{"version":1,"lectureId":"01M14TZSZRQC5V822RN2T74YRJ","attempt":1,"publication":{"slug":"driven-oscillators-resonance-phase-damping-and-quality-factor","title":"Driven Oscillators: Resonance, Phase, Damping, and Quality Factor","subject":"physics","summary":"A first-year physics treatment of the driven harmonic oscillator built around a live frequency sweep. Beginning with a child on a swing and the natural frequency of a free mass-spring system, the lecture shows amplitude rising near resonance while phase lag moves from almost zero to a quarter cycle near the peak and toward half a cycle above it. Damping then lowers, broadens, and shifts the response peak. Quality factor connects that frequency selectivity to the duration of free ring-down, before a tuned mass damper and Taipei 101 show how the same physics is used honestly in structural engineering.","metaDescription":"Watch a live frequency sweep reveal resonance and phase, then connect damping and quality factor to ring-down and tuned structural control.","transcript":"A child on a swing already knows the central idea of resonance. One push can be gentle, yet a sequence of pushes can make the motion large if each push arrives at the useful part of the swing. Let the child swing freely first. The motion repeats with a rhythm set mainly by the swing itself. Nobody has to prescribe that rhythm from outside. A push in the direction of motion adds energy. The same push at the wrong time may slow the child instead. Timing matters because the swing has a natural rhythm of its own. To isolate that rhythm, replace the swing by the simplest oscillator: a mass m attached to a spring of stiffness k. Displacement x is measured from the dashed equilibrium line. Pull the mass to the right. The spring pulls left. Move the mass to the left, and the spring pulls right. In either case the force points back toward equilibrium. Hooke's law writes that restoring force as minus k x. The minus sign records the reversal we just watched: force and displacement point in opposite directions. Newton's second law says mass times acceleration equals the total force. With no driver and no damping, the spring force is the only force in this one-dimensional model. Divide by the mass and collect everything on the left. We obtain x double prime plus k over m times x equals zero. A sinusoid solves this equation. Its amplitude A and starting phase phi depend on how we release the mass, but the angular frequency does not. Substitution gives the natural angular frequency, omega nought, equal to the square root of k over m. A stiffer spring raises it. A larger mass lowers it. The period is two pi divided by omega nought. That is the time for one complete free oscillation, and therefore the timing an outside force must confront. So before any driving force appears, the oscillator already owns a preferred time scale. Resonance begins when the rhythm imposed from outside approaches this natural rhythm. Now attach an external force that oscillates sinusoidally. Its maximum strength is F nought, and omega tells us how rapidly the applied force repeats. There are now two frequencies to keep separate. Omega nought is fixed by the mass and spring. Omega is chosen by the driver, and we are free to change it. Far below the natural frequency, the force changes slowly. The mass has time to follow, so displacement and force are nearly in step. At the ideal undamped resonance, the usual steady amplitude does not exist. Each correctly timed cycle adds energy, and the amplitude grows roughly in proportion to time. That is the mathematical version of pushing the swing at the useful moment. The driver keeps doing positive work instead of returning the energy it supplied on the previous cycle. A real experiment always loses a little energy, even before we add a deliberate damper. We will use a small damping ratio for a finite steady response, then increase it later. Here is the central experiment. We change the driving frequency slowly enough for the transient motion to settle. The upper yellow point reads amplitude, and the lower one reads phase lag. The response has the same frequency as the driver, but its amplitude A and lag delta depend on frequency. The ratio r compares the driving frequency with the natural frequency. Begin well below resonance. The amplitude is modest, and the phase lag is close to zero. The oscillator follows the slowly changing force. Now sweep upward. The amplitude climbs sharply as omega approaches omega nought. The driver is repeatedly adding energy at nearly the rhythm the oscillator prefers. At the resonance region, the displacement lags the force by about ninety degrees, one quarter of a cycle. The force is then well placed to feed energy into the velocity. Continue above resonance. The amplitude falls again, while the phase lag keeps increasing. The oscillator can no longer reverse quickly enough to follow the driver. Well above resonance, the lag approaches one hundred eighty degrees, or half a cycle. Force and displacement are then almost opposite. The phase relation is easier to read as paired waveforms. Gray is the driving force and blue is the displacement. Below resonance, their peaks are nearly aligned. Near the resonance peak, the blue displacement reaches its maximum one quarter cycle after the gray force. That is a phase lag of pi over two. Far above resonance, the blue response is nearly inverted. A force maximum occurs close to a displacement minimum, which is the half-cycle limit. The live sweep has therefore shown two linked changes. Amplitude rises and falls around resonance, while phase moves smoothly from almost zero, through a quarter cycle, toward half a cycle. Now increase the damping deliberately. The new term b x prime opposes velocity, so it removes mechanical energy whenever the mass moves. The dimensionless damping ratio zeta compares that loss with the mass and spring scales. It lets different oscillators be compared on the same frequency graph. The blue curve has light damping. Its response rises into a tall, narrow peak, so a small change in driving frequency produces a large change in amplitude. Increase the damping to the green curve. The peak is lower and broader. Energy is removed more quickly, so less of it can accumulate from one cycle to the next. With still stronger damping, the red response has no distinct resonance peak at all. The system still responds, but it no longer selects one sharply amplified driving frequency. Damping also shifts the maximum. For the standard viscous model, the peak occurs at omega nought times the square root of one minus two zeta squared, while that expression describes a distinct peak. So the frequency of maximum amplitude lies slightly below the undamped natural frequency. More damping pushes it farther down until the peak itself ceases to be a useful feature. Quality factor packages this behavior into one number. For light damping, Q is approximately omega nought divided by the resonance bandwidth delta omega. A narrow bandwidth makes Q large. Only frequencies close to omega nought drive a large response. A broad bandwidth makes Q smaller and the resonance less selective. For the same lightly damped oscillator, Q is also approximately one over twice zeta. Increasing damping therefore lowers Q at the same time that it flattens the response. The second reading of Q appears after the driver is switched off. The oscillator continues to ring, but damping makes its amplitude envelope decay exponentially. The blue ring-down has Q equal to eight. Many oscillations remain visible because the amplitude changes only a little during each cycle. The red ring-down has Q equal to two. It loses a much larger fraction of its energy per cycle, so the motion disappears quickly. For light damping, the amplitude behaves approximately like A nought times exponential minus omega nought t over two Q. A larger Q therefore means a longer decay time. A useful estimate is that the number of clearly significant cycles in the ring-down scales like Q divided by pi. This is not a sharp stopping rule, but it connects the abstract number to something visible. Quality factor therefore answers two versions of the same energy question. How narrowly does the oscillator accept energy from a driver, and how slowly does it give stored energy away after the driver stops? Finish with a structure that engineers do not want to resonate strongly. This sketch compresses one important sideways mode of a tall building under an external force into the coordinate X. Suspend a second mass inside the structure. Choose its natural frequency near the troublesome building frequency, so it responds strongly