WEBVTT

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A child on a swing already knows
the central idea of resonance.

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One push can be gentle,
yet a sequence of pushes

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can make the motion large if each push
arrives at the useful part of the swing.

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Let the child swing freely first.

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The motion repeats with a rhythm
set mainly by the swing itself.

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Nobody has to prescribe
that rhythm from outside.

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A push in the direction
of motion adds energy.

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The same push at the wrong
time may slow the child instead.

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Timing matters because the swing
has a natural rhythm of its own.

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To isolate that rhythm, replace
the swing by the simplest oscillator:

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a mass m attached
to a spring of stiffness k.

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Displacement x is measured
from the dashed equilibrium line.

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Pull the mass to the right.

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The spring pulls left.

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Move the mass to the left,
and the spring pulls right.

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In either case the force
points back toward equilibrium.

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Hooke's law writes
that restoring force as minus k x.

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The minus sign records
the reversal we just watched:

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force and displacement point
in opposite directions.

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Newton's second law says mass times
acceleration equals the total force.

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With no driver and no damping,

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the spring force is the only force
in this one-dimensional model.

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Divide by the mass
and collect everything on the left.

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We obtain x double prime plus
k over m times x equals zero.

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A sinusoid solves this equation.

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Its amplitude A and starting phase
phi depend on how we release the mass,

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but the angular frequency does not.

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Substitution gives the natural
angular frequency, omega nought,

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equal to the square root of k over m.

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A stiffer spring raises it.

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A larger mass lowers it.

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The period is two pi
divided by omega nought.

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That is the time for one
complete free oscillation,

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and therefore the timing
an outside force must confront.

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So before any driving force appears,

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the oscillator already
owns a preferred time scale.

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Resonance begins when the rhythm imposed

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from outside approaches
this natural rhythm.

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Now attach an external force
that oscillates sinusoidally.

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Its maximum strength is F nought,

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and omega tells us how rapidly
the applied force repeats.

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There are now two
frequencies to keep separate.

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Omega nought is fixed
by the mass and spring.

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Omega is chosen by the driver,
and we are free to change it.

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Far below the natural frequency,
the force changes slowly.

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The mass has time to follow, so
displacement and force are nearly in step.

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At the ideal undamped resonance,
the usual steady amplitude does not exist.

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Each correctly timed cycle adds energy,

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and the amplitude grows
roughly in proportion to time.

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That is the mathematical version
of pushing the swing at the useful moment.

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The driver keeps doing

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positive work instead of returning the
energy it supplied on the previous cycle.

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A real experiment always
loses a little energy,

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even before we add a deliberate damper.

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We will use a small damping ratio
for a finite steady response,

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then increase it later.

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Here is the central experiment.

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We change the driving frequency slowly
enough for the transient motion to settle.

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The upper yellow point reads amplitude,
and the lower one reads phase lag.

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The response has the same
frequency as the driver,

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but its amplitude A and lag
delta depend on frequency.

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The ratio r compares the driving
frequency with the natural frequency.

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Begin well below resonance.

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The amplitude is modest,
and the phase lag is close to zero.

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The oscillator follows
the slowly changing force.

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Now sweep upward.

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The amplitude climbs sharply
as omega approaches omega nought.

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The driver is repeatedly adding energy at
nearly the rhythm the oscillator prefers.

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At the resonance region,

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the displacement lags the force by about
ninety degrees, one quarter of a cycle.

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The force is then well placed
to feed energy into the velocity.

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Continue above resonance.

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The amplitude falls again,
while the phase lag keeps increasing.

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The oscillator can no longer reverse
quickly enough to follow the driver.

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Well above resonance,

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the lag approaches one hundred
eighty degrees, or half a cycle.

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Force and displacement
are then almost opposite.

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The phase relation is easier
to read as paired waveforms.

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Gray is the driving force
and blue is the displacement.

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Below resonance,
their peaks are nearly aligned.

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Near the resonance peak,

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the blue displacement reaches its maximum
one quarter cycle after the gray force.

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That is a phase lag of pi over two.

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Far above resonance, the blue
response is nearly inverted.

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A force maximum occurs close
to a displacement minimum,

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which is the half-cycle limit.

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The live sweep has therefore
shown two linked changes.

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Amplitude rises and falls
around resonance, while phase moves

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smoothly from almost zero, through
a quarter cycle, toward half a cycle.

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Now increase the damping deliberately.

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The new term b x prime opposes velocity,

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so it removes mechanical
energy whenever the mass moves.

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The dimensionless damping
ratio zeta compares that loss

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with the mass and spring scales.

