Direction cosine matrix of rolling disk on circular ring

In summary, the conversation discusses finding the direction cosine matrix [BN] in terms of angle phi, given a vector in the B frame and finding the corresponding vector in the N frame. The relevant equations include the direction cosine matrix of a homogeneous system and the use of Euler Angles. The conversation also mentions the use of matrices B10, B21, B20, T10, T21, and T20 to represent the movement of three rigid solids or reference systems.
  • #1
QuantumLollipop
8
0
Hey all,
I'm stuck on this problem and not sure how to proceed/if I'm in the right direction.

Problem: One reference frame N sits at the origin (inertial frame) while another frame, B, describes a disk rolling on a circular ring about the other frame. Picture below

upload_2015-3-4_16-22-8.png
(A) find the direction cosine matrix [BN] in terms of angle phi
(B) Given a vector in the B frame find the vector in the N frame

Relevant Equations: The direction cosine matrix of a homogeneous system such as this is a 4x4 matrix whose elements are: top left ([BN], rotation matrix as if centered at origin)3x3, top right (a vector from the origin of N frame to origin of B frame)3x1, bottom left (0)1x3, bottom right (1)1x1 = [NB]4x4
upload_2015-3-4_16-30-18.png
Attempt:
The position vector was relatively easy to compute:
r(N)=L*cos(Φ) n1^ + L*sin(Φ) n2^ + r n3^ (I hope that's right)

Next, to complete the full matrix I need to find the direction cosine matrix NB which I believe is computed as if B were sitting at the origin.

I'm having difficulty with this part, I assume the only way to find this is using the 3 consecutive Euler Angles. My trouble is finding which Euler Angles to use for the rotation. Any help would be appreciated. Apologies for the crude equations, I was having troubles with the equation editor and couldn't figure out how to make matrices. Thanks
 
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  • #2
The problem can be modelling as the figure, where there are three rigid solids or reference systems: 0 is the laboratory, 1 is an auxiliary solid which rotates about the vertical axe z with respect to 0, this rotation is modelling as a motor of rotation its axis is fixed to 0 and, finally, 2 is the disk which rotate with respect to 1 . This movement is modelling as a motor which housing is fixed to the before motor housing, and its shaft is fixed to the disk. I'm using the same formulation as used in the the robotic field.
At the initial position, origin of the rotation angles, the three coordinates systems of the three solids are coincidentes.
After this, is easy to find the matrixes Bij,(3x3) base change, as the Tij (4x4) coordinate change matrix

B10=[[Cφ,-Sφ,0],[Sφ,Cφ,0],[0,0,1]]
B21=[[Cθ,0,Sθ],[0,1,0],[-Sθ,0,Cθ]]
B20=B10.B21T10=[[Cφ,-Sφ,0,0],[Sφ,Cφ,0,0],[0,0,1,0],[0,0,0,1]]
T21=[[Cθ,0,Sθ,0],[0,1,0,0],[-Sθ,0,Cθ,0],[0,0,0,1]]
T20=T10.T21
According to your fig.; L.φ=-r.θ

(A) B20
(B) {r}0=B20.{r}2

I hope you can understand my bad english. Sorry.
 

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1. What is the direction cosine matrix of a rolling disk on a circular ring?

The direction cosine matrix of a rolling disk on a circular ring is a 3x3 matrix that represents the orientation of the disk relative to the ring. It is used to describe the rotational motion of the disk on the ring.

2. How is the direction cosine matrix calculated?

The direction cosine matrix is calculated using the orientation of the disk and the ring. This can be done using trigonometric functions such as sine and cosine. The specific equations used will depend on the orientation of the disk and the ring.

3. What is the significance of the direction cosine matrix in this scenario?

The direction cosine matrix is significant because it allows us to describe the orientation and rotation of the disk on the ring in a mathematical way. This can be useful for analyzing the motion of the disk and predicting its future behavior.

4. How does the direction cosine matrix change as the disk rolls on the ring?

As the disk rolls on the ring, the direction cosine matrix will change accordingly to reflect the changing orientation and rotation of the disk. This matrix will be constantly updated as the disk moves, providing a dynamic representation of its motion.

5. Are there any limitations to using the direction cosine matrix for a rolling disk on a circular ring?

Yes, there are some limitations to using the direction cosine matrix for this scenario. It assumes a perfect circular ring and a perfectly rolling disk, which may not be the case in real-world situations. Additionally, external forces such as friction and air resistance may affect the motion of the disk, making the calculations less accurate.

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