where the building needs help. When the tower moves one way, the auxiliary mass can move the other way. That counter-motion reduces the response of the main structural mode. The useful quantity is relative motion. It drives the connecting damper, which converts part of the mechanical energy into heat instead of letting that energy remain in the sway. This is intentional resonance. Engineers tune the smaller oscillator so that it accepts motion near a selected structural frequency, then use damping to dispose of energy. The red curve is a simple model of the structural response without the auxiliary mass. Near its natural frequency, one large resonance peak dominates. The blue curve includes the tuned mass and its damping. The single large peak is split into two smaller peaks, and the response near the original resonance is greatly reduced. The device does not cancel every possible motion. It targets a chosen range, and its performance depends on tuning, mass ratio, available travel, damping, and the modes present in the real structure. A famous real example is Taipei 101. High in the tower hangs a pendulum-like steel tuned mass damper with a mass of about six hundred sixty metric tonnes. Suspending the mass high places it where an important sway mode has large motion. When the building moves, the mass develops relative motion rather than simply riding with the floors. Viscous dampers connected to that motion dissipate energy. The system is intended especially to reduce wind-driven accelerations and sway, helping both structural response and occupant comfort. The honest limit matters. One tuned mass cannot control every structural mode or every possible earthquake input. Engineers still need the full structure, safety margins, travel limits, and other protective systems. The connection back to the swing is timing and energy. A well-timed push feeds an oscillator. A tuned counterweight creates an opposing motion at a selected frequency, and damping removes energy before the main structure builds a dangerous response. Carry four ideas away. Natural frequency sets the preferred rhythm. A live sweep reveals resonance and phase together. Quality factor connects peak sharpness with ring-down. And carefully tuned damping can turn resonance from a hazard into an engineering tool.","watch":{"version":1,"scenes":[{"title":"The Natural Rhythm","start":0,"end":143.2581875,"objects":{"child":"a Circle [yellow] drawn in swing (center=((1.7 * sin(swing_phase)), ((1.8 - (1.7 * cos(swing_phase))) + …, radius=0.24, filled=True)","clock":"a VariableNumber (initial_value=0.2)","equilibrium":"a Line [gray] drawn in oscillator (start=(0.0, -1.25), end=(0.0, 1.25), dashed=True)","free_work":"a Derivation [text] that says \"$F_s &= -k x \\ m x'' &= -k x \\ x'' + frac(k, m) x &= 0 \\ x(t) &= A cos(omega_0 t + phi) \\ omega_0 &= sqrt(frac(k, m)) \\ T_0 &= frac(2 pi, omega_0)$\"","head_free":"a Heading that says \"The Oscillator Chooses Its Own Frequency\"","head_swing":"a Heading that says \"A Push at the Right Moment\"","mass":"a Polygon [yellow] drawn in oscillator (vertices=((((1.7 * cos(clock)) - 0.55), -0.55), (((1.7 * cos(clock)) + 0…, fill_opacity=0.45)","mass_name":"a Point [text] labelled \"m\" drawn in oscillator (location=((1.7 * cos(clock)), 0.0), show_marker=False)","origin":"a Point [gray] labelled \"x=0\" drawn in oscillator (location=(0.0, -1.0), show_marker=False)","oscillator":"a Figure (x_range=(-3.6, 3.6), y_range=(-1.6, 1.6))","push":"a Vector [green] labelled \"upright(\"push\")\" drawn in swing (start=(2.2, 0.1), end=(1.35, 0.1))","restoring":"a Vector [red] labelled \"F_s\" drawn in oscillator (start=((1.7 * cos(clock)), 0.9), end=((0.25 * (1.7 * cos(clock))), 0.9))","rope":"a Line [blue] drawn in swing (start=(0.0, 1.8), end=((1.7 * sin(swing_phase)), (1.8 - (1.7 * cos(swing_phase)))))","seat":"a Line [blue] drawn in swing (start=(((1.7 * sin(swing_phase)) - 0.45), ((1.8 - (1.7 * cos(swing_ph…, end=(((1.7 * sin(swing_phase)) + 0.45), ((1.8 - (1.7 * cos(swing_ph…)","spring":"a Line [blue] labelled \"k\" drawn in oscillator (start=(-3.0, 0.0), end=(((1.7 * cos(clock)) - 0.55), 0.0))","support":"a Line [gray] drawn in swing (start=(-1.5, 1.8), end=(1.5, 1.8))","swing":"a Figure (x_range=(-2.5, 2.5), y_range=(-1.5, 2.2))","swing_phase":"a VariableNumber (initial_value=0.35)","timing":"a Text [text] that says \"A well-timed push adds energy. A poorly timed push can oppose the motion.\"","wall":"a Line [gray] drawn in oscillator (start=(-3.0, -1.1), end=(-3.0, 1.1))"},"beats":[{"start":0,"say":"A child on a swing already knows the central idea of resonance. One push can be gentle, yet a sequence of pushes can make the motion large if each push arrives at the useful part of the swing.","live":[],"does":[[0,"head_swing is shown on the screen, written out."],[0.673,"swing is shown on the screen, written out."],[0.673,"support is shown on the screen, written out."],[0.673,"rope is shown on the screen, written out."],[0.673,"seat is shown on the screen, written out."],[0.673,"child is shown on the screen, written out."]]},{"start":12.326,"say":"Let the child swing freely first. The motion repeats with a rhythm set mainly by the swing itself. Nobody has to prescribe that rhythm from outside.","live":["swing","head_swing","support","rope","seat","child"],"does":[[13.638,"rope is redrawn as the numbers it depends on change."],[13.638,"seat is redrawn as the numbers it depends on change."],[13.638,"child is redrawn as the numbers it depends on change."],[13.638,"swing_phase ticks to 2.791592653589793."],[16.262,"rope is redrawn as the numbers it depends on change."],[16.262,"seat is redrawn as the numbers it depends on change."],[16.262,"child is redrawn as the numbers it depends on change."],[16.262,"swing_phase ticks to 6.633185307179586."]]},{"start":22.3185,"say":"A push in the direction of motion adds energy. The same push at the wrong time may slow the child instead. Timing matters because the swing has a natural rhythm of its own.","live":null,"does":[[22.702,"push is shown on the screen, written out."],[24.304,"timing is shown on the screen, written out."],[26.858,"push is hidden from the screen."],[33.3825,"head_swing is hidden from the screen — left the board."],[33.3825,"swing is hidden from the screen — left the board."],[33.3825,"support is hidden from the screen — swing left the board."],[33.3825,"rope is hidden from the screen — swing left the board."],[33.3825,"seat is hidden from the screen — swing left the board."],[33.3825,"child is hidden from the screen — swing left the board."],[33.3825,"timing is hidden from the screen — left the board."]]},{"start":33.9825,"say":"To isolate that rhythm, replace the swing by the simplest oscillator: a mass m attached to a spring of stiffness k. Displacement x is measured from the dashed equilibrium line.","live":[],"does":[[33.9825,"head_free is shown on the screen, written out."],[33.9825,"oscillator is shown on the screen, written out."],[33.9825,"wall is shown on the screen, written out."],[39.242000000000004,"mass is shown on the screen, written out."],[39.242000000000004,"mass_name is shown on the screen, written out."],[41.076,"spring is shown on the screen, written out."],[46.602000000000004,"equilibrium is shown on the screen, written out."],[46.602000000000004,"origin is shown on the screen, written out."]]},{"start":48.920500000000004,"say":"Pull the mass to the right. The spring pulls left. Move the mass to the left, and the spring pulls right. In either case the force points back toward equilibrium.","live":["oscillator","head_free","wall","equilibrium","origin","spring","mass","mass_name"],"does":[[51.742000000000004,"restoring is shown on the screen, written out."],[52.067,"spring is redrawn as the numbers it depends on change."],[52.067,"mass is redrawn as the numbers it depends on change."],[52.067,"mass_name is redrawn as the numbers it depends on change."],[52.067,"clock ticks to 3.3415926535897933."],[59.068,"spring is redrawn as the numbers it depends on change."],[59.068,"mass is redrawn as the numbers it depends on change."],[59.068,"mass_name is redrawn as the numbers it depends on change."],[59.068,"restoring is redrawn as the numbers it depends on change."],[59.068,"clock ticks to 6.483185307179586."]]},{"start":60.898500000000006,"say":"Hooke's law writes that restoring force as minus k x. The minus sign records the reversal we just watched: force and displacement point in opposite directions.","live":["oscillator","head_free","wall","equilibrium","origin","spring","mass","mass_name","restoring"],"does":[[63.453,"oscillator moves to a new place on the board."],[63.453,"free_work is shown on the screen, written out."],[65.72800000000001,"free_work (the \"-k x\" part) is emphasized."],[72.17200000000001,"free_work (the \"-k x\" part) is no longer emphasized."]]