# Rigid body mechanics and coordinate frames

• I
• AbsoluteUnit
In summary, AUI figured out that you need to include the inertial force (a.k.a fictitious force) when working in a non-inertial frame, so the equations of motion in the B-frame would be:aBx = (1/m)*FBx + ψ' vBy
AbsoluteUnit
Hello all,

I have some issues understanding the inertial-frame (or global-frame, G-frame) versus the body-frame (B-frame) when it comes to simulating the motion of a rigid body in 2 dimensions (planar body mechanics) in a system of ODEs. I have been self-learning from textbooks on simulating rigid body motion, so I sometimes misunderstand concepts, since some books are a bit vague when they assume the reader already have the technical knowledge to make the necessary connections. I hope someone can clear up any misunderstandings that I have here.

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Given a state space, Q and its gradient Q' describing the planar mechanics of a rigid body:

Q = [ VBx(t), VBy(t), ψ(t) ]T ... (1)
Q' = [ aBx(t), aBx(t), ψ'(t) ]T ... (2)

Where
Bx := Quantity expressed in body-frame (B-frame) coordinates​
V := Velocity​
a(t) := Acceleration​
ψ, ψ'(t) := Body angular velocity and acceleration (respectively)​

The values of a(t) and ψ'(t) are being generated by some process in the B-frame at each time point (t) as forces and moments, e.g. vehicle accelerating/braking and steering.

Essentially,
• Some books (seem to) suggest that it is possible to perform integration directly in the B-frame on Q'(t) (ref. (2)) to solve for Q(t) (ref. (2)) at every time point, t. Then, the G-frame positions of the rigid body, (X,Y,ψ) can be obtained by rotating Q to the G-frame coordinates via the transformation:
VGx = VBx*cos(ψ) - VBy*sin(ψ)
VGy = VBx*sin(ψ) + VBy*cos(ψ)​
Then integrate VGx, VGy to get (X,Y) in G-frame.​
In short:
Fully solve the ODE in the B-frame first, then rotate the velocities to the G-frame and integrate to obtain the G-frame positions: (X,Y,ψ)​

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Part of me thinks that this interpretation of mine doesn't really make sense, because Newton's First Law only applies directly in the G-frame, but not in the B-frame.

I suspect that for every time point, t, one should instead:
1. Compute the forces and moments that give rise to aBx(t), aBx(t), and ψ'(t)
2. Transform (aBx(t), aBx(t), ψ'(t)) to G-frame coordinates
3. Solve for VGx, VGy, ψ'(t)
So that at every time point, t, the forces are applied in the Inertial frame (instead of the body frame).

Is that the correct way to think about this? Or is there a technicality that I am not getting here and both methods are actually equivalent?

AU

I figured it out, I forgot about the inertial force (a.k.a fictitious force): ψ' vBy, ψ' vBx, that you need to include when working in a non-inertial frame, so actually:

FBx = mB(aBx - ψ' vBy)
FBy = mB(aBy + ψ' vBx)
So it is possible to write the equation of motion in the B-frame as:

aBx = (1/m)*FBx + ψ' vBy
aBy = (1/m)*FBy - ψ' vBx
Then it would be possible to solve the ODE in the B-frame. I also realize the original question was impossible to diagnose since I did not include expressions for FBx and FBy. That was my bad.

Derp-derp-derp.

Anyway cheers.

vanhees71 said:
Thanks! I skimmed through and it looks like really good reading material, I'll have to set aside some time to look into it since I need to improve my understanding of the subject.

In the meantime, between my last post and now I went ahead and did some simulation experiments with both scenarios and this is what I think is going on (also wanted to make a correction to my 2nd post).
------------

[1] The B-frame equations of motion in the form:
aBx = (1/m)*FBx + ψ' vBy
aBy = (1/m)*FBy - ψ' vBx
I think this describes the accelerations from the perspective of an observer in the B-frame.

[2] The first set of equations from my 2nd post:
FBx = m(aBx - ψ' vBy)
FBy = m(aBy + ψ' vBx)
Describes the forces acting on the rigid-body, but rotated to the B-frame (fixed to the rigid-body long/lat axes).

[3] The issue with integrating wrt time in the equations of motion in the B-frame is that the B-frame rotations are not static, so the infinitesimal quantities added to to (X,Y) at every time point, t, are not necessarily spatially coherent with (X,Y) from previous time points (i.e. (X(t),Y(t)) are oriented differently from (X(t-1),Y(t-1)).

Hence, if the forces acting on the rigid-body are computed in the B-frame, it has to be first rotated into an inertial frame before the integration step such that (X,Y) are coherent between each time step.

I suspect that it is possible to solve the ODE in the B-frame, but that would require the additional effort of tracking relative changes in the rigid-body's orientation between t and (t-1)... Which is probably a problem that is relevant to air and marine navigation (sans GPS), but a bit off-topic to what I am doing. Still an interesting problem nonetheless.

Thanks and best,
AU

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

A rigid body is a theoretical object that is considered to be perfectly rigid, meaning that it does not deform or change shape under the influence of external forces. It is often used in mechanics to simplify the analysis of motion and forces in a system.

## 2. How are coordinate frames used in rigid body mechanics?

Coordinate frames are used in rigid body mechanics to define the position and orientation of a rigid body in space. They provide a reference point for measuring the motion and forces acting on the body, and can be used to track the body's movement and calculate its velocity and acceleration.

## 3. What is the difference between a fixed and a moving coordinate frame?

A fixed coordinate frame is stationary and does not move with the rigid body, while a moving coordinate frame is attached to the body and moves with it. In rigid body mechanics, it is common to use both fixed and moving frames to analyze the motion and forces of a system.

## 4. How are forces and moments represented in rigid body mechanics?

Forces and moments are represented as vectors in rigid body mechanics. Forces are typically represented by arrows pointing in the direction of the force, while moments are represented by a vector perpendicular to the plane of rotation and pointing in the direction of rotation.

## 5. What is the principle of rigid body mechanics?

The principle of rigid body mechanics states that the net force and net moment acting on a rigid body must be equal to the body's mass times its acceleration and the body's moment of inertia times its angular acceleration, respectively. This principle is used to analyze the motion and forces of a rigid body in a system.

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