Vibrations in Mechanical Engineering: Analysis and Control
Every mechanical system vibrates. Some vibrations are harmless. Others shake machines apart. Understanding vibrations is essential for mechanical engineers because vibration causes fatigue failure, generates noise, reduces precision, and creates discomfort. The discipline of vibration analysis provides the tools to predict, measure, and control these motions.
Mechanical vibration is the oscillatory motion of a body about an equilibrium position. It occurs when a system is displaced from equilibrium and restoring forces try to return it. The study of vibrations ranges from simple single-degree-of-freedom systems to complex multi-body assemblies with thousands of modes.
Single-Degree-of-Freedom Systems
The simplest vibrating system consists of a mass, a spring, and a damper. This single-degree-of-freedom system captures most of the fundamental concepts of vibration theory.
Free Vibration
Free vibration occurs when a system is disturbed and then allowed to oscillate without external excitation. The natural frequency depends on the stiffness and mass. Higher stiffness increases natural frequency. Higher mass decreases it.
Undamped systems oscillate forever at their natural frequency. Real systems have damping that causes oscillations to decay. The damping ratio determines how quickly the amplitude decreases. Critically damped systems return to equilibrium in the shortest time without oscillating.
Forced Vibration
Forced vibration occurs when an external excitation drives the system. The excitation can be harmonic, periodic, or random. The response depends on the excitation frequency relative to the natural frequency.
At low excitation frequencies, the response is stiffness-dominated. At high frequencies, the response is mass-dominated. Near the natural frequency, resonance occurs, and the response amplitude can be many times the static deflection.
Resonance
Resonance is the most important concept in vibration engineering. When the excitation frequency matches a natural frequency, the amplitude grows rapidly. Without damping, the amplitude would increase without bound. Even with damping, resonance can produce forces that damage structures and injure people.
The Tacoma Narrows Bridge collapse in 1940 is the most famous example of resonance. Wind-induced vortex shedding excited a torsional mode of the bridge, causing oscillations that grew until the bridge failed.
Multi-Degree-of-Freedom Systems
Real systems have multiple masses connected by multiple springs and dampers. They have multiple natural frequencies and mode shapes.
Modal Analysis
Modal analysis finds the natural frequencies and mode shapes of a system. Each mode shape describes the relative displacement of each mass when the system vibrates at that natural frequency.
The first mode typically has all masses moving in phase. Higher modes have nodes — points that remain stationary. A mode shape can be excited only if the external force distribution matches the shape.
Modal analysis is the starting point for all dynamic analysis. The Finite Element Analysis guide covers how FEA programs perform modal analysis on complex geometries.
Mode Superposition
Any vibration response can be expressed as a combination of mode shapes. The contribution of each mode depends on the excitation. This principle allows engineers to reduce complex analyses to a few dominant modes.
Damping Mechanisms
Damping dissipates vibrational energy. Without damping, all systems would oscillate indefinitely.
Viscous Damping
Viscous damping occurs when a fluid is forced through a small gap. Automotive shock absorbers use viscous damping. The damping force is proportional to velocity. Most vibration analysis assumes viscous damping because the mathematics are straightforward.
Structural Damping
Structural damping, or hysteretic damping, occurs within solid materials as they deform. The energy dissipation per cycle is independent of frequency. Rubber and polymers have high structural damping. Metals have very low structural damping.
Coulomb Damping
Coulomb damping, or dry friction damping, occurs when two surfaces slide against each other. The damping force is constant, independent of velocity magnitude. Coulomb damping is nonlinear and difficult to analyze.
Vibration of Continuous Systems
Real structures are not discrete masses connected by springs. They are continuous systems with distributed mass and stiffness.
String Vibration
A stretched string supports transverse waves. The natural frequencies are integer multiples of the fundamental frequency. This harmonic relationship gives musical instruments their pleasing sound.
Beam Vibration
The vibration of beams is described by the Euler-Bernoulli beam equation. The natural frequencies depend on the boundary conditions. A cantilever beam has different natural frequencies than a simply supported beam.
The mode shapes of a vibrating beam are the characteristic shapes that the beam assumes at each natural frequency. The first mode has no nodes. The second mode has one node. Higher modes have more nodes.
Plate and Shell Vibration
Plates and shells have two-dimensional vibration patterns. Their natural frequencies depend on the in-plane dimensions, thickness, boundary conditions, and material properties. Plate vibration is important in aircraft panels, ship hulls, and floors.