},{"start":72.772,"say":"Newton's second law says mass times acceleration equals the total force. With no driver and no damping, the spring force is the only force in this one-dimensional model.","live":null,"does":[[73.12,"free_work is shown on the screen, written out."],[74.664,"mass is indicated — a transient flash."],[77.358,"restoring is indicated — a transient flash."]]},{"start":84.9,"say":"Divide by the mass and collect everything on the left. We obtain x double prime plus k over m times x equals zero.","live":null,"does":[[85.248,"free_work is shown on the screen, written out."]]},{"start":95.601,"say":"A sinusoid solves this equation. Its amplitude A and starting phase phi depend on how we release the mass, but the angular frequency does not.","live":null,"does":[[96.14699999999999,"free_work is shown on the screen, written out."],[98.968,"free_work (the \"A\" part) is emphasized."],[100.524,"free_work (the \"A\" part) is no longer emphasized."],[100.524,"free_work (the \"phi\" part) is emphasized."],[104.366,"free_work (the \"phi\" part) is no longer emphasized."]]},{"start":106.5805,"say":"Substitution gives the natural angular frequency, omega nought, equal to the square root of k over m. A stiffer spring raises it. A larger mass lowers it.","live":null,"does":[[111.991,"free_work is shown on the screen, written out."],[114.59100000000001,"free_work (the \"k\" part) is emphasized."],[117.227,"free_work (the \"k\" part) is no longer emphasized."],[117.227,"free_work (the \"m\" part) is emphasized."],[118.6315,"free_work (the \"m\" part) is no longer emphasized."]]},{"start":119.2315,"say":"The period is two pi divided by omega nought. That is the time for one complete free oscillation, and therefore the timing an outside force must confront.","live":null,"does":[[119.75399999999999,"free_work is shown on the screen, written out."],[124.607,"clock ticks to 6.483185307179586."]]},{"start":130.9195,"say":"So before any driving force appears, the oscillator already owns a preferred time scale. Resonance begins when the rhythm imposed from outside approaches this natural rhythm.","live":null,"does":[[135.273,"A box is drawn around free_work."],[142.21652083333333,"free_work is hidden from the screen — left the board."],[142.21652083333333,"head_free is hidden from the screen — left the board."],[142.21652083333333,"oscillator is hidden from the screen — left the board."],[142.21652083333333,"wall is hidden from the screen — oscillator left the board."],[142.21652083333333,"equilibrium is hidden from the screen — oscillator left the board."],[142.21652083333333,"origin is hidden from the screen — oscillator left the board."],[142.21652083333333,"spring is hidden from the screen — oscillator left the board."],[142.21652083333333,"mass is hidden from the screen — oscillator left the board."],[142.21652083333333,"mass_name is hidden from the screen — oscillator left the board."],[142.21652083333333,"restoring is hidden from the screen — oscillator left the board."]]}]},{"title":"Sweeping Through Resonance","start":143.2581875,"end":342.17925,"objects":{"amp_axes":"an Axes (x_range=(0.0, 2.1), y_range=(0.0, 7.0), x_ticks_every=0.5)","amp_curve":"a FunctionPlot [blue] labelled \"zeta=0.08\" drawn in amp_axes (function=<function>, x_range=(0.0, 2.1))","amp_pointer":"a PlotPoint [yellow] drawn in amp_axes (target='amp_curve', x=<VariableNumber frequency = 1.85>)","axes":"an Axes (x_range=(0.0, 6.283185307179586), y_range=(-1.15, 1.15), x_ticks_every=3.141592653589793)","axes_2":"an Axes (x_range=(0.0, 6.283185307179586), y_range=(-1.15, 1.15), x_ticks_every=3.141592653589793)","axes_3":"an Axes (x_range=(0.0, 6.283185307179586), y_range=(-1.15, 1.15), x_ticks_every=3.141592653589793)","drive_force":"a Vector [green] labelled \"F_0 cos(omega t)\" drawn in mechanism (start=(1.5, 0.0), end=(2.9, 0.0))","drive_mass":"a Polygon [yellow] labelled \"m\" drawn in mechanism (vertices=((0.4, -0.55), (1.5, -0.55), (1.5, 0.55), (0.4, 0.55)), fill_opacity=0.42)","drive_note":"a Panel that says \"The driving angular frequency $omega$ belongs to the external force. 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Its maximum strength is F nought, and omega tells us how rapidly the applied force repeats.","live":[],"does":[[143.2581875,"head_drive is shown on the screen, written out."],[143.2581875,"mechanism is shown on the screen, written out."],[143.2581875,"drive_wall is shown on the screen, written out."],[144.1291875,"drive_spring is shown on the screen, written out."],[144.6631875,"drive_mass is shown on the screen, written out."],[145.68418749999998,"undamped_equation is shown on the screen, written out."],[151.8841875,"drive_force is shown on the screen, written out."]]},{"start":154.2491875,"say":"There are now two frequencies to keep separate. Omega nought is fixed by the mass and spring. Omega is chosen by the driver, and we are free to change it.","live":["undamped_equation","mechanism","head_drive","drive_wall","drive_spring","drive_mass","drive_force"],"does":[[155.34018749999998,"drive_note is shown on the screen, written out."],[157.8481875,"undamped_equation (the \"omega\" part) is emphasized."],[165.11568749999998,"undamped_equation (the \"omega\" part) is no longer emphasized."]]},{"start":165.7156875,"say":"Far below the natural frequency, the force changes slowly. The mass has time to follow, so displacement and force are nearly in step.","live":["drive_note","undamped_equation","mechanism","head_drive","drive_wall","drive_spring","drive_mass","drive_force"],"does":[[168.36318749999998,"drive_force is indicated — a transient flash."]]},{"start":176.1376875,"say":"At the ideal undamped resonance, the usual steady amplitude does not exist. Each correctly timed cycle adds energy, and the amplitude grows roughly in proportion to time.","live":null,"does":[[176.1376875,"resonant_growth is shown on the screen, written out."],[187.3061875,"resonant_growth (the \"t\" part) is indicated — a transient flash."]]},{"start":188.80068749999998,"say":"That is the mathematical version of pushing the swing at the useful moment. The driver keeps doing positive work instead of returning the energy it supplied on the previous cycle.","live":["drive_note","undamped_equation","resonant_growth","mechanism","head_drive","drive_wall","drive_spring","drive_mass","drive_force"],"does":[[193.7701875,"drive_force is indicated — a transient flash."]]},{"start":199.4431875,"say":"A real experiment always loses a little energy, even before we add a deliberate damper. We will use a small damping ratio for a finite steady response, then increase it later.","live":null,"does":[[207.8141875,"resonant_growth is indicated — a transient flash."],[211.1581875,"drive_note is hidden from the screen — left the board."],[211.1581875,"head_drive is hidden from the screen — left the board."],[211.1581875,"mechanism is hidden from the screen — left the board."],[211.1581875,"drive_wall is hidden from the screen — mechanism left the board."],[211.1581875,"drive_spring is hidden from the screen — mechanism left the board."],[211.1581875,"drive_mass is hidden from the screen — mechanism left the board."],[211.1581875,"drive_force is hidden from the screen — mechanism left the board."],[211.1581875,"resonant_growth is hidden from the screen — left the board."],[211.1581875,"undamped_equation is hidden from the screen — left the board."]]},{"start":211.7581875,"say":"Here is the central experiment. We change the driving frequency slowly enough for the transient motion to settle. The upper yellow point reads amplitude, and the lower one reads phase lag.","live":[],"does":[[211.7581875,"head_sweep is shown on the screen, written out."],[216.1701875,"slow_note is shown on the screen, written out."],[219.7691875,"amp_axes is shown on the screen, written out."],[220.03618749999998,"amp_pointer is shown on the screen, written out."],[220.96518749999998,"amp_curve is shown on the screen, drawn."],[222.2881875,"phase_axes is shown on the screen, written out."],[223.21718749999997,"phase_curve is shown on the screen, drawn."],[223.58918749999998,"phase_pointer is shown on the screen, written out."]]