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I'm trying to analyze this mass-spring system -- i.e. write down the differential equation governing it.

enter image description here

As you can see, there is a block of mass $m_1$ attached to a wall by an ideal spring of spring constant $k_1$. On the other side the block is attached by another ideal spring, of spring constant $k_2$, to a motor which is driving the system with arbitrary force $F(t)$.

I'm really just not sure how to account for the motor. Whenever it pulls (or pushes) on the $k_2$ spring, both the $k_2$ and $k_1$ springs expand or contract, but not by the same amount.

If the motor weren't there, the equation for a change in position of the block should be $\sum F = -k_1x -k_2x \implies m\ddot x + (k_1 + k_2)x =0$.

However with the motor the change in position is dictated by the function $F(t)$ and I'm not sure how to account for that. Will it just be that the motor will add an additional force to the spring -- i.e. $m\ddot x + (k_1 + k_2)x -F(t) =0$? That doesn't seem right because it doesn't take into account that the stretching of the springs is entirely due to $F(t)$.

I'm just not sure how to set up this problem. How should I be thinking about this?

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Think of the function generator producing prescribed displacements $X(t)$.

What is the extension of the first spring? It is $x$, so its restoring force is $-k_1 x$. Now what is the extension of the second spring? It is $X(t)-x$, with its restoring force $-k_2 (X(t)-x)$.

What is missing from the diagram is the direction of positive forces and displacements. I assumed it is to the right. So a positive $X(t)$ stretches the second spring reducing its tension. This is similar to the first spring where a positive displacement reduces the tension.

If the force of the generator is prescribed, it should be equal to $F(t) = -k_2 (X(t)-x)$. As a result the equation of motion depends on $k_1$ only because the forces acting on the block are $$m \ddot{x} = F - k_1 x$$

Think of it like this. A massless spring only transfers force, and so whatever force is applied on one end of it, it gets transfered to the block regardless of the extension of the second spring.

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So this is a bit tricky actually and perhaps the original question makes this more clear.

The problem I'm having when thinking about this is what exactly does the function generator do. Does it take the original force of the string and add some force $F(t)$ on top of it, or does it essentially fix the force on one side of the spring? So if the spring is pulling on the force generator with 20N left and the $F(t)$ says to pull 5N to the right, does the force then go to 25 newtons or does the force generator 'give way' until it's only pulling with a force of 5N?

In the case where the force generator fixes the total force at once side, Newton's law would say the force on one side of the spring is opposite the force on the other and the $k_2x$ term disappears from the equation. It's almost replaced in a way just by $F(t)$.

In the case where the function generator just adds a force $F(t)$ to the system, you simply add it onto the differential equation just like you have.

Since the problem goes through the trouble of adding a second spring, I would go with the interpretation that the function generator simply gives another added force to the system. We can even think of a simple system that does this. It's hard to imagine a function generator that just adds a force. But imagine if the function generator just oscillated and changed positions according to some function $d=f(t)$. Since it's attached to spring $k_2$, it's easy to see this would just add a force $-k_2f(t)$ to the system.

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