Mechanical Vibrations

Everything we have learned about second-order differential equations converges here. The mass-spring-damper system is not merely an academic exercise. It is the foundation for understanding how buildings respond to earthquakes, how car suspensions absorb road bumps, how musical instruments produce sound, and why bridges can catastrophically fail when soldiers march across them in step.

This chapter applies the techniques from previous chapters to a single, profoundly important physical system.

The Mass-Spring-Damper System

Imagine a mass attached to a spring, sliding on a surface with friction. Newton's second law tells us that $F = ma$, where $F$ is the net force, $m$ is the mass, and $a$ is acceleration.

Three forces act on the mass:

Spring force: By Hooke's Law, a spring exerts a restoring force proportional to its displacement from equilibrium: $F_s = -ky$, where $k$ is the spring constant (stiffness) and $y$ is displacement. The negative sign means the force opposes displacement.

Damping force: Friction or viscous drag opposes velocity: $F_d = -cy'$, where $c$ is the damping coefficient. This force resists motion.

External force: There may be an external driving force $F(t)$ applied to the mass.

Applying Newton's second law:

Since acceleration is $y''$, we get:

This is the equation of mechanical vibrations. Every term has physical meaning:

• $my''$ is the inertial term (mass times acceleration)
• $cy'$ is the damping term (resistance to velocity)
• $ky$ is the spring term (restoring force)
• $F(t)$ is the external forcing term

Adjust the mass, damping, and stiffness to see how the system responds. Notice how different parameter combinations produce qualitatively different behaviors. The position-time graph on the right shows the corresponding mathematical solution.

Natural Frequency and Damping Ratio

Two dimensionless quantities characterize the behavior of any mass-spring-damper system.

The natural frequency is:

This is the frequency at which the system would oscillate if there were no damping. It depends only on stiffness and mass: stiffer springs and lighter masses oscillate faster.

The damping ratio is:

This dimensionless number determines whether the system oscillates or not. Its value classifies the behavior into four distinct regimes.

Free Vibration: The Four Cases

When there is no external forcing ($F(t) = 0$), we have free vibration. The homogeneous equation is:

The characteristic equation $mr^2 + cr + k = 0$ has roots:

The discriminant $c^2 - 4mk$ determines everything. In terms of the damping ratio:

This leads to four cases:

Undamped ($\zeta = 0$)

With no damping, the roots are purely imaginary: $r = \pm i\omega_0$. The solution is:

The mass oscillates forever at the natural frequency, never losing energy. This is an idealization. Real systems always have some damping.

Underdamped ($0 < \zeta < 1$)

The roots are complex: $r = -\zeta\omega_0 \pm i\omega_d$, where $\omega_d = \omega_0\sqrt{1 - \zeta^2}$ is the damped natural frequency. The solution is:

The mass oscillates, but each swing is smaller than the last. The exponential envelope $e^{-\zeta\omega_0 t}$ governs the decay. This is the most common behavior in mechanical systems: a tuning fork, a guitar string, a diving board after someone jumps off.

Critically Damped ($\zeta = 1$)

The discriminant is exactly zero, giving a repeated root $r = -\omega_0$. The solution is:

The system returns to equilibrium as fast as possible without oscillating. This is the design goal for many engineering applications. Car shock absorbers are tuned to be near critical damping: you want the car to settle quickly after a bump without bouncing.

Overdamped ($\zeta > 1$)

Two distinct negative real roots give:

where both $r_1$ and $r_2$ are negative. The system returns to equilibrium without oscillating, but more slowly than critical damping. Opening a door with a stiff pneumatic closer demonstrates overdamped motion.

All four systems start from the same initial displacement and zero initial velocity. Watch how they return to equilibrium. The undamped case oscillates forever. The underdamped case oscillates while decaying. The critically damped case returns fastest without overshooting. The overdamped case sluggishly creeps back.

Physical Intuition for Damping

Think of the damping ratio as measuring the competition between two tendencies:

• The spring wants to restore equilibrium, which creates oscillation
• The damper wants to resist motion, which prevents oscillation

When damping is light ($\zeta < 1$), the spring dominates. The system oscillates, though the oscillations decay.

When damping is heavy ($\zeta > 1$), the damper dominates. Motion is so sluggish that the system cannot build up enough momentum to overshoot equilibrium.

At the critical point ($\zeta = 1$), these effects perfectly balance. The system returns to equilibrium in the minimum time without overshooting.

Forced Vibration

Now suppose an external force is applied: $F(t) = F_0\cos(\omega t)$, a periodic force with amplitude $F_0$ and frequency $\omega$. The equation becomes:

This is nonhomogeneous. The general solution is the sum of the homogeneous solution (which decays due to damping) and a particular solution.

The transient response is the homogeneous solution. It depends on initial conditions and decays over time.

The steady-state response is the particular solution. It persists indefinitely and oscillates at the forcing frequency $\omega$:

where the amplitude $A$ and phase $\phi$ depend on the forcing frequency.

The solid curve shows the total response, which starts with transient oscillations that eventually die out. The dashed curve shows the steady-state response that remains. After sufficient time, only the steady state matters.

The Steady-State Amplitude

For forced vibration with $F(t) = F_0\cos(\omega t)$, the steady-state amplitude is:

where $r = \omega/\omega_0$ is the ratio of forcing frequency to natural frequency.

