Teaching Vibration to university undergraduates

This paper provides a description of an undergraduate course on vibration, given to second and third year students at Bristol university in the UK. The course, and my teaching philosophy, were developed over more than 30 years. The lectures were given in two 20-hour courses which were supported by an equal number of examples classes. Students were provided with a series of question sheets which contained questions from previous examination papers. In addition, laboratory classes were provided to give the students “hands on” experience on how to excite, control, measure, and to interpret various vibrating systems. The first set of lectures began with the analysis of a single degree of freedom system, adding different forcing functions and more degrees of freedom. The second set of lectures introduced continuous systems, consisting of bars, beams, and plates. The limitations of reality, particularly boundary conditions, was emphasized. Wherever possible, some artifact was taken to the lecture amplify the mathematics. I have interleaved into the presentation some of my teaching philosophy and how it is important in a heavily mathematical subject such as vibration to teach rather than to try and impress the students as to how clever is their lecturer. Finally, if the lecturer does not enjoy giving the lectures, the students will not receive that “extra” which distinguishes a good lecture from a bad lecture, and also distinguishes a good lecture from reading a text book.

tools during his/her career. The first question to be asked about the teaching of any part of any subject is "Why should we teach this"? Questions as to when, how, and how deep will surely follow. Some subjects, such as electronics and computer science have moved very quickly in the last few years while subjects such as dynamics, elasticity, plasticity, and thermodynamics have hardly changed, while they are still necessary to understand modern engineering. But some people have odd perspectives. I was once asked by a lecturer in electronics why we still taught steam in our thermodynamics courses. He had no concept as to how electricity was generated via steam turbines and to gain a tiny increase in efficiency had major economic implications. Vibrations, being a major subset of dynamics, is important in all aspects of engineering where structural forms can be subject to some form of excitation. It is therefore necessary to understand how structures respond and what is the nature of the excitation. These days, complex structures can be analysed using computer programs such as finite element [FE] analysis. However, to understand the limitations and pitfalls of these programs, the basic building blocks on which they are based must be understood. This also includes efficient approximate methods, such as Rayleigh and Stodola, for obtaining solutions. In vibrations, real systems are modelled so that we can understand why a structure or system responds in the way it does to some sort of excitation. We teach simple [lumped] systems where the mass, stiffness and damping are separate, and systems such as bars, beams, and plates where the mass and stiffness are "distributed". In my own teaching, the lumped systems were in year 2 and the distributed systems in year 3. Both lumped and distributed systems can quickly become very complicated mathematically, except for very simple cases. Without a thorough understanding of those simple systems , FE and similar models cannot be created and only the unwise use them without a good knowledge of the underlying mechanics. There is an unfortunate tendency in universities to teach only that which can be examined. It leads to lots of mathematical manipulations which are readily marked in examinations. However, the useful part of engineering is not just solving algebra but understanding the limits of such algebra we have applied to a real engineering situation. Unfortunately, this "useful" material is not easily taught and examined as it requires a maturity of experience which undergraduates do not have. However, the lecturer ought to have some relevant experience from his/her research, experience in industry, or consulting. Imparting experience forms a useful means of breaking up a solid hour of mathematics and allows some of the slower members of the class to catch up. All classes will have a range of ability and my experience of teaching means it is not easy to keep everyone happy all the time. Some of the "whiz kids" can get impatient with deviations from straight line mathematics as all they are interested in is doing well in the examinations; such people do not always become good engineers. At a dinner, I asked a student from another engineering Department which lecturer was generally thought of [by the students] as the cleverest in his Department. His answer was Dr X. This surprised me as Dr X had a very light research record, so I asked why he was the cleverest. He said that Dr X's lectures were so difficult to follow that he must be very clever as only he seemed to be able to understand them. In contrast, there is also the danger of making lectures too easy to understand so students lose concentration and hence the thread of the lecture. For this reason, it is vital to be able to see the faces of the students as this will give a good indication as to whether they are bored or baffled. It is also important to encourage students to ask questions during a lecture since for every bold student, there will be ten timid ones who would have liked to have asked that question but did not have the courage to do so. Above all, with a subject such as vibrations, where there is a substantial mathematical content, the lecturer must communicate and not just try to look clever.

What do we teach and how to get good outcomes
All universities and colleges will have different resources and will allocate different weighting to their menu of courses. I will describe what I did at Bristol University when I taught there on Vibrations for a period of over 30 years. I gave 20 lectures, each of 50 minutes duration, in years 2 and 3. While I am not trying to write another book, it will be necessary to describe in some detail what was taught.

Teaching programme
In Year 2, an introductory lecture was used to outline the course and to talk generally about why engineers need to understand what happens due to unwanted vibration. Problems of fatigue, noise, discomfort, and component malfunction were described. The mathematical representation of vibration was introduced using a vector, u0, rotating at an angular frequency ω such that the projection on a horizontal plane is u0 cos ωt where t is the time from some datum as shown in Figure 1. The angular frequency ω [rad/sec] [=2πF] is used for convenience; F is the frequency and has units 1/time or, scientifically, Hertz [Hz] and is such that ω = 2πF. It is easily shown that for a vector rotating at an angular velocity ω, the period of one oscillation is τ such that ωτ = 2π and that Fτ = 1. The variation of amplitude with time is given in Figure 2.

