Vibration Analysis and Wave Behavior in Engineering Systems
(Level 4)
So far: motion, forces, rotation. All treated as if things happen once and you solve for the result. Level 4 asks: what if the system doesn't stop? What if it bounces back, oscillates, repeats? What if disturbances travel through the system instead of just sitting there?
Vibrations mean things move back and forth. Waves mean disturbances propagate. Both show up constantly in machines, structures, vehicles, acoustics, and failure modes. If you don't understand them, you can't predict resonance, noise, fatigue, or stability.
What You'll Learn
Simple Harmonic Motion and Oscillatory Behavior
Many common systems behave as masses attached to springs. For example, imagine pushing a mass on a spring, then releasing it. It oscillates back and forth. It returns to its rest state, or equilibrium position, over and over again. Before releasing the mass from rest, it is displaced from its equilibrium position. As the mass moves away from the equilibrium position, it experiences a restoring force that pulls the mass back to the equilibrium. However, once the mass reaches its equilibrium position and begins moving in the opposite direction, it actually overshoots the equilibrium position, returns to the other side, changes direction and continues to oscillate past the equilibrium position.
Simplest case: the restoring force is proportional to the displacement. F = -kx. Combining with F = ma we obtain the following
Solution: Sines and cosines. The system oscillates forever (in the idealized, frictionless case).
Amplitude (A), angular frequency (ω), and phase (φ) are defined parameters for this sort of wave. The oscillatory behavior defined by these parameters is called simple harmonic motion (SHM). Any linear restoring force produces simple harmonic motion.
Stiffness k = 100 N/m Mass m = 4 kg
ω = √(k/m) = √(100/4) = 5 rad/s
Period of the function (in seconds per cycle) T = 2π/ω = 1.26 s/cycle.
SHM is everywhere – from simple springs and pendulums, to more complex things like bridges and electric circuits (e.g., LC oscillators). So, you should become familiar with this important concept of physics because the analysis of vibration depends on it.
Natural Frequency of Mechanical Systems
Every system has a 'native' frequency at which it likes to oscillate. This natural frequency is determined by the system's physical properties - its stiffness and its inertia. In general, stiffer systems have higher natural frequencies (oscillating more quickly) and more massive systems have lower natural frequencies (oscillating more slowly).
This is not an arbitrary choice. It is determined by the differential equation and in fact follows directly from it. The system does not "oscillate at ωn" - it has to - the physics of the system forces it to do so.
System A: m = 1 kg --> 10 rad/s, √(100/1)
At the input of System B a constant acceleration is applied. The mass m of System B is 4 kg, thus the natural angular velocity ω_n is 5 rad/s.
If you double the mass, you cut the frequency in half. Things that are heavier oscillate more slowly.
Why This Matters Later
There is one frequency at which driving your system will cause problems for you. It is the frequency at which your system responds most strongly, and even a small input can cause a large response: resonance. All physical systems have natural frequencies at which they tend to respond best. Be aware of these frequencies and don't operate on them unless you intend to.
Damping and Energy Dissipation in Mechanical Systems
An ideal spring-mass system oscillates forever; real systems do not. The oscillations of such a system are attenuated by friction, air drag, internal damping in the mass, etc. Damping refers to this effect.
Physics -- Motion -- damping term -- add a damping term (proportional to velocity)
ζ (zeta) is the damping ratio, which is the damping relative to critical damping. There are three cases.
The system still has some oscillations, but the amplitude is decreasing with time. Most physical systems can be described well within this regime. Examples range from car suspension systems over swinging buildings down to vibrating machines that settle back into position.
No oscillation. The system returns to its equilibrium position as fast as possible without overshooting. Use this profile for door closers, instruments and other applications in which you want as fast return as possible without bouncing.
Note the flat top and the slow return to equilibrium. The system is underdamped since there is no overshoot but it is too heavily damped since the return to equilibrium is slow, or slower than that for a critically damped system.
Damping has a profound impact on the vibration amplitudes of devices, their noise levels, their wear and the ring or non-ring response. It is essential to either design or retrofit a proper amount of damping in many engineering systems. Too little damping causes systems to be unhealthy, undergoing unacceptable levels of vibration, resonant peaking, premature wear and sudden failure. On the other hand, too much damping results in a slow system response which is often less efficient and also has its problems.
Forced Vibrations and Resonance Effects
Free vibration refers to a system which has been displaced from its static position and then released and allowed to perform oscillations. Forced vibration is defined as a system which is subjected to an external periodic force and as a result performs oscillations at the frequency of the external force, not necessarily the natural frequency of the system.
Equation of motion with external force F₀ cos(ωt):
Steady-state solution: the system oscillates at the frequency of the forcing ω, and its amplitude depends on the distance of ω from ω_n. Far away from ω_n the system shows a small response. Close to ω_n the response is large. At the limit, when ω is equal to ω_n, the system is in resonance.
