MDME: MANUFACTURING, DESIGN, MECHANICAL ENGINEERING

# Vibration (Modal Analysis)

Vibration analysis is called "Model Analysis" because the analysis involved identifying the different modes of vibration. The damping effect is not included in the FEA calculations, which means that even the smallest exciting force would produce infinite vibration amplitude. So modal analysis applies it's own 'forces' to vibrate the component in the various modes.

Lecture Notes empty.pdf empty.one

### Vibration (Modal Analysis)

Modal analysis in FEA looks similar to static analysis except there are no forces!

Because there is no damping, the tinyest oscillating force (at the critical frequency) might vibrate the object to destruction. So Inventor is not looking at forces and stresses, but vibration frequencies. It finds the different vibration modes, which are the different ways the thing can vibrate). So vibration analysis is called modal analysis.

Here are some different vibration modes for an I beam.   Vertical vibration Sideways vibration Torsional vibration

Modal analysis looks at the different modes of vibration and the frequencies that generate them. The goal for the designer is often to make sure the product is stiffened so that the natural frequencies of the components deliberately avoid the excitation frequencies during operation. e.g. If a car gearbox has natural frequencies close to the common frequencies of the meshing gears it will make more noise. Understanding Modal Analysis by CADLearning

Suspension bridge over Tacoma Narrows. Collapses due to modal vibrations that matched the wind speed.

### Free Vibration theory

Ever heard a jet engine get up to speed on the runway? You may notice that the engines go through stages where they sound very loud (vibration), but then as they speed up even faster the engine gets quieter.

It was designed like that on purpose. If they stiffen up the supports, this will raise the natural frequency and the engine could be making maximum noise at operating speed! That would be much more annoying.

These loud periods are near significant natural frequencies, where vibration is exaggerated. Once the engine gets near the operating speed, the natural frequencies are so much lower that (hopefully), the engine does not create such amplified vibrations.

##### Natural Frequency

The equation for the natural frequency of vibration is: Where;

fn = undamped natural frequency (Hz)

k = spring stiffness (N/m)

m = mass (kg)

Note! This equation will not work in mm!

This shows that in order to raise the natural frequency, we can either increase the stiffness or reduce the mass. Stiffness can be increased by selecting a stiffer material, or making it geometrically thicker and shorter.

Reducing the mass is achieving by lighter materials and by trimming less efficient parts of the design. Image: Wikipedia

There are many examples of natural frequency - both good and bad.

• Musical instruments are deliberately designed to vibrate on the natural frequency of the note. For example, a piano string, where the mass of the string and the length increases as notes get lower. The string vibrates at its natural frequency.
• An unbalanced or out-of-shape car tyre might vibrate excessively at certain speeds - where the tyre and suspension stiffness matches the mass to give a natural frequency according to the equation.
• A noisy CNC motor or fan might run quieter at a slightly different speed. When running a stepper-motor driven CNC the resonant frequencies can easily be heard as louder vibrations at certain feedrates.
• Many components in a car (e.g. gearbox, exhaust mounts) are designed with natural frequencies that avoid the excitation frequencies of vibration, such as pulses from gear teeth, bearings, pistons, rotating loads and random impacts. (Compared to excitation, the gearbox casing will have a higher natural freq, and exhaust mounts will have lower natural frequency).
• Pushing a kid on a swing. The excitation force (the parent) is at the same frquency as the pendulum (kid on the swing). That way a small push can eventually produce a very large swing.
• A ship rolling in waves. The natural frequency of the ship's roll can sometimes be excited by a particular wave frequency (from the side). In this case, the captain may change speed or even course in order to avoid the excitation frequency matching the natural frequency.
• Most sound generating things are related to the equation (k/m)0.5. Speaker cones and cases, bells, drums, a triangle, the air mass in organs, flutes whistles and other wind instruments, the strings in strings instruments like pianos, violins and guitars. Even the sound of hitting a metal plate makes a tone that is determined by the equation (k/m)0.5
• Room acoustics - where certain frequencies are louder than others. This is because those frequencies have a wavelength that is a multiple of the dimensions of the room. This can be fixed by changing the dimensions, tilting the walls, and (of course) adding damping to reduce the amplitude of oscillations.
• An earthquake - a single sudden slip at a fault can turn into a series of oscillations some distance away. The frequency of these oscillations is caused by the natural frequency of the geology - the stiffness and mass of the rocks. A building should be built to NOT vibrate near this frequency
• A bell - A single strike of the hammer causes the bell to vibrate sinusoidally for a period of time. A good bell has less damping and so will vibrate for a longer time.
• Discussion - Squeaky hinge or brakes, cruise control, stick slip vibrations

### Resonance

Resonance refers to the exaggerated response to vibrations at or near the natural frequency (resonant frequency).

A familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo produce smaller arcs. This is because the energy the swing absorbs is maximized when the pushes are "in phase" with the swing's natural oscillations, while some of the swing's energy is actually extracted by the opposing force of the pushes when they are not.

Resonance occurs widely in nature, and is exploited in many manmade devices. It is the mechanism by which virtually all sinusoidal waves and vibrations are generated. Many sounds we hear, such as when hard objects of metal, glass, or wood are struck, are caused by brief resonant vibrations in the object. Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale, such as electrons in atoms. Other examples are: The response of a system where the input frequency varies from the natural frequency. Wikipedia

Question: There is a square root in the equation, so why does halving the length of a cantilever cause the frequency to double?

The stiffness of a beam is inversely proportional to the deflection under a known force. Consider a cantilever when it vibrates (its centre of mass) like a simple cantilever deflection = wL3/3EI.

When L is halved, the deflection is reduced 23 = 8 times. So stiffness increases by 8 times. k' = 8k

Mass m is reduced 2 times (obviously).

So f' = (k'/m')0.5 = (8k/2m)0.5 = (4)0.5 = 2 f

Makes sense then doesn't it? The same happens with stringed instruments. It doesn't matter whether a cantilever or a simply supported beam, the deflection is still L3 and mass is still L1 which gives (L2)0.5 = L.

### Modal Analysis by FEA (Inventor)

The following videos give a quick introduction using INVENTOR for the modal analysis of a "tuning fork", a simple device designed to vibrate at a particular frequency when the forks are struck. The set frequency gives a certain pitch of sound, and is used for tuning musical instruments. (Before smart phones of course)

"Modes of vibration" refers to the different types of vibration that can be set up in the same object. For example, when a guitar string is struck, the usual vibration mode is with a half-wave over the full length of the string, (harmonic 1 below).

However, the same string can produce a note one octave higher by creating a node at the centre. This makes the string vibrate twice as fast, which doubles the frequency. The full-length string vibrating with a node in the middle is called the 2nd harmonic. There are more harmonics, each with increasing frequency... The 2nd harmonic is the open "C" note on a brass instrument (the lowest normal open note). A bugle (trumpet with no valves) can only play a tune like "The Last Post", which is played using these harmonics - c",g",c"',e"',g"'. These are all geometric subsets of the fundamental length of the instrument.

Vibrations are rarely this simple however. Unlike a piano string or air in a pipe, most engineering components are complex shapes, which means there is usually many vibration modes.

A string is 1 dimensional. The next level of complexity is 2 dimensions, which is a flat plane, like the skin of a drum. But even a simple drum skin has an surprisingly large number of possible vibration modes. Here's a few of the more common ones. (Note: The frequency of oscillation is not exactly correct here, otherwise the last one would be going so fast it would be a blur). Since frequency is square root of stiffness/mass, the more complex oscillations have both higher stiffness and lower mass in the deformation. So the simplest vibrations - called the fundamental - like the first one, have the lowest fequency. More complex modes have higher frequencies (since they are moving smaller mass and the membrane acts more stiff).

 1st harmonic 2nd harmonic 3rd harmonic Peak at centre   Peaks off centre on one axis   Peaks off centre on two axes   https://commons.wikimedia.org/wiki/Category:Drum_vibration_animations Note: The frame rate on these Wikipedia animations have been adjusted to match the natural frequencies more closely.

For a 3D object, there can be many modes of vibration - up/down, sideways and even torsion. And this is only in the fundamental (first harmonic) mode.

### Damping

Damping is the frictional resistance to motion that reduces the amplitude of oscillation. It works just like shock absorbers on a car. Without them, the car will bounce around wildly on a bumpy road. Even without shock absorbers (e.g. a box trailer with leaf springs), there is usually enough friction to dampen the oscillations to some extent.

Damping is essential near the natural frquency, otherwise the amplitude will increase without limit.

##### Damping Ratio

In engineering, the damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance.

The damping ratio is a parameter, usually denoted by ζ (zeta), where;

ζ = actual damping coefficient / critical damping coefficient = c / cc, where; • Undamped: ζ = 0 An imaginary perfect oscillating system with no friction.
• Underdamped: ζ < 1 Most spring/mass systems are underdamped if they oscillate.
• Overdamped: ζ > 1 Massive resistance which slows the mass (e.g. a door closing mechanism)
• Critically damped: ζ= 1 This turns out to be useful when we want the mass to come back to position quickly but without oscillations (e.g., a high performance shock absorber with exactly zero rebound).

