What they are

• Every vibrating system, be it mechanical, electrical or acoustic, has particular modes of oscillation in which all its component parts vibrate in harmonic motion, and with the same frequency. These are precisely the normal modes of the system.
• Therefore, each normal mode corresponds to a particular frequency (called a natural frequency of the system). The converse, however, does not hold: several modes, in which the oscillators have different phase relations with each other, can correspond to the same frequency. Typical examples are the "degenerate" modes of square membranes (see normal modes of a rectangular membrane) and the acoustic resonances of a room (see standing waves in 3D).
• Each mode is therefore uniquely characterized
  1. by its natural frequency (or eigenvalue of the mode), and
  2. by the shape of the oscillation, that is, by the relative amplitudes and phases with which all the oscillators oscillate (or eigenvector of the mode).

Since all the elements oscillate at the same frequency, their mutual phase differences remain unchanged throughout the motion.

• As a rule, a physical system does not oscillate according to just one of its normal modes, but according to a combination of them (superposition). The task of many of the methods of investigation developed in physics and engineering is precisely to identify all the normal modes of a system.

Why do we care about them

The great importance of these modes lies in some of their important properties:

1. Normal modes are independent of each other: the total energy of an oscillating system, however complex, is simply equal to the sum of the energies of the individual normal modes that make up the oscillation. The energy that belongs to a particular normal mode is never transferred to another mode. They are completely independent of one another.
   • Note that, if a physical system is made up of several elementary oscillators, and these exchange energy with each other, their motion will not in general be a simple harmonic motion: the amplitude and frequency of the motion can change in time, because the component oscillators can exchange energy, and so at every instant, even if the total energy stays the same, the energy of each of the components can fluctuate.
   • If instead the system is in one of its normal modes, all its parts behave as if they were independent oscillators. They keep their oscillation constant in frequency and amplitude, and do not exchange energy with the others.
   • See the case of two coupled oscillators for a simple, practical demonstration of this property.
   • This property has important consequences not only in classical mechanics, but also in quantum mechanics.
2. Any mode of vibration of the system, however complicated, can be described by suitably combining its normal modes.
   • This statement is not obvious, and its proof is an important theorem.
   • See also the discussions on the superposition principle and on Fourier's theorem.
3. Once the normal modes are known, combining them to obtain new possible oscillations is simple: it is enough to add them, each with its own amplitude and phase.
   • There is, however, a flip side: the elementary "normal oscillators" may not coincide with the oscillators that physically make up the system. In particular, the coordinates of the normal oscillations are, in general, linear combinations of the coordinates of the oscillators of the system. See again the case in which the system consists of only two oscillators. The "normal coordinates" are in this case the sum and the difference of the oscillations of the individual parts.
4. Normal modes are "simple" for another reason too: because all the parts of the system move in simple harmonic motion.
   • Consequently, a normal mode is essentially defined by a single frequency, amplitude and phase, regardless of the number of elementary oscillators that make up the system.
5. The frequencies of the normal modes correspond to the particular resonance frequencies of the system, which indicate the most efficient way to exchange energy with the system without dissipating it.
   • Whatever the nature of the physical system under examination, this fact allows us to "see" its dynamics clearly. In the case of a building, this corresponds to revealing its weak points. In the case of other systems, these could actually be strong points (think of musical instruments, which produce the most sound when they are excited at frequencies corresponding to those of one of their normal modes).
6. Each different way in which a physical system can acquire or give up energy is called a degree of freedom of the system. Since normal modes are independent of each other, it is clear that the number of degrees of freedom of a system is equal to the number of its normal modes.

In summary, the study of normal modes allows us to break down an oscillatory motion, however complex, into simple harmonic components, revealing the internal dynamics of the system.

A mechanical example

The following diagram shows:

1. on the left, a "real" mechanical system (a house).
2. Next to it, the diagram of a mathematical model of it based on mass-spring systems. At this point the system is described by a set of properties (the masses m₁, m₂, ..., the elastic constants k₁, k₂, ... and the frictions c₁, c₂, ...). However, at this level the dynamics is intricate, and to calculate the motion one has to solve a system of many coupled equations. It is not evident what the natural frequencies are, nor what happens to the system when it is brought into resonance with one of these frequencies.
3. On the right, instead, a modal analysis has been carried out: the normal modes have been identified, and each is characterized by three constants M_i, K_i, C_i (different from those of the model in the previous point). Now the modes of oscillation of the system are all independent of each other, and so we can predict exactly what happens at each frequency simply by superposing the resonances of the normal modes.
4. The general solution for a given external excitation will be the superposition of the individual motions of the normal oscillators for that excitation.

physical model	dynamical model	normal modes
Modal analysis makes it possible to separate a system made of coupled oscillators into simple uncoupled oscillators. The compound motion is the sum of the motions of the normal oscillators, each with its own amplitude and phase.

