How to Apply Rotation Constraints in a Linkage System Equation of Motion?

[Ortafux]

Hi guys! I have a question on applying constraint on Linkage systems. Assumed that there is a two dimensional one-bar linkage, one end can only rotate and one end is free (Such as the figure above, please neglect the damper-spring system if you want).
This link can rotate only 180 degrees, not 360 degrees. But when I solve the equations of motion of this system, the animation shows that the system can rotate freely 360 degrees.
I want to know that how it is possible to apply this constraint (rotating only between 0 degree and 180 degrees) in the equation of motion of this link ?
Is it correct to solve the equation of motion and in the end, only consider the value between 0 degree and 180 degrees?!

I would be grateful if you could help me with this problem.

 

[Dr.D]

There is no way to include the constrain in equation of motion. Assuming that you have correctly solved the equation of motion, it is certainly legitimate to only look at the interval 0<=theta<=pi.

You speak of the animation as though it was something that happened automatically. How did you solve the equation of motion.

By the way, the equation of motion that you would have developed for the motion before the bar hits the wall is no longer valid once the bar does hit the wall. Thus, if you want to be really fussy about the mathematics, you need to substitute a completely different equation of motion for theta>pi.

 

[Ortafux]

There is no way to include the constrain in equation of motion. Assuming that you have correctly solved the equation of motion, it is certainly legitimate to only look at the interval 0<=theta<=pi.

You speak of the animation as though it was something that happened automatically. How did you solve the equation of motion.

By the way, the equation of motion that you would have developed for the motion before the bar hits the wall is no longer valid once the bar does hit the wall. Thus, if you want to be really fussy about the mathematics, you need to substitute a completely different equation of motion for theta>pi.

First of all thank you for your help.
you asked about the way I have solved the equation of motion. I firstly derived the non-linear EOM (Equation Of Motion) and then using numerical methods (like runge-kutta method), got the results and time response (Theta Vs. Time). finally using mathematical softwares ( such as MATHEMATICA) and derived EOM, I plotted the animation of this system.

Do you have any idea to use any equation for Theta>pi ?

 

[OldEngr63]

When theta = pi, the equation of motion used for theta<=0 ceases to apply.

At theta = pi, you have to write an impact equation, describing the event of impacting the wall. This will result in a new set of conditions, with theta-dot < 0, and you can then describe the downward swing of the bar until it strikes another constraint.

 

[Ortafux]

When theta = pi, the equation of motion used for theta<=0 ceases to apply.

At theta = pi, you have to write an impact equation, describing the event of impacting the wall. This will result in a new set of conditions, with theta-dot < 0, and you can then describe the downward swing of the bar until it strikes another constraint.

Thank you for your reply.

You mean I have to derive the EOM first, after that consider theta-dot < 0 for impact conditions ?! How is it possible to combine this constraint using Lagrangian method ( Or even Lagrangian multiplier method for constraints) ?

 

[Danger]

This is why I prefer Vise-Grips and hammers over math when designing things. Just weld a backstrap to the fulcrum above the pivot point on the side opposite your desired arc. When the bar hits it, it ain't going any farther.
By the way, whatever "C" is, you do know that it's not going to experience a straight downward force, right? The connecting rod will just bind or push it sideways.
I know that isn't what you asked about, but it's all that I can determine from the sketch.

 

[OldEngr63]

Actually, there is only one equation of motion, that applies no matter what the sign of theta-dot is. The difficulty is that this equation of motion does not apply at the instant of impact. Thus,
(1) the first solution must be stopped at the instant of impact,
(2) the impact description must be processed to produce new initial conditions,
(3) the same ODE solution is then continued, but with the new initial conditions.

For get Lagrange multipliers, etc. That is just spinning your wheels on this problem.

 

[Ortafux]

Actually, there is only one equation of motion, that applies no matter what the sign of theta-dot is. The difficulty is that this equation of motion does not apply at the instant of impact. Thus,
(1) the first solution must be stopped at the instant of impact,
(2) the impact description must be processed to produce new initial conditions,
(3) the same ODE solution is then continued, but with the new initial conditions.

For get Lagrange multipliers, etc. That is just spinning your wheels on this problem.

That's a great approach for this problem. Thank you so much.