When forcing frequency matches natural frequency (ω ≈ ω_n), amplitude spikes. Without damping, amplitude would grow to infinity (in theory). With damping, amplitude is limited but still huge.
The response amplitude is inversely proportional to the damping ratio. The damping also has a pronounced effect on the shape of the transfer function. Low damping corresponds to a massive resonance peak, high damping results in a flatter peak.
This is why you either tune down the resonance or add damping. Tacoma Narrows Bridge - wind vortex shedding at the bridge's natural frequency with very low damping resulted in massive oscillations leading to its failure. A washing machine on spin being locked into resonance is another common example of this. Engine mounts and suspension tuning is all about trying to avoid resonance, and when that's not possible, how to control it in structures.
Resonance is not always bad. Musical instruments rely on resonance, as well as radio detectors, sensors and filters for selectivity. However, in structural and mechanical systems, unexpected resonance is a common failure mode. Understanding the natural frequencies of a system and avoiding coincidence with forcing frequencies is key to performance and reliability.
Wave Propagation in Physical Systems
Vibrations are called local oscillations or fluctuations. Waves are called oscillations or fluctuations which are in motion, i.e. which travel. If you pluck a guitar string at one end of it, the disturbance will not remain local; it will soon travel the entire length of the string.
Governing equation (one-dimensional wave equation):
Here, u represents displacement (or pressure, or voltage—whichever is waving) and c is the wave speed. We have introduced partial derivatives because both spatial variation (x) and time variation (t) are to be taken into account.
Wavelength λ: distance between peaks
Frequency f: oscillations per second
Speed c: how fast the wave propagates
Related by: c = fλ
Before delving into acoustics of air, it's worth noting how sound travels through other mediums as a wave - as a pressure wave (longitudinal) where areas of high and low pressure travel through a medium in a sequential fashion. This concept applies to vibrations in beams and plates as well, except in these cases the motion is transverse and the beam or plate undergoes a bending action. It's also relevant to note that stress waves propagate through solids at a speed that varies based on the physical properties of the material. For example, striking a hammer against a metal bar does not cause the whole bar to react immediately - there is a time delay as the stress wave travels through the bar at a speed determined by metal's elastic properties.
Waves in nature reflect, refract, interfere and diffract. The occurrence of standing waves requires boundary conditions which restrict the wavelengths that are allowed to fit within a given dimension. Every acoustic resonance in ducts, every vibration mode of structures and every transmission line in mechanical systems employs waves.
Hit a part with a hammer and see what happens. The force of the hammer won't stay in one place - it will travel in waves, bounce off surfaces, interfere with other waves, set up complex patterns of vibration. Studying waves means studying how energy and disturbances travel through real systems comprised of connected objects instead of isolated point masses.
Worked Example: Identifying System Properties from the Equation of Motion
The natural frequency is $\omega_n = 3\,s^{-1}$.
The general solution of this second-order linear differential equation is $y = 5\cos(3t)$. The coefficient 25 is the square of the amplitude (magnitude) of the oscillation.
For Simple Harmonic Motion (SHM), the typical equation of motion is given by: d²x/dt² + ω²x = 0. By matching patterns with this typical equation, we can find ω² = 25. This means ω = 5 rad/s which is known as the natural frequency of the system. For example, the system will complete one cycle every 2π/5 ≈ 1.26 seconds.
Natural frequency: ω = 5 rad/s
A second-order linear ordinary differential equation (ODE) with constant coefficients always has solutions that are sinusoids. The two constants of integration of a second-order equation account for the two independent initial conditions, such as initial position and initial velocity, which determine the particular solution.
x(t) = A sin(ωt + φ); equivalently x(t) = C cos(5t + φ), where C and ω are constants related to each other.
This is for a spring-mass system: ω² = k/m. Larger value → faster oscillations. Larger value → stiffer system OR lighter mass. Smaller value → slower oscillations. The coefficient completely determines the time scale of motion. Change 25 to 100? Frequency doubles. Change to 6.25? Frequency halves.
Physical Meaning - It is the ratio of the restoring force for a system to its mass, i.e. the ratio of the stiffness to the inertia. This ratio is equal to the natural frequency of the system. The same math describes all oscillating systems - springs, simple pendulums, electrical circuits, structural modes. Same difference.
The differential equation tells you everything you need to know about the system's behaviour. Read the coefficients carefully, extract the natural frequency, and then solve for the motion. If there are no coefficients, there is no restoring force and hence no oscillation. But if the coefficients are wrong, then your predictions will be wrong, and your design will fail. Getting the equation of motion right is half the battle.
What You Should Take From Physics Level 4
- Vibrations arise from restoring forces and inertia
- Natural frequency depends on stiffness and mass
- Damping controls oscillation decay and resonance peaks
- Waves propagate disturbances through continuous media
Apply Vibration Analysis in Specializations
Vibration and resonance analysis is essential in:
- Structural Analysis & FEA - Perform modal analysis and predict structural vibrations
- Automotive & Transportation - Design suspension systems and reduce NVH (noise, vibration, harshness)