# Does Natural Frequency = Resonant Frequency?

Well, are these two terms synonomous?

Yes, they are interchangeable. Resonance occurs when the input frequency matches the system’s natural frequency.

Thanks for clarifying, but I was under the impression that resonance referred to all multiples of the natural frequency, too.

Usually the two terms are synonymous, but it can get very confusing. I blew out a job interview many years ago by asking this very question of the Technical Director, and he didn’t know. I wasn’t trying to be awkward, I was genuinely looking for enlightenment, and given that he was asking me about loop stability margins I thought he might know.

The best explanation I’ve heard so far is that resonant frequencies are a subset of natural frequencies. If you like, a system may have several natural frequencies, and the ones with the strongest response are known as resonant frequencies.

Knowing the undamped natural frequency can be useful in some systems, and adding damping will lower the actual resonant frequency.

As QED said, for mechanical or electrical system the resonant frequency is the natural frequency. There is another definition of “resonant” as in “a resonant tone” meaning that the tone is full and rich. Such a tone contains a lot of overtones, harmonics, of the fundamental frequency.

An oscillatory system can’t be excited into resonance by multiples of the natural frequency because they are phased wrong. For example suppose you give a playground swing a push and then give it another push as it swings back toward you. That tends to stop it because you are pushing in the wrong direction. You have to wait until the swing is going away again, or one full back and forth cycle. You can also wait two full cycles, or 3 or n, because then, again, the swing is going away from you and your push adds to its velocity. This fact is used in electrical engineering to make frequency multipliers. An amplifier is arranged so that it supplies pulses of current to a resonant circuit. The resonant circuit can be tuned to the fundamental pulse frequency, twice that for a frequency doubler, three times that for a frequency trippler, etc.

A resonant frequency is one in which the oscillations of the system build upon each other. In other words, if you are osciallting a system at a frequency other than a resonant frequency then part of your stimulus is wasted becuase it opposes the oscillation. There is a very good demonstration that can be done of this. If you have a rope and begin oscillating one side the oscillations will start out small. As the frequency of the oscillations near the first resonant oscillation the “waves” will increase in size. Once you reach the first resonant frequency the wave is at its maximize size. In the first case there will be one wave going up and down.

As you continue to increase the frequency the waves die out as you move from the resonant frequency. Eventually as you begin to approach the second resonant frequency the waves begin to get bigger. Except this time instead of a single wave going up and down you have two waves moving opposite of each other. Moving past the second resonant frequency the waves get smaller again until you reach the 3rd resonant frequency where they get larger again but with 3 waves going. This process repeats until you reach the physical limits of the system.

Basically a resonant frequency is a frequency in which the input builds upon itself.

Natural frequencies on the other hand are the frequency that the system “wants” to oscillate at. Typically this is for a system that can be modelled as a spring and a mass. For example if you have a spring and displace a mass it will always oscillate at its natural frequency no matter how far you displace it or how much initial velocity has. In the case of a mass-spring system the resonant frequency are equal to each other. However, when you add in a dampener (which all real world systems have) the resonant frequency becomes a multiple of the natural frequency.

The number of resonant frequencies is equal to the number of springs and masses in a system. So a one mass one spring system has one resonant and naturla frequency. A system that looks like: wall-spring-mass-spring-mass has two resonant and natural frequencies. A system that looks like wall-spring-mass-spring-mass-spring-mass has 3 and so forth.

So in short: Resonant frequency is a frequency in which the oscillations build upon each other and the displacement gets bigger and bigger. Natural frequency is the frequency that the system will oscillate at any given initial conditions.

At least thats my take on the matter.

One thing I forgot to add, in the case of multiple mass-spring systems becuase its impossible to get initial conditions exact the system will resonate in some combination of the two natural frequencies. If it were possible to exactly fix initial conditions you could get the system to oscillate at the two different frequiencies.

This is a bugbear of mine: the term is (or, at least, should be) resonance frequency, ie. the frequency at which resonance occurs (when a periodic force causes maximum amplitude of vibration). I’ve always thought that calling the frequency itself ‘resonant’ introduces confusion straight from the off.

Natural frequencies, on the other hand, is IMO a nice way of describing those frequencies at which resonance occurs (and yes, there will always be many such frequencies, since any object has many different possible patterns of vibration).

So there is no difference between them, except a pet peeve. And just to clarify a little, resonance only occurs with periodic forces (specifically, when the forcing frequency matches the natural frequency). A brief impulsive force cannot be said to induce resonance. For example, when the opera singer finds the right note, the wine glass resonates: the forcing frequency equals a natural frequency (usually the fundamental), vibration amplitude hits maximum, and resonance occurs. However, merely tapping the wine glass with your finger might “sound resonant”, but this is rather sloppy use of language. One has merely initiated the natural frequencies of the wine glass, which decay away slowly because the system has low damping (high Q).

The word for resonances other than the fundamental is partials. Partials are only harmonic if the resonance frequency is exactly 2, 3, 4 … (an integer multiple) times the fundamental frequency.