},{"start":224.8506875,"say":"The response has the same frequency as the driver, but its amplitude A and lag delta depend on frequency. The ratio r compares the driving frequency with the natural frequency.","live":["slow_note","amp_axes","phase_axes","head_sweep","amp_curve","amp_pointer","phase_curve","phase_pointer"],"does":[[225.30318749999998,"steady is shown on the screen, written out."],[232.5711875,"steady is shown on the screen, written out."],[234.0801875,"steady (the \"omega\" part) is emphasized."],[235.7061875,"steady (the \"omega\" part) is no longer emphasized."],[235.7061875,"steady (the \"omega_0\" part) is emphasized."],[237.1801875,"steady (the \"omega_0\" part) is no longer emphasized."]]},{"start":237.7801875,"say":"Begin well below resonance. The amplitude is modest, and the phase lag is close to zero. The oscillator follows the slowly changing force.","live":null,"does":[[240.8451875,"steady is shown on the screen, written out."],[241.5071875,"amp_pointer is indicated — a transient flash."],[242.5521875,"steady is shown on the screen, written out."],[243.7011875,"phase_pointer is indicated — a transient flash."]]},{"start":248.8636875,"say":"Now sweep upward. The amplitude climbs sharply as omega approaches omega nought. The driver is repeatedly adding energy at nearly the rhythm the oscillator prefers.","live":null,"does":[[249.58418749999998,"amp_pointer is redrawn as the numbers it depends on change."],[249.58418749999998,"phase_pointer is redrawn as the numbers it depends on change."],[249.58418749999998,"frequency ticks to 0.82."],[253.79818749999998,"amp_pointer is redrawn as the numbers it depends on change."],[253.79818749999998,"phase_pointer is redrawn as the numbers it depends on change."],[253.79818749999998,"frequency ticks to 1.0."]]},{"start":261.4226875,"say":"At the resonance region, the displacement lags the force by about ninety degrees, one quarter of a cycle. The force is then well placed to feed energy into the velocity.","live":null,"does":[[265.48618749999997,"phase_pointer is indicated — a transient flash."],[267.04218749999995,"steady (the \"delta\" part) is emphasized."],[272.6846875,"steady (the \"delta\" part) is no longer emphasized."]]},{"start":273.2846875,"say":"Continue above resonance. The amplitude falls again, while the phase lag keeps increasing. The oscillator can no longer reverse quickly enough to follow the driver.","live":null,"does":[[273.63318749999996,"amp_pointer is redrawn as the numbers it depends on change."],[273.63318749999996,"phase_pointer is redrawn as the numbers it depends on change."],[273.63318749999996,"frequency ticks to 1.45."],[279.0541875,"amp_pointer is redrawn as the numbers it depends on change."],[279.0541875,"phase_pointer is redrawn as the numbers it depends on change."],[279.0541875,"frequency ticks to 1.85."]]},{"start":284.9836875,"say":"Well above resonance, the lag approaches one hundred eighty degrees, or half a cycle. Force and displacement are then almost opposite.","live":null,"does":[[285.65718749999996,"amp_pointer is indicated — a transient flash."],[289.18618749999996,"phase_pointer is indicated — a transient flash."],[294.26018750000003,"amp_axes is hidden from the screen — left the board."],[294.26018750000003,"amp_curve is hidden from the screen — amp_axes left the board."],[294.26018750000003,"amp_pointer is hidden from the screen — amp_axes left the board."],[294.26018750000003,"head_sweep is hidden from the screen — left the board."],[294.26018750000003,"phase_axes is hidden from the screen — left the board."],[294.26018750000003,"phase_curve is hidden from the screen — phase_axes left the board."],[294.26018750000003,"phase_pointer is hidden from the screen — phase_axes left the board."],[294.26018750000003,"slow_note is hidden from the screen — left the board."],[294.26018750000003,"steady is hidden from the screen — left the board."]]},{"start":294.8601875,"say":"The phase relation is easier to read as paired waveforms. Gray is the driving force and blue is the displacement. Below resonance, their peaks are nearly aligned.","live":[],"does":[[294.8601875,"head_phase is shown on the screen, written out."],[297.5881875,"axes is shown on the screen, written out."],[299.0401875,"low_drive is shown on the screen, written out."],[300.9321875,"low_reply is shown on the screen, written out."],[305.3671875,"axes moves to a new place on the board."],[305.3671875,"low_caption is shown on the screen, written out."]]},{"start":306.8846875,"say":"Near the resonance peak, the blue displacement reaches its maximum one quarter cycle after the gray force. That is a phase lag of pi over two.","live":["axes","low_caption","head_phase","low_drive","low_reply"],"does":[[307.58118749999994,"axes_2 is shown on the screen, written out."],[308.7191875,"peak_reply is shown on the screen, written out."],[310.9951875,"peak_caption is shown on the screen, written out."],[312.4811875,"peak_drive is shown on the screen, written out."]]},{"start":317.2606875,"say":"Far above resonance, the blue response is nearly inverted. A force maximum occurs close to a displacement minimum, which is the half-cycle limit.","live":["axes","low_caption","axes_2","peak_caption","head_phase","low_drive","low_reply","peak_drive","peak_reply"],"does":[[317.8991875,"axes_3 is shown on the screen, written out."],[318.87418749999995,"high_reply is shown on the screen, written out."],[321.55618749999996,"high_drive is shown on the screen, written out."],[325.53918749999997,"high_caption is shown on the screen, written out."]]},{"start":327.6016875,"say":"The live sweep has therefore shown two linked changes. Amplitude rises and falls around resonance, while phase moves smoothly from almost zero, through a quarter cycle, toward half a cycle.","live":["axes","low_caption","axes_2","peak_caption","axes_3","high_caption","head_phase","low_drive","low_reply","peak_drive","peak_reply","high_drive","high_reply"],"does":[[338.36418749999996,"peak_caption is indicated — a transient flash."],[339.8381875,"high_caption is indicated — a transient flash."],[341.13758333333334,"axes is hidden from the screen — left the board."],[341.13758333333334,"low_drive is hidden from the screen — axes left the board."],[341.13758333333334,"low_reply is hidden from the screen — axes left the board."],[341.13758333333334,"axes_2 is hidden from the screen — left the board."],[341.13758333333334,"peak_drive is hidden from the screen — axes_2 left the board."],[341.13758333333334,"peak_reply is hidden from the screen — axes_2 left the board."],[341.13758333333334,"axes_3 is hidden from the screen — left the board."],[341.13758333333334,"high_drive is hidden from the screen — axes_3 left the board."],[341.13758333333334,"high_reply is hidden from the screen — axes_3 left the board."],[341.13758333333334,"head_phase is hidden from the screen — left the board."],[341.13758333333334,"high_caption is hidden from the screen — left the board."],[341.13758333333334,"low_caption is hidden from the screen — left the board."],[341.13758333333334,"peak_caption is hidden from the screen — left the board."]]}]},{"title":"Damping and Quality Factor","start":342.17925,"end":532.2701041666667,"objects":{"band":"a Line [yellow] labelled \"Delta omega\" drawn in q_axes (start=(0.92, 4.4), end=(1.08, 4.4))","cycles":"a Math [text] that says \"$N_(upright(\"cycles\")) approx frac(Q, pi)$\"","damping_note":"a Text [text] that says \"Damping removes mechanical energy and makes the steady resonance finite.\"","equation":"a Math [text] that says \"$m x'' + b x' + k x = F_0 cos(omega t)$\"","head_damping":"a Heading that says \"More Damping, Less Resonance\"","head_q":"a Heading that says \"One Number, Two Readings\"","heavy_curve":"a FunctionPlot [red] labelled \"zeta=0.80\" drawn in response_axes (function=<function>)","high_ring":"a FunctionPlot [blue] labelled \"Q=8\" drawn in ring_axes (function=<function>)","light_curve":"a FunctionPlot [blue] labelled \"zeta=0.08\" drawn in response_axes (function=<function>)","light_peak":"a PlotPoint [blue] drawn in response_axes (target='light_curve', x=0.9935793878699376)","low_ring":"a FunctionPlot [red] labelled \"Q=2\" drawn in ring_axes (function=<function>)","medium_curve":"a FunctionPlot [green] labelled \"zeta=0.30\" drawn in response_axes (function=<function>)","medium_peak":"a PlotPoint [green] drawn in response_axes (target='medium_curve', x=0.9055385138137417)","peak_condition":"a Math [text] that says \"$zeta < frac(1, sqrt(2)) quad upright(\"for a distinct peak\")$\"","peak_line":"a Math [text] that says \"$omega_r = omega_0 sqrt(1-2 zeta^2)$\"","q_axes":"an Axes (x_range=(0.6, 1.4), y_range=(0.0, 7.0), x_ticks_every=0.2)","q_curve":"a FunctionPlot [blue] drawn in q_axes (function=<function>)","q_formula":"a Math [text] that says \"$Q approx frac(omega_0, Delta omega) approx frac(1, 2 zeta)$\"","quality_note":"a Panel that says \"For light damping, a large $Q$ means a narrow resonance and a slow decay after the driver is removed.