, since this is a coupled system, there is now more than one mode of oscillation, and the system will resonate at more than one frequency. In particular, we might expect resonance at ω=(Sqrt[3])(Subscript[ω, 0]).

a) Solve this system using NDSolve and plot the trajectory of (Subscript[x, 1])(t) until a steady state is reached.  Use a driving frequency of  ω=(Subscript[ω, 0]).  Try integrating out to t=150 s.  If you get an error that indicates you have exceeded the maximumm number of iterations, try using the option MaxSteps in NDSolve (look up NDSolve for help if needed).

Give an estimate of the steady state amplitude in this case:  TYPE ANSWER HERE.

b) Solve this system and plot the trajectory of (Subscript[x, 1])(t) until a steady state is reached, this time using an intermediate driving frequency of  ω=(((Sqrt[3])(Subscript[ω, 0])+(Subscript[ω, 0]))/2).  It may be easier to determine the steady-state amplitude if you don't plot from t=0, but rather from some later time.  You may need to integrate for a long time.

Give an estimate of the steady state amplitude in this case:

c) Solve this system and plot the trajectory of (Subscript[x, 1])(t) until a steady state is reached, using a possibly resonant driving frequency of  ω=(Sqrt[3])(Subscript[ω, 0]).

Give an estimate of the steady state amplitude in this case:

What is the evidence that there is more than one resonant frequency?