Solution Found!
You can make the Fourier series solution for a
Chapter 5, Problem 5.51(choose chapter or problem)
You can make the Fourier series solution for a periodically driven oscillator a bit tidier if you don't mind using complex numbers. Obviously the periodic force of Equation (5.90) can be written as f = Re(g), where the complex function g is cx) g(t) = fneinwt. n=0 Show that the real solution for the oscillator's motion can likewise be written as x = Re(z), where z(t),Ecneinwt n=0 and cn = fn (02 n2 co2 + 2i finco 0 This solution avoids our having to worry about the real amplitude An and phase shift 5, separately. (Of course An and 8n are hidden inside the complex number Ca.)
Questions & Answers
QUESTION:
You can make the Fourier series solution for a periodically driven oscillator a bit tidier if you don't mind using complex numbers. Obviously the periodic force of Equation (5.90) can be written as f = Re(g), where the complex function g is cx) g(t) = fneinwt. n=0 Show that the real solution for the oscillator's motion can likewise be written as x = Re(z), where z(t),Ecneinwt n=0 and cn = fn (02 n2 co2 + 2i finco 0 This solution avoids our having to worry about the real amplitude An and phase shift 5, separately. (Of course An and 8n are hidden inside the complex number Ca.)
ANSWER:Step 1 of 3
The equation of motion for the driven oscillator is;
Here is the position of the oscillator, is the natural frequency, is the damping constant, and is the periodic driving force with period .
The compact form of the equation of motion for the driven oscillator is;
Here, is the linear differential operator.
The linear differential operator is;