Free vibration, Single degree of freedom system [1DoF]
The first few lectures considered a single degree of freedom system consisting of a mass, spring, and a viscous damper, as shown in Figure 4. By considering the forces acting on the mass, the basic equation: can be derived. In my notation, u,v, and w are displacements in the x, y, and z directions, m is the mass, f is the dashpot constant and k is the spring stiffness. The solution to Equation 1 is of the form: The special case of c = 1 has a response given by: Note that Equation 1 was developed for a horizontally moving mass where gravity is not considered. But if the mass is hanging vertically, the position it sits at rest, the equilibrium position, must be defined and oscillation defined from that datum. Otherwise, the equations become rather messy, and it is easy to make mistakes.
The students know what a mass and a spring look like, but few will have met a viscous dashpot. Having worked on my own cars, I had a damper or two in my garage, so these were handed round the class to get the "feel" of how they acted. As most students are interested in cars, they all know the terms "damper" and "shock absorber" and use them interchangeably. Now is the opportunity to illustrate the lectures on single degree of freedom systems without any excitation term. A damper serves to reduce any oscillation of the car body relative to the wheels. Until the mid 1930s, most cars used friction dampers. These consisted of friction discs compressed by a central bolt. If the bolt was too tight, the damper was locked solid and so did not work. If the bolt was too slack, there was little frictional resistance and so little damping. These friction dampers were replaced by oil filled telescopic viscous dampers which have a stronger force in one direction than the other for a given velocity. Why? This is where the shock absorber behaviour is revealed. A 1000kg car will have each wheel supporting about 250kg, so the spring will normally have a compressive load of 2500N. If the wheel hits a raised bump in the road, the wheel is forced upwards, compressing the spring and damper. The force in spring cannot be avoided [hard or soft springs…] but the force in the damper is proportional to the dashpot constant. To reduce the force transmitted to the vehicle body, this dashpot constant should ideally be zero, or as small as possible. Now consider the opposite case where the wheel encounters a downward depression or hole in the road. A typical wheel might have a combined mass [rim plus tyre] of 15kg for a 1000kg vehicle. We now have a force of 2500N acting on a 15kg mass. The spring will accelerate the wheel towards the bottom of the hole where it will suddenly stop, causing a severe impulse on the car body. This is where the shock absorber function comes into action. If the damper constant is high, the downwards acceleration of the wheel, and hence the impulse, will be reduced. This is why the dashpot constants in the two directions are different. Of course, for practical considerations, there has to be a compromise, and note that modern telescopic dampers can be much more complicated than a simple piston in a tube. Equation 1 and its solution are very useful in illustrating not only the need to understand vibration, but also how this affects everyday events. Explaining to the students that the damping in a typical modern car is near to critical and this can be seen by pushing a car up and down and then seeing how long it takes to come to rest, is left to the discretion of the lecturer… It was also pointed out that viscous damping is a mathematical convenience. Apart from certain polymers and oil based viscous dampers, material damping is hysteretic and not frequency dependent, but is often amplitude dependent. Also, much structural damping is due to friction which is certainly not viscous.

Forced vibration, 1 DoF
Next comes forced vibration where there are two cases, neither of which is exactly followed in practice. In the first, a harmonic force of magnitude P acts on the mass. This magnitude of this force is usually kept constant and it is assumed that the force can move through any distance as the mass moves, even at resonance. In this case, we have: From the forces acting on the mass shown in Figure 5 and applying Newton's second law, the equation of motion is given by: The solution of which in the steady state is given by the Frequency Response Function, FRF: In practice, the force is often due to a rotating out of balance or a reciprocating component such that the magnitude of the force is proportional to ω 2 . This is particularly important at frequencies above resonance. The second case in forced vibration is where there is abutment [earthquake] excitation as shown in Figure 6. Here, it is assumed that whatever is causing the excitation is such that any reaction back to the abutment has no effect on its ability to shake the vibrating system. Now, we have: The relative motion between the abutment and the mass is ur such that: The responses u and ur are the Frequency Response Functions. These should be sketched out so that resonance can be defined and the characteristic differences between the different forms of excitation and response [displacement, velocity or acceleration] can be pointed out. An important, and often overlooked, feature is Transmissibility, T, which defines the force transmitted from the vibrating system to the support, and thence to its surroundings. For the case of an excitation of the form P cos ωt [see Figure 5] we have that the force transmitted to the support, FT, is given by: and it can be shown that: Which is of the same form as the FRF for abutment excitation. The concept of Transmissibility leads to another demonstration and an introduction to acoustics. I made a box about 500mm cube which was open on one side. It was made from 20mm thick chipboard and lined with 20 mm thick carpet underfelt. The open side was supported on 10mm diameter rubber tube which provided a seal when the box was rested on it. The excitation source was a small electric "buzzer" which sat on a 100 gm mass. Some soft foam and some light coil springs completed the inventory. When held in the hand, the buzzer can usually be heard by the class. However, when placed on a desk, it couples well because the desk acts as a sounding board [as in a violin]. But when we introduce the soft foam or springs between the buzzer and the desk, the sound is much reduced. We now see that soft springs are good for reducing the force transmitted to the desk, which can be seen in the variation of T with frequency above the resonance region where r >> 1. If the box is placed on the desk with the buzzer inside and isolated from the desk, all is silent. If the foam or springs are removed, the buzzer can now be clearly heard as it is exciting the desk and so by-passing the box. If we now put the buzzer in the box on the foam, all is silent again until the box is tilted with a 50mm or so gap towards the students. Now, the buzzer is clearly heard. So, what has been learned? First and foremost, isolation from the surroundings reduces the energy transmitted. Second, a simple acoustic enclosure is very effective at reducing airborne acoustic propagation, provided the source [buzzer] is isolated so that it cannot by-pass the box. Third, even a small opening in the enclosure easily allows sound to escape. At the other end of the scale, I showed a rubber block which contained 2mm thick steel sheets which was used as a building support where earthquakes were prevalent. The steel sheets were placed in the horizontal direction; this provided a stiff vertical support but was flexible to the [mainly] horizontal motion which is so dangerous in earthquakes.

Two or more degrees of freedom
Systems with two or more degrees of freedom [DOF] were also taught. While two degrees of freedom could be solved "by hand", systems with three or more degrees of freedom were tediously complex and best left to a computer. Also, even with 2 DOF, it was difficult to solve the equations if damping was added. A 2 DOF system, without damping, is shown in Figure 7. Note that m2 can either be connected by one spring [k2] or by another which is terminated in an abutment.
If we know k1, k2, k3 and m1, m2, we can determine 1 and 2. There will be a different mode shape, u2/u1, at each natural frequency. Because there are now two natural frequencies, it can be shown that: 1st mode shape , = = nd mode shape  The constants A, B, C, D are determined from the initial conditions. A typical response of one of the masses to an initial displacement might be of the form: If this signal is electronically filtered, we would see that it has two natural frequencies at 3 and 7 rad/sec as shown in Figure 9. Note that my physical model was not as in the diagram above but needed an additional spring of stiffness k1 attached to another abutment on the right. By giving the base a sharp push, the two cantilevers could be made to move together in the first mode of vibration. By holding them apart and releasing, the second mode was demonstrated. By moving the spring to different positions, thereby changing its effective stiffness, it could be shown that the frequency of the first mode was independent of the spring stiffness while that of the second mode was not. But now we come to the interesting part. By attaching a 100gm magnet to one of the strips, displacing and then releasing, it could be shown that energy was transferred from one cantilever to the other and back again such that each in turn might be instantaneously at rest at one time or another or, by suitable positioning of the magnet, only one might come to rest instantaneously. It was easy to move the magnet and to give different initial conditions, creating many ways in which the system could vibrate, all of which could be described by the equations above. A colleague said he could model my system on his computer and demonstrate it on a screen, but it would be much less interesting or convincing to the students. Forced vibration with a 2 DOF system is straightforward [providing damping is not introduced] .
Using P cos ωt as the excitation applied to mass 1, it can be shown that the motion of the two masses is given by: Inspection shows that resonance will occur when the excitation frequency ω coincides with either of the two natural frequencies. An important result is when ω 2 = k2/m2. This defines the detuned frequency where the first mass is at rest while the second mas s moves to exactly balance the excitation. In certain cases, this is a very useful phenomenon and can be used in practice with great effect. If damping is present, u1 is never zero but the frequency range over which the vibration amplitude is reduced is widened. The case with damping is sometimes referred to as a vibration absorber. I was consulted by a former student to see if it was possible to control the vibration transmission from a large shaker [using contra-rotating out of balance masses] used to crush refractory powder in a ball mill. The transmitted vibration was causing damage to the walls of the building and suggested solutions were very expensive or unfeasible. Fortunately, the electric motor ran at a constant speed, so by designing my detuner to oscillate at this frequency, the transmitted vibration was reduced almost to zero, much to the amazement of the technicians who ran the ball mill. A year later, it was still working perfectly. Colleagues who worked on machine tool vibration used this same principle [usually with damping] to stop machines "chattering" when the machine tool vibrates so that the cutting tool leaves unwanted marks on the workpiece.

Torsional vibration
Torsional vibration is equivalent to linear vibration except that k  torsional stiffness The main complication occurs in geared systems as shown in Figure 11.
The reduction ratio = n = r3/r2 = -2/3 To simplify the solution, we let I3 be zero. Considering the moment acting on I1 and applying Newton's second law, we have: F r Since I3 = 0, the net torque acting on 3 is zero, so: We have 5 equations and 5 unknowns, so we can eliminate 4 unknowns to get a single equation in say  2 , which will give the frequency equation. After some manipulation, we have: / This has two positive roots, 1 and 2. The mode shapes can be determined by substituting  1 and 2 in turn into the above equations.

Approximate numerical solutions of Rayleigh and Stodola
Students were introduced to Rayleigh's method for obtaining a numerical estimate of the lowest natural frequency of a structure. Rayleigh's method is based on energy. For a linear system, with no damping, the total energy at any time is constant, Kinetic Energy + Stored Energy is a constant, i.e. KE + SE = R, so maximum K.E. = maximum S.E. (SE = 0) (KE = 0) For a system of masses m1 m2 ... with displacements u1 u2 ... vibrating at some natural is the true natural frequency according to that mode shape. However, if we guess the mode shape, i will not be exact. Rayleigh's theorem tells us that a reasonable approximation to u1 u2 ... up will give a good approximation to i. Three points to note are: (1) An incorrect guess will be equivalent to applying constraints to the sys tem, leading to an increase in SE. Thus, for the fundamental frequency, Rayleigh will always give a high value of . The better the guess, the nearer is  to the true natural frequency.
(2) We always use strain energy and not gravitational energy since oscillation about the equilibrium position removes the gravitational term.
(3) A good guess is often the static deflected shape.   n -0 + 'error in mode shape' Figure 12: Illustration of the frequency error as a function of mode shape with Rayleigh's method A method of improving the accuracy of the frequency prediction is to use Stodola's iterative method which uses the frequency predicted by Rayleigh's method, together with the equation of motion, to improve the guessed mode shape. This step can be repeated as required. A simple example using the system in Figure 7 with equal masses and springs can produce a very accurate value for the mode shape and frequency in just a few steps.
In year 3, the topics covered were on systems with distributed mass and stiffness, such as bars, beams, and plates. The differential equations were different from those used for the discrete systems taught in year 2, and particular attention was paid to boundary conditions, since these describe how the vibrating system interacted with its surroundings. Because the vibration changes with position as well as time, we need to use partial differential equations. Because of the more complicated mathematics involved, damping was not introduced, nor were forcing functions. The aim was to establish natural frequencies and mode shapes , since these are what are needed in practice. Introducing time functions was by-passed and it was assumed that we always had steady state sinusoidal oscillation. Transient situations will occur, of course, and can be dealt with by using the mathematics developed if so needed.

Axial Vibration
I started the course with the axial vibration of uniform prismatic bars. In practice, the treatment was restricted to bars which were slender, in which the length/diameter ratio was at least 10. Transverse [radial] motion is caused by Poisson's ratio coupling so the "length" parameter really relates to the wavelength. At higher frequencies, the wavelength becomes shorter so that the simple equations need to be interpreted with care. A schematic of a uniform bar in axial vibration is given in Figure 13 where the symbols have their usual meaning. We consider an element of the bar at distance x from one end; the arrows shown define the positive x direction. The stress, strain, and displacement will Similarly, the force acting on face 2 in the positive x direction is : By subtracting the forces on face 1 from those on face 2, and noting that a positive value of ∂u/∂x represents a tensile strain, we can show that the net force acting on the element is Note that we must now use partial derivatives since u is a function of both x and t. By applying Newton's second law, we have: Which, in the limit, gives us the equation of motion for axial vibration, It is useful to note that E/ρ = c 2 where c is the speed of extensional waves in the bar. The general solution of Equation 4 is: . Where: u = (P cosx + Q sinx) (T cos t + U sin t) = 2/c  = frequency (Hz) = 2 The boundary conditions give the constants P, Q, T, U. Normally, T and U are not considered as most cases concern steady state oscillation, but they are there if needed. A bar may be fixed at one end or free. It might also be terminated by a spring or a [point] mass. Apart from the free condition, all the other boundary conditions are impossible to achieve in practice. For instance, a steel bar cannot be attached to a block of infinite stiffness, springs have mass as well as stiffness, and masses cannot be made of a material of infinite density so cannot be "point" masses. However, it is useful to look at the ideal situations as these give a guide to the likely natural frequencies and mode shapes to be expected. Fixed no movement The force on the mass is in the negative direction of x which gives: If we have a bar fixed at one end with a mass at the other, we have: There must be a continuity of force at the junction, so:

 
Note that it is MUCH better to specify x1 and x2, u1 and u2, in opposite directions as in Figure  20. If you don't believe me, try the alternative! Because u1, and x1 are positive in the same direction, as are u2 and x2, then: And both indicate tension. At the join, we have: Such a system, in which the two half bars are of equal length, is used in a resonant wave guide for high power ultrasonics. It can be shown that the ratio of the amplitudes of vibrati on at the two ends are inversely proportional to the areas of the bars. At resonance, each of the two bars is a quarter wavelength long. In reality, the conditions at the join with respect to force are mathematically incorrect and the sharp change in section is smoothed by a radius to avoid fatigue failure. Nonetheless, it works, and I used such a device resonating at 11.6 kHz for over two years when measuring the damping of metals at cyclic stresses up to their fatigue limit.
So the net torque acting on the element is: Using Newton's second law in torsional rotation and noting that the moment of inertia of the element is Jρ.δx gives our equation of motion: which is very similar to that for axial vibration. Flexural vibration of beams Flexural vibration is somewhat more complicated than axial vibration. I used the Bernoulli-Euler theory for the bending of beams in which plane sections remain plane and perpendicular to the neutral axis. It is necessary to adopt and rigidly adhere to a strict and consistent si gn convention. For this reason, the sign convention was attached to the equation sheet issued to the students early in the course and made available to them in the exam. Other conventions exist and it is a matter of choice which is used. It is essential that positive and negative faces are clearly defined; in our convention, a positive face is where the outward going normal is in the positive x direction. The displacements u, v, w, are in the x, y, z directions. We have defined the y direction as positive downwards and the z direction as directed into the page. where ρA ω 2 = EIα 4 and I is the second moment of area. Appropriate mathematical manipulation leads us to the solution of this partial differential equation of the form: Where the constants PQRSTU are defined by the boundary conditions and ω = 2 πF and F is the frequency of oscillation. [I have F for frequency and F for shear force as there is an unfortunate clash of accepted nomenclature]. T and U will only be called on if there is a need for a transient vibration solution, but this will only be a tiny minority of cases as we are mainly interested in mode shapes and natural frequencies. Because we have 4 constants PQRS, we need 4 equations which can usually be obtained from the boundary conditions. At a free end, the bending moment, M, and the shear force, F, will be zero, so ∂ 2 v/∂x 2 =0 and ∂ 3 v/∂x 3 = 0 At a fixed end, there is no deflection and no slope, so u = 0 and ∂v/∂x = 0 At a pinned [simply supported] end, there is no deflection and no moment, so u = 0 and ∂ 2 v/∂x 2 =0 Of these boundary conditions, the free end is easily obtained, but the other two are difficult, if not impossible, to achieve in practice. For a beam vibrating freely (no end restraints), the frequency equation is: The solution of this equation cannot be obtained explicitly and needs to be solved numerically. Figure 24 gives the mode shapes and the nodal positions for the first five modes of vibration. The constant B is used to define the natural frequency using the equation: where is the length of the beam, d is its thickness, and  E c  . Other boundary conditions, such as a mass at the end of a beam or some position along it, can be incorporated by using the bending moment and shear force terms from the bending of the beam together with Newton's second law. But note that since point masses do not exist, the results will only be approximate. It is very important that signs are carefully observed, and it is understood what [shear force or bending moment] is acting on which face. I give below an example for a mass at the end of a beam. In Figure 23, consider the element of length δx as a point mass. The positive shear force on the negative face [of the mass] is acting in the negative x direction. Using Newton's second law, we can write: -F = m ∂ 2 v/∂t 2 But since F = -EI [∂ 3 v/∂x 3 ]l and ∂ 2 v/∂t 2 = -ω 2 vl Then, EI [∂ 3 v/∂x 3 ]l = -m ω 2 vl I also introduced the class to the concepts of additional terms to the Bernoulli -Euler equation for a flexurally vibrating beam to allow for shear and rotary inertia. These terms are needed if the beams are thick or for predicting the frequencies of higher modes.
where w is the deflection in the z direction. This differential equation can only be solved explicitly for certain conditions. If the plate sides are simply supported (hinged), then the solution is: , which has some interesting characteristics. As a tapered flat sheet of steel terminated in a wooden handle, it has a variety of natural frequencies. The saw has 3 edges free and the fourth has an uncertain termination and is heavily damped because of the handle and how it is held. However, when the blade is bent into an "S" shape, the damping is very small and it is possible to play musical notes when stroking one of the long free edges with a 'cello bow, or even a wooden rod. On one occasion, the students filmed me playing the saw [sitting on a chair which was on a table] and sent it to their non-engineering friends to show that engineering lectures were not boring! Rayleigh's method for continuous systems Continuous systems, such as rods, beams, and plates, present some challenging , and often impossible, problems to solve by conventional algebra. Numerical solutions based on Rayleigh's method are particularly useful in predicting natural frequencies, especially the fundamental frequency. The same principles apply as with lumped systems. In particular, the better the guessed mode shape, the more accurate is the frequency prediction. Knowledge of Rayleigh's method can be tested in examinations, so it is worth teaching both for knowledge and assessment purposes. Using a fixed-free bar in axial vibration as an example, it can show that by using a ¼ sine wave as the mode shape, the exact natural frequency is obtained. This is not surprising but is a useful demonstration of how Rayleigh's method works. But a linear mode shape where the displacement is proportional to x gives an error of only 10.25%, even though it violates the boundary condition [zero strain] at the free end of the bar. On the other hand, using the statically deflected shape, in which the bar hangs under gravity from the fixed end, gives an error of only 0.635% as this mode shape satisfies the boundary conditions at both ends. Of course, the algebraic computation is much greater than for the linear option. But whereas the case with a mass on the end of a bar needed and graphical/numerical solution, an example with a linear mode shape and a mass equal to that of a bar could be solved in a few lines with an error of less than 1%.
For flexural vibration, a cantilever was used as an example. Note that there are now 4 boundary conditions to satisfy. A variety of mode shapes was used. The statically deflected shape gave an error of less than 1%, a cosine-based shape had an error of 1.5% while a parabola gave an error of over 25%.
