I Integrating function of force with respect to angle

1. Apr 3, 2017

Brian

I have a function of force with respect to a certain angle, θ. I also have the angle of the force, Ψ, with respect to the same angle. Plug in θ, get the force and Ψ. This function is determined by a table of data: F(θ) and Ψ(θ)

This force is acted upon a beam at a distance L from a pivot at one end. The beam has a skew angle of Φ. If I find the force that is normal to the skew angle, Φ, the resultant force is: Fr(θ) = F(θ) ⋅ cos(Ψ(θ)) ⋅ sin(Φ). Note: The force always points vertically (see top view below) The angle Φ changes as a result of the force on the beam.

I need to find the angular displacement of Φ as a function of θ.

Using Newtons second law, I know that Fr ⋅ L = I ⋅ α
If mass moment of intertia is constant, Fr(θ) ⋅L = I ⋅ α(θ)
Thus α(θ) = (Fr(θ) ⋅L) / I

Now my next thought was to integrate α(θ) with respect to the angle θ to find ω(θ), as I know my initial ω is zero. Once I find ω(θ), I can integrate again with respect to θ to find Φ(θ), as I know my initial Φ and that the rotation only takes place around a single axis, the pivot point.

First, I have doubts that this is the correct way to find Φ(θ). I'm unsure whether the pivot can be considered a single axis of rotation.

Second, I am having trouble integrating:
(F(θ) ⋅ cos(Ψ(θ)) ⋅ sin(Φ)⋅L)/I.
with respect to θ, as F(θ) and Ψ(θ) are tables of values, not equations. I could use the trapezoidal method to find the integrals of both tables, call them F'(θ) and Ψ'(Φ), but how would these fit into integrating the equation above?

Finally, I realize that I will have Φ on both sides of the equation, and thus Φ will be a function of itself. This reflects on my assumption that the pivot is a "single" axis.

Below is a rough sketch of the "beam" to which the force is being applied.
http://imgur.com/aKniN5Z
http://imgur.com/a/hsBoT (if embed isnt working)
The angle θ, for all intents and purposes, could be thought of as time, as it is the independent variable. I could convert this angle into time (as it is determined by a different rod rotating at a known angular speed) but I need to solve for the skew angle as a function of the given angle and the functions F(θ) and Ψ(θ) are different during different angular speeds.

If integrating the angular velocity is incorrect, could I integrate the regular velocity to find the displacement of the point at which the force is contacting the beam, and then use trig to find the resultant change in angle?

Major Edit: Was using F = ma instead of τ = Iα

Last edited: Apr 3, 2017
2. Apr 8, 2017

PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.