The quantity $F_0/k$ is the static deflection: how much the spring would compress under a constant force $F_0$. The denominator is the dynamic magnification factor. It tells us how much larger the dynamic response is compared to the static response.

When $r$ is small (slow forcing), the denominator approaches 1, and the response is nearly the static deflection. The mass essentially follows the forcing.

When $r$ is large (fast forcing), the denominator grows as $r^2$, and the amplitude decreases. The forcing oscillates so fast that the mass cannot keep up.

The interesting behavior is in between.

Resonance

When the forcing frequency approaches the natural frequency ($r \to 1$), the denominator becomes $2\zeta$. For light damping, this is small, making the amplitude very large:

This is resonance: a small periodic force can produce enormous oscillations when applied at the right frequency.

Slide the forcing frequency toward the natural frequency and watch the amplitude explode. The red coloring indicates dangerous operating conditions.

Resonance is beautiful and terrible. It allows a child on a swing to build up large oscillations with small pushes timed to the swing's natural frequency. It allows opera singers to shatter wine glasses by singing at the glass's resonant frequency. It also caused the Tacoma Narrows Bridge to twist itself apart in 1940 when wind gusts matched a torsional resonance of the structure.

The Frequency Response

Engineers visualize the steady-state behavior using a frequency response plot, which shows amplitude versus forcing frequency for different damping values.

Notice the key features:

• Low-frequency behavior: At low frequencies, the response approaches the static deflection regardless of damping
• Resonant peak: Near $\omega = \omega_0$, lightly damped systems show a sharp peak. Heavily damped systems have a broader, lower peak
• High-frequency rolloff: At high frequencies, all curves decay. The forcing is too fast for the system to respond
• Effect of damping: More damping flattens and broadens the resonant peak, reducing the maximum amplitude

This plot is essential in engineering design. If a machine operates near a resonant frequency, even small periodic forces (from rotating parts, for instance) can cause destructive vibrations.

Real-World Applications

The mass-spring-damper model appears throughout engineering:

Vehicle suspension: A car's shock absorbers are dampers connecting the wheels to the chassis. They should be near critically damped: too little damping causes bouncing; too much makes the ride harsh.

Earthquake engineering: Buildings are large mass-spring-damper systems. Engineers design them so their natural frequencies do not coincide with typical earthquake frequencies. Some buildings have tuned mass dampers: enormous pendulums near the top that absorb oscillation energy.

Musical instruments: A guitar string is an underdamped oscillator. When plucked, it vibrates at its natural frequency (the note) while the amplitude decays. The body amplifies this vibration and shapes the sound.

Seismometers: Instruments that measure ground motion are damped oscillators calibrated to respond to a wide range of frequencies.

Why Damping is Usually Desirable

In free vibration, damping makes the system return to equilibrium. Without damping, the slightest disturbance would cause oscillations that persist forever.

In forced vibration near resonance, damping limits the amplitude. Without damping, resonance would cause the amplitude to grow without bound. Any system driven at its natural frequency would eventually destroy itself.

The only time we want minimal damping is when we want sustained oscillation: in clocks, tuning forks, or laser cavities. Even then, some energy must be supplied to overcome unavoidable losses.

Solving Forced Vibration Problems

Here is the procedure:

1. Identify $m$, $c$, $k$, and the forcing function $F(t)$
2. Compute $\omega_0 = \sqrt{k/m}$ and $\zeta = c/(2\sqrt{mk})$
3. For the transient: solve the homogeneous equation $my'' + cy' + ky = 0$ using previous techniques
4. For the steady state: if $F(t) = F_0\cos(\omega t)$, use the amplitude formula and phase formula
5. Add transient and steady state for the complete solution
6. Apply initial conditions to determine constants

Example: A 2 kg mass on a spring with $k = 200$ N/m and $c = 4$ Ns/m is driven by $F(t) = 10\cos(8t)$ N. Find the steady-state response.

First, compute the parameters:

• $\omega_0 = \sqrt{200/2} = 10$ rad/s
• $\zeta = 4/(2\sqrt{2 \cdot 200}) = 4/40 = 0.1$ (underdamped)
• $r = 8/10 = 0.8$

The steady-state amplitude is:

The phase angle is:

So the steady-state response is:

The Broader Picture

The mass-spring-damper system is a universal model. The same equation describes:

• Electrical RLC circuits, where inductance plays the role of mass, resistance acts as damping, and capacitance provides the restoring force
• Acoustic resonators, from organ pipes to the human vocal tract
• Molecular vibrations in spectroscopy
• Even abstract economic models of markets responding to shocks

Whenever you see oscillation and decay, whenever you encounter resonance, wherever periodic forcing meets a system with inertia and restoring force, the mathematics of this chapter applies.

Key Takeaways

• The mass-spring-damper equation $my'' + cy' + ky = F(t)$ describes countless physical systems
• The natural frequency $\omega_0 = \sqrt{k/m}$ sets the oscillation rate without damping
• The damping ratio $\zeta = c/(2\sqrt{mk})$ classifies behavior: undamped, underdamped, critically damped, overdamped
• Critical damping ($\zeta = 1$) is often the engineering target: fastest return to equilibrium without overshoot
• Resonance occurs when the forcing frequency matches the natural frequency, causing large amplitudes for lightly damped systems
• Damping is usually desirable: it prevents runaway oscillations and limits resonance
• The frequency response curve summarizes steady-state behavior across all forcing frequencies