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^{-1}x, 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 λ_{1}, λ_{2}<0 are negative numbers, and e^{Āt} → infinity for t → infinity if eigenvalues λ_{1}, λ_{2}>0. If λ_{1}, λ_{2} 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

By defining

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

where

Then

The solution of x̄_{1}, x̄_{2} 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

for λ_{1}=5.4772i

for λ_{2}=-5.4772i

we can then show that

where V=[ϑ_{1} ϑ_{2}]. Define x̄= V^{-1}x,

Substitute ω=5.4772 and x̄_{1}(0) into (7) we will find the solution of x̄_{1}. Substitute ω=5.4772 and x̄_{2}(0) we will find the solution of x̄_{2}. The solution of the original system x=Vx̄ can then be found. For x_{1}(0)=-0.2, x_{2}(0)=0, the solution of x_{1}=x_{1}(0)cos(ωt)=-0.2cos(5.4772t) and x_{2}=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

Defining

then state space representation of mass spring damper model is

By adding a damper there is an extra term associated with c ⁄ m ẋ_{2} 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 x̄_{1 is }

Since σ<0, x̄_{1} → 0 as time approaches to infinity, same applies to x̄_{2}. 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**

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