A couple of examples

1. Irregularities in the road transmit to every vehicle forces that could shake its parts apart, not to mention the discomfort of the passengers. Everyone has seen (at least in some western film) the axles and wheels of a horse-drawn carriage travelling at speed disintegrate because of a pothole or a stone. If this does not happen all the time to our cars too, it is thanks to the suspension, which is nothing but a strongly dissipative oscillating system, able to absorb the stresses and dissipate their energy as heat.
2. Everyone knows that in a washing machine (especially in somewhat older models) the spin cycle produces no oscillations when the machine is fully loaded, while if the load is smaller the clothes tend to gather on one side, producing jolts and rhythmic knocks.

What do these phenomena have to do with resonance?

Vibrations and resonance

Engines, like all mechanical machines, contain parts rotating at precise speeds, that is, with precise rotation frequencies. In many cases it is possible to tell how fast an engine is turning simply by listening to the sound it produces (try with the washing machine, and don't forget the 2π). We all know that the higher the pitch of the sound, the higher the rotation speed, and this connection is sometimes used, perhaps unconsciously, to change gear while driving.

• The case that interests us as a model is that of a shaft on which a cylinder is mounted (it could be the drum of the washing machine, but also the gear system of an engine shaft).

• Let us first consider the free rotating system (as in the washing machine). In the ideal case the cylinder has a perfectly homogeneous mass, that is, evenly distributed. It therefore has a constant angular momentum parallel to the axis of the gear, and the forces that the gear exerts on the shaft are perfectly balanced. Each point of the disc corresponds to a diametrically opposite point, and the two centrifugal forces exerted on the shaft by these points correspond to a pull in the plane of the gear. The forces acting on the shaft have zero resultant in the plane, and so the cylinder rotates on the axis without strain.

• If, however, we add a small imperfection, or an extra mass on one side only of the disc (the small load of washing), things change... or rather, the music begins. The force exerted by the extra mass on the shaft is no longer balanced by that of an equal mass placed symmetrically with respect to the axis. The axis is "pulled" towards the unbalancing mass by a force whose intensity is proportional to the square of the angular velocity with which the mass rotates and to its distance from the axis (centrifugal force).

In the washing machine the motor is subject to this oscillating force only during the spin cycle, so it is not particularly harmful, but a structural imbalance (that is, one that is always present) in an engine rotating at high speed can cause the mechanism to seize or break. The construction must therefore be accurate, and internal friction must be suitably reduced by means of lubricants. Here, however, we are not interested in studying the wear of the system, but its vibration.

Vibration isolation

Granted that engineers work hard to build the parts of a Ferrari engine in the best possible way, in a washing machine we cannot prevent the load from being unbalanced. The same holds for all machines in which the gears are not free, but are subject to complex systems of forces coming from other mechanisms, constraints and external loads. In short: no matter how sophisticated they are, all mechanical systems are subject to vibrations.

By studying the phenomenon of resonance, however, we can try to isolate vibrations, so that they are transmitted to the outside as little as possible.

In a washing machine, or in a car, isolation is obtained by means of a suitable system of springs and dampers that "suspend" the drum, so that part of the energy that sets it oscillating is turned into potential energy of the springs, and then dissipated in internal friction.

More generally, isolating the system requires the following steps:

1. study of the normal modes of the system, which reveals its natural frequencies, that is, the frequencies at which the system can go into resonance
2. study of the frequency response of the system, which allows us to identify the frequencies most responsible for harmful or annoying vibrations
3. study of the resonances, to determine exactly what the characteristics of the suspension system that ensures the desired isolation must be.

In real systems this whole process can be very complex, and involves both experimental measurements and complex calculations, but in the simplest systems we can solve everything with paper and pencil. Here is an example.

Our first isolation

Let us put what we know about resonance to the test, and look for a way to isolate the simplest oscillating system. We can imagine precisely the case of the washing machine, somewhat simplified, inserting the asymmetric motor in a support and mounting the whole on an elastic suspension with friction.

Suppose the system has only one degree of freedom, that is, it can move only vertically. In other words, we model the system as a simple harmonic oscillator. The theory of this system is set out in detail on the pages on harmonic motion and on the impedance of the harmonic oscillator. Here we only need to recall that the oscillator is characterized by a single resonance frequency close to

${\displaystyle 2\pi f_0=\sqrt{\frac{k}{m}}}$

and that this frequency indicates the "weak" point of the system. An external force also oscillating at the frequency f₀, even if of small intensity, can make the system perform large oscillations (see the graph of the resonance of the single oscillator).

We have only three ways of acting on the resonance: changing the mass, changing the elastic constant (e.g. by adding further springs in series or in parallel), or changing the friction coefficient. In the three cases, the graphs represent the maximum force F that the oscillator transmits to the base when the oscillator is subjected to a force of maximum amplitude ${\displaystyle F_0}$ and frequency f.

variable mass	variable spring

In the two previous cases we observe that the resonance frequency of the system shifts

• towards smaller values when the oscillating mass is increased,
• towards higher values when the elastic constant, that is, the stiffness of the suspension spring, is increased (in practice by adding more springs in parallel)

The shape of the resonance curve, apart from the "rigid" shift, is not changed.

variable friction

The different curves in the graph represent the behaviour of the system for increasing values of the friction coefficient. We observe that the response of the system, that is, the force transmitted to the base, is strongly damped at the resonance frequency of the system stays the same for any value of the damping at the frequency ${\displaystyle \sqrt{2}{f}_0}$ increases slightly for frequencies above ${\displaystyle \sqrt{2}{f}_0}$

From this we have deduced three elementary ways of isolating the system, once its resonance frequency is known. Obviously the calculation is less simple for systems that can oscillate in several different ways (systems with more degrees of freedom), but it is based on the same principles. The diagram of the three-wheeled van alongside, for example, although still far from the complexity of a real vehicle, has two degrees of freedom because, besides the vertical motion of its centre of mass g, it can rotate around the centre of mass, always remaining in the plane of the drawing. By adjusting the parameters k₁, r₁, k₂ and r₂, given the mass m and the moment of inertia of the vehicle about its centre of mass, the oscillations that could compromise the stability of the vehicle's attitude while moving can be damped.

Links

• For the connection between rotation, harmonic motion and oscillations, see the page on harmonic motion.
• Response functions are described in impedance and, in the various particular cases, in mechanical impedance, electrical impedance, acoustic impedance