\"","response_axes":"an Axes (x_range=(0.0, 2.1), y_range=(0.0, 7.0), x_ticks_every=0.5)","ring_axes":"an Axes (x_range=(0.0, 12.0), y_range=(-1.1, 1.1), x_ticks_every=2.0)","ring_formula":"a Math [text] that says \"$A(t) approx A_0 exp(-frac(omega_0 t, 2Q))$\"","tex":"a Tex [text] that says \"Frequency response\"","tex_2":"a Tex [text] that says \"Free ring-down\"","zeta_line":"a Math [text] that says \"$zeta = frac(b, 2 sqrt(k m))$\""},"beats":[{"start":342.17925,"say":"Now increase the damping deliberately. The new term b x prime opposes velocity, so it removes mechanical energy whenever the mass moves.","live":[],"does":[[342.17925,"head_damping is shown on the screen, written out."],[342.17925,"equation is shown on the screen, written out."],[345.36025,"equation (the \"b x'\" part) is emphasized."],[348.14725000000004,"damping_note is shown on the screen, written out."],[349.12225,"equation (the \"b x'\" part) is no longer emphasized."]]},{"start":351.98575000000005,"say":"The dimensionless damping ratio zeta compares that loss with the mass and spring scales. It lets different oscillators be compared on the same frequency graph.","live":["equation","damping_note","head_damping"],"does":[[353.80825000000004,"zeta_line is shown on the screen, written out."],[361.33125,"response_axes is shown on the screen, written out."]]},{"start":362.66325,"say":"The blue curve has light damping. Its response rises into a tall, narrow peak, so a small change in driving frequency produces a large change in amplitude.","live":["equation","zeta_line","damping_note","response_axes","head_damping"],"does":[[362.66325,"light_curve is shown on the screen, drawn."],[365.51925000000006,"light_peak is shown on the screen, written out."],[369.10625000000005,"light_peak is indicated — a transient flash."]]},{"start":373.74625000000003,"say":"Increase the damping to the green curve. The peak is lower and broader. Energy is removed more quickly, so less of it can accumulate from one cycle to the next.","live":["equation","zeta_line","damping_note","response_axes","head_damping","light_curve","light_peak"],"does":[[373.74625000000003,"medium_curve is shown on the screen, drawn."],[377.45025000000004,"medium_peak is shown on the screen, written out."],[377.45025000000004,"light_curve is indicated — a transient flash."],[377.89125,"medium_curve is indicated — a transient flash."]]},{"start":385.03925000000004,"say":"With still stronger damping, the red response has no distinct resonance peak at all. The system still responds, but it no longer selects one sharply amplified driving frequency.","live":["equation","zeta_line","damping_note","response_axes","head_damping","light_curve","light_peak","medium_curve","medium_peak"],"does":[[387.26825,"heavy_curve is shown on the screen, drawn."],[388.45225000000005,"peak_condition is shown on the screen, written out."]]},{"start":396.88925,"say":"Damping also shifts the maximum. For the standard viscous model, the peak occurs at omega nought times the square root of one minus two zeta squared, while that expression describes a distinct peak.","live":["equation","zeta_line","peak_condition","damping_note","response_axes","head_damping","light_curve","light_peak","medium_curve","medium_peak","heavy_curve"],"does":[[398.79325000000006,"peak_line is shown on the screen, written out."],[402.38025000000005,"light_peak is indicated — a transient flash."],[402.63625,"medium_peak is indicated — a transient flash."]]},{"start":411.04925000000003,"say":"So the frequency of maximum amplitude lies slightly below the undamped natural frequency. More damping pushes it farther down until the peak itself ceases to be a useful feature.","live":["equation","zeta_line","peak_line","peak_condition","damping_note","response_axes","head_damping","light_curve","light_peak","medium_curve","medium_peak","heavy_curve"],"does":[[414.84525,"peak_line (the \"sqrt(1-2 zeta^2)\" part) is emphasized."],[423.26275000000004,"damping_note is hidden from the screen — left the board."],[423.26275000000004,"equation is hidden from the screen — left the board."],[423.26275000000004,"head_damping is hidden from the screen — left the board."],[423.26275000000004,"peak_condition is hidden from the screen — left the board."],[423.26275000000004,"peak_line is hidden from the screen — left the board."],[423.26275000000004,"response_axes is hidden from the screen — left the board."],[423.26275000000004,"light_curve is hidden from the screen — response_axes left the board."],[423.26275000000004,"light_peak is hidden from the screen — response_axes left the board."],[423.26275000000004,"medium_curve is hidden from the screen — response_axes left the board."],[423.26275000000004,"medium_peak is hidden from the screen — response_axes left the board."],[423.26275000000004,"heavy_curve is hidden from the screen — response_axes left the board."],[423.26275000000004,"zeta_line is hidden from the screen — left the board."],[423.26275000000004,"peak_line (the \"sqrt(1-2 zeta^2)\" part) is no longer emphasized."]]},{"start":424.46275,"say":"Quality factor packages this behavior into one number. For light damping, Q is approximately omega nought divided by the resonance bandwidth delta omega.","live":[],"does":[[424.46275,"head_q is shown on the screen, written out."],[429.79125000000005,"q_formula is shown on the screen, written out."],[433.30925,"q_axes is shown on the screen, written out."],[433.30925,"q_curve is shown on the screen, drawn."],[433.94825000000003,"band is shown on the screen, written out."]]},{"start":435.87125000000003,"say":"A narrow bandwidth makes Q large. Only frequencies close to omega nought drive a large response. A broad bandwidth makes Q smaller and the resonance less selective.","live":["q_formula","q_axes","head_q","q_curve","band"],"does":[[436.41725,"band is indicated — a transient flash."],[436.80025,"q_formula (the \"Delta omega\" part) is emphasized."],[447.86375000000004,"q_formula (the \"Delta omega\" part) is no longer emphasized."]]},{"start":448.46375,"say":"For the same lightly damped oscillator, Q is also approximately one over twice zeta. Increasing damping therefore lowers Q at the same time that it flattens the response.","live":null,"does":[[452.65525,"q_formula (the \"frac(1, 2 zeta)\" part) is emphasized."],[459.58625000000006,"q_formula (the \"frac(1, 2 zeta)\" part) is no longer emphasized."]]},{"start":460.18625000000003,"say":"The second reading of Q appears after the driver is switched off. The oscillator continues to ring, but damping makes its amplitude envelope decay exponentially.","live":null,"does":[[462.71725000000004,"ring_formula is shown on the screen, written out."],[466.04925000000003,"ring_axes is shown on the screen, written out."],[468.80025,"high_ring is shown on the screen, drawn."]]},{"start":471.22325,"say":"The blue ring-down has Q equal to eight. Many oscillations remain visible because the amplitude changes only a little during each cycle.","live":["q_formula","q_axes","ring_formula","ring_axes","head_q","q_curve","band","high_ring"],"does":[[473.60325000000006,"high_ring is indicated — a transient flash."]]},{"start":480.61225,"say":"The red ring-down has Q equal to two. It loses a much larger fraction of its energy per cycle, so the motion disappears quickly.","live":null,"does":[[481.09925000000004,"low_ring is shown on the screen, drawn."],[488.42525,"low_ring is indicated — a transient flash."]]},{"start":489.89625,"say":"For light damping, the amplitude behaves approximately like A nought times exponential minus omega nought t over two Q. A larger Q therefore means a longer decay time.","live":["q_formula","q_axes","ring_formula","ring_axes","head_q","q_curve","band","high_ring","low_ring"],"does":[[497.97725,"ring_formula (the \"2Q\" part) is emphasized."],[502.67875000000004,"q_formula moves to a new place on the board."],[502.67875000000004,"ring_formula moves to a new place on the board."],[502.67875000000004,"q_axes is hidden from the screen — left the board."],[502.67875000000004,"q_curve is hidden from the screen — q_axes left the board."],[502.67875000000004,"band is hidden from the screen — q_axes left the board."],[502.67875000000004,"ring_axes is hidden from the screen — left the board."],[502.67875000000004,"high_ring is hidden from the screen — ring_axes left the board."],[502.67875000000004,"low_ring is hidden from the screen — ring_axes left the board."],[502.67875000000004,"ring_formula (the \"2Q\" part) is no longer emphasized."]]