Correct equations of motion for object?

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Discussion Overview

The discussion revolves around the equations of motion for an object that rotates around an axis while bouncing on a piece of yarn. Participants explore the dynamics of the system, including the effects of gravity and yarn torque, and the conservation of energy in the simulation of this motion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes their equations of motion, which include gravity torque and yarn torque, but notes an issue with energy conservation when both torques are considered.
  • Another participant questions the concept of yarn torque, suggesting it may relate to resistance from stretching the yarn, and proposes analyzing the system without gravity as a simple harmonic oscillator (SHO).
  • A later reply acknowledges the complexity of the problem and inquires about the participant's familiarity with Lagrange's method for deriving equations of motion (EOMs).
  • One participant identifies a flaw in the modeling of the yarn's interaction with the object, specifically regarding momentum changes during collisions, and expresses uncertainty about whether addressing this flaw will resolve the energy conservation issue.
  • Another participant emphasizes the importance of momentum conservation during collisions to ensure the simulation's accuracy.
  • One participant suggests that the general equations of motion for a rigid body can be described by Euler's equations, but notes potential complications due to the choice of coordinate system.
  • Another participant proposes using Lagrangian mechanics to derive the equations of motion, given certain assumptions about the system's constraints and characteristics.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the yarn torque and its impact on energy conservation, with no consensus reached on the correct equations of motion or modeling approach. The discussion remains unresolved regarding the best method to accurately simulate the system.

Contextual Notes

Participants mention limitations in their understanding of advanced topics such as Lagrangian mechanics and Euler's equations, which may affect their ability to derive or apply the correct equations of motion. There are also unresolved assumptions regarding the modeling of the yarn and the interactions involved.

Simon666
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Hello,

I am currently trying to simulate an object that can rotate around an axis bouncing on a piece of yarn. I have as equations of motion for the object:

I d²Theta/dt² = Gravity_Torque + Yarn_Torque

With the gravity torque alone this system is perfectly conserving energy: the rotating lever will rotate down from horizontal to horizontal at the other side and back. However, if I also put in the yarn torque, this is only directed upwards, so in my simulations the lever will bounce back from the yarn higher than the position it left from and hence energy is continuously added and my object will bounce back higher and higher. How to solve this issue, what are the correct equations of motion? I'm not so good at physics.

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The object can't bounce off with more energy it arrived with. I don't know what you mean by yarn torque. Is this the resistance caused by stretching the yarn ?

You should try analysing the system as if there were no gravity, in which case it is like a SHO with a period of free fall in the middle.

It's not a trivial problem. Do you know how to work out EOMs with Lagrange's method ?
 
Mentz114 said:
You should try analysing the system as if there were no gravity, in which case it is like a SHO with a period of free fall in the middle.

It's not a trivial problem. Do you know how to work out EOMs with Lagrange's method ?
Sorry, I'm being a bit overwhelmed. No I don't know those things. I do suspect the equations of motion used for the object are alright but the interaction with the yarn is at fault. The yarn is discretized in a number of 0-dimensional nodes. I have discovered a flaw in my modeling though. I have modeled the effect of collision of the nodes with the object: they can bounce off of the object but then the object should get a momentum change in the opposite direction, which I have not modeled yet. That I have not modeled yet. I don't know though whether that will suffice to make the system conserve energy. I'll get back here when it doesn't.
 
they can bounce off of the object but then the object should get a momentum change in the opposite direction,
Yes, it is vital that objects bouncing off each other should conserve momentum or your simulation is bound to fail.
 
I don't understand your example well enough to give you the equations of motion for it, but the general equations of motion for a rigid body can be described by Euler's equations .

Euler's equations use body-fixed coordinates. It looks to me like you might be using absolute space coordinates, in which case you can't use the above directly as written.

The topic is discussed in Goldstein's "Classical mechanics", but since it's a graduate level textbook and you say your math is not that great, it might be a bit advanced.

Unfortunately I don't understand the exact coordinate choice you've made to be able to give you any specifics.

If your body is constrained to rotate only in a plane, I.e. if you assume the yarn and the body are always coplanar, you might be able to generate the equations of motions by using Lagrangian mechanics.

Using a Lagrangian approach, if the yarn doesn't stretch and has some constant length L, the state of a planar system should be able to be described by two angles, the angle between the yarn and the vertical, and the angle between the end of your rod and the yarn, along with characteristics of the problem such as the length of the yarn, and the moment of inertia (i.e. mass, length, width) of the rod.
 

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