How can we build the model that describes the system exactly? Or, which is the same thing: how can we determine the parameters M_i, K_i, C_i? Many methods of physics and engineering have been developed to answer this question. The line of thought is almost always the same:

1. theoretical methods: one has to form an idea of the elementary constituents of the system (the individual oscillators) and of their interactions (couplings), derive from them a system of equations containing the parameters, and solve them to compare the actual motion of the system with the one predicted theoretically.
2. Experimental methods: one has to investigate the resonances of the system, which typically means exciting the system in a suitable way and measuring its response, to compare it with the model of point 1.

Modal analysis highlights the resonances of the system: the frequency response of our building model might look like the one shown in the figure. Its interpretation is immediate: each peak corresponds to the resonance of a particular normal mode. The four modes of vibration can have completely different characteristics, and they are completely determined once, besides the natural frequencies, the oscillations of the elements are also known. For example, two modes could be bending modes, one a torsion mode and one a compression mode. In the three cases the building would be traversed by elastic waves of different kinds (flexural, torsional and longitudinal), which would stress it differently.

• The application in this case is evident: knowing, for example, that the building withstands torsion poorly, we could design a damper that suppresses the torsion peak, making the structure more robust in a targeted way.

Density of modes and dimensionality of the system

Many properties of physical systems depend on the distribution of the normal modes in frequency space, that is, on how many modes exist at each given frequency.

For example, in the case of vibrating systems, it is essential to distinguish between systems in which the modes form a continuous distribution and those in which the modes accumulate at certain frequencies, separated by wide non-resonant gaps. Indeed, the response of the system will be completely different. In the first case we will have a fairly uniform response at all frequencies, while in the second the system will tend to respond violently (that is, to go into resonance) only to excitations aimed at precise frequencies. If the system is not excited at a single frequency, as happens with impulsive excitations (a sharp blow), systems of the second kind will behave as a filter, extracting from the excitation only the components in resonance with one of the modes of the system.

From the acoustic point of view too, the distribution of the modes plays a fundamental role and, in particular, we want to stress here how some important characteristics of the sound that can be obtained from a vibrating system depend on its dimensionality.

Here we speak of dimensionality, and not of size, referring to whether the system is able to oscillate in one, two or three dimensions. As an example, let us fix the basic length of the system at 10 m, and study the distribution of the normal modes for a one-dimensional system (e.g. a 10 m string), a two-dimensional one (a 10 m×10 m membrane) and a three-dimensional one (a 10 m×10 m×10 m room).

1D system	2D system	3D system
This distribution is characteristic of strings or pipes, and shows a series of peaks with frequencies in integer ratios. The theory of Western music was based (at least originally) on systems like these. The corresponding instruments emit well-defined, stable tones. Each resonance frequency corresponds to a single normal mode. The density of modes therefore stays constant.	This distribution is characteristic of plates or membranes. Its structure can vary considerably depending on the ratio between the sides of the system, but it is generally characterized by the presence of modes in non-harmonic ratios. "Two-dimensional" instruments do not necessarily produce tones of definite pitch, precisely for this reason. The music of East Asia relies on them more.	This distribution is characteristic of rooms, although the details of its structure depend strongly on the ratio between the lengths of the sides. Compared with the other two, it has a higher density for the same length of the basic element (10 m), and hence a less resonant, more uniform response. The number of modes increases with the square of the frequency.

In conclusion, we only note that the analysis of normal modes refers only to the stationary states of the system, that is, those in which the oscillations have reached a steady state and do not substantially change their characteristics over time. Many other acoustic and dynamic properties of systems depend instead on transient regimes, which cannot be analysed in terms of normal modes alone, or on nonlinear effects, which are examined elsewhere.

In-depth study and links

Normal modes are not a property of elastic systems alone, but are common to all oscillating systems, be they electrical, acoustic, mechanical, macroscopic (that is, described by classical physics) or quantum (that is, described by quantum physics).

• See the sections on resonance and on standing waves for a more complete view of normal modes.
• See also the Questions and answers on resonance
• To find out how important the modes of vibration of musical instruments are for the sound they produce, see the section on the physics of musical instruments and the Questions and answers on musical instruments.

Follow these links to see and hear the normal modes of some vibrating objects that are particularly important for musical instruments.

1. normal modes of a string
2. normal modes of a circular membrane
3. normal modes of a rectangular membrane
4. coupled oscillators