What are eigenvalues in a physical system? Eigenvalues describe the stability of a system and often associate with linear algebra. One way to understand eigenvalues is to show their use in describing engineering systems, such as the mass spring damper system described in this post. It shows that by adding a damper to a mass spring system, the eigenvalues of the system moves from the imaginary axis to the left side of the complex plane, thus stabilizes the system to an equilibrium. To begin with, we will briefly introduce eigenvalues before associating them with mass spring damper system.
1.What are eigenvalues?
We call a real or complex number λ an eigenvalue of a n × n real matrix A if there exists a non-zero vector x such that Ax=λx. We can find eigenvalues by solving
In order for equation (1) to have non-zero solution x, the matrix A-λI must be singular or have determinant det(λI-A)=0.
2. Eigenvalues in dynamical systems
Now lets see what the eigenvalues are in a second order dynamical system. Consider linear time invariant state space equation:
A matrix transformation exists x̄=V-1x, where V-1 is a non-singular matrix. (2) becomes
3. Eigenvalues in the solution of dynamical system
The solution of (3) is
It is easy to see that the exponential term in (4) converges to zero e Āt → 0 for t → infinity if eigenvalues λ1, λ2<0 are negative numbers, and eĀt → infinity for t → infinity if eigenvalues λ1, λ2>0. If λ1, λ2 are zeros or on the imaginary axis of the complex plane, see figure 2, then oscillation will occur for nonzero initial conditions x̄ (0) or bounded input u. Now let’s see what the eigenvalues are in a mass spring damper system.
4. Eigenvalues in mass spring model
In our case, a simple undamped mass spring system moves only along the vertical direction. By applying Newton’s second law of motion, we obtain
then state space representation of mass spring system (5) is
Putting (6) into the form of (2)
It is known that eigenvalues of A are all on the imaginary axis. Suppose the eigenvalues of A are
then the transformation of matrix A becomes
The solution of x̄1, x̄2 are
Using Euler’s formula
For a mass spring constant k=30 N/m and mass weight of m=1 kg, then the system matrix A for (6) is
the eigenvalues of A are ∓ 5.477i. Define a diagonal matrix with eigenvalues on diagonal axis
we find eigenvectors ϑ such that
we can then show that
where V=[ϑ1 ϑ2]. Define x̄= V-1x,
Substitute ω=5.4772 and x̄1(0) into (7) we will find the solution of x̄1. Substitute ω=5.4772 and x̄2(0) we will find the solution of x̄2. The solution of the original system x=Vx̄ can then be found. For x1(0)=-0.2, x2(0)=0, the solution of x1=x1(0)cos(ωt)=-0.2cos(5.4772t) and x2=1.0954sin(5.4772t), see figure 3.
The eigenvalues of mass spring system are plotted in figure 2 below.
5. Eigenvalues in mass spring damper model
Adding a damping force into the mass spring system in figure 1, the mass spring damper system looks like in figure 4.
The viscous damping force is
the equation of motion becomes
then state space representation of mass spring damper model is
By adding a damper there is an extra term associated with c ⁄ m ẋ2 in (8) compared to non-damper model (6).
Putting (8) into the form of (2)
The eigenvalues of A are on the left side of the complex plane. Suppose then eigenvalues of A are
where σ<0 is a negative number. Then
The solution of x̄1 is
Since σ<0, x̄1 → 0 as time approaches to infinity, same applies to x̄2. Since x̄ → 0, x → 0 also.
The eigenvalues of (8) with damping constant c=2 N∙s/m are -1∓ 5.39i. Their plot in the complex plane is shown in figure 5.
Since the eigenvalues are on the left side of the complex plane, the system is stable.
The time response of the mass spring damper model is
From the above analysis we can see that by adding a damper, the spring mass system has its eigenvalues moved from imaginary axis to the left side of the complex plane. The resulting eigenvalues stabilizes the mass spring damper model. It shows that eigenvalues associate with the natural frequency of the mass spring system.
This post is just one example of showing physical meaning of eigenvalues in real system. Eigenvalue is a basic system concept and is often used for stability analysis. They are also used in google web search engines to identify links’ impact factors, in signal processing for musical instruments, and many more every day applications.
Written by: Xiaoran Han – Project Engineer