Essential Concepts
- Second-order constant-coefficient differential equations can be used to model spring-mass systems.
- An examination of the forces on a spring-mass system results in a differential equation of the form [latex]mx^{\prime\prime}+bx^{\prime}+kx=f(t)[/latex], where [latex]m[/latex] represents the mass, [latex]b[/latex] is the coefficient of the damping force, [latex]k[/latex] is the spring constant, and [latex]f(t)[/latex] represents any net external forces on the system.
- If [latex]b=0[/latex], there is no damping force acting on the system, and simple harmonic motion results. If [latex]b\ne{0}[/latex], the behavior of the system depends on whether [latex]b^{2}-4mk>0[/latex], [latex]b^{2}-4mk=0[/latex], or [latex]b^{2}-4mk<0[/latex].
- If [latex]b^2-4mk>0[/latex], the system is overdamped and does not exhibit oscillatory behavior.
- If [latex]b^2-4mk=0[/latex], the system is critically damped. It does not exhibit oscillatory behavior, but any slight reduction in the damping would result in oscillatory behavior.
- If [latex]b^2-4mk<0[/latex], the system is underdamped. It exhibits oscillatory behavior, but the amplitude of the oscillations decreases over time.
- If [latex]f(t)\ne{0}[/latex], the solution to the differential equation is the sum of a transient solution and a steady-state solution. The steady-state solution governs the long-term behavior of the system.
- The charge on the capacitor in an [latex]RLC[/latex] series circuit can also be modeled with a second-order constant-coefficient differential equation of the form [latex]L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)[/latex], where [latex]L[/latex] is the inductance, [latex]R[/latex] is the resistance, [latex]C[/latex] is the capacitance, and [latex]E(t)[/latex] is the voltage source.
Key Equations
- Equation of simple harmonic motion
[latex]x^{\prime\prime}+\omega^{2}x=0[/latex] - Solution for simple harmonic motion
[latex]x(t)=c_{1}\cos{(\omega{t})}+c_{2}\sin{(\omega{t})}[/latex] - Alternative form of solution for SHM
[latex]x(t)=A\sin{(\omega{t}+\phi)}[/latex] - Forced harmonic motion
[latex]mx^{\prime\prime}+bx^{\prime}+kx=f(t)[/latex]
- Charge in a RLC series circuit
[latex]L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)[/latex]
Glossary
- RLC series circuit
- a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an RLC series circuit
- simple harmonic motion
- motion described by the equation [latex]x(t)=c_{1}\cos{(\omega{t})}+c_{2}\sin{(\omega{t})}[/latex] as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely
- steady-state solution
- a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution
