Mechanical Vibrations

Mechanical vibrations are the oscillatory motions that occur when a machine or structure is displaced from equilibrium and allowed to move under the action of restoring forces. They show up everywhere: a car body responding to road irregularities, a washing machine shaking during spin, a turbine rotor with slight unbalance, or a building swaying under wind loads. In engineering practice, vibrations are not only about comfort and noise. They directly affect fatigue life, accuracy, reliability, and safety.

A good vibration analysis connects physical intuition with mathematical models: how mass, stiffness, and damping shape motion; how external forcing can amplify response; and how multi-degree-of-freedom systems can be understood through modes and modal analysis.

Core concepts: mass, stiffness, damping, and natural frequency

Most mechanical vibration problems can be traced back to three ingredients:

• Mass (inertia) resists changes in motion.
• Stiffness provides restoring forces that pull the system back toward equilibrium.
• Damping dissipates energy, typically through friction, material hysteresis, or fluid effects.

For a basic single-degree-of-freedom (SDOF) model with displacement $x(t)$, mass $m$, stiffness $k$, and viscous damping $c$, the equation of motion under an external force $F(t)$ is:

$$m\ddot{x}+c\dot{x}+kx=F(t)$$

Two parameters are foundational:

• Natural circular frequency: ${\omega }_n=\sqrt{k/m}$
• Damping ratio: $\zeta =\frac{c}{2m{\omega }_n}$

These quantities govern how the system responds when disturbed, whether the vibration dies out quickly, persists, or grows dangerously under forcing.

Free vibration: what happens after a disturbance

Free vibration occurs when a system is displaced and released, with no continuing external force (so $F(t)=0$). The behavior depends strongly on damping.

Undamped free vibration

With $c=0$, the motion is purely oscillatory:

• The system vibrates indefinitely at its natural frequency ${\omega }_n$.
• Energy alternates between kinetic (mass motion) and potential (spring deformation).

This is a useful idealization for understanding resonance and modal behavior, but real machines always have some damping.

Damped free vibration

With damping present, energy is dissipated and the amplitude decays over time. For most mechanical systems, damping is underdamped ($0<\zeta <1$), producing an oscillation whose envelope shrinks exponentially. The oscillation occurs at the damped natural frequency:

$${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$$

In practical terms, damping determines how quickly a system “settles” after a transient event such as a gear engagement shock, a valve closure, or a machine start-up.

Forced vibration: steady-state response and resonance

Forced vibration occurs when an external excitation continuously drives the system. Common sources include periodic forces from rotating machinery, reciprocating mechanisms, or cyclic loads in structures.

A particularly important case is harmonic forcing:

$$F(t)=F_0\sin (\omega t)$$

After initial transients decay, the system reaches a steady-state oscillation at the forcing frequency $\omega$, not the natural frequency. The key question becomes: how large is the amplitude, and how does it vary with frequency?

Resonance and the frequency response

As the forcing frequency approaches the natural frequency ($\omega \approx {\omega }_n$), the response amplitude can increase dramatically. This is resonance, and it is one of the central concerns in vibration engineering.

• With very low damping, resonance can produce extremely large amplitudes.
• With higher damping, the peak amplitude is reduced and the resonance curve broadens.

Resonance is not an abstract concept. It is why rotating equipment has critical speeds, why vehicle suspensions must avoid amplifying road inputs, and why structural designers check dynamic response under wind and machinery loads.

Phase and dynamic behavior

Forced vibration is also characterized by a phase difference between force and displacement:

• At low frequencies, displacement tends to be nearly in phase with force.
• Near resonance, phase shifts rapidly.
• At high frequencies, displacement can lag substantially, and inertia dominates.

This phase behavior matters in control, isolation design, and diagnosis. For example, a phase shift observed in vibration measurements can indicate whether the response is stiffness-controlled, mass-controlled, or near resonance.

Rotating unbalance: a dominant real-world excitation

One of the most common causes of forced vibration in machinery is rotating unbalance. If a rotor’s mass center is offset from its geometric axis by an eccentricity $e$, rotation produces a sinusoidal centrifugal force that grows with the square of speed:

• Force magnitude scales as $me{\omega }^2$.

