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Note: more than 2 lectures, §5.3 in [EP], not in [BD]

While we did say that we will usually only look at ﬁrst order systems, it is sometimes more convenient to study the system in the way it arises naturally. For example, suppose we have 3 masses connected by springs between two walls. We could pick any higher number, and the math would be essentially the same, but for simplicity we pick 3 right now. Let us also assume no friction, that is, the system is undamped. The masses are , , and and the spring constants are , , , and . Let be the displacement from rest position of the ﬁrst mass, and and the displacement of the second and third mass. We make, as usual, positive values go right (as grows, the ﬁrst mass is moving right). See Figure 3.11.

This simple system turns up in unexpected places. For example, our world really consists of many small particles of matter interacting together. When we try the above system with many more masses, we obtain a good approximation to how an elastic material behaves. By somehow taking a limit of the number of masses going to inﬁnity, we obtain the continuous one dimensional wave equation (that we study in § 4.7). But we digress.

Let us set up the equations for the three mass system. By Hooke’s law we have that the force acting on the mass equals the spring compression times the spring constant. By Newton’s second law we have that force is mass times acceleration. So if we sum the forces acting on each mass and put the right sign in front of each term, depending on the direction in which it is acting, we end up with the desired system of equations.

We deﬁne the matrices

We write the equation simply as

At this point we could introduce 3 new variables and write out a system of 6 ﬁrst order equations. We claim this simple setup is easier to handle as a second order system. We call the displacement vector, the mass matrix, and the stiﬀness matrix.

Exercise 3.6.1: Repeat this setup for 4 masses (ﬁnd the matrices and ). Do it for 5 masses. Can you ﬁnd a prescription to do it for masses?

As with a single equation we want to “divide by .” This means computing the inverse of . The masses are all nonzero and is a diagonal matrix, so computing the inverse is easy:

This fact follows readily by how we multiply diagonal matrices. As an exercise, you should verify that .

Let . We look at the system , or

Many real world systems can be modeled by this equation. For simplicity, we will only talk about the given masses-and-springs problem. We try a solution of the form

We compute that for this guess, . We plug our guess into the equation and get

We divide by to arrive at . Hence if is an eigenvalue of and is a corresponding eigenvector, we have found a solution.

In our example, and in other common applications, has only real negative eigenvalues (and possibly a zero eigenvalue). So we study only this case. When an eigenvalue is negative, it means that is negative. Hence there is some real number such that . Then . The solution we guessed was

By taking the real and imaginary parts (note that is real), we ﬁnd that and are linearly independent solutions.

If an eigenvalue is zero, it turns out that both and are solutions, where is an eigenvector corresponding to the eigenvalue 0.

Exercise 3.6.2: Show that if has a zero eigenvalue and is a corresponding eigenvector, then is a solution of for arbitrary constants and .

Theorem 3.6.1. Let be an matrix with distinct real negative (or zero) eigenvalues we denote by , and corresponding eigenvectors by , , …, . If is invertible (that is, if ), then

is the general solution of

for some arbitrary constants and . If has a zero eigenvalue, that is , and all other eigenvalues are distinct and negative, then the general solution can be written as

We use this solution and the setup from the introduction of this section even when some of the masses and springs are missing. For example, when there are only 2 masses and only 2 springs, simply take only the equations for the two masses and set all the spring constants for the springs that are missing to zero.

Example 3.6.1: Suppose we have the system in Figure 3.12, with , , , and .

The equations we write down are

or

We ﬁnd the eigenvalues of to be (exercise). We ﬁnd corresponding eigenvectors to be and respectively (exercise).

We check the theorem and note that and . Hence the general solution is

The two terms in the solution represent the two so-called natural or normal modes of oscillation. And the two (angular) frequencies are the natural frequencies. The ﬁrst natural frequency is 1, and second natural frequency is 2. The two modes are plotted in Figure 3.13.

Let us write the solution as

The ﬁrst term,

corresponds to the mode where the masses move synchronously in the same direction.

The second term,

corresponds to the mode where the masses move synchronously but in opposite directions.

The general solution is a combination of the two modes. That is, the initial conditions determine the amplitude and phase shift of each mode.

Example 3.6.2: We have two toy rail cars. Car 1 of mass 2 kg is traveling at 3 ^{m}/_{s} towards the second rail car of mass 1 kg. There is a bumper on the second rail car that engages at the moment the cars hit (it connects to two cars) and does not let go. The bumper acts like a spring of spring constant . The second car is 10 meters from a wall. See Figure 3.14.

We want to ask several questions. At what time after the cars link does impact with the wall happen? What is the speed of car 2 when it hits the wall?