},{"start":503.87875,"say":"A useful estimate is that the number of clearly significant cycles in the ring-down scales like Q divided by pi. This is not a sharp stopping rule, but it connects the abstract number to something visible.","live":["q_formula","ring_formula","head_q"],"does":[[504.78425,"cycles is shown on the screen, written out."],[514.08425,"quality_note is shown on the screen, written out."]]},{"start":517.63275,"say":"Quality factor therefore answers two versions of the same energy question. How narrowly does the oscillator accept energy from a driver, and how slowly does it give stored energy away after the driver stops?","live":["q_formula","ring_formula","head_q","cycles","quality_note"],"does":[[523.82125,"A box is drawn around q_formula."],[527.38525,"A box is drawn around ring_formula."],[531.2284375,"cycles is hidden from the screen — left the board."],[531.2284375,"head_q is hidden from the screen — left the board."],[531.2284375,"q_formula is hidden from the screen — left the board."],[531.2284375,"quality_note is hidden from the screen — left the board."],[531.2284375,"ring_formula is hidden from the screen — left the board."]]}]},{"title":"A Tower with a Counterweight","start":532.2701041666667,"end":716.1985,"objects":{"bare_response":"a FunctionPlot [red] labelled \"upright(\"without damper\")\" drawn in response_axes (function=<function>)","cable":"a Line [gray] drawn in model (start=(<VariableNumber tower_motion = 0.32>, 4.65), end=((-1.5 * tower_motion), 3.22))","case_bob":"a Circle [yellow] labelled \"upright(\"about 660 tonnes\")\" drawn in case_figure (center=(0.0, 4.0), radius=0.52, filled=True)","case_figure":"a Figure (x_range=(-2.5, 2.5), y_range=(-0.3, 6.0))","case_ground":"a Line [gray] drawn in case_figure (start=(-2.0, 0.0), end=(2.0, 0.0))","case_tower":"a Polygon [blue] drawn in case_figure (vertices=((-0.65, 0.0), (0.65, 0.0), (0.38, 5.5), (-0.38, 5.5)), fill_opacity=0.18)","closing":"a Block [text] that says \"Every oscillator has a natural frequency. A slow sweep reveals amplitude and phase together. Damping lowers $Q$, broadens resonance, and shortens ring-down. Tuned counter-motion can protect a larger oscillator.\"","controlled_response":"a FunctionPlot [blue] labelled \"upright(\"with tuned mass\")\" drawn in response_axes (function=<function>)","counterweight":"a Circle [yellow] labelled \"m_d\" drawn in model (center=((-1.5 * tower_motion), 2.8), radius=0.42, filled=True)","damper":"a Line [red] labelled \"b_d\" drawn in model (start=(<VariableNumber tower_motion = 0.32>, 1.3), end=((-1.5 * tower_motion), 2.38), dashed=True)","energy":"a Math [text] that says \"$upright(\"relative motion\") arrow.r upright(\"damping\") arrow.r upright(\"energy loss\")$\"","external_force":"a Vector [green] labelled \"F(t)\" drawn in model (start=(-2.0, 4.4), end=(-0.9, 4.4))","facts":"a Table [text] that says \"Feature Physical role Steel pendulum about 660 metric tonnes High suspension large motion in an important sway mode Viscous dampers remove relative-motion energy Honest limit reduces selected motion, not every disturbance\" (rows=(('Feature', 'Physical role'), ('Steel pendulum', 'about 660 me…, header=True)","ground":"a Line [gray] drawn in model (start=(-2.0, 0.0), end=(2.0, 0.0))","head_case":"a Heading that says \"Taipei 101\"","head_model":"a Heading that says \"Let a Second Oscillator Take the Motion\"","head_response":"a Heading that says \"One Large Peak Becomes Two Smaller Peaks\"","left_cable":"a Line [gray] drawn in case_figure (start=(-0.28, 5.15), end=(-0.28, 4.45))","left_damper":"a Line [red] drawn in case_figure (start=(-0.8, 3.0), end=(-0.45, 3.7), dashed=True)","model":"a Figure (x_range=(-2.5, 2.5), y_range=(-0.3, 5.8))","model_note":"a Panel that says \"An auxiliary oscillator tuned near a structural mode can absorb motion from that mode.\"","response_axes":"an Axes (x_range=(0.6, 1.4), y_range=(0.0, 18.0), x_ticks_every=0.2)","response_note":"a Text [text] that says \"The auxiliary resonance divides one large response peak into two smaller peaks while its damping removes energy from relative motion.\"","right_cable":"a Line [gray] drawn in case_figure (start=(0.28, 5.15), end=(0.28, 4.45))","right_damper":"a Line [red] drawn in case_figure (start=(0.8, 3.0), end=(0.45, 3.7), dashed=True)","tower":"a Polygon [blue] drawn in model (vertices=(((tower_motion - 0.55), 0.0), ((tower_motion + 0.55), 0.0), ((…, fill_opacity=0.2)","tower_motion":"a VariableNumber (initial_value=0.25)","tower_name":"a Point [blue] labelled \"X\" drawn in model (location=(<VariableNumber tower_motion = 0.32>, 5.2), show_marker=False)","tuning":"a Math [text] that says \"$omega_d approx sqrt(frac(k_d, m_d)) approx omega_0$\""},"beats":[{"start":532.2701041666667,"say":"Finish with a structure that engineers do not want to resonate strongly. This sketch compresses one important sideways mode of a tall building under an external force into the coordinate X.","live":[],"does":[[532.2701041666667,"head_model is shown on the screen, written out."],[532.2701041666667,"model is shown on the screen, written out."],[532.2701041666667,"ground is shown on the screen, written out."],[532.8741041666667,"tower is shown on the screen, written out."],[541.1861041666667,"external_force is shown on the screen, written out."],[542.9281041666667,"tower_name is shown on the screen, written out."]]},{"start":544.3291041666666,"say":"Suspend a second mass inside the structure. Choose its natural frequency near the troublesome building frequency, so it responds strongly where the building needs help.","live":["model","head_model","ground","tower","tower_name","external_force"],"does":[[545.2691041666667,"counterweight is shown on the screen, written out."],[545.2691041666667,"cable is shown on the screen, written out."],[548.4971041666666,"model_note is shown on the screen, written out."],[549.6121041666667,"tuning is shown on the screen, written out."]]},{"start":555.4481041666667,"say":"When the tower moves one way, the auxiliary mass can move the other way. That counter-motion reduces the response of the main structural mode.","live":["model","model_note","tuning","head_model","ground","tower","tower_name","external_force","counterweight","cable"],"does":[[556.1211041666667,"tower is redrawn as the numbers it depends on change."],[556.1211041666667,"tower_name is redrawn as the numbers it depends on change."],[556.1211041666667,"counterweight is redrawn as the numbers it depends on change."],[556.1211041666667,"cable is redrawn as the numbers it depends on change."],[556.1211041666667,"damper is redrawn as the numbers it depends on change."],[556.1211041666667,"tower_motion ticks to -0.35."]]},{"start":565.3706041666667,"say":"The useful quantity is relative motion. It drives the connecting damper, which converts part of the mechanical energy into heat instead of letting that energy remain in the sway.","live":null,"does":[[567.1701041666666,"tower is redrawn as the numbers it depends on change."],[567.1701041666666,"tower_name is redrawn as the numbers it depends on change."],[567.1701041666666,"counterweight is redrawn as the numbers it depends on change."],[567.1701041666666,"cable is redrawn as the numbers it depends on change."],[567.1701041666666,"damper is redrawn as the numbers it depends on change."],[567.1701041666666,"tower_motion ticks to 0.32."],[570.0151041666667,"damper is shown on the screen, written out."],[572.4291041666667,"energy is shown on the screen, written out."],[573.1841041666667,"damper is indicated — a transient flash."]]},{"start":577.5691041666666,"say":"This is intentional resonance. Engineers tune the smaller oscillator so that it accepts motion near a selected structural frequency, then use damping to dispose of energy.","live":["model","model_note","tuning","energy","head_model","ground","tower","tower_name","external_force","counterweight","cable","damper"],"does":[[584.6051041666667,"tuning (the \"omega_0\" part) is emphasized."],[589.2951041666666,"tuning moves to a new place on the board."],[589.2951041666666,"energy is hidden from the screen — left the board."],[589.2951041666666,"head_model is hidden from the screen — left the board."],[589.2951041666666,"model is hidden from the screen — left the board."],[589.2951041666666,"ground is hidden from the screen — model left the board."],[589.2951041666666,"tower is hidden from the screen — model left the board."],[589.2951041666666,"tower_name is hidden from the screen — model left the board."],[589.2951041666666,"external_force is hidden from the screen — model left the board."],[589.2951041666666,"counterweight is hidden from the screen — model left the board."],[589.2951041666666,"cable is hidden from the screen — model left the board."],[589.2951041666666,"damper is hidden from the screen — model left the board."],[589.2951041666666,"model_note is hidden from the screen — left the board."],[589.2951041666666,"tuning (the \"omega_0\" part) is no longer emphasized."]]