Initial conditions will initiate all natural frequencies (unless, say, the force occurs precisely at a node of one or more modes): an object can’t not vibrate at its natural frequencies. Resonance is about the frequency of the force, not the object’s vibration modes.

I’d say they aren’t quite synonomous: more like different meanings, same value.

The natural frequency is the frequency the system will oscillate at if displaced from equilibrium and released. E.g., stretch a spring and let it go. The frequency it’s vibrating at is the natural frequency.

The resonant frequency is the frequency with which you can drive the system such that resonance occurs. (I.e., the frequency at which the system will gain the most energy from being driven.) E.g., shake a spring back and forth with your hand. The frequency at which you should do this to get the biggest oscillations is the resonant frequency.

Typically, these frequencies have the same value. (I can’t think of any case where they wouldn’t.) So, if the natural frequency of the system is 10 Hz, the resonant frequency is 10 Hz. But just because they have the same value doesn’t mean they have the same meaning. Natural frequency is the frequency the system tends to oscillate at on its own, whereas resonant frequency is the frequency you should drive the system at for maximum effect.

For a mass-damper-spring system the natural frequency is (k/m)[sup]1/2[/sup] and the resonant frequency is (k/m)(1-2C[sup]2[/sup])[sup]1/2[/sup] where C=c/(2*(mk)[sup]1/2[/sup]). k=spring constant, m=mass and c=dampening constant.

Resonant frequencies and natural frequencies often have the same value but they are fundamentally different things. For example, a pipe with one end open and one closed does not have a natural frequency but it has many resonant frequencies. The resonant frequencies in this case are those whose wavelengths fit inside the pipe. This page has two good diagrams of what I am talking about. Basically what happens is that at an open end you have maximum displacement and at a closed end you have zero displacement. Resonant frequiencies are those that result in wavelengths that fit those conditions.

treis, ‘natural frequency’ does not only apply to systems in which damping is zero. All real systems or objects have some damping - that would mean that no real system has natural frequencies! The natural frequencies (ie. resonance frequecies) of an organ pipe depend on whether it is open or closed, but it is simply not the case that one has both kinds while the other has only one kind.

Thats true and I put the wrong formula down. The damped natural frequency is Wd=Wn*(1-C[sup]2[/sup])[sup]1/2[/sup]

Natural frequencies are not the same as resonance frequencies:

Natural frequency: The frequency a system will oscillate at given any input

Resonance frequency: A frequency in which the responses of a system build upon each other.

Something like a rope hanging from two posts doesn’t have a natural frequency becuase it won’t oscillate at any initial conditions. It does have resonance frequencies becuase you can oscillate it at a frequency in which the responces build upon themselves.

The resonance of a spring-mass or capacitance-inductance system isn’t the same phenomenon as the vibration modes of a string.

The former is a result of the storage of energy in first one element, the spring or the capacitance, and then the other, the mass or the inductance. This swapping back and forth of the energy has a characteristic frequency depending upon the magnitudes of the elements of the system.

The vibration of a string depends upon the velocity of propagation of a wave in the string. The correct frequencies for a given length are those that result in standing waves in the string. The string is analagous to an electrical transmission line.

Here is a short discourse on thevibration of a string that is fixed at both ends.

As can be seen, the natural frequency of vibration is some whole number multiple of the ratio of the wave velocity in the string to twice the length. The velocity, √(τ/p), is a function of the tension in the string, τ, and the mass/unit length, p.

First of all, a rope hanging from two posts is a really bad model for a vibratory system, as the rope obviously exhibits both considerable damping and multimodal behavior, i.e. it won’t just vibrate at integer multiples of its length but at any length that can be considered locally constrained (which will change instant to instant) and longitudinally as well as transversely.

Let us instead consider a taut guitar string with no damping as the ideal case. Guitar strings, as we know, all vibrate at a particular frequency for a given length. Normally, a string that is constrained against translation at both ends will display three different modes of vibration; in the up-and-down direction, side-to-side, and about its axis (like a jump rope). However, in our idealized example we can assume that the string vibrates in only one dimension in plane as the up-and-down and side-to-side are identical (owing to symmetry) and we can dismiss the axial rotation because the string is so taut that any rotational vibration occurs at much higher frequencies than the other modes. (We’ll talk about this more in a second.)

When you pluck the string, it will vibrate at its first natural frequency, which is also called the fundamental frequency. This vibration is nondestructive; that is, the wave doesn’t break off into little wavelets at the endpoints that come back and interfere with each other; hence, it resonantes. If you apply a periodic impulse that is in phase with this wave, like the child in a swing who keeps getting pushed, the system will resonate, or grow larger in amplitude (the extent to which the string deflects.) More on this in a second.

However, any slight asymmetry (plucking it more at the top or bottom) will also introduce other frequencies at which the string tends to vibrate; in the ideal case, these are exact multiples of the fundmental frequency (which is an eigenvalue). These are all natural frequencies. However, the amplitude is greatest at the first natural frequency, and decreases with increasing natural frequency number. Although the difference in amplitude for a natural frequency depends upon various parameters of the system (and for real systems, the damping constant or function, as well as nonlinearity outside of a linear elastic range), in general the reduction in damping is greater than linear proportion; often is is more like a geometric sequence. So in most systems, identifying the first three natural frequencies is sufficent to predict resonance behavior of the system.

