example: four masses on springs 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