Vibration Measurement and Data Analysis
Accelerometer Selection
Piezoelectric accelerometers are the most common vibration sensors. Charge-mode accelerometers require external signal conditioning. Integrated electronics piezoelectric accelerometers contain built-in amplifiers. MEMS accelerometers are small, inexpensive, and suitable for low-frequency measurements.
Frequency Response Functions
The FRF is the ratio of response to excitation in the frequency domain. It completely characterizes the linear dynamic behavior of a structure. FRF measurements are the basis for experimental modal analysis.
Operational Deflection Shapes
ODS analysis measures the actual vibration pattern of a structure under operating conditions. Unlike mode shapes, which are properties of the structure, ODS shows how the structure deforms at specific frequencies during operation.
Vibration Control
Engineers use several strategies to control unwanted vibration.
Isolation
Vibration isolation reduces the transmission of vibration from a source to a receiver. Machine mounts isolate equipment from building vibrations. Vehicle suspensions isolate passengers from road vibrations.
The transmissibility of an isolation system depends on the frequency ratio and damping. Isolation is effective only when the excitation frequency exceeds the natural frequency by a factor of at least 1.4. Below this ratio, the isolator amplifies the vibration.
Absorbers
Tuned vibration absorbers add a secondary mass-spring system that vibrates out of phase with the primary system. The absorber cancels the primary vibration at the tuning frequency. Absorbers are used on power lines, bridges, and machine tools.
Damping Treatments
Constrained-layer damping applies a viscoelastic layer between two metal sheets. The shear deformation of the viscoelastic layer dissipates energy. This treatment is common in automotive body panels and aircraft skins.
Measurement and Signal Processing
Vibration measurement is essential for condition monitoring and experimental modal analysis.
Accelerometers
Piezoelectric accelerometers are the most common vibration sensors. They produce a charge proportional to acceleration. Modern MEMS accelerometers provide digital output at low cost.
Vibration Isolator Design
Effective isolator design requires selecting the correct stiffness and damping. The natural frequency of the isolated system should be well below the excitation frequency. The transmissibility curve shows that isolation begins at a frequency ratio above the square root of two.
Wire rope isolators provide high damping for shock isolation. Elastomeric mounts provide good vibration isolation with moderate damping. Air springs offer adjustable natural frequencies for precision equipment.
Frequency Analysis
Piezoelectric accelerometers are the most common vibration sensors. They produce a charge proportional to acceleration. Modern MEMS accelerometers provide digital output at low cost.
Frequency Analysis
The Fourier transform converts a time-domain vibration signal to the frequency domain. The frequency spectrum reveals the dominant frequencies present. Fast Fourier transform analyzers perform this calculation in real time.
Condition Monitoring
Vibration monitoring detects developing faults in rotating machinery. Bearing faults produce characteristic frequency signatures. Imbalance and misalignment cause vibration at one times and two times the rotational speed. The Automotive Engineering guide discusses NVH analysis in vehicle development.
Rotor Dynamics
Rotating machinery presents unique vibration challenges.
Critical Speeds
A critical speed occurs when the rotational speed matches a natural frequency. The shaft deflection grows rapidly at critical speeds. Rotors are designed to operate between critical speeds or to pass through them quickly during startup and shutdown.
Imbalance
Mass imbalance is the most common source of vibration in rotating machinery. Balancing procedures add or remove mass to align the principal axis with the rotational axis. Single-plane balancing is sufficient for short rotors. Long rotors require two-plane balancing.
Frequently Asked Questions
What is the difference between natural frequency and resonance? Natural frequency is a property of the system — the frequency at which it prefers to vibrate when disturbed. Resonance is the condition that occurs when the excitation frequency matches a natural frequency, causing large amplitude oscillations.
How do engineers avoid resonance in design? Engineers design the natural frequencies to be far from expected excitation frequencies. If that is not possible, they add damping to limit resonance amplitudes or use vibration absorbers.
What causes vibration in rotating machinery? Common causes are mass imbalance, misalignment of shafts, bearing faults, gear meshing, aerodynamic forces on fan blades, and electromagnetic forces in motors.
Can vibration ever be useful? Yes. Vibratory feeders move parts along a track. Ultrasonic cleaners use high-frequency vibration. Vibratory compactors densify soil and concrete. Vibration is also used in material testing to determine dynamic properties.