How Can the Spring Equation for a Chain of Springs be Solved?

[MathematicalPhysicist]

There is a chain of springs, such that q_k is the displacement of the kth mass in this chain.

We have the next equation of Newton second law, where T is the tension force:
$$\frac{d^2q_k}{dt^2}=T(q_{k+1}-q_k)-T(q_k-q_{k-1})$$
Let Q_k=q_k+1-q_k

I showed that if Q(t)_k=Q(k-ct)=Q(s)

then we get the next ODE:
(1)$$c^2 Q''(s)=T(s+1)-2T(s)+T(s-1)$$
I need to show that the solution is:
$$Q(s)=\int_{s-1}^{s+1} (1-|s-z|)T(z)dz$$
by direct integration i.e integrating (1) or by Fourier transform.

I tried both but I get stuck, by direct integration I get:

$$c^2Q(s)= \int^{s} \int^{x} [T(z+1)-2T(z)+T(z-1)] dz dx$$
don't know how to proceed from here, can anyone give a helping hand? (-:

 

[MathematicalPhysicist]

I also tried the Fourier transform I got to:
$$Q(s)=FT(FT^-1(T(s))*\frac{sin^2(w/2)}{(w/2)^2})$$

I tried mathematica to solve this, but I am not sure how to write it with an implicit function as FT^-1(T(s)).

 

[ross_tang]

I wonder about equation (1). Since:

$$\frac{d^2Q_k}{\text{dt}^2}=T\left(Q_{k+1}\right)-2T\left(Q_k\right)+T\left(Q_{k-1}\right)$$

So, we have:

$$c^2 Q''(s)=T(Q(s+1))-2T(Q(s))+T(Q(s-1))$$

instead of

$$c^2 Q''(s)=T(s+1)-2T(s)+T(s-1)$$

Please clarify.

 

[LCKurtz]

What are the equations if you have k = 2 springs? Using your equations you would have displacements q₀, q₁, q₂, and q₃ involving 4 displacements and only two equations.

 

[blue_raver22]

I would say that the spring equation described here is a representation of the motion of a chain of springs, where qk represents the displacement of the kth mass in the chain. The equation is derived from Newton's second law, where T is the tension force acting on the springs. By setting Qk=qk+1-qk, we can simplify the equation and obtain an ODE (1) that represents the motion of the chain.

The goal is to show that the solution to this ODE is given by Q(s)=\int_{s-1}^{s+1} (1-|s-z|)T(z)dz, where s is a constant. This can be done through direct integration or by using Fourier transform. However, both methods seem to have led to a dead end.

In this case, it may be helpful to consider consulting with other experts in the field or conducting further research to find alternative methods for solving this ODE. It is also important to carefully check all calculations and assumptions made in the process to ensure accuracy and validity of the solution. Science is a collaborative and ever-evolving field, and seeking help and further exploration is a crucial part of the scientific process.