EN4: Dynamics and Vibrations

 

Division of Engineering
Brown University

 

6. Vibrations

This is the last topic to be covered in EN4. It is arguably the most useful part of the course – almost all engineers will need to solve a vibration problem at some point in their careers. It is also the most challenging part of the course, for a number of reasons. Firstly, we will cover (at high speed) some new and quite difficult mathematics. Secondly, to solve vibration problems, we will use almost all the techniques developed in the rest of the course. Thirdly, we will, for the first time, use mathematical models to make engineering decisions. Finally, we will cover the material in a manner that differs significantly from the textbook. For these reasons, we strongly recommend that you attend classes for the rest of the semester, even if you have been able to avoid doing so up to now.

6.1 Overview of issues in controlling vibrations

Vibration is a continuous cyclic motion of a structure or a component.

Generally, engineers try to avoid vibrations, because vibrations have a number of unpleasant effects:

Cyclic motion implies cyclic forces. Cyclic forces are very damaging to materials.

Even modest levels of vibration can cause extreme discomfort;

Vibrations generally lead to a loss of precision in controlling machinery.

Examples where vibration suppression is an issue include:

 

Structural vibrations. Most buildings are mounted on top of special rubber pads, which are intended to isolate the building from ground vibrations. The picture below shows rubber pads being laid on top of the foundations of a building under construction.

 

 

 

No vibrations course is complete without a mention of the Tacoma Narrows suspension bridge. This bridge, constructed in the 1940s, was at the time the longest suspension bridge in the world. Because it was a new design, it suffered from an unforseen source of vibrations. In high wind, the roadway would exhibit violent torsional vibrations, as shown in the picture below.

 

 

This was the ultimate consequence of the vibrations (to the credit of the designers, the bridge survived for an amazingly long time before it finally failed):

 

It is thought that the vibrations were a form of self-excited vibration known as `flutter.’ A similar form of vibration is known to occur in aircraft wings. Interestingly, the new cable stayed bridges that were constructed in the past 10 years or so also suffer from a new vibration problem: the cables are very lightly damped and vibrate badly in high winds (this is a resonance problem, not flutter). At present, there is no known cure, and this is a serious problem.

Vehicle suspension systems need little explanation…

Precision Machinery: The picture below shows one example of a precision instrument. It is essential to isolate electron microscopes from vibrations. A typical transmission electron microscope is designed to resolve features of materials down to atomic length scales. If the specimen vibrates by more than a few atomic spacings, it will be impossible to see! This is one reason that electron microscopes are always located in the basement – the basement of a building vibrates much less than the upper floors.

 

 

Here is another precision instrument that is very sensitive to vibrations.

 

The picture shows features of a typical hard disk drive. It is particularly important to prevent vibrations in the disk stack assembly and in the disk head positioner, since any relative motion between these two components will make it impossible to read data. The disk stack assembly has some very interesting vibration characteristics (which fortunately for you, is beyond the scope of this course).

Vibrations are not always undesirable, however. On occasion, they can be put to good use. Examples of beneficial applications of vibrations are

Ultrasonic probes, both for medical application and for nondestructive testing;

 

The picture shows a medical application of ultrasound: it is an image of someone’s colon. This type of instrument can resolve features down to a fraction of a millimeter, and is infinitely preferable to exploratory surgery. Ultrasound is also used to detect cracks in aircraft and structures.

Musical instruments and loudspeakers are a second example of systems which put vibrations to good use.

Finally, most mechanical clocks use vibrations to measure time.

 

Vibration Measurement

When faced with a vibration problem, engineers generally start by making some measurements to try to isolate the cause of the problem.

There are two common ways to measure vibrations:

(a) An accelerometer is a small electro-mechanical device that gives an electrical signal proportional to its acceleration. The picture shows a typical 3 axis accelerometer.

(b) A displacement transducer is similar to an accelerometer, but gives an electrical signal proportional to its displacement.

Displacement transducers are generally preferable if you need to measure low frequency vibrations; accelerometers behave better at high frequencies.

The most common procedure is to mount three accelerometers at a point on the vibrating structure, so as to measure accelerations in three mutually perpendicular directions. The velocity and displacement are then deduced by integrating the accelerations.

 

6.2 Features of a Typical Vibration Response

The picture below shows a typical signal that you might record using an accelerometer or displacement transducer

Important features of the response are

 The signal is often (although not always) periodic: that is to say, it repeats itself at fixed intervals of time. Vibrations that do not repeat themselves in this way are said to be random. All the systems we consider in this course will exhibit periodic vibrations.

 The PERIOD of the signal, T, is the time required for one complete cycle of oscillation, as shown in the picture.

 The FREQUENCY of the signal, f, is the number of cycles of oscillation per second. Cycles per second is often given the name Hertz: thus, a signal which repeats 100 times per second is said to oscillate at 100 Hertz.

 The ANGULAR FREQUENCY of the signal, , is defined as . We specify angular frequency in radians per second. Thus, a signal that oscillates at 100 Hz has angular frequency 200 radians per second.

 Period, frequency and angular frequency are related by

 The PEAK-TO-PEAK AMPLITUDE of the signal, A, is the difference between its maximum value and its minimum value, as shown in the picture

 The AMPLITUDE of the signal is generally taken to mean half its peak to peak amplitude. Engineers sometimes use amplitude as an abbreviation for peak to peak amplitude, however, so be careful.

 The ROOT MEAN SQUARE AMPLITUDE or RMS amplitude is defined as



 

6.3 Harmonic Oscillations

Harmonic oscillations are a particularly simple form of vibration response. Consider the spring-mass system shown below (you will only see the spring-mass system if your browser supports Java).

 

If the spring is perturbed from its static equilibrium position, it vibrates (press `start’ to watch the vibration). We will analyze the motion of the spring mass system soon. We will find that the displacement of the mass from its static equilibrium position, , has the form

Here, is the amplitude of the displacement, is the frequency of oscillations in radians per second, and (in radians) is known as the `phase’ of the vibration. Vibrations of this form are said to be Harmonic.

Typical values for amplitude and frequency are listed in the table below

	Frequency /Hz	Amplitude/mm
Atomic Vibration
Threshold of human perception	1-8
Machinery and building vibes	10-100
Swaying of tall buildings	1-5	10-1000

 

We can also express the displacement in terms of its period of oscillation T

The velocity and acceleration of the mass follow as

Here, is the amplitude of the velocity, and is the amplitude of the acceleration. Note the simple relationships between acceleration, velocity and displacement amplitudes.

Experiment with the Java applet shown until you feel comfortable with the concepts of amplitude, frequency, period and phase of a signal.

Surprisingly, many complex engineering systems behave just like the spring mass system we are looking at here. To describe the behavior of the system, then, we need to know three things (in order of importance):

(1) The frequency (or period) of the vibrations

(2) The amplitude of the vibrations

(3) Occasionally, we might be interested in the phase, but this is rare.

So, our next problem is to find a way to calculate these three quantities for engineering systems.

We will do this in stages. First, we will analyze a number of freely vibrating, conservative systems. Second, we will examine free vibrations in a dissipative system, to show the influence of energy losses in a mechanical system. Finally, we will discuss the behavior of mechanical systems when they are subjected to oscillating forces.