Resonance Simulator

What Is the Resonance Simulator?

How to use the resonance physics simulator

1. Read the eight readings before you touch anything. The four cards are already answering for a 2.0 kg block on a 200 N/m spring pushed at its own frequency: Natural frequency  f₀ 1.592 Hz, Drive frequency  f 1.592 Hz, Amplitude  A 100.0 mm and Quality factor  Q 10.00. The compact row under them reads 10.00 mm, 10.00, 90.00 degrees and 1.588 Hz.
2. Notice what is missing from the panel: there is no force control. The push is a fixed 2.00 N at every setting, and the sentence strip under the drawing says so outright. That single restriction is what makes the Amplitude card worth reading: 100.0 mm of swing from a push that bends this spring 10.00 mm when it just leans on it. The resonance in physics guide is where that mechanism is argued out cycle by cycle; this panel only ever shows you the result.
3. Now drag Damping (N s/m) and nothing else, and keep your eye on the first card. Take it to 0.4 N s/m and the Amplitude card reads 500.0 mm; take it to 20.0 N s/m and it reads 10.00 mm. Through both of those the Natural frequency card has not moved off 1.592 Hz, and neither has its sub-line.
4. Then move Drive frequency (x f₀) instead, and watch the opposite happen. This slider is a multiple of the natural frequency rather than a frequency in hertz, so 1.00 is exactly on resonance whatever the first two sliders say. Push it to 2.00 and the amplitude collapses from 100.0 to 3.326 mm — while Quality factor sits there reading 10.00 in both states.
5. Read the Phase lag cell while you do it. It is the one reading that marks the natural frequency exactly, and it is savage about it: 78.63 degrees at 0.99 x f0, 90.00 degrees at 1.00, 101.3 degrees at 1.01. One step of the slider either side, and the quarter-cycle lag is gone.
6. Move Mass (kg) and Spring constant (N/m) to see the first card finally respond. Four times the stiffness takes it from 1.592 to 3.183 Hz; four times the mass takes it the other way to 0.7958 Hz. These are the only two controls it answers to. The spring constant calculator is where the stiffness you type in here is measured from a load and an extension.
7. Read the drawing in two halves, and read the scale bar first. The upper strip is the block at its steady swing; the lower panel is the whole response curve with your drive marked on it by a filled dot. The strip auto-fits, so the block never looks much bigger or smaller than it did — the short bar underneath it prints the millimetres that make the picture honest.
8. Press Pause motion when you would rather read than watch. The motion is already running when the page loads, so the button reads Pause motion rather than asking to be started, and pressing it changes no reading at all. Reset puts all four sliders back to 2.0 kg, 200 N/m, 2.0 N s/m and 1.00 x f0 and leaves the motion running.

Dragging is the right way to find out which readings a control is allowed to move, and a poor way to hit an exact figure — the mass slider steps in tenths of a kilogram, the spring in tens of newtons per metre and the damping in fifths. When you already know the stiffness and the mass and want the frequency typed out to four figures, the resonance frequency calculator does that step, and solves it backwards for the mass or the spring as well. It has no drawing and no millimetres; this panel has no unit menus.

Worked example: change one thing at a time

Nothing in the three tables below was worked out here. Each row is a setting the lab was put into and then read back from, cell by cell, so every string in them is the panel’s own. Each table moves exactly one slider down its column and pins the other three, which is what makes the column that changes diagnostic.

One: the damping slider, with everything else pinned

The mass stays at 2.0 kg, the spring at 200 N/m and the drive at 1.00 x f0 in all seven rows. Only the damper changes, from the floor of its slider to its ceiling.

Read the second column down the page first. 1.592 Hz in all seven rows, from the lightest damping the slider offers to a hundredfold more of it. The damper is the control that looks most like it ought to change the frequency, and it is the one control that cannot.

Now read the amplitude column beside it. 500.0, 100.0, 33.33, 10.00, 7.143, 7.042, 5.000 mm — a factor of 100 from the same spring, the same mass and the same 2.00 N. Rows one and four are the pair worth pressing one after the other: Light damping and Heavy damping differ in nothing but the damper, and the swing between them differs by 50.

The last three rows are where the panel stops reporting a peak. At 28.0 N s/m Peak at still prints 0.2251 Hz, which looks like an answer; its magnification is 1.000, the same as a drive of nothing at all, so there is nothing there to find. The slider stop between that row and the next, 28.2 N s/m, is still inside the threshold at a damping ratio of 0.7050 and still prints a frequency, 0.1228 Hz. Only at 28.4 N s/m — two steps from 28.0, 0.4 N s/m more damping — does the reading become the word none, and it stays that word to the end of the slider.

