How is the Formula for Torsion Derived?

In summary, the conversation is about the derivation of the formula for torsion, specifically the formula [tex] \tau = \frac {\left( \begin{array}{ccc} \dot{x} & \ddot{x} & \dddot{x}\\\dot{y}& \ddot{y}& \ddot{y} \\\dot{z} & \ddot{z} & \dddot{z}\end{array} \right)} {|v \times a|^2}. The person asking the question is familiar with another formula for torsion, [tex] \tau = -\frac {dB} {dS} \cdot N [/itex], but does not know how to derive
  • #1
thechunk
11
0
Is anyone familiar with the derivation for this formula for torsion.
[tex]
\tau = \frac {\left( \begin{array}{ccc} \dot{x} & \ddot{x} & \dddot{x}\\\dot{y}& \ddot{y}& \ddot{y} \\\dot{z} & \ddot{z} & \dddot{z}\end{array} \right)} {|v \times a|^2}
[/tex]
I know of expressing torsion as [tex] \tau = -\frac {dB} {dS} \cdot N [/itex], but I do not know how to derive the former. My teacher said it could be derived with knowledge from our multivariable class however my textbook reads that the derivation is found in more advanced texts. The numerator in the first formula looks like [tex] (v \times a) \cdot a' [/itex], but I do not know where to go from there. Any ideas?
 
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  • #2
Anyone know the answer to this question?

I'm totally lost on it.
 
  • #3


The formula for torsion, as you have mentioned, can be expressed as [tex] \tau = -\frac {dB} {dS} \cdot N [/itex], where B is the binormal vector, S is the arc length parameter, and N is the normal vector. This formula is derived using the Frenet-Serret formulas, which describe the behavior of a curve in three-dimensional space.

To derive the formula [tex] \tau = \frac {\left( \begin{array}{ccc} \dot{x} & \ddot{x} & \dddot{x}\\\dot{y}& \ddot{y}& \ddot{y} \\\dot{z} & \ddot{z} & \dddot{z}\end{array} \right)} {|v \times a|^2} [/itex], we can start with the definition of torsion, which is the rate of change of the binormal vector with respect to arc length:

[tex] \tau = \frac {d \vec{B}} {dS} [/itex]

Using the Frenet-Serret formulas, we can express the derivative of the binormal vector as:

[tex] \frac {d \vec{B}} {dS} = \kappa \vec{T} + \tau \vec{N} [/itex]

Where [tex] \kappa [/itex] is the curvature of the curve and [tex] \vec{T} [/itex] is the tangent vector. We can also express the tangent vector and the normal vector in terms of the velocity and acceleration vectors:

[tex] \vec{T} = \frac {\vec{v}} {|v|} [/itex]

[tex] \vec{N} = \frac {\vec{v} \times \vec{a}} {|v \times \vec{a}|} [/itex]

Substituting these expressions into the derivative of the binormal vector, we get:

[tex] \frac {d \vec{B}} {dS} = \kappa \frac {\vec{v}} {|v|} + \tau \frac {\vec{v} \times \vec{a}} {|v \times \vec{a}|} [/itex]

Taking the magnitude of both sides and rearranging, we get:

[tex] \tau
 

Related to How is the Formula for Torsion Derived?

1. What is torsion and why is it important?

Torsion is the twisting or bending of a structural element due to an applied torque. It is important to understand torsion because it can greatly affect the stability and strength of structures, especially in engineering and architecture.

2. What is the equation for torsion?

The equation for torsion is T = k * θ, where T is the torque applied, k is the torsional stiffness of the material, and θ is the angle of twist.

3. How is the torsional stiffness of a material determined?

The torsional stiffness, or shear modulus, of a material is determined by performing torsion tests on samples of the material and measuring the resulting angle of twist under a known applied torque. The ratio of torque to angle of twist gives the torsional stiffness value.

4. What is the difference between pure and warping torsion?

Pure torsion occurs when a structural element is subjected to a torque that is applied along its longitudinal axis, causing it to twist. Warping torsion, on the other hand, occurs when the applied torque causes a change in the cross-sectional shape of the element, resulting in both twisting and bending.

5. How does the distribution of shear stress change in a torsion member?

In a torsion member, the shear stress is maximum at the outer surface and decreases linearly towards the center. This distribution is known as the shear stress distribution curve and is important in determining the maximum shear stress a member can withstand before failure.

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