Determining the Eigenfrequencies and Normal Modes of Coupled Pendula

While the motion of a system of coupled oscillators can be very complex, the system’s eigenfrequencies ω and normal mode motion can be determined by applying a fairly simple procedure.  A system’s eigenfrequencies and normal mode motion refers to the natural resonant frequencies of the system and the pattern of motion in which the system undergoes when all parts move with the same frequency, respectively.  To determine the eigenfrequencies and normal mode motion of a system of coupled oscillators:

1.  Formulate the lagrangian equations for the system’s kinetic energy T and potential energy U.
2. Construct A_ij and m_ij tensors in n x n arrays to describe how the system’s various components are coupled.
3.   Form and solve the characteristic equation for ω.  The characteristic equation is given by
   
4. Determine the eigenvectors of the matrix A_ij – ω²m_ij describing the normal mode motion of the system.

Example 1: Two Pendula

Given a system consisting of two pendula of equal length L and mass m connected by a spring of force constant k, we can determine the eigenfrequencies ω and describe the normal mode motion of the system.  Figure 1 illustrates the system.

Figure 1:  This figure shows the system of coupled pendula.  

Image Credit: Classical Dynamics of Particles and Systems by Thornton and Marion

As our pendula only move about the Θ direction and kinetic energy of an object is described by the general equation

where m is mass and v is velocity, the total kinetic energy of our system is

Similarly, as the potential energy of a pendulum and a spring are described by the general equations

and

where g is gravity and x is the distance from equilibrium the spring is either stretched or compressed, the total potential energy of our system is

If we assume our pendula will only undergo small oscillations, we can simplify our potential energy equation using the approximations

and

 Thus, our potential energy is

Next, we construct our A_ij and m_ij tensors.  The components for each of these tensors is determined by the relations

 and

where q_i and q_j represent the respective generalized coordinates.  In this case, the generalized coordinates are simply Θ₁ and Θ₂.  Thus, our tensors are

and 

Next, we form and solve the characteristic equation for ω. Our characteristic equation is given by

meaning we must set the determinant of matrix A-ω²m equal to 0 and solve for ω.  We find the eigenfrequencies of the system are

and

Finally, after solving for the eigenvectors of the system, we determine the normal modes of the system to be

and

This means that our pendula can obtain constant oscillatory motion two ways: by swinging together at equal velocities or by swinging exactly opposite eachother, see Figure 2.

Figure 2:  This figure illustrates the normal modes of the coupled pendula.  

Image Credit: Classical Dynamics of Particles and Systems by Thornton and Marion

Example 2: Three Pendula

Given a similar system to that of example 1 but containing an additional pendulum, we can use the same process to determine the new system’s eigenfrequencies and normal modes.  Below are the respective functions and tensors.

In this case, the pendula can swing together, the outer two can swing in opposite directions while the middle remains stationary, or the outer two can swing together while the middle swings in the opposite direction with an increased amplitude.