Normal Modes

advertisement
Normal Modes
example: four masses on springs
Four masses on springs
•
Find a physical description of a system that might look like this:
•
We will use:
– Newton’s laws
– Vectors
– Matrices
– Linearity (superpositions)
– Complex numbers
–
–
–
–
Differential equations
Exponential functions
Eigenvalues
Eigenvectors
Problem: masses on springs (I)
•
We consider four masses connected to springs with spring constant k and
their motion restricted to one spatial dimension.
•
Step 1: Write down Newton’s law for the motion of the masses
Problem: masses on springs (I)
•
Step 2: Combine the degrees of freedom into a vector and write the
equations of motion as a matrix equation
•
Step 3: Use a complex exponential as the ansatz for the solution to this
equation
Problem: masses on springs (II)
•
Step 4: Substitute this ansatz into the equation of motion
•
Step 5: Solve the eigenvalue equation for eigenvalues 2 and eigenvectors v
We do not need the negative
frequency solutions since we only
consider the real part as physically
relevant
Problem: masses on springs (III)
•
Find the eigenvectors (here not normalized) from the corresponding
homogeneous equations
•
Step 6: The general solution is then given by a superposition of all these
normal modes with complex amplitudes A1, A2, A3, A4 chosen to meet the
initial conditions:
•
If the system is in one of these normal modes (i.e. all Ai zero except An) all
masses will oscillate with the same frequency n=n(k/m)1/2 and constant
amplitude ratios defined by vn .
Problem: masses on springs (IV)
•
Visualization of the normal modes
Download
Related flashcards

Cybernetics

25 cards

Create Flashcards