Composition of 2 simple harmonic motions with different angular veloci

[TheDoctor46]

Homework Statement

I have 2 simple harmonic motions and I want to compose them on the same axis. So:

x1 = A1*sin(ω1*t+θ1)
x2 = A2*sin(ω2*t+θ2)

The goal is to find the resultant motion of these 2 in the form:
X = A*sin(ω*t+Θ), so to find A,ω and Θ as functions of A1,A2,ω1,ω2,θ1 and θ2.

Homework Equations

I believe there is an analytical way and a graphical way(Fresnel graph) to solve this .



Thanks,
Radu

 

[dauto]

Use the trigonometric formulas for the addition and/or subtraction of two sines.

 

[Simon Bridge]

What do you understand by "composition"?
http://www.mathsisfun.com/sets/functions-composition.html

Do you mean you want to find X(t)=x1(x2(t))?

 

[dauto]

What do you understand by "composition"?
http://www.mathsisfun.com/sets/functions-composition.html

Do you mean you want to find X(t)=x1(x2(t))?

They probably mean superposition X(t)=x1(t) + x2(t)

 

[TheDoctor46]

The idea is to find the resultant motion of these two harmonic motions.

 

[dauto]

Have you tried my suggestion on post #2?

 

[TheDoctor46]

The thing is that x1+x2 is not a solution for the differential equation, because the two motions do not have the same pulsation(angular velocity).

 

[Simon Bridge]

The idea is to find the resultant motion of these two harmonic motions.

There are many ways that two harmonic motions can be combined.
You used the work "composition" to describe how they are combined.

How are they to be combined?

The thing is that x1+x2 is not a solution for the differential equation, because the two motions do not have the same pulsation(angular velocity).

In English ##\omega## is "angular velocity" or "angular frequency".
This is the first time you have mentioned a differential equation.
Please provide all information that is important to the problem.

Note: if x1 and x2 are solutions to the same linear DE, then x1+x2 is also a solution.

Please understand:
(1) If you leave out important information, we cannot help you.
There is no way to tell of a suggestion is any good if we don't know what conditions the solutions have to satisfy.

(2) If you do not answer questions, we cannot help you.
The questions we ask you are not arbitrary or rhetorical - the answers to the questions help us to know how best to help you.

 

[gneill]

If the two harmonic motions are to be combined additively and they do not have the same angular frequency then the resultant cannot be represented by a single sinusoid. Consider the Fourier transform of the combined (added) functions; both frequencies will be represented in the power spectrum as separate components. As a consequence the inverse transform must contain terms for each, too.

Things might be more interesting if the motions were directed along different axes. Then the combined motion might be represented by a 2D Lissajous figure.

 

[dauto]

The thing is that x1+x2 is not a solution for the differential equation, because the two motions do not have the same pulsation(angular velocity).

You will have to allow the amplitude of the combined motion to be a function of time. Have you tried my suggestion from post #2?