With plates, boundary conditions other than all simply supported generally have to be solved numerically using beam functions and Rayleigh's method. These days, computer solutions using finite element analysis are normally used to obtain natural frequencies and mode shapes. In practice, circular plates supported on razor blades are the nearest real situation to the mathematics. Using such a support system for rectangular plates poses problems of rotation at the corners which can be partially solved by omitting the supports near to the corners. Non-linear vibration A topic which is important when it comes to testing resonant systems concerns non-linearity. This has been extensively covered by many authors and quickly slips into complex mathematics if other than a qualitative system is considered. In effect, there are softening springs and stiffening springs. With softening springs, the resonant frequency decreases with amplitude, while the opposite is true for stiffening springs. With a linear system, there is only one response amplitude for a given frequency. However, in non-linear systems, depending on the level of damping there can be three possible amplitudes near to resonance. As the driving frequency is increased towards resonance for a softening spring, the amplitude of vibration will suddenly increase [jump] from A to B as shown in Figure 25, and then decrease without reaching the resonant amplitude. When the frequency is decreased from above resonance, the resonant amplitude will be reached and then decrease as the frequency is reduced before there is a sudden jump downwards from C to D. The part of the response curve from A to C can never be found in practice. A stiffening spring slopes the other way and also gives the jump phenomenon. For what it is worth, one of the few laboratory practicals I can remember from undergraduate days concerned the flexural vibration of a simply-supported beam which was excited at its mid point by an out of balance mass driven by an electric motor. Because the "simple supports" were very firmly clamped knife edges, deflection of the beam induced tensile forces which gave a stiffening characteristic. I spent much of my Easter vacation trying to find out why my experimental results did not conform to the expected Frequency Response Function for a linear system and to explain what had happened. Frustrating, but I learned a lot! Figure 25: Frequency response function for a softening spring Beyond a simple, qualitative description of nonlinearity and how it affects the resonant response, the algebra ascends [descends?] into hideous complexity which is completely unjustified for an undergraduate course. Examples classes Most courses will provide questions following the lectures so that students can test themselves on their understanding our the course.
In year 2, there was a one-hour examples class to support each lecture. In these classes, which usually followed the lecture, the students were issued with sheets of questions which were typical [usually actual] exam questions. However, it is important to grade the questions so that the first few are rather easier to give students confidence in tackling the paper. Attendance at the classes was optional and no records were kept on attendance. I was always at these classes to provide help to the students, two or three minutes being enough for me to see a difficulty and to help resolve it. From time to time, I tried using postgraduates [PhD students or post docs] to assist me, but I found that they spent too much time and often caused more confusion. As the classes were rarely more than 30 students [optional attendance], I preferred to be the lone assistant. There was, of course, a significant amount of self-help between the students themselves. In year 2, the numerical answers were given out, but not the worked solutions. A set of typical examples sheet is given in the Supplementary Information: [SI_Continuous_systems_example_sheets,SI_Lumped_parameter_Introductory_Example_sheets_1A&2A,SI_Lumped_parameter_Examples_sheets].
In year 3, the Department did not timetable examples classes, but I was available in my office to help at a specified time. I gave out worked solutions but only much after the question sheets. I found that if such solutions were made available with the question sheets, there was a tendency to skim through the solution but not to understand the basics. As always, the students have a wide range of ability and motivation, and have their own agendas and timescales. That's life, but sometimes people have to be protected from themselves. For both courses, the exams consisted of a three hour paper following completion of the lecture course. The students were not allowed books or notes but were issued with the same crib sheet they had been working with during the year. Exam papers from previous years were available and students were issued with the answers and some worked solutions. Laboratory Classes One of my professors, G F C Rogers, a thermodynamicist who wrote a well-known book with Y R Mayhew, challenged me when I joined the Department to introduce at least one new laboratory experiment each year. Rogers was a great believer in the benefit of laboratory experiments in developing the understanding of engineering principles. There were many discussions as to whether the laboratory classes should come after the relevant lecture or before it. The conclusion was that there were benefits both ways and realistic problems of space, equipment, and the timetable meant that some students had the lectures before, and some after and it did not seem to make a lot of difference in the long run.
To support my vibration lectures, I created a series of laboratory classes [most of the other courses did the same]. These were to help the students to see the lectures from a new perspective. The classes were staffed mainly by PhD students [from my own group] but I had a wandering wizard role to check that all was going smoothly and to add a few words of [I hope] enrichment. Lab sheets were issued to help the students and to guide them into what they might discover from their experience. The students worked in groups of 4 as there was only limited equipment available. While the lazy student might sometimes slip into the background, it was excellent practice for team working. A couple of typical lab sheet is given in the Supplementary information for each part of the course [SI_2nd_year_vibrations_lab_2DoF_sheets and SI_3rd-year-vibrations-lab-plates_sheets]. At one stage, I added an "applied" part to the standard experiment. In one case, I had a Mini exhaust system, excited by a rotating out of balance device, which the students examined for resonances and where it was best to mount the rubber suspension so as to minimise transmission to the vehicle body; it turned out that the manufacturer had got it right. In another example, two car doors were examined for vibration response. One door was just the metal [body in white] and the other was fully trimmed inside. Timetable constraints eventually saw the end of these additions… Sadly, academic time is increasingly being consumed by the demands of research and pointless administration. Also, undergraduate laboratories need space which is not used for the whole year. Consequently, undergraduate laboratory classes are being slowly dropped from the timetable in many universities. We always need to change with the times and my own experimental programme was new once and often replaced earlier experiments. The laboratory classes are an essential part of bringing an understanding of the mathematical content of the lecture courses. The course lecturer cannot run each experiment, but must be seen, even as a wandering wizard, several times during the class.

