Mass on a Stick Model: Solve for Realistic Results

[Integral]

I have need of a mathematical model of a mass at the end of a stick, pivoting about the opposite end. What I have done so far is not giving realistic results, would appreciate it if some one could point out my error.

Basic model :

$$\tau = I \ddot {\theta}$$

$$I = m r^2$$

The system is being pushed by an air cylinder (force F) with the fixed end mounted a distance L directly below the pivot, the moving end of the cylinder is mounted a distance r along the pivot arm from the pivot point. With $\phi$ the angle between the cylinder and the pivot arm. So I have :

$$\tau = r F sin( \phi)$$

My variable of interest will be the angle between the pivot arm and the line defined by the pivot and the fixed end of the cylinder, call this angle $\theta$

By the law of sines I get

$$\frac {Sin(\phi)} {L} = \frac {sin(\theta)} {x}$$
where x is the length of the cylinder.

I get x in terms of $\theta$ from the law of cosines

$$x^2 = L^2 + r^2 -2lr cos(\theta)$$

The differential equation is:

$$\ddot{ \theta } = \frac { \tau } {I}$$

See the attachment for a diagram.

 

[Integral]

Ok, The above model is correct. For the last week I have been running this through an excel spreadsheet using a Runga Kutta method, the results were not correct, the system simply did not respond the way I KNEW it should.

My error? using the fricking English measurement system. I was using lbs where I needed slugs. So I was throwing 32 times the mass I thought I was. That sort of slowed things down. I finally made the simulation work by changing everything to metric, Newtons for force, kg for mass and meters everywhere, It worked great. It was then that I seriously began to figure out the lbs mass lbs force issue.

 

[sir-pinski]

Thank you for sharing your model and the steps you have taken so far. It seems that you have correctly set up the basic model for a mass on a stick system, including the moment of inertia and the torque equation. However, as you have mentioned, your results are not giving realistic results.

One possible error in your model could be the assumption that the force from the air cylinder is acting at a distance r from the pivot point. This may not be the case, as the force may be applied at a different point along the pivot arm, depending on the setup of the system.

Additionally, it may be helpful to consider the effects of friction and other external forces on the system, as these can greatly affect the movement and stability of the mass on a stick.

To improve the realism of your model, you could also consider incorporating more complex equations, such as those for rotational motion and dynamics, as well as taking into account the properties of the stick and the mass itself. It may also be helpful to experiment with different values for your variables and see how they affect the results.

Overall, it is important to continuously review and refine your model to ensure that it accurately reflects the real-life system. Consulting with experts in the field or conducting physical experiments can also help to validate and improve your model. Keep up the good work and don't be afraid to make adjustments as needed.