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It lets different oscillators
be compared on the same frequency graph.

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The blue curve has light damping. Its
response rises into a tall, narrow peak,

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so a small change in driving frequency
produces a large change in amplitude.

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Increase the damping to the green curve.

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The peak is lower and broader.

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Energy is removed more quickly,

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so less of it can accumulate
from one cycle to the next.

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With still stronger damping,

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the red response has no distinct resonance
peak at all. The system still responds,

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but it no longer selects one
sharply amplified driving frequency.

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Damping also shifts the maximum.

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For the standard viscous model, the peak
occurs at omega nought times the square

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root of one minus two zeta squared, while
that expression describes a distinct peak.

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So the frequency

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of maximum amplitude lies slightly
below the undamped natural frequency.

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More damping pushes it farther
down until the peak itself ceases

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to be a useful feature.

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Quality factor packages
this behavior into one number.

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For light damping,

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Q is approximately omega nought divided
by the resonance bandwidth delta omega.

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A narrow bandwidth makes Q large.

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Only frequencies close to omega
nought drive a large response.

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A broad bandwidth makes Q smaller
and the resonance less selective.

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For the same lightly damped oscillator, Q
is also approximately one over twice zeta.

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Increasing damping therefore lowers Q

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at the same time that it
flattens the response.

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The second reading of Q appears
after the driver is switched off.

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The oscillator continues to ring,

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but damping makes its amplitude
envelope decay exponentially.

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The blue ring-down has Q equal to eight.

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Many oscillations remain visible

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because the amplitude changes
only a little during each cycle.

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The red ring-down has Q equal to two.

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It loses a much larger
fraction of its energy per cycle,

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so the motion disappears quickly.

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For light damping, the amplitude
behaves approximately like A nought

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times exponential minus
omega nought t over two Q.

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A larger Q therefore
means a longer decay time.

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A useful estimate is that the number

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of clearly significant cycles
in the ring-down scales like Q

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divided by pi.

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This is not a sharp stopping rule,

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but it connects the abstract
number to something visible.

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Quality factor therefore answers two
versions of the same energy question.

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How narrowly does the oscillator
accept energy from a driver,

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and how slowly does it give stored
energy away after the driver stops?

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Finish with a structure that engineers
do not want to resonate strongly.

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This sketch compresses one important

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sideways mode of a tall building under
an external force into the coordinate X.

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Suspend a second mass
inside the structure.

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Choose its natural frequency near
the troublesome building frequency,

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so it responds strongly
where the building needs help.

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When the tower moves one way,
the auxiliary mass can move the other way.

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That counter-motion reduces
the response of the main structural mode.

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The useful quantity is relative motion.

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It drives the connecting
damper, which converts part

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of the mechanical energy into heat instead
of letting that energy remain in the sway.

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This is intentional resonance.

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Engineers tune the smaller oscillator
so that it accepts motion near a selected

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structural frequency, then use
damping to dispose of energy.

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The red curve is a simple model

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of the structural response
without the auxiliary mass.

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Near its natural frequency,
one large resonance peak dominates.

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The blue curve includes
the tuned mass and its damping.

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The single large peak
is split into two smaller peaks,

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and the response near the original
resonance is greatly reduced.

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The device does not
cancel every possible motion.

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It targets a chosen range, and its
performance depends on tuning, mass ratio,

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available travel, damping, and the
modes present in the real structure.

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A famous real example is Taipei 101.

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High in the tower hangs a pendulum-like
steel tuned mass damper with a mass

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of about six hundred sixty metric tonnes.

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Suspending the mass high places it where
an important sway mode has large motion.

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When the building moves,

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the mass develops relative motion
rather than simply riding with the floors.

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Viscous dampers connected
to that motion dissipate energy.

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The system is intended especially to
reduce wind-driven accelerations and sway,

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helping both structural
response and occupant comfort.

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The honest limit matters.

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One tuned mass cannot
control every structural mode

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or every possible earthquake input.

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Engineers still need the full structure,

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safety margins, travel limits,
and other protective systems.

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The connection back
to the swing is timing and energy.

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A well-timed push feeds an oscillator.

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A tuned counterweight creates an opposing
motion at a selected frequency,

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and damping removes energy before the main
structure builds a dangerous response.

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Carry four ideas away.

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Natural frequency sets
the preferred rhythm.

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A live sweep reveals
resonance and phase together.

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Quality factor connects
peak sharpness with ring-down.

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And carefully tuned damping
can turn resonance from a hazard

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into an engineering tool.