Conclusions
I hope the reader will find the above useful as a basis for his/her own teaching of a course in vibration. My lecture course was developed over many years and you see above the finished product. It fitted into the time allowed by the timetable and was aimed at stretching the intellect of the students, while not trying to blind them with difficult mathematics. You can add to it or subtract from it to suit your enthusiasm, experience, and the time available. But, above all, you must work to enjoy your presentation and always give it 100% effort. Do not be afraid to deviate and recount [briefly] some relevant experience or to tell a joke. Yes, I did sit in a bar in New Orleans watching an out of balance ceiling fan and wondering how I could make it into an exam question, and the Ogden Nash contribution to the compatibility of forces and displacements is left to your literary research.

UNIVERSITY OF BRISTOL Department of Mechanical Engineering
Notes which may be used in the Vibrations 2M examination.

 
(rad / sec) = 2 (Hz)(cycles / sec).  Two-degrees-of-freedom: Transient: Positive forces and moments acting on positive and negative faces 1. An axially-vibrating system consists of n rigid masses, m, connected by n -1 springs each of stiffness k. If n >> 1, this system may be modelled as a uniform bar of length l and crosssectional area A made from a material of density  and Young's modulus E.
An ore-train consisting of 100 trucks, each weighing 30 tonnes is close-coupled such that the buffers are in a state of compression. Each truck has a pair of buffers at each end. Each buffer has an effective spring stiffness of 1.2 x 10 6 N/m.
Assuming that the buffers have sufficient pre-compression that they do not separate, calculate the fundamental longitudinal natural frequency of the system.
(Ans: 3.1623 x 10 -2 Hz) 2. (Fig. Q2) A steel bar of length 250 mm and radius 10 mm is vibrating in its fundamental axial free/free mode of vibration, such that the amplitude at a free end is 0.1 mm. Calculate the frequency of vibration and the cyclic axial strain amplitude at the mid-point of the bar.
The bar is horizontal and it is observed that if a loop of light wire is put on the bar near either end, as shown in Fig. Q2, it moves along the bar and stops at the mid-point. Explain why this should happen.

Fig. Q2
3. The transverse vibration model of a portal frame milling machine is shown in Fig. Q3. The columns are pin-jointed at one end and bolted rigidly to a stiff, heavy cross beam at the other end. The cross beam has twice the mass of each column and prevents any significant rotation of the column end faces.
Derive the frequency equation for the structure and determine the fundamental natural frequency. Each column has the parameters ascribed to it in Fig. Q3.
(Ans: 2 = l(tan l -tanh l); l = 1.19;  = ) 4. A helicopter rotor blade may be regarded as a uniform beam which is simply supported at one end and free at the other. The equivalent beam is 4.57 m long with a second moment of area of 8.33 x 10 -8 m 4 and a cross-sectional area of 2.9 x 10 -3 m 2 . The rotor blade is made of an aluminium alloy for which Young's modulus is 71 GPa and the density if 2.63 x 10 3 kg/m 3 .
Find the natural frequencies of transverse vibration in the first and second modes.
(Ans: 3.272 Hz, 10.603 Hz.) 5. A cantilever is driven in flexural vibration by an electrical coil mounted at its free end and moving in the field of a magnet. The coil has been so designed that its mass is sufficiently small that it may be neglected but unfortunately this resulted in the moment of inertia of the coil being by no means negligible.
For the beam, Young's modulus is E, the density is  its length is l and the second moment of inertia is I. The moment of inertia of the coil about an axis through its centre of gravity (which coincides with the end of the beam) perpendicular to the plane of vibration is J.
Determine the frequency equation for this system. If the frequency equation of a cantilever with a completely free end is and this has as its first solution that l = 1.875, indicate whether the addition of the coil will lead to an increase or a decrease in fundamental frequency of the cantilever. A two-stage rocket has been modified to place a large space-laboratory in orbit. The payload may be regarded as a rigid mass attached to the second stage, as shown in Fig. Q1. The mass m of the payload is equal to that of the second stage.
Use Rayleigh's method to determine the fundamental frequency of axial vibration after lift-off. Credit will be given for using a realistic mode shape. (Ans: A ship at sea can vibrate in several modes. One mode is usually laterally as a free-free uniform beam. Using Rayleigh's method, determine the fundamental frequency of lateral vibration for a free-free uniform beam of length  which has a flexural rigidity EI and a uniform weight of  per unit length. The lateral deformation can be described approximately by the equation where b is the maximum lateral deflection at mid ship. Sketch the mode shape.  Ans: 1 2 = k/m (rad/s) 2 ; 2 2 = k(2 + 3n)/2m (rad/s) 2 (u1/u2)1 = 1 ; (u1/u2)2 = -2 2. Determine the natural frequencies and the corresponding mode shapes of the fixed-fixed, two degrees of freedom system sketched in Fig. Q2. (u1/u2)2 = -9.744 3. A system vibrating axially can be represented by two discrete masses 2 kg and 4 kg, attached to separate abutments by springs of stiffness 1.10 6 N/m and 3.10 6 N/m respectively, and connected to each other by a spring of stiffness 2.10 6 N/m. If the deflection of the connecting spring is 0.5 mm when the system is vibrating in its second mode, what is the amplitude of displacement of the 2 kg mass?

