Foced Vibrations - beats

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In summary, the conversation discusses the equation of motion for a spring-mass system with no damping and a periodic external force. The general solution and a specific solution for when the mass is initially at rest are provided. The conversation then focuses on a step where trigonometric identities are used to simplify the equation. By writing out the equations for A+B and A-B, and using the angle addition formula for cos(A±B), it is shown how the equation simplifies to its final form.
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jellicorse
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I am trying to work though an example on this topic in my book and have reached a point that I am not sure about. I was wondering if anyone could help me clear this up.

The equation of motion for a spring-mass system with no damping and a periodic external force is

[tex]mu'' + ku = Fcos\omega t[/tex]

The general solution to this is:

[tex] u = Acos\omega_0t + B sin\omega_0 t +\frac{F}{m(\omega_0^2+\omega^2)}cos\omega t[/tex]

If the mass is initially at rest, so that u(0)=0 and u'(0)=0, then the solution to this equation is

[tex]u=\frac{F}{m(\omega_0^2-\omega^2)}(cos\omega t -cos\omega_0 t)[/tex]


I have managed to follow it to here but I can not see how they have completed the next step:

"making use of the trigonometric identities for cos(A[itex]\pm[/itex]B) with [itex]A=(\omega_0+\omega)t/2[/itex] and [itex]B=(\omega_0-\omega)t/2[/itex] we can write the equation in the form":

[tex]u=\left[\frac{2F}{m(\omega_0^2-\omega^2)}sin\frac{(\omega_0-\omega)t}{2}\right]sin\frac{(\omega_0+\omega)t}{2}[/tex]

I can not see how the penultimate equation becomes the final equation here; can anyone tell me how this works?
 
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  • #2
Write out A+B and A-B.
Then write out the angle addition formula for cos(A±B).
You'll see how it works.
 
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  • #3
Thanks a lot for the tip, Oliver. I've got it now.
 

1. What are forced vibrations?

Forced vibrations are vibrations that occur in a physical system when it is subjected to an external periodic force. This force can be applied continuously or intermittently and can vary in frequency, amplitude, and direction.

2. How do forced vibrations differ from natural vibrations?

Natural vibrations occur in a physical system without any external force being applied. They are caused by the system's own inherent characteristics, such as its mass, stiffness, and damping. Forced vibrations, on the other hand, are caused by an external force acting on the system.

3. What is the relationship between forced vibrations and beats?

Forced vibrations can result in the phenomenon of beats, which is the periodic variation in amplitude that occurs when two vibrations with slightly different frequencies are superimposed on one another. This creates a pattern of alternating constructive and destructive interference, resulting in the characteristic "beating" sound.

4. How are beats used in music?

Beats are used in music to create a rhythmic effect and to tune instruments. Musicians can adjust the frequency of their instrument to match the frequency of the beat, resulting in a harmonious sound. Beats are also used in music production to create unique and interesting sound effects.

5. How can forced vibrations and beats be controlled or manipulated?

Forced vibrations and beats can be controlled or manipulated by changing the frequency, amplitude, or direction of the external force, or by adjusting the natural frequency of the system. This can be achieved through various methods such as changing the physical properties of the system, adding dampers or springs, or using electronic control systems.

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