I'm trying to figure out the dynamics of the system below:
Two masses connected to an inverted parabola, free to slide. They are connected by a spring of constant . Both masses are identical in every way and are of same weight (); they slide down/up the parabola at the same time.
The natural length of the spring is ; gravitational acceleration ; and the frictional coefficient of the parabola is .
So, I'm trying to find out the dynamics of the system using code.
Please, if you can, provide a source code that simulates the dynamics of the system.
The physics behind the simulation:
There are 3 forces acting on the block, tangential to the ramp.
The tangential gravitational force is given by:
Where is the unit tangent vector and is the angle of the tangent.
Tangential spring force can be resolved using ; it is given by .
And then we have the frictional force which acts opposite to the movement of the mass. To calculate this we have to calculate the normal forces.
There are two forces which contribute to the total normal force; gravitational and spring.
The normal force by gravitation on a ramp is given by .
The spring part of the normal force is orthogonal to the tangent, and can also be resolved using :
Once these normal forces are calculated, we can calculate the frictional force:
So, total force:
So guys, the main problem is that the tangential spring force is too small and the gravitational tangential force always pulls the object down and it never goes back up or undergoes oscillatory motion. The tangential ramp angle goes off to 90 degrees, which I don't think it should be doing; the spring doesn't seem to be bringing it back up. As a result of the ramp angle increasing to 90 degrees, there is almost no spring force to bring it back into oscillatory motion.
The question I would like to ask is:
Does the above system oscillate? Is the code producing a realistic result at all?