7.3 Applications  (Page 9/13)

Key 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
  
  $mx\text{″}+b{x}^{\prime }+kx=f(t),$
  
  where $m$ represents the mass, $b$ is the coefficient of the damping force, $k$ is the spring constant, and $f(t)$ represents any net external forces on the system.
• If $b=0,$ there is no damping force acting on the system, and simple harmonic motion results. If $b\ne 0,$ the behavior of the system depends on whether $b^2-4mk>0,$ $b^2-4mk=0,$ or $b^2-4mk<0.$
• If $b^2-4mk>0,$ the system is overdamped and does not exhibit oscillatory behavior.
• If $b^2-4mk=0,$ the system is critically damped. It does not exhibit oscillatory behavior, but any slight reduction in the damping would result in oscillatory behavior.
• If $b^2-4mk<0,$ the system is underdamped. It exhibits oscillatory behavior, but the amplitude of the oscillations decreases over time.
• If $f(t)\ne 0,$ 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 RLC series circuit can also be modeled with a second-order constant-coefficient differential equation of the form
  
  $L\frac{d^{2}q}{d{t}^2}+R\frac{dq}{dt}+\frac{1}{C}q=E(t),$
  
  where L is the inductance, R is the resistance, C is the capacitance, and $E(t)$ is the voltage source.

Key equations

• Equation of simple harmonic motion
  $x\text{″}+{\omega }^{2}x=0$
• Solution for simple harmonic motion
  $x(t)=c_1\text{cos}\left(\omega t\right)+c_2\text{sin}\left(\omega t\right)$
• Alternative form of solution for SHM
  $x(t)=A\text{sin}\left(\omega t+\varphi \right)$
• Forced harmonic motion
  $mx\text{″}+b{x}^{\prime }+kx=f(t)$
• Charge in a RLC series circuit
  $L\frac{d^{2}q}{d{t}^2}+R\frac{dq}{dt}+\frac{1}{C}q=E(t)$