This explains an important diagnostic pattern: unbalance vibration often increases rapidly with speed and is strongest at the rotational frequency (often called “1X” in condition monitoring). Unbalance is managed through balancing procedures, precision manufacturing, and maintenance practices such as cleaning deposits from fan blades.

Unbalance becomes especially critical when operating near a system’s natural frequencies, since even modest forcing can trigger resonance.

Vibration isolation: reducing transmitted forces and motion

Vibration isolation aims to prevent vibration from traveling from a source (a machine) to a receiver (a foundation, a building, a sensitive instrument), or vice versa. The simplest isolator is a spring and damper supporting the vibrating mass.

Isolation design revolves around the frequency ratio $r=\omega /{\omega }_n$:

• If $r<\sqrt{2}$, the system tends to amplify motion (the isolator behaves more like a rigid connection).
• If $r>\sqrt{2}$, isolation becomes effective, and transmitted vibration decreases as frequency increases.

Damping plays a nuanced role:

• Near resonance, damping is beneficial because it limits peak response.
• At higher frequencies, excessive damping can increase transmitted force, reducing isolation effectiveness.

In practice, isolator selection balances start-up and shut-down resonance behavior, steady operating speed, load variability, and environmental constraints. Examples include elastomer mounts for motors, air springs for precision equipment, and tuned isolators for HVAC systems.

Multi-degree-of-freedom systems: why modes matter

Real machines and structures rarely move in just one coordinate. A vehicle body can bounce, pitch, and roll. A machine tool structure can flex in multiple directions. These are multi-degree-of-freedom (MDOF) systems, described by vector equations:

$$\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{Kx}=\mathbf{f}(t)$$

Here, $\mathbf{x}(t)$ contains multiple displacements (or generalized coordinates), and $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}$ represent mass, damping, and stiffness matrices.

Natural frequencies and mode shapes

For undamped free vibration, solutions exist where the system vibrates in characteristic patterns called mode shapes, each associated with a natural frequency. In a given mode, all coordinates oscillate at the same frequency with fixed relative amplitudes.

These modes provide a powerful mental model:

• Low-frequency modes often involve large, global motions (like a frame swaying).
• Higher-frequency modes tend to be more localized (like a panel or bracket vibrating).

Modal analysis in engineering practice

Modal analysis is the framework that uses modes to simplify and interpret MDOF behavior. Instead of solving a tightly coupled system directly, the response is represented as a sum of modal contributions. This is valuable because:

• A few dominant modes often capture most of the response in the frequency range of interest.
• It clarifies which structural features influence problematic resonances.
• It supports testing and validation, such as experimental modal analysis using impact hammers or shakers.

Modal thinking is also central to troubleshooting. If a machine shows excessive vibration at a particular frequency, identifying the associated mode helps locate the weak stiffness path, insufficient support, or compliance in connections.

Practical guidance: preventing vibration problems

Vibration issues usually come from a mismatch between excitation and structural dynamics. Common mitigation strategies align with the concepts above:

• Shift natural frequencies by changing stiffness or mass (stiffen a support, add a brace, reduce overhang).
• Increase damping where appropriate (constrained-layer damping, friction interfaces, viscous dampers).
• Reduce excitation (balance rotors, align couplings, reduce gear transmission error, isolate reciprocating forces).
• Use isolation intelligently by selecting an isolator natural frequency well below operating speed.
• Avoid operating near resonance during steady operation, and manage run-up and coast-down through critical speeds.

Why mechanical vibrations deserve rigorous attention

Mechanical vibrations are not only a theoretical topic. They are a daily design constraint and a diagnostic tool. Understanding free and forced vibration, damping, resonance, vibration isolation, rotating unbalance, and modal analysis equips engineers to design quieter machines, extend component life, improve product feel, and prevent failures that begin as small oscillations but end as cracked welds, loosened fasteners, or damaged bearings.

When vibration is treated as a first-class design parameter, performance and reliability improve together, and the system behaves predictably across real operating conditions.