At resonance without damping the system will continue to vibrate indefinately. If the input continues to reinforce the vibration, the amplitude will increase until the system either reaches a nonlinear state (where the resonance behavior changes) or it’ll tear itself apart, like the infamous Tacoma Narrows Bridge. So resonance always occurs at a natural frequency, typically (but not always, depending on the input) the fundamental frequency.

So far we’ve been talking about this guitar string with one degree of freedom (1DOF). Real systems aren’t like this, of course; even a taut, elastic guitar string with minimal damping is going to display the aforementioned three modes of vibration (3DOF). As previously discussed, the first two modes are identical, and the frequency of the third is so much higher (and of lower strain energy amplitude) than the others that it generally isn’t a concern.

In a more complex system–say, for something I’m familiar with, a rocket booster stack–you’ll have many, many modes of vibration, because each section will have a different vibratory length and characteristic response, as well as damping. The overall system will have a resonance frequency–and it usually seems to be irritatingly near either a critical input, like the nozzle characteristic or a turbopump frequency, or the resonance frequency of a delicate subsystem, like the guidence and control computer–which is not readily predictable from the overall system dimensions. Instead, you have to model each stage, interstage, payload adaptor module, et cetera as it’s own spring, damper, and mass system and beat on it (mathematically speaking) to figure out what the characteristic response of the system is. In this case, the system has many “natural” frequencies, although most of them aren’t really resonant as the subsystems will tend to interfere or damp out the vibration; you’re left with a handful of frequencies–hopefully just two or three–at which the system does resonante, although these are not necessarily the natural frequencies of any particular stage (though they’re usually pretty close). So in this case, the resonance frequencies are a subset of the system “natural” frequencies, and you use these numbers to isolate out any problem vibrations. They’re also used to tell the guidence system what to expect so that it doesn’t get freaked out by small oscollations in sensor inputs.

So, to summarize: for a simple, ideal system, the natural and resonance frequencies are the same; resonance occurs at the natural frequencies, though typically only the three lowest frequencies are of concern. For a more complex system, we refer to the resonance frequencies as those frequencies at which the system displays the greatest amplitude of vibration, although these may not match up to the natural frequencies of any of the components of the system. These may not be the three lowest “natural” frequencies of the overall system, either, though they do tend to be toward the lower frequency range. Real systems also have damping, which will both tend to alter the undamped natural frequency (usually shifting it slightly higher) and progressively reduce or control the amplitude, and real systems can only vibrate so much before they start behaving nonlinearly or coming apart like a AMC Gremlin at 75mph.

There are a lot of expensive hardback texts on mechanical vibrations (which is probably the easiest model for the average person to conceptualize about resonance) but I think J.P. Den Hartog’s Mechanical Vibrations is about the best introduction I’ve read on this topic, especially the first two chapters. Any linear systems or partial differential equations text should have a basic mathematical introduction to vibratory systems. Feynman’s Lectures has several chapters (21,23, 24) on vibrations and resonance, but reviewing them they seem to be geared more toward circuits and electromagnetism, which makes sense in the context of the lectures but isn’t so helpful conceptually. Chapters 47 through 51 deal with harmonics and waves in sound (Feynman being a famous bongoist) but again, don’t detail a mechanical example that the layman could readily visualize.

Anyway, for the simple answer: yes, resonance occurs at a natural frequency; in a simple system, it usually occurs at the first natural (fundamental) frequency.

Stranger

Correction, fourth paragraph, fourth sentence:

Stranger

Sometime! If the excitation is a sweep-sine signal, the resonance is passed after the natural frequency is passed, even for undamped systems, irrespective of the sign of the sweep ration. For harmonic excitation, the frequencies fit.

This is incorrect, strictly speaking, at least in modern music physics theory dating from Hertz (yes, that Hertz). The first partial is in fact the fundamental frequency. “Overtone” is a slippery translation of “upper” (German “Ober”), the “upper partials.”

Sadly, especially for Pythagoreans/Just tuning guys, who live by integer division, not the “Über” partials.

Haven’t studied this in decades, but…
resonant frequencies would be the natural frequency and any multiples?

If you open one of those new upright front-loading washing machines it is full of (very) heavy weights attached to a multiple of springs of different lengths. The idea as I understand is to have multiple competing natural frequencies to help damp each other out so the machine does not walk across the floor if unbalance during the spin cycle. This means the system’s frequencies have a low (wide) Q?

Yes, the confusion is that when there is only one resonant frequency in the system,
its the natural frequency too, they are the same.

But if there are multiple, the natural frequency might not be the lowest resonant frequency.

Probably is, but possibly isn’t.

The natural frequency is the frequency it resonates when stimulated with a broad spectrum, eg from an impulse… … which in mechanical terms means hitting it with a hammer, for example. (or the edges of square waves.)