Two: the spring and the mass, which are the only controls the first card answers to

The damper stays at 2.0 N s/m and the drive at 1.00 x f0. Rows one to three move the spring alone; rows four and five move the mass alone, around the same middle row.

Rows one to three are a factor of sixteen in stiffness and a factor of four in frequency. 50, 200, 800 N/m give 0.7958, 1.592, 3.183 Hz. The static deflection goes the other way and faster — 40.00, 10.00, 2.500 mm — because a stiffer spring bends less under the same fixed push.

Rows four and five do it with the mass and reach the same two frequencies. 0.5 kg gives 3.183 Hz and 8.0 kg gives 0.7958 Hz, matching rows three and one exactly — and the static deflection stays at 10.00 mm in both, because the spring has not been touched. Two different controls, the same frequency, a different panel underneath it. The free, undriven version of this oscillator is next door at the simple harmonic motion simulator, and the motion itself is simple harmonic motion.

Watch the quality factor column while you do it, because it moves for a reason that is easy to miss. Nothing changed the damper, yet Q goes 5.000, 10.00, 20.00 down the spring rows and 5.000, 20.00 on the mass rows. The damping ratio is the coefficient divided by twice the square root of the mass times the spring, so raising either of those raises that root and lowers the ratio — and Q is one over twice the ratio, which is why it climbs as the ratio falls.

Three: the drive slider, with the oscillator left alone

Now the mass, the spring and the damper are all pinned at 2.0 kg, 200 N/m and 2.0 N s/m, and only the drive moves, from the bottom of its slider to the top.

The phase column is the one to read, and it is the reason this table exists. It climbs from 0.5787 to 177.9 degrees across the slider and passes through 90.00 degrees at exactly one place, 1.00 x f0. Unlike the peak, that crossing does not move when you change the damper, which is why it is the honest signature of the natural frequency.

The amplitude column tells the other half of the story. Below resonance the block barely amplifies anything — 10.10 mm at 0.10 x f0, against a static deflection of 10.00 mm. Above it the response dies off faster than it grew: 3.326 mm at twice the natural frequency and 1.249 mm at three times it, both less than simply pushing and holding.

Rows four to six are one step of the slider either side of resonance. 0.99, 1.00, 1.01 give 9.903, 10.00, 9.711 for the magnification — barely a change. Now press Light damping and repeat it: the same three positions give 35.62, 50.00, 35.09. A tall peak is a narrow one, and one hundredth of the natural frequency is enough to fall off it.

Formula and symbol reference

The lab prints its own relations under the drawing, in the fixed definition line, in plain ASCII so that no radical sign and no subscript glyph is ever needed. Written out the same way, with sqrt spelled and every subscript an underscore:

f0 = (1/2pi) sqrt(k/m) is the natural frequency; damping ratio = c / (2 sqrt(m k)) and Q = 1 / (2 damping ratio) are the two dimensionless numbers; x_st = F0 / k is the static deflection with F0 = 2.00 N fixed; and with r = f / f0 the response is M = 1 / sqrt((1 - r^2)^2 + (2 damping ratio r)^2) with A = M x_st. Put r = 1 into that and everything but the damping term disappears, which is where M = Q comes from.

The spring constant is an input here and this panel never measures it. If you do not have a figure for the spring in front of you, the spring constant lab is where a load and an extension become a stiffness, and the Hooke’s law calculator is the force-extension relation it rests on. The amplitude this panel reports is an output and only an output; the amplitude guide is where the height of a vibration is a subject in its own right.

Fourteen rows follow and only four of them are something you drag. The rest are cards, cells of the compact row, or quantities the panel names without giving you a control for. For the sliders the right-hand column gives the end stops and the step; for everything else it gives the strings the lab prints and the span it prints them over.

If you read one row twice, make it the pair at the bottom. Phase lag and Peak at are the two readings that disagree about where resonance is, and the panel prints them side by side rather than choosing for you.

The physics: three claims the panel lets you check

The first is that the natural frequency belongs to the oscillator and not to the person pushing it. Two sliders set it and the other two cannot touch it, which is a stronger statement than a formula makes on paper, because you can try to break it. Hold the spring and the mass still, sweep the damping and the drive wherever you like, and the first card’s sub-line stays byte-identical at f0 = (1/2pi) sqrt(k/m) — set by the spring and the mass only.

