Integrating function of force with respect to angle

[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: F_r(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 F_r ⋅ L = I ⋅ α
If mass moment of intertia is constant, F_r(θ) ⋅L = I ⋅ α(θ)
Thus α(θ) = (F_r(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 isn't 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

 

[eljose79]

Thank you for sharing your problem with us. I understand your concerns and I would be happy to provide some guidance.

Firstly, your approach to finding the angular displacement of Φ as a function of θ seems reasonable. However, I would like to clarify a few points.

Regarding your doubts about whether the pivot can be considered a single axis of rotation, the answer is yes. In the context of this problem, the pivot is the fixed point around which the beam rotates, and it can be considered as a single axis of rotation.

Next, let's focus on your trouble with integrating (F(θ) ⋅ cos(Ψ(θ)) ⋅ sin(Φ)⋅L)/I with respect to θ. As you have correctly stated, F(θ) and Ψ(θ) are tables of values, not equations. In this case, you can use numerical integration methods such as the trapezoidal method to approximate the integrals. This will give you F'(θ) and Ψ'(θ) as functions of θ, which can then be used in the integration process.

As for your concern about having Φ on both sides of the equation, this is a common issue in mathematics and can be solved by using numerical methods or by using a solver to find the roots of the equation. This will give you the value of Φ as a function of θ.

Finally, it is important to note that you should use the equation τ = Iα instead of F = ma in this problem, as you are dealing with rotational motion. This will give you the correct relationship between the force and the angular acceleration.

In conclusion, your approach seems valid, but it may require some numerical methods to solve the integrals and equations involved. I hope this helps and I wish you the best of luck in solving your problem.