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Unit 8: Mechanical and Electrical Vibrations

A mass on a spring, damped, forced, and pushed to resonance — the three root cases from Unit 6 made physical, plus the RLC circuit that obeys the same equation.

10 topics · 36 questions

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3.7 Setting Up the Spring-Mass Equation

  1. How it worksHow do Hooke's law and Newton's second law combine into m x'' + gamma x' + k x = F(t)?
  2. Worked exampleBuild the differential equation for a 2 kg mass on a spring with spring constant 8 N/m and no damping or forcing.
  3. Going furtherWhat happens to the spring-mass equation once the mass hangs so that gravity also stretches the spring?
  4. Common mistakeWhy doesn't a stiffer spring, meaning a bigger spring constant k, automatically mean slower oscillation?
Your turn1 of 2
Build the differential equation for a 4 kg mass on a spring with spring constant 15 N/m and no damping or forcing: m x'' + kx = 0. Find the natural frequency omega0 = √(k/m).

3.7 Free Undamped Vibrations

  1. How it worksHow does the natural frequency omega0 = √(k/m) fall straight out of the spring-mass equation's characteristic roots?
  2. Worked exampleFind the natural frequency and period of a 4 kg mass on a spring with k = 16 N/m.
  3. Common mistakeWhy doesn't pulling the mass back farther before releasing it change how fast it oscillates?
Your turn1 of 2
Find the period T = 2·π/omega0 of a 4 kg mass on a spring with k = 18 N/m.

3.7 Solving a Free Undamped Vibration Problem

  1. Worked exampleSolve x'' + 9x = 0 for a mass released from x(0) = 2 with zero initial velocity, and write the answer as R cos(omega t - delta).
  2. Worked exampleFind the amplitude R and phase angle delta for x(0) = 1, x'(0) = 3 on x'' + 4x = 0.
  3. Going furtherWhat happens to the amplitude R when the same mass is released from rest but from twice the initial displacement?
Your turn1 of 2
Find the amplitude R = √(x02 + (v0/omega)2) for x'' + 42 x = 0 with x(0) = 3 and x'(0) = 6.

3.8 The Three Cases of Free Damped Vibration

  1. How it worksHow does the sign of gamma2 - 4mk split damped vibration into three completely different behaviors?
  2. Worked exampleFind whether x'' + 5x' + 4x = 0, x'' + 4x' + 4x = 0, and x'' + x' + 4x = 0 are overdamped, critically damped, or underdamped.
  3. Going furtherWhat happens to a mass's motion as the damping coefficient gamma is slowly increased from zero past the critical value?
  4. Ties togetherHow do the three damping cases here connect to the same three root cases from a constant-coefficient characteristic equation?
Your turn1 of 2
For m x'' + 7x' + 5x = 0 with m = 2, find the discriminant gamma2 - 4mk.

3.8 Underdamped Motion

  1. How it worksHow does an exponentially decaying envelope multiplying a cosine produce underdamped motion's shrinking oscillations?
  2. Worked exampleSolve x'' + 2x' + 5x = 0 with x(0) = 1, x'(0) = 0, and identify the decay rate and quasi-frequency.
  3. Common mistakeWhy isn't the quasi-period of underdamped motion the same as the natural period with no damping at all?
  4. Going furtherWhat happens to underdamped motion's quasi-frequency as the damping coefficient climbs closer to the critical value?
Your turn1 of 2
Solve x'' + (2·2)x' + (22+42)x = 0 with x(0) = 4 and x'(0) = 0. (Its characteristic roots are -2 plus or minus 4i.) Find x(t).

3.8 Critically Damped and Overdamped Motion

  1. How it worksHow does a repeated real root produce the fastest possible non-oscillating return in critically damped motion?
  2. Worked exampleSolve x'' + 4x' + 4x = 0 with x(0) = 1, x'(0) = 1, and describe the motion.
  3. Common mistakeWhy isn't critically damped motion just an overdamped motion that happens to be a little bit faster?
Your turn1 of 2
Solve x'' + (3·2)x' + 32 x = 0 with x(0) = 2 and x'(0) = 4, given that -3 is a repeated root. Find x(t).

3.9 Forced Vibrations Without Damping and Beats

  1. How it worksHow does adding two cosines with close but different frequencies produce the slow beat envelope on top of the fast oscillation?
  2. Worked exampleSolve x'' + 25x = 4 cos(4t) with x(0) = 0, x'(0) = 0, and identify the beat frequency.
  3. Going furtherWhat happens to the beat pattern as the forcing frequency is dialed closer and closer to the natural frequency?
Your turn1 of 2
For x'' + 5.2272 x = 8cos(4.084t) with x(0)=0, x'(0)=0 (w is not equal to w0), find the beat frequency (w0-w)/2.

3.9 Resonance in Forced, Damped Vibrations

  1. How it worksHow does the steady-state amplitude formula spike as the forcing frequency approaches the system's natural frequency?
  2. Worked exampleFind the forcing frequency that maximizes the steady-state amplitude of x'' + 2x' + 25x = F0 cos(omega t).
  3. Common mistakeWhy doesn't resonance require zero damping to make a system's amplitude climb dramatically?
  4. Going furtherWhat happens to the resonance peak's height and sharpness as the damping coefficient shrinks toward zero?
  5. Ties togetherHow does mechanical resonance here connect to the amplitude blow-up an RLC circuit shows at its resonant frequency?
Your turn1 of 2
For x'' + 3x' + 23x = F0 cos(omega t), find the forcing frequency omegar = √(k - gamma2/2) that maximizes the steady-state amplitude.

3.7 The RLC Circuit as a Mechanical Analogy

  1. How it worksHow do inductance, resistance, and inverse capacitance play the roles of mass, damping, and spring stiffness?
  2. Worked exampleFind the differential equation for charge in an RLC circuit with L = 1 henry, R = 4 ohms, and C = 0.25 farad, with no applied voltage.
  3. Going furtherWhat happens to the circuit's behavior when the resistance R is increased past the critical damping value?
  4. Worked exampleFind whether an RLC circuit with L = 1 henry, R = 6 ohms, and C = 0.25 farad is overdamped, critically damped, or underdamped.
Your turn1 of 2
For an RLC circuit with L = 2 henry, R = 6 ohms, and C = 0.473 farad, find the discriminant R2 - 4L/C that determines its damping case.

3.8 Energy in a Vibrating System

  1. How it worksHow does damping steadily drain energy out of a vibrating system until it settles at equilibrium?
  2. Worked exampleFind the total energy of a 1 kg mass on a spring with k = 9 N/m at x(0) = 1, x'(0) = 0, assuming no damping.
  3. Going furtherWhat happens to the energy-versus-time graph of a damped oscillator compared to an undamped one?
Your turn1 of 2
Find the total energy E = (1/2)mv2 + (1/2)kx2 of a 2 kg mass on a spring with k = 11 N/m, at x(0) = 3, x'(0) = 1, assuming no damping.