OK, let us ﬁrst set the system up. Let be the time when the two cars link up. Let be the displacement of the ﬁrst car from the position at , and let be the displacement of the second car from its original location. Then the time when is exactly the time when impact with wall occurs. For this , is the speed at impact. This system acts just like the system of the previous example but without . Hence the equation is

or

We compute the eigenvalues of . It is not hard to see that the eigenvalues are 0 and (exercise). Furthermore, eigenvectors are and respectively (exercise). Then and by the second part of the theorem we ﬁnd our general solution to be

We now apply the initial conditions. First the cars start at position 0 so and . The ﬁrst car is traveling at 3 ^{m}/_{s}, so and the second car starts at rest, so . The ﬁrst conditions says

It is not hard to see that . We set and in and diﬀerentiate to get

So

Solving these two equations we ﬁnd and . Hence the position of our cars is (until the impact with the wall)

Note how the presence of the zero eigenvalue resulted in a term containing . This means that the carts will be traveling in the positive direction as time grows, which is what we expect.

What we are really interested in is the second expression, the one for . We have . See Figure 3.15 for the plot of versus time.

Just from the graph we can see that time of impact will be a little more than 5 seconds from time zero. For this we have to solve the equation . Using a computer (or even a graphing calculator) we ﬁnd that seconds.

The speed of the second cart is . At the time of impact (5.22 seconds from ) we get . The maximum speed is the maximum of , which is 4. We are traveling at almost the maximum speed when we hit the wall.

Suppose that Bob is a tiny person sitting on car 2. Bob has a Martini in his hand and would like not to spill it. Let us suppose Bob would not spill his Martini when the ﬁrst car links up with car 2, but if car 2 hits the wall at any speed greater than zero, Bob will spill his drink. Suppose Bob can move car 2 a few meters towards or away from the wall (he cannot go all the way to the wall, nor can he get out of the way of the ﬁrst car). Is there a “safe” distance for him to be at? A distance such that the impact with the wall is at zero speed?

The answer is yes. Looking at Figure 3.15, we note the “plateau” between and . There is a point where the speed is zero. To ﬁnd it we need to solve . This is when or in other words when and so on. We plug in the ﬁrst value to obtain . So a “safe” distance is about 7 and a quarter meters from the wall.

Alternatively Bob could move away from the wall towards the incoming car 2 where another safe distance is and so on, using all the diﬀerent such that . Of course is always a solution here, corresponding to , but that means standing right at the wall.

Finally we move to forced oscillations. Suppose that now our system is

(3.3) |

That is, we are adding periodic forcing to the system in the direction of the vector .

As before, this system just requires us to ﬁnd one particular solution , add it to the general solution of the associated homogeneous system , and we will have the general solution to (3.3). Let us suppose that is not one of the natural frequencies of , then we can guess

where is an unknown constant vector. Note that we do not need to use sine since there are only second derivatives. We solve for to ﬁnd . This is really just the method of undetermined coeﬃcients for systems. Let us diﬀerentiate twice to get

Plug and into equation (3.3):

We cancel out the cosine and rearrange the equation to obtain

So

Of course this is possible only if is invertible. That matrix is invertible if and only if is not an eigenvalue of . That is true if and only if is not a natural frequency of the system.

Example 3.6.3: Let us take the example in Figure 3.12 with the same parameters as before: , , , and . Now suppose that there is a force acting on the second cart.

The equation is

We solved the associated homogeneous equation before and found the complementary solution to be

The natural frequencies are 1 and 2. As 3 is not a natural frequency, we try . We invert :

Hence,

Combining with the general solution of the associated homogeneous problem, we get that the general solution to is

The constants , , , and must then be solved for using the initial conditions.

Note that given force , we write the equation as to get the units right. Then we write . The term in is in units of force per unit mass.

If is a natural frequency of the system resonance occurs because we will have to try a particular solution of the form

That is assuming that the eigenvalues of the coeﬃcient matrix are distinct. Next, note that the amplitude of this solution grows without bound as grows.

Exercise 3.6.4 (challenging): Let us take the example in Figure 3.12 with the same parameters as before: , , and , except for , which is unknown. Suppose that there is a force acting on the ﬁrst mass. Find an such that there exists a particular solution where the ﬁrst mass does not move.

Note: This idea is called dynamic damping. In practice there will be a small amount of damping and so any transient solution will disappear and after long enough time, the ﬁrst mass will always come to a stop.

Exercise 3.6.5: Let us take the Example 3.6.2, but that at time of impact, cart 2 is moving to the left at the speed of 3 ^{m}/_{s}. a) Find the behavior of the system after linkup. b) Will the second car hit the wall, or will it be moving away from the wall as time goes on? c) At what speed would the ﬁrst car have to be traveling for the system to essentially stay in place after linkup?

Exercise 3.6.6: Let us take the example in Figure 3.12 with parameters , . Does there exist a set of initial conditions for which the ﬁrst cart moves but the second cart does not? If so, ﬁnd those conditions. If not, argue why not.

Exercise 3.6.102: Suppose there are three carts of equal mass and connected by two springs of constant (and no connections to walls). Set up the system and ﬁnd its general solution.

Exercise 3.6.103: Suppose a cart of mass 2 kg is attached by a spring of constant to a cart of mass 3 kg, which is attached to the wall by a spring also of constant . Suppose that the initial position of the ﬁrst cart is 1 meter in the positive direction from the rest position, and the second mass starts at the rest position. The masses are not moving and are let go. Find the position of the second mass as a function of time.