Two masses coupled with a spring

[Silken]

Hi

Homework Statement

Given is a system containing to masses m1 and m2 which are connected by a spring with spring constant k.Oscillation and translation is restricted to one dimension only.

Homework Equations

euqation of motion

The Attempt at a Solution

So, I can find out the equation of motion, which I done

$$m_{1} \ddot x_{1}=-k(x_{1}-x_{2})$$
$$m_{2} \ddot x_{2}=k(x_{1}-x_{2})$$

Now we solved it to find the equation of motion with a method, I don't understand. First of all: I got the solution, but I don't understand why we did this, would be awesome if someone could explain it to me.
First of all we wrote the 2 equations with the help of a matrix. Afterwards we were looking for the Eigenvalues (which are the frequency/frequencies <- I hope that's right). But after this step I don't understand what happens. We multiply the matrix with a vector, find the norm and finally get to the solution. I understand the final solution but not the 'way' of how I get there. The solution looks like

$$\vec x_{1}(t)=\vec v_{1}*(At+B)$$
$$\vec x_{2}(t)=\vec v_{2}*(Ccos( \omega t)+Dsin(\omega t))$$

which gives us the translation and the oscillation. I hope my text isn't too confusing as I'm just not sure about what 'happens' here to get to the solution. At our university, or rather at my study course, we don't get taught matrix calculus so I have trouble understanding this problem.

Thanks for your help in advance

 

[anathema]

!

Hi there! I can see that you have already done a great job in finding the equation of motion for this system. Now, to understand the method used to solve it, let's break it down step by step.

Firstly, writing the equations in matrix form is a common technique used to solve systems of equations. It allows us to represent multiple equations in a more compact and organized way. In this case, we have two equations and two unknowns (x1 and x2), so we can write them in matrix form as:

m1 * x1'' + k * x1 = -k * x2
k * x1 - m2 * x2'' = -k * x1

Next, we need to find the eigenvalues of the matrix. These eigenvalues represent the natural frequencies of the system, which is the frequency at which the system will oscillate without any external forces acting on it. To find the eigenvalues, we need to solve the characteristic equation of the matrix, which is given by:

det(A - λI) = 0

Where A is the matrix, λ is the eigenvalue, and I is the identity matrix. Solving this equation will give us two eigenvalues, which we can then use to find the eigenvectors.

The eigenvectors represent the direction in which the system will oscillate at the corresponding eigenfrequency. These eigenvectors are represented by the vectors v1 and v2 in your solution. It is important to note that these eigenvectors are orthogonal (perpendicular) to each other, which means that the two masses will oscillate in different directions.

Next, we need to find the general solution to the system of equations. This is where the vector multiplication and norm come in. Multiplying the matrix by a vector will give us a new vector, which represents the position of the masses at any given time t. This vector can be written as a linear combination of the eigenvectors, which is represented by the constants A, B, C, and D in your solution.

Finally, we can use the initial conditions (such as the initial positions and velocities of the masses) to solve for the constants and get the specific solution to the system.

I hope this helps to clarify the method used to solve this problem. Matrix calculus may seem intimidating at first, but with practice, it will become easier to understand. Keep up the good work!