Show me a derivation of the BKE

  • Thread starter abercrombiems02
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    Derivation
In summary, The BKE, or Basic Kinematic Equation, is a formula used in advanced dynamics to describe the relationship between the inertial frame (represented by 'e') and the working frame (represented by 'u'). This equation includes the angular velocity (represented by 'omega') and the cross product operator (represented by 'X'). In some cases, a linear velocity term may also be included, but for simplicity, it is assumed to be zero. While the solution to this equation for a simple single rotation problem may seem obvious, a formal proof exists and can be found in old dynamics class notes or through a geometrical approach. However, it may take some time to write out the proof or scan in the notes for verification.
  • #1
abercrombiems02
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For any of you advanced dynamics people, can you please show me a derivation of the BKE. (Basic Kinematic Equation)

The BKE is...

e d/dt(Vector) = u d/dt(Vector) + omega between e and u X (Vector)

sometimes an additional term is added which is the linear velocity between the frames but for some simplicity let's assume this is zero.

e is the inertial frame
and u is the working frame

omega is the angular velocity
X is the cross product operator

I know just by looking at a simple single rotation problem, the solution is trivial but does anyone know of a formal proof?
 
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  • #2
I do, but I think it will take me a month of Sundays to write here. Let me see if I can scan in my old dynamics notes from school. Give me a bit!
 
  • #3
Thanks, take your time, I'm in no rush!
 
  • #4
Here we go. I had to scan it in from my Dynamics class notes. The pdf is a bit rough on the quality side. If you can't read it, let me know and I'll e-mail you the scan.
 
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  • #5
thanks alot! This works perfectly. I actually found a more geometrical way to prove it with the few extra hours i had to spare today. Unfortunately, I don't have a scanner to show anyone to ensure correctness. Perhaps when I get back to school I can scan it in and have someone verify that my method is correct. Thanks again!
 

1. What is the BKE equation?

The BKE, or Boltzmann kinetic equation, is a mathematical equation that describes the dynamics of a system of particles in terms of their microscopic behavior. It is often used in statistical mechanics to analyze the behavior of gases, plasmas, and other systems of particles.

2. What does the BKE equation represent?

The BKE equation represents the relationship between the distribution function of a system, which describes the probability of finding particles with certain properties at a given time, and the forces acting on those particles. It provides a way to calculate the evolution of the system over time.

3. How is the BKE derived?

The BKE is derived from the Boltzmann transport equation, which describes the evolution of the distribution function over time. The BKE is obtained by making certain assumptions and simplifications, such as assuming the particles are point-like and neglecting their interactions with each other.

4. What are the main applications of the BKE equation?

The BKE equation is used in many fields of physics, including gas dynamics, plasma physics, and astrophysics. It is also used in engineering applications, such as designing gas turbines and analyzing the behavior of rarefied gases in vacuum systems.

5. Are there any limitations to the BKE equation?

While the BKE is a powerful tool for analyzing the behavior of systems of particles, it has some limitations. It assumes that the particles are in thermal equilibrium and that their interactions can be described by simple collisions. In reality, many systems do not meet these assumptions, so the BKE may not accurately describe their behavior.

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