Ans:
3. 0.31 mm Teaching Vibration to university undergraduates R. D. Adams Journal on Teaching Engineering, 1:1 (2021) 40-93 84 4. A machine may be modelled as the two degree of freedom system sketched in Fig. Q4. The excitation is equivalent to a force P cos t (where P = 50N) acting on the 10 kg mass as shown. Determine the resonant frequencies 1 and 2 and the amplitude of motion of the 10 kg mass at a frequency of (1 + 2)/2. What is the value of the detuned frequency?

Supplementary information: SI_2nd_year_vibrations_lab_2DoF_sheets
Second Year Vibrations Laboratory

AIMS
To allow you to explore and investigate the vibration characteristics of simple one and two degree of freedom (DoF) systems. Also, to allow you to become familiar with standard vibration measurement equipment. The lab will take approximately 2 1 /2 hours to complete.

REPORTING REQUIREMENTS
(1) Details of the measurements taken and calculations made should be included in your lab book. This will need to be marked by the lab demonstrator before you leave.
(2) Your report should include the following in the discussion; For the l DoF system, discuss the agreement between the experimental results and the theoretical prediction. Is the experimental system a simple spring-mass-damper system?
In the light of the results of the 2 DoF system, discuss how a second mass and spring can be added to a one degree of freedom system to reduce vibration at a given frequency (see Appendix 2 for details of the theory).

OVERVIEW
Although it may be possible to analyse the complete dynamic response of a system, this often leads to complex analysis and the production of large amounts of data. Even if the full dynamic response is required, a first step in any vibrational analysis is to attempt to model the system as either a one or a two degree of freedom system. In this way, much physical insight can be gained and the results act as a useful check on the full results produced later.
When modelling a real system, simplifying assumptions are made. For example, a distributed mass maybe considered as a lumped mass, the effect of damping may be ignored, a non-linear spring may be assumed to be linear over a limited range of motion, and the possible directions of motion restricted. As with any modelling, there is a compromise between simplicity and accuracy. Figure 1(a) shows the apparatus used in this experiment. This a close approximation to a one degree of freedom system which is shown schematically in Figure 1(b) and consists of a Mass (acceleration proportional to net external force), a Spring (force proportional to displacement), and a Damper (force proportional to velocity). At low frequency (low acceleration), when the force required to accelerate the mass is low, the spring stiffness dominates the motion ('stiffness controlled'). At high frequency (high acceleration), the mass dominates ('mass controlled'). Between these 'high' and 'low' frequency regions, the mass and stiffness cancel each other out when they are opposite in perhaps the most popular method of solution today is to use a finite element package, or some other PC-based solution.

Figure 1
Mode shapes of a free-free square plate (the values of B are shown for each mode).
A series of nodal patterns for a free-free square plate as predicted by a finite element model are shown in Figure 1. The constant B is used to define the natural frequency F for each mode using the equation, where, c is the velocity of extensional waves    E  , d is the plate thickness, is the side length, and v is Poisson's ratio. For non-square (rectangular) plates, the natural frequencies are related to the side lengths a and b. As for the square plates, the frequencies have to be predicted by numerical techniques as there is no explicit relationship for w(x, y) which fits the differential equation of motion and the boundary conditions (except for simply supported edges).
When a>>b, the plate approximates to a beam and the solution is as in the Vibrating Beams experiment. Between the beam and square plate lies an interesting transition zone, in which we need to answer the question 'What is a beam and what is a plate?'