},{"start":589.8951041666667,"say":"The red curve is a simple model of the structural response without the auxiliary mass. Near its natural frequency, one large resonance peak dominates.","live":["tuning"],"does":[[589.8951041666667,"head_response is shown on the screen, written out."],[590.3831041666666,"response_axes is shown on the screen, written out."],[590.3831041666666,"bare_response is shown on the screen, drawn."],[597.8361041666667,"bare_response is indicated — a transient flash."]]},{"start":600.5841041666666,"say":"The blue curve includes the tuned mass and its damping. The single large peak is split into two smaller peaks, and the response near the original resonance is greatly reduced.","live":["tuning","response_axes","head_response","bare_response"],"does":[[601.0481041666667,"controlled_response is shown on the screen, drawn."],[604.9721041666667,"bare_response is indicated — a transient flash."],[605.6691041666667,"response_note is shown on the screen, written out."],[610.1731041666667,"controlled_response is indicated — a transient flash."]]},{"start":611.6671041666666,"say":"The device does not cancel every possible motion. It targets a chosen range, and its performance depends on tuning, mass ratio, available travel, damping, and the modes present in the real structure.","live":["tuning","response_note","response_axes","head_response","bare_response","controlled_response"],"does":[[625.1236041666666,"head_response is hidden from the screen — left the board."],[625.1236041666666,"response_axes is hidden from the screen — left the board."],[625.1236041666666,"bare_response is hidden from the screen — response_axes left the board."],[625.1236041666666,"controlled_response is hidden from the screen — response_axes left the board."],[625.1236041666666,"response_note is hidden from the screen — left the board."],[625.1236041666666,"tuning is hidden from the screen — left the board."]]},{"start":626.3236041666667,"say":"A famous real example is Taipei 101. High in the tower hangs a pendulum-like steel tuned mass damper with a mass of about six hundred sixty metric tonnes.","live":[],"does":[[626.3236041666667,"head_case is shown on the screen, written out."],[627.5191041666667,"facts is shown on the screen, written out."],[630.7471041666666,"case_figure is shown on the screen, written out."],[630.7471041666666,"case_ground is shown on the screen, written out."],[630.7471041666666,"case_tower is shown on the screen, written out."],[632.7671041666666,"facts is shown on the screen, written out."],[635.7971041666667,"case_bob is shown on the screen, written out."],[635.7971041666667,"left_cable is shown on the screen, written out."],[635.7971041666667,"right_cable is shown on the screen, written out."]]},{"start":638.6496041666667,"say":"Suspending the mass high places it where an important sway mode has large motion. When the building moves, the mass develops relative motion rather than simply riding with the floors.","live":["case_figure","head_case","case_ground","case_tower","case_bob","left_cable","right_cable"],"does":[[638.9981041666666,"facts is shown on the screen, written out."],[647.4731041666666,"case_bob is indicated — a transient flash."]]},{"start":651.4861041666667,"say":"Viscous dampers connected to that motion dissipate energy. The system is intended especially to reduce wind-driven accelerations and sway, helping both structural response and occupant comfort.","live":null,"does":[[651.8341041666666,"left_damper is shown on the screen, written out."],[651.8341041666666,"right_damper is shown on the screen, written out."],[653.9821041666667,"facts is shown on the screen, written out."]]},{"start":664.8691041666666,"say":"The honest limit matters. One tuned mass cannot control every structural mode or every possible earthquake input. Engineers still need the full structure, safety margins, travel limits, and other protective systems.","live":["case_figure","head_case","case_ground","case_tower","case_bob","left_cable","right_cable","left_damper","right_damper"],"does":[[665.9261041666666,"facts is shown on the screen, written out."],[668.6771041666666,"facts (the \"reduces selected motion\" part) is emphasized."],[680.1016041666667,"facts (the \"reduces selected motion\" part) is no longer emphasized."]]},{"start":680.7016041666667,"say":"The connection back to the swing is timing and energy. A well-timed push feeds an oscillator. A tuned counterweight creates an opposing motion at a selected frequency, and damping removes energy before the main structure builds a dangerous response.","live":null,"does":[[682.7221041666667,"closing is shown on the screen, written out."],[686.1931041666667,"closing (the \"natural frequency\" part) is emphasized."],[688.0621041666667,"closing (the \"Tuned counter-motion\" part) is emphasized."],[688.0621041666667,"closing (the \"natural frequency\" part) is no longer emphasized."],[691.6731041666667,"closing (the \"Damping\" part) is emphasized."],[691.6731041666667,"closing (the \"Tuned counter-motion\" part) is no longer emphasized."],[696.2476041666666,"closing (the \"Damping\" part) is no longer emphasized."]]},{"start":696.8476041666667,"say":"Carry four ideas away. Natural frequency sets the preferred rhythm. A live sweep reveals resonance and phase together. Quality factor connects peak sharpness with ring-down. And carefully tuned damping can turn resonance from a hazard into an engineering tool.","live":["closing","case_figure","head_case","case_ground","case_tower","case_bob","left_cable","right_cable","left_damper","right_damper"],"does":[[699.6921041666667,"closing (the \"Every oscillator\" part) is indicated — a transient flash."],[703.1861041666666,"closing (the \"slow sweep\" part) is indicated — a transient flash."],[706.2401041666667,"closing (the \"Damping\" part) is indicated — a transient flash."],[713.6581041666666,"closing (the \"Tuned counter-motion\" part) is indicated — a transient flash."],[715.1568333333333,"case_figure is hidden from the screen — left the board."],[715.1568333333333,"case_ground is hidden from the screen — case_figure left the board."],[715.1568333333333,"case_tower is hidden from the screen — case_figure left the board."],[715.1568333333333,"case_bob is hidden from the screen — case_figure left the board."],[715.1568333333333,"left_cable is hidden from the screen — case_figure left the board."],[715.1568333333333,"right_cable is hidden from the screen — case_figure left the board."],[715.1568333333333,"left_damper is hidden from the screen — case_figure left the board."],[715.1568333333333,"right_damper is hidden from the screen — case_figure left the board."],[715.1568333333333,"closing is hidden from the screen — left the board."],[715.1568333333333,"facts is hidden from the screen — left the board."],[715.1568333333333,"head_case is hidden from the screen — left the board."]]}]}]},"durationSeconds":716,"chapters":[{"title":"The Natural Rhythm","startSeconds":0,"narration":"A child on a swing already knows the central idea of resonance. One push can be gentle, yet a sequence of pushes can make the motion large if each push arrives at the useful part of the swing. Let the child swing freely first. The motion repeats with a rhythm set mainly by the swing itself. Nobody has to prescribe that rhythm from outside. A push in the direction of motion adds energy. The same push at the wrong time may slow the child instead. Timing matters because the swing has a natural rhythm of its own. To isolate that rhythm, replace the swing by the simplest oscillator: a mass m attached to a spring of stiffness k. Displacement x is measured from the dashed equilibrium line. Pull the mass to the right. The spring pulls left. Move the mass to the left, and the spring pulls right. In either case the force points back toward equilibrium. Hooke's law writes that restoring force as minus k x. The minus sign records the reversal we just watched: force and displacement point in opposite directions. Newton's second law says mass times acceleration equals the total force. With no driver and no damping, the spring force is the only force in this one-dimensional model. Divide by the mass and collect everything on the left. We obtain x double prime plus k over m times x equals zero. A sinusoid solves this equation. Its amplitude A and starting phase phi depend on how we release the mass, but the angular frequency does not. Substitution gives the natural angular frequency, omega nought, equal to the square root of k over m. A stiffer spring raises it. A larger mass lowers it. The period is two pi divided by omega nought. That is the time for one complete free oscillation, and therefore the timing an outside force must confront. So before any driving force appears, the oscillator already owns a preferred time scale. Resonance begins when the rhythm imposed from outside approaches this natural rhythm."