Forced oscillations and ressonance

[williamsal]

Hi friends, I will be right to the point.

On the book "Mechanics" by Landau & Lifgarbagez, chapter "Small Oscillations", section "Forced Oscillations":
1. What is the meaning of the term beta (phase constant) on the expression for the driven force, F(t) = f cos(gamma t + beta), how it relates to the initial configuration of the system? And why almost all other textbooks just write F(t) = f cos(gamma t) without the phase constant, in this case, "beta"?

2. How can we get, step by step, the expression for the linear dependence of the amplitude in the ressonance case as described by the equation 22.5. I read dozens of texts and books like Goldstein, Symon, Marion, etc but none could give a clear guidance on how to get this expression.


Thans
Williams

 

[torquil]

Hi friends, I will be right to the point.

On the book "Mechanics" by Landau & Lifgarbagez, chapter "Small Oscillations", section "Forced Oscillations":
1. What is the meaning of the term beta (phase constant) on the expression for the driven force, F(t) = f cos(gamma t + beta), how it relates to the initial configuration of the system? And why almost all other textbooks just write F(t) = f cos(gamma t) without the phase constant, in this case, "beta"?

2. How can we get, step by step, the expression for the linear dependence of the amplitude in the ressonance case as described by the equation 22.5. I read dozens of texts and books like Goldstein, Symon, Marion, etc but none could give a clear guidance on how to get this expression.

1) Admitting an arbitrary beta allows one to develop forumlas valid in cases for which the force at t=0 does not vanish. If others assume beta=0 they have probably just done it for purposes of simplification.

2) I'm not going to do it here, but it is simply a case of using L'Hopitals rule for determining the value of a limit that approaches 0/0. What they are doing is to determine the expression 22.4 in the limit where gamma -> omega. The limit can be obtained by differentiating with respect to gamma independently in the numerator and denominator, and then take the limit gamma -> omega. For more info, look up L'Hopitals rule e..g on Wikipedia.