Chapter 3.5, Problem 6E

Interpretation Introduction

Interpretation:

Solve the equation analytically for all $\text{ε > 0}$. Show there are two widely separate time scales for $\text{ ε}\ll 1$ and estimate that the time scales in terms of $\text{ ε}$. Plot a graph of the solution $\text{ x (t) for ε}\ll 1$ and show different time scales on it. Give electrical and mechanical analog for the system.

Concept Introduction:

For some limiting condition, e.g. limit of strong damping, the term contains highest order derivative is neglected from the system equation, such limit is called as singular limit.

For second order differential equation, the solution is a linear combination of two roots, so its trajectory is traced by two parts i.e. it consists of two different time scales; one is initially drifting rapidly and gets slower exponentially and one time scale drifts slowly(no rapid transient).

Solution of the second order homogeneous differential equations is

If roots are imaginary ${\text{x (t) =A e}}^{{\text{λ}}_{\text{+}}\text{t}}{\text{ + B e}}^{{\text{λ}}_{\text{-}}\text{t}}{\text{ = Ae}}^{\text{λt}}{\text{cosμt + Be}}^{\text{λt}}\text{sinμt}$

If roots are real and distinct ${\text{x (t) = c}}_{\text{1}}{\text{e}}^{{\text{λ}}_{\text{1}}\text{x}}{\text{ + c}}_{\text{2}}{\text{e}}^{{\text{λ}}_{\text{2}}\text{x}}$

If roots are real and identical ${\text{x(t) = c}}_{\text{1}}{\text{xe}}^{\text{-λx}}{\text{+ c}}_{\text{2}}{\text{e}}^{\text{-λx}}$

Source free series RLC circuit: Resistor, Inductor, and Capacitor are connected in the series without any source, current in a circuit flows due to the initial stored energy in capacitor and inductor.

Damped harmonic oscillator: A spring-mass system dipped in to a viscous fluid is mechanical damped harmonic oscillator. Viscous fluid exerts damping force on the spring-mass harmonic oscillator.