Link with spring-mass-damper system

[trojsi]

From this mechanical part (attached), I need to derive a differential equation relating mass 'x' position to hinge 'y' position.

$\ddot{x}m + b\dot{x} + cx = F$

The link l would be rotating. I am confused about considering the displacement of x with reference to y. Would this change the differential equation by replacing x with (x-y) ?

I derived an equation to relate 'y' with $\theta$ ;

$y = lsin(\theta)$

Finally I also need to derive a differential equation from the previous two to relate the loading torque on the shaft for some rotation theta.

$\tau = (\ddot{x}m + b\dot{x} + cx)lcos(\theta)$

I would really appreciate if you can give me some hints on my work. thanks

 

[Simon Bridge]

Damped-driven harmonic motion with feedback?
Is the servo "active" (it is being driven by a motor and the second mass is just a counter-balance?) ... the it is just providing the f(t) in the standard equations.

 

[trojsi]

I need to investigate some tight control on an open loop modelling of a servo motor linked with a spring mass damper system.
The linkage with spring mass damper is stable with a counterbalance on the other side. The aim is to shift the mass (by the servo) of the s-m-d system and eliminate oscillations as much as possible. The servo is modeled as having a feedback loop with theta. I will also be implementing a PID loop in the servo controller but the spring-mass-damper is open loop.

I already derived the transfer functions and state space for the servo which is simply a DC motor with a 10:1 gearbox.

My main confusion right now are the equations in my previous post. I need these in order to simulate everything together in simulink. attached

 

[Simon Bridge]

Gotcha:

From this mechanical part (attached), I need to derive a differential equation relating mass 'x' position to hinge 'y' position.

$\ddot{x}m + b\dot{x} + cx = F$

The link l would be rotating. I am confused about considering the displacement of x with reference to y. Would this change the differential equation by replacing x with (x-y) ?

The F in the above relation is being applied at y - y moving up and down is what is generating it.

I derived an equation to relate 'y' with $\theta$ ;

$y = l\sin(\theta)$

If $\theta$ is expected to be small enough that you can discount lateral movement then $\sin(\theta)\approx \theta$
considering the idea is to minimize oscillations, that seems reasonable.

measure x from some equilibrium/rest position and y likewise ... then you'll see better how they interact.

 

[trojsi]

If $\theta$ is expected to be small enough that you can discount lateral movement then $\sin(\theta)\approx \theta$
considering the idea is to minimize oscillations, that seems reasonable.

I cannot assume this. The system has to remain non linear that is why it will be simulated via simulink. Dont worry, I will not be working any calc with a non linear system.

The F in the above relation is being applied at y - y moving up and down is what is generating it.
measure x from some equilibrium/rest position and y likewise ... then you'll see better how they interact.

Do you mean $y-y_{o}$ ? Then how would the differential equation be?