Normal Modes
example: four masses on springs
Four masses on springs
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Find a physical description of a system that might look like this:
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We will use:
– Newton’s laws
– Vectors
– Matrices
– Linearity (superpositions)
– Complex numbers
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Differential equations
Exponential functions
Eigenvalues
Eigenvectors
Problem: masses on springs (I)
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We consider four masses connected to springs with spring constant k and
their motion restricted to one spatial dimension.
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Step 1: Write down Newton’s law for the motion of the masses
Problem: masses on springs (I)
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Step 2: Combine the degrees of freedom into a vector and write the
equations of motion as a matrix equation
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Step 3: Use a complex exponential as the ansatz for the solution to this
equation
Problem: masses on springs (II)
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Step 4: Substitute this ansatz into the equation of motion
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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)
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Find the eigenvectors (here not normalized) from the corresponding
homogeneous equations
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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:
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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)
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Visualization of the normal modes