An electrical analog circuit that is equivalent to damping driven harmonic oscil

Question

An electrical analog circuit that is equivalent to damping driven harmonic oscillator is the series LCD circuit,i.e an AC voltage source is connected to a resistor, inductor an capacitor. The sum of the voltages around the closed loop is zero, hence V - iR - L *di/dt - q/C = 0. Express this equation as 2nd order inhomogeneous ODE with driving term V =Vnot Cos omega t. Identify the term analogous to spring force and the damping force. What is the natural frequency, omga not ? What is the damping factor, gamma ? What are possible solution for q(t)?

Explanation / Answer

we have V-iR-L*i'-q/C=0. Let (q-VC)=Q We have. Q'*R+Q''*L+Q'/C=0 where Q'=i=q' so we have. Q/C+Q'*R+Q''*L=0. Q/(LC)+Q'*R/L+Q''=0. we have this equation is similar to x*k + k*x'+x''*m=0. so gamma = R/2. and spring force is Q/C and damping force similar to Q'*R. --------- Natural frequency. w=sqrt(1/LC) ------- with the equation. Q*w^2+2*gamma*Q'+Q''=0. we have 3 possible solution for overdamped, critical damped and under damped

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