Damping is a force that opposes the velocity of the object, and if proportional to the velocity, like a viscous damper (dashpot). Then F = -cv , where c is the damping coefficient, given in units of newton-seconds per meter (Ns/m).

In practice, when the system is underdamped, an easier way to find the damping ratio is; Where x1/x2 is the ratio between any two successive amplitudes. This is easily determined from direct optical measurement (e.g. a video). Image: Wikipedia:
Underdamped
spring–mass system The effect of Damping Ratio on the oscillations of a sping/mass system. Wikipedia

Above: Damping reduces amplitude. It also changes the resonant frequency (reducing it slightly below the natural frequency). See the blue curve on the earlier diagram Resonance Transmissibility.

## The Report

1. HAND CALCULATION: Calculate the natural frequency of the aluminium cantilever rig.
This applies to a cantilever where the mass is the cantilever itself. Below are the equations for the natural frequency of the first three modes of vertical vibration. http://iitg.vlab.co.in/?sub=62&brch=175&sim=1080&cnt=1
Where
E = modulus of Elasticity (Pa)
I = second moment of area (m4)
L = Length (m)
m = mass per unit length (kg/m)
ω = circular natural frequency (rads/s)
To convert ω to frequency f use the following equation: Note: Use base units! i.e. all units must be in metres, not mm. The mm trick does not work here because the undamped natural frequency relation does not work in mm. By converting to mm, the stiffness k will decrease but mass m will stay the same. Therefore mm will give a false (lower) frequency.

Note: A mass-less cantilever with a load on the end...
By comparison, the natural frequency of a cantilevered uniform beam (with negligible beam weight) with a load on the free end is given by: 2. FEA MODAL ANALYSIS: Model the cantilever and use a Solid Modeller (e.g. Inventor) to solve the fundamental frequency. Discuss the effect of mesh size and comment on the convergence of this system. (Use a simple cantilever of matching dimensions, Al 6061 and a simple fixed constraint at the mounting end face)

Simulation of cantilever beam using Autodesk Inventor

3. REAL RESULTS: Experimental determination of the natural frequency: There are a number of ways to measure the natural frequency of the cantilever beam rig. The simplest way is to video the beam. Like this...

This video shows 50 oscillations of the beam, recorded in 1026 frames (slowed down from 240 to 25 FPS).
From this you can compute the natural frequency of the beam in this fundamental (1st harmonic) mode. What is it?
4. COMPARE RESULTS: Comment on the comparison between each of the 3 methods and discuss the accuracy. Identify any sources of error that may cause differences in the results between the three methods.
5. 2ND AND 3RD HARMONICS: Repeat hand calculation and computer simulation for the 2nd and 3rd harmonics (in the vertical mode). (Don't bother trying to get it to vibrate like that in the test rig, this would need some electric oscilators attached)
6. ADDED MASS: Use an Inventor assembly to add a 2kg mass to the end of the beam (you need to start a new Stress Analysis project). Use modal analysis to measure the frequency of the first harmonic. Explain what what has happened to the frequency based on the Natural Frequency equation. 7. DAMPING (See video before attempting this)
(a) What is the damping of the cantilever caused by?
(b) Determine the damping ratio ζ (zeta) for the experimental cantilever beam. Where:
x1 = peak at start of two consecutive cycles
x2 = peak at end of two consecutive cycles
δ = damping decrement

Use the following image to determine the amplitude ratio in successive oscillations. The dimensions are measured from the centre of oscillation to the maximum downward deflection. Click here for larger image. The dimensions are measured on the screen (you might do better yourself on a large screen. Just zoom up on your screen and measure with a ruler).
(c) Using the damping ratio, roughly determine the magnification of the driving amplitude. In other words, if the oscillator has enough force to cause 1mm of deflection, how many mm will it reach while running (for some time)? Use the following plot as a guide; This is a log graph in the vertical axis. You may get a damping ratio that is smaller than the minimum in this chart ζ = 0.01.
See bottom of page for more advanced transmissibility plots.

• https://en.wikipedia.org/wiki/Finite_element_method
• https://www.newport.com/t/fundamentals-of-vibration Nice page but the transmissibilty formula does not seem to work.
• https://www.engproguides.com/hvac-supportive-knowledge.html This is a nice page covering similar content. It also has the equation for drawing your own transmissibility plots for any damping ratio ζ. Here is a chart plotted from the equation above. (Similar amplitude magnification near resononance for the low damping ratios, but the higher damping coefficients don't seem to match too well. Oh well, we have a low damping coeff so should be OK).  