The second is the one the lab exists for: at the natural frequency the amplitude is the static deflection multiplied by exactly the quality factor. On the opening oscillator that is 10.00 mm becoming 100.0 mm; with the damper at 0.4 N s/m it is the same 10.00 mm becoming 500.0 mm. No force appears in that statement at all, which is why there is no force slider to find.

That is also why the damping is the engineering question and the push is not. Doubling the push would double the static deflection and the swing together and change nothing about the multiplier. Halving the damper doubles the swing from the same push. The panel puts both numbers in front of you — Static deflection and Magnification, side by side in the compact row — so the claim is arithmetic rather than rhetoric.

The third claim is the awkward one, and it needs two readings to state. The frequency at which the response is largest is not the natural frequency: Peak at reads 1.588 Hz while the first card reads 1.592 Hz, and the gap widens as the damper goes up — 1.591, 1.588, 1.555, 1.125 Hz down the damping table. What stays exactly in place is the 90.00 degrees of phase lag.

Push the damper further and the peak does not merely move; it dies. The response curve stops having a maximum anywhere, and Peak at prints the word none instead of a frequency. An object with a whole ladder of resonances rather than one behaves very differently, which is the subject of the standing waves and harmonics guide and is drawn by the standing waves simulator.

Where the resonance simulator’s readings break down

Every slider position this lab can reach is answered, and the arithmetic is not where it can let you down. The drawing is, and so is the model behind it. Nine limits sit under the readings on this page; six are choices the panel makes on purpose, and three are bounds on the physics rather than on the panel.

The block on screen does not move at the frequency you set

One drawn cycle takes the same two seconds at every setting, and the lab says so in its own accessible description of the canvas. The reachable drive frequency runs from 0.01125 Hz at the bottom corner of the grid to 47.75 Hz at the top: a real-time animation would alias into noise at one end and stand still at the other, and either way it would be claiming a rate the lab has no business claiming. What the motion shows is the size of the swing and where the block is within one cycle — which is enough to see the quarter-cycle lag at resonance, and nothing more.

The strip auto-fits, so the drawn swing is never a measurement

The amplitude spans more than four decades across the sliders: the settings behind this page alone run from 0.1387 mm to 7071 mm, and no fixed scale is legible over that range. The strip therefore scales itself so the travel uses about seventy per cent of the room it has, which means the block looks about equally far from the rest position whether the panel says 500.0 mm or 7.042 mm. The short bar below it prints the millimetres it represents, and that printed value is the only honest way to read a distance off the picture.

The static-deflection mark disappears instead of being pinned to the edge

The fainter dashed mark in the strip sits at the static deflection, and its distance from the rest position as a fraction of the drawn travel is one over the magnification. That runs over more than four decades, so at light damping the mark is hidden inside the block itself and above resonance with heavy damping it is several travels off the end of the strip. In that case it is left out rather than clamped to the frame, because a mark drawn where the quantity is not would be a reading a careful visitor could take and be wrong about.

The magnification axis is logarithmic, so a shallow-looking curve is not a small one

The guide lines are the powers of ten, and the labelled set is always an unbroken run upwards from the bottom. That is a necessity rather than a flourish: the magnification runs from 0.02500 to 353.6 over the settings behind this page, and on a linear axis every heavily damped case would draw as the same flat line. The cost is that equal vertical distances mean equal ratios, so the tall, narrow peak of the lightly damped case looks far gentler on the screen than the amplitude card says it is.

A peak that formally exists is not a peak anyone can see

Press Just inside the threshold, where the damping ratio is 0.7000: Peak at reads 0.2251 Hz, which looks like a usable answer. Its magnification is 1.000 to four significant figures, identical to the magnification the curve draws at zero drive, so there is no bump there at all. The caption under the curve refuses the claim in the same breath as printing the number, and it does it by comparing the two printed values rather than at a hand-picked damping ratio.

The scene stops naming a case as soon as you touch a slider

The title above the drawing reads a preset’s name only when all four sliders match one of the seven stored value sets. Nudge any one of them off and it prints A driven mass on a spring instead. That is a refusal rather than a fault: every card carries on answering exactly as before, and the title is declining to call an arbitrary setting by a name that would be false.

