# Rigid body dynamics question

• alextex
In summary, the conversation discusses the calculation of torque on a rigid body using the formula t = r X f, where r is in the body-fixed system and there is uncertainty about which coordinate system f should be expressed in. It is later clarified that all calculations should be done in the same coordinate system. The conversation then shifts to discussing the formula for torque on a rigid body expressed in the body-fixed frame, which is given by t = -0.5*S(q)*(dU/dq), where U is potential, q is a quaternion, and S(q) is a matrix. The formula is derived using the relationship dq/dt = 0.5*S(q)w, where w is the angular velocity.

#### alextex

Hi!
I have been working on a rigid body subject quite a long. But till now there is an unresolved question for me. When we calculating the torque acting on the rigid body we use the following definition of the torque: t = r X f, X - is a cross product. So if I calculate the torque in a body-fixed system I use r in the body-fixed system too, but for force f - I am not sure. In which coordinate system should it be expressed - in the body-fixed or in space-fixed. Are this forces different in both systems? Sorry, for probably stupid questions. Thanks

Sorry, I've found - we really need to do all in the same coordinate system.

But now, I have some other question - probably, more interesting. The torque acting on the rigid body (expressed in the body-fixed frame) is given by: t = -0.5*S(q)*(dU/dq), gde U - potential, q - is a quaternion, ans S(q) - is a matrix such that: dq/dt = 0.5*S(q)w, where w - is a 4-dimensional angular velocity. Can anybody explain how to get that formula for the torque?

Hello there!

First of all, there is no such thing as a "stupid question" in science. It's important to ask questions and seek clarification in order to fully understand a concept.

To answer your question, the force should be expressed in the same coordinate system as the position vector r. This is because the torque is a vector that is perpendicular to both the position vector and the force vector. Therefore, in order to accurately calculate the torque, both vectors should be in the same coordinate system.

In a body-fixed system, the force and position vectors will have different components than in a space-fixed system. However, the physical force acting on the body will be the same regardless of the coordinate system used to express it. This is because forces are independent of the reference frame in which they are observed.

I hope this helps to clarify your understanding of torque in rigid body dynamics. Keep asking questions and exploring the subject - that's what being a scientist is all about!

## 1. What is a rigid body in dynamics?

A rigid body is a theoretical concept in mechanics that refers to an object that maintains its shape and size, even when subjected to external forces. This means that the distances between all of its particles remain constant.

## 2. How is the motion of a rigid body described?

The motion of a rigid body is described using three main parameters: translation, rotation, and inertia. Translation refers to the movement of the body as a whole, rotation refers to the movement of the body about a fixed axis, and inertia refers to the resistance of the body to changes in its motion.

## 3. What is the difference between a rigid body and a deformable body?

A rigid body maintains its shape and size, while a deformable body can undergo changes in shape and size when subjected to external forces. This means that the distances between particles in a deformable body can change, unlike in a rigid body.

## 4. How are forces and moments applied to a rigid body?

Forces and moments can be applied to a rigid body at any point along its surface. A force is a push or pull applied to the body, while a moment is a rotational force applied about a fixed axis. These external forces and moments can cause changes in the motion of the body.

## 5. How is the motion of a rigid body calculated?

The motion of a rigid body is calculated using Newton's laws of motion and the principles of dynamics. This involves analyzing the external forces and moments acting on the body and using equations to calculate its acceleration, velocity, and position over time.