Damper and Spring in series?

[jackycheun]

1. Please help. How should I derive the equation for the damper and spring at right side of the mass(as shown in attach pic)? Can I combine both together as Fd + Fs or I need to have set a point between the spring and damper? Thanks.

 

[CEL]

1. Please help. How should I derive the equation for the damper and spring at right side of the mass(as shown in attach pic)? Can I combine both together as Fd + Fs or I need to have set a point between the spring and damper? Thanks.



View attachment 17877

If the damper and the spring are in series, the force applied is the same on both of them and propagates to the mass. This force will cause a change of length in the spring and a variation of the velocity in the damper.
I prefer to make an analogy with electric circuits. Force is analog to current, velocity is analog to voltage, displacement is analog to magnetic flux, mass is analog to capacitance, elastic constant is analog to the inverse of inductance and viscous friction is analog to conductance.
So, in the same way that a current in series with a resistor and an inductor will give origin to voltages Ri and $$L\frac{di}{dt}$$, the force in series with a damper and a spring will give origin to velocities $$\frac{1}{b}f$$ and $$\frac{1}{k}\frac{df}{dt}$$

 

[Mene]

I would recommend approaching this problem using the principles of Newton's Laws of Motion. First, we must define the forces acting on the mass: the force from the damper (Fd) and the force from the spring (Fs). These forces are acting in opposite directions, so they can be combined as Fd - Fs.

Next, we can use the equation F = ma, where F is the net force, m is the mass, and a is the acceleration. By substituting Fd - Fs for the net force, we get:

ma = Fd - Fs

We also know that the force from the spring can be represented as Fs = -kx, where k is the spring constant and x is the displacement from the equilibrium position. Therefore, we can rewrite the equation as:

ma = Fd + kx

Finally, we can use the equation for the force from the damper, Fd = -cv, where c is the damping coefficient and v is the velocity. By substituting this into the equation, we get:

ma = -cv + kx

This is the equation for a damped harmonic oscillator, which can be solved using techniques from differential equations. Therefore, to derive the equation for the damper and spring in series, we do not need to set a point between them. We can simply combine the forces and use the principles of Newton's Laws to solve the problem.

I hope this helps and good luck with your derivation. Remember to always approach problems in a systematic and logical manner, using the fundamental principles of science.