},{"title":"Sweeping Through Resonance","startSeconds":143.2581875,"narration":"Now attach an external force that oscillates sinusoidally. Its maximum strength is F nought, and omega tells us how rapidly the applied force repeats. There are now two frequencies to keep separate. Omega nought is fixed by the mass and spring. Omega is chosen by the driver, and we are free to change it. Far below the natural frequency, the force changes slowly. The mass has time to follow, so displacement and force are nearly in step. At the ideal undamped resonance, the usual steady amplitude does not exist. Each correctly timed cycle adds energy, and the amplitude grows roughly in proportion to time. That is the mathematical version of pushing the swing at the useful moment. The driver keeps doing positive work instead of returning the energy it supplied on the previous cycle. A real experiment always loses a little energy, even before we add a deliberate damper. We will use a small damping ratio for a finite steady response, then increase it later. Here is the central experiment. We change the driving frequency slowly enough for the transient motion to settle. The upper yellow point reads amplitude, and the lower one reads phase lag. The response has the same frequency as the driver, but its amplitude A and lag delta depend on frequency. The ratio r compares the driving frequency with the natural frequency. Begin well below resonance. The amplitude is modest, and the phase lag is close to zero. The oscillator follows the slowly changing force. Now sweep upward. The amplitude climbs sharply as omega approaches omega nought. The driver is repeatedly adding energy at nearly the rhythm the oscillator prefers. At the resonance region, the displacement lags the force by about ninety degrees, one quarter of a cycle. The force is then well placed to feed energy into the velocity. Continue above resonance. The amplitude falls again, while the phase lag keeps increasing. The oscillator can no longer reverse quickly enough to follow the driver. Well above resonance, the lag approaches one hundred eighty degrees, or half a cycle. Force and displacement are then almost opposite. The phase relation is easier to read as paired waveforms. Gray is the driving force and blue is the displacement. Below resonance, their peaks are nearly aligned. Near the resonance peak, the blue displacement reaches its maximum one quarter cycle after the gray force. That is a phase lag of pi over two. Far above resonance, the blue response is nearly inverted. A force maximum occurs close to a displacement minimum, which is the half-cycle limit. The live sweep has therefore shown two linked changes. Amplitude rises and falls around resonance, while phase moves smoothly from almost zero, through a quarter cycle, toward half a cycle."},{"title":"Damping and Quality Factor","startSeconds":342.17925,"narration":"Now increase the damping deliberately. The new term b x prime opposes velocity, so it removes mechanical energy whenever the mass moves. The dimensionless damping ratio zeta compares that loss with the mass and spring scales. It lets different oscillators be compared on the same frequency graph. The blue curve has light damping. Its response rises into a tall, narrow peak, so a small change in driving frequency produces a large change in amplitude. Increase the damping to the green curve. The peak is lower and broader. Energy is removed more quickly, so less of it can accumulate from one cycle to the next. With still stronger damping, the red response has no distinct resonance peak at all. The system still responds, but it no longer selects one sharply amplified driving frequency. Damping also shifts the maximum. For the standard viscous model, the peak occurs at omega nought times the square root of one minus two zeta squared, while that expression describes a distinct peak. So the frequency of maximum amplitude lies slightly below the undamped natural frequency. More damping pushes it farther down until the peak itself ceases to be a useful feature. Quality factor packages this behavior into one number. For light damping, Q is approximately omega nought divided by the resonance bandwidth delta omega. A narrow bandwidth makes Q large. Only frequencies close to omega nought drive a large response. A broad bandwidth makes Q smaller and the resonance less selective. For the same lightly damped oscillator, Q is also approximately one over twice zeta. Increasing damping therefore lowers Q at the same time that it flattens the response. The second reading of Q appears after the driver is switched off. The oscillator continues to ring, but damping makes its amplitude envelope decay exponentially. The blue ring-down has Q equal to eight. Many oscillations remain visible because the amplitude changes only a little during each cycle. The red ring-down has Q equal to two. It loses a much larger fraction of its energy per cycle, so the motion disappears quickly. For light damping, the amplitude behaves approximately like A nought times exponential minus omega nought t over two Q. A larger Q therefore means a longer decay time. A useful estimate is that the number of clearly significant cycles in the ring-down scales like Q divided by pi. This is not a sharp stopping rule, but it connects the abstract number to something visible. Quality factor therefore answers two versions of the same energy question. How narrowly does the oscillator accept energy from a driver, and how slowly does it give stored energy away after the driver stops?"},{"title":"A Tower with a Counterweight","startSeconds":532.2701041666667,"narration":"Finish with a structure that engineers do not want to resonate strongly. This sketch compresses one important sideways mode of a tall building under an external force into the coordinate X. Suspend a second mass inside the structure. Choose its natural frequency near the troublesome building frequency, so it responds strongly where the building needs help. When the tower moves one way, the auxiliary mass can move the other way. That counter-motion reduces the response of the main structural mode. The useful quantity is relative motion. It drives the connecting damper, which converts part of the mechanical energy into heat instead of letting that energy remain in the sway. This is intentional resonance. Engineers tune the smaller oscillator so that it accepts motion near a selected structural frequency, then use damping to dispose of energy. The red curve is a simple model of the structural response without the auxiliary mass. Near its natural frequency, one large resonance peak dominates. The blue curve includes the tuned mass and its damping. The single large peak is split into two smaller peaks, and the response near the original resonance is greatly reduced. The device does not cancel every possible motion. It targets a chosen range, and its performance depends on tuning, mass ratio, available travel, damping, and the modes present in the real structure. A famous real example is Taipei 101. High in the tower hangs a pendulum-like steel tuned mass damper with a mass of about six hundred sixty metric tonnes. Suspending the mass high places it where an important sway mode has large motion. When the building moves, the mass develops relative motion rather than simply riding with the floors. Viscous dampers connected to that motion dissipate energy. The system is intended especially to reduce wind-driven accelerations and sway, helping both structural response and occupant comfort. The honest limit matters. One tuned mass cannot control every structural mode or every possible earthquake input. Engineers still need the full structure, safety margins, travel limits, and other protective systems. The connection back to the swing is timing and energy. A well-timed push feeds an oscillator. A tuned counterweight creates an opposing motion at a selected frequency, and damping removes energy before the main structure builds a dangerous response. Carry four ideas away. Natural frequency sets the preferred rhythm. A live sweep reveals resonance and phase together. Quality factor connects peak sharpness with ring-down. And carefully tuned damping can turn resonance from a hazard into an engineering tool."}]}}