Nothing here is a statement about the build-up

Every reading is the steady state: the swing the oscillator has settled into, not the swing it has reached so far. Nothing on the panel counts cycles and no control starts a run or ends one — the motion loops rather than building up to anything. How long a real object would take to arrive at the figure in the Amplitude card is a separate calculation, and this lab makes none of it. No ring-up or ring-down figure appears anywhere here.

One mass, one spring, one mode, and a loss proportional to speed

Real objects break all four assumptions. A panel, a beam or a casing has a whole series of natural frequencies, each with its own shape and its own damping; a coil spring compressed until its turns touch stiffens as it deflects, so its natural frequency starts to depend on the amplitude; and dry friction, internal losses in a metal and acoustic radiation obey different laws from a dashpot. The driven damped oscillator described on the Wikipedia harmonic oscillator page is the idealisation this panel implements, and the amplitude is exactly where it is most likely to fail.

The famous bridge is not an example of anything this panel draws

Flutter is what the 1940 Tacoma Narrows failure is attributed to, not a drive at a natural frequency. Wikipedia’s account of the 1940 bridge records both halves of that: many physics textbooks still present the event as elementary forced mechanical resonance, and the collapse is attributed instead to aeroelastic flutter — an oscillation that excites itself out of the structure’s own movement in a steady wind. Billah and Scanlan set the correction out in the American Journal of Physics in 1991. There is no slider on this panel that could produce it.

Where resonance is actually used

Deciding whether a drive sits too near a mounting’s own frequency

This is the forward question and the reason the panel opens where it does. A mass you know on a mounting whose stiffness you know gives a frequency to keep the driving machinery away from, and the drive slider tells you what it costs to be close. On the opening oscillator a drive at 1.00 x f0 is magnified 10.00 times, while the same oscillator at 2.00 x f0 is magnified 0.3326 — less than if you pushed and held.

Retuning by changing the mass rather than fighting the response

The mass slider is the cheap control in practice: a spring is usually fixed by the installation and a lump of ballast is not. Drag it from 2.0 kg to 8.0 kg with everything else held and the first card halves, 1.592 to 0.7958 Hz, while the static deflection stays at 10.00 mm. Notice what comes with it, because it is the trap: the quality factor doubles from 10.00 to 20.00, so the retuned mounting is sharper than the one you started with.

Choosing how much damping is worth paying for

Press the first five presets in order and watch one column: 50.00, 10.00, 1.000, 0.7143, 0.7042. That is the whole design trade in five clicks, from one spring and one mass. It is also where diminishing returns show up: the first jump removes a factor of five, while the last removes almost nothing, because by then there is no resonance left to suppress.

Reading a measured resonance backwards

If you have shaken something and found where it responded most, you have the peak rather than the natural frequency, and the panel shows you how far apart those can be. At light damping the gap is negligible, 1.591 against 1.592 Hz. At a damping ratio of 0.5000 it is not: the peak sits at 1.125 Hz while the natural frequency is still 1.592 Hz. A phase measurement is the better one, because the 90.00 degrees crossing does not move.

Recognising the same panel somewhere it has no spring in it

Every resonance has this shape: something with a frequency of its own, something driving it, and a quality factor deciding how much it answers. Swap out the two ingredients and the algebra survives the change. A swinging pendulum puts a length and the strength of gravity where this panel has a spring and a mass, which is the pendulum period calculator; a tuned electrical circuit puts a capacitance and an inductance there instead. Neither has a slider here and no figure on this page transfers to either.

Where to go next

A panel can show you which readings a control is allowed to move, and that is the end of what a panel is for. Resonance in Physics is where the subject itself is written out: what resonance is, how a well-timed push adds energy on every cycle, where the peak really sits, the everyday cases, the four misconceptions and eight worked problems with every step shown. The calculator that turns a stiffness and a mass into a frequency to four figures heads the related tools below.

For the motion itself rather than the response to it, the simple harmonic motion guide is the companion, with the simple harmonic motion lab and its calculator beside it. The amplitude guide is where the size of a vibration is the subject instead of an output.

Nearer the inputs this panel asks for, the spring constant lab and its calculator are where a stiffness is measured, and the Hooke’s law calculator is the relation underneath both. The frequency formula guide is the conversion for anyone arriving with a period rather than a frequency.

Further out, the standing waves and harmonics guide is the case of one object with a whole ladder of resonances, drawn by the standing waves lab, and the pendulum period formula guide is the same shape of algebra on a different system. Beyond those there is the simulation library, an index of calculators, and the blog for everything written out at length.