Math behind stiffness constant and damping constant in joints in Simscape Multibody

Hello,

I am experimenting with gimbal joints and wanted to clarify how exactly the stiffness constant aand damping constant parameters in this joint work. What is the equation behind the scenes?

I have a gimbal joint with a custom torque feedback in all the three axes as the figure below.

The torque feeding into the joint is a function of joint angle in that particular axis. This is done through the Matlab function block.

This setup compiles without errors.

However, I notced that without any damping (C_Gimbal = 0) in the joint, the output angle is very jittery and oscillaroty. When I give a small amount of damping (C_Gimbal = 0.1), the system is smooth and non oscillatory under load - as expected.

I am unsure of what exactly is the math behind this scenario, since I am using a custom torque input in the joint. How does damping fint into the equation.

ultimately, I want to put C_Gimbal = 0 and integrate this damping component in the torque function itself. I tried

and the result is non smooth and very oscillatory. For the differentiation, I tried both 'Derivative' and 'Discrete Derivative' blocks.

Any insights in this would be very helpful. Thank you.

Answer 1

[akshatsood]

Hi @Siril TD,

I recognize your interest in understanding the mathematical principles underlying the calculation of stiffness and damping constants for a gimbal joint, as well as seeking clarification on how variations in the damping constant affect oscillations. Below, I will share my insights into the behavior you are observing.

In systems like gimbal joints, which can have complex dynamics, both stiffness and damping play crucial roles in controlling behavior. When you are dealing with a custom torque feedback mechanism, understanding how these parameters interact becomes even more critical.

Stiffness (K): This parameter is related to the joint's resistance to angular displacement. In a simplified model, the torque due to stiffness could be represented as , where is the torque due to stiffness, K is the stiffness constant, and θ is the joint angle. This model assumes a linear relationship between the torque and the angle, which might not always be the case in complex systems but serves as a good starting point for understanding.

Damping (C): Damping represents resistance to angular velocity. It's a force that opposes the motion and is proportional to the rate of change of the angle (angular velocity). The torque due to damping can be represented as , where is the torque due to damping, C is the damping constant, and is the angular velocity.

In your setup, when you apply a custom torque feedback based on the joint angle and possibly its derivatives, you are essentially controlling the dynamics directly. However, the absence of damping i.e., means there's no inherent mechanism to counteract the energy added by your torque function, leading to oscillations or jittery motion as the system continuously overshoots and corrects.

When you introduce a small damping factor , it helps to remove energy from the system, counteracting the oscillations and leading to a smoother response. This damping effect is beneficial for achieving stability, especially under dynamic loads or when the system is perturbed.

I hope this helps.