Velocity from position vector in rotating object

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SUMMARY

The discussion focuses on calculating the change in velocity for the center of a rim concerning the contact patch of a tire with camber. The key equation referenced is equation 2.6, which derives from the position vector in equation 2.5. The confusion arises around the terms dj/dt = -wz i and dk/dt = -γ' / cos² γ j, particularly the use of the i unit vector and the derivation of k as -tanγ. Clarification on these mathematical derivations is sought to enhance understanding.

PREREQUISITES
  • Understanding of rotational dynamics and camber effects in tire mechanics
  • Familiarity with vector calculus and unit vector notation
  • Knowledge of angular velocity and its relation to linear motion
  • Ability to interpret and derive equations from physical models
NEXT STEPS
  • Study the derivation of position vectors in rotational systems
  • Learn about the effects of camber on tire performance and dynamics
  • Explore angular velocity and its impact on linear velocity in rotating objects
  • Review vector calculus, focusing on unit vectors and their applications in physics
USEFUL FOR

Students in mechanical engineering, automotive engineers, and anyone studying dynamics of rotating objects and tire mechanics.

Nikstykal
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Q1.PNG

Homework Statement


I am trying to solve for change in velocity for the center of a rim with respect to the contact patch of a tire that has some degree of camber. The equation finalized is shown in the image below, equation 2.6.
http://imgur.com/a/oHucp

Homework Equations

The Attempt at a Solution


I understand how to get the position vector shown in 2.5. The first part of 2.6 is just deriving 2.5 with respect to h. The 3rd and 4th terms are what confuse me. In regards to dj/dt = -wz i, I understand that the change in j with respect to time is directly related to the yaw moment (wz) but what is the mathematical reasoning for using the i unit vector? Further, the 4th term demonstrates that dk/dt = -γ' / cos2 γ j, showing that k = -tanγ.
Sorry for the improper notation, was hoping to get further insight into how these terms are being derived.
 
Last edited:
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I can't see your figure, and without that, there is no good way to reply to you. Please get the figure into the post.
 
Dr.D said:
I can't see your figure, and without that, there is no good way to reply to you. Please get the figure into the post.
Sorry about that, should be fixed.
 

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