M067 · Vibration design

Tuned-Mass Damper Design

Can a small tuned mass keep a structure from shaking hard at resonance? With careful tuning and damping, it can. Tuned near the structure's own frequency, the small mass splits the single tall resonance into two lower peaks, and the right damping in the absorber brings them to similar heights. Check the response across the whole frequency band, not at one frequency, and make sure the absorber has room to swing.

Two carts are joined by a spring and a dashpot in parallel, shown in three dimensions. Release the structure by hand, watch both masses move and compare the exact force-response curves with a measurement over three cycles. Then search for tuning and damping that keep the relative stroke within a limit.

Acquired time $0\,\mathrm{s}$

Primary displacement $0\,\mathrm{m}$

Primary velocity $0\,\mathrm{m\,s^{-1}}$

Mechanical energy $0\,\mathrm{J}$

Integrated signed actuator work $0\,\mathrm{J}$

Independent viscous dissipation $0\,\mathrm{J}$

Independent energy balance defect $0\,\mathrm{J}$

Position change with half step · first four cycles $0\,\mathrm{m}$

Actual integration step $0\,\mathrm{ms}$

Model domain / record status $\text{Ready}$

Primary spring force $0\,N$

Primary acceleration $0\,\mathrm{m/s^2}$

Force actuator power $0\,W$

Total viscous power $0\,W$

Attachment force on primary $0\,N$

Absorber acceleration $0\,\mathrm{m/s^2}$

Absolute absorber displacement $0$

Signed relative stroke $0$

Exact steady primary amplitude $0$

Exact steady relative stroke amplitude $0$

Bare structure steady amplitude $0$

Three-cycle acquired primary amplitude $0$

Acquired harmonic residual RMS $0$

Sampled band maximum primary amplitude $0$

Sampled band maximum relative stroke $0$

Constrained search result $0$

Set up a vibration and record its response

Can a small tuned mass keep a structure from shaking hard at resonance?

Physical force vectorsEvery force on both masses is drawn to one common scale at each instant.Velocity vectorsBlue arrows; rightward is positive. Velocity has its own scale, separate from force.Recent acquired pathsShows only the last two natural periods; future positions are never drawn.Primary mass

M

1.00\,\mathrm{kg}

Primary stiffness

K

16.00\,\mathrm{N\,m^{-1}}

Absorber mass ratio

\mu=m/M

0.08\,

Absorber tuning ratio

\omega_a/\omega_0

0.93\,

Primary damping ratio

\zeta_0

0.02\,

Absorber damping ratio

\zeta_a

0.15\,

Drive force amplitude

F_0

0.30\,\mathrm{N}

Drive to bare natural rate

\omega/\omega_0

1.00\,

Primary release displacement

x_0

0.08\,\mathrm{m}

Acquisition cycles

N

32.00\,

Allowed steady relative stroke

s_{\max}

0.40\,\mathrm{m}

Maximum integration step

\Delta t_{\max}

10.00\,\mathrm{ms}

Inspect acquired record

t/t_{\mathrm{end}}

0.00\,\%

Orbit the apparatus. Set the release point in Motion & response, or move the sliders with the keyboard. Switching to Energy & power keeps the acquired cursor and any compatible maps. Each fine step advances at most two milliseconds.

The model has two linear horizontal degrees of freedom driven by a harmonic force. It does not model earthquakes or base motion. The absorber position is measured from the ground, and its stroke is measured relative to the primary mass. The exact curves are steady references reached after infinite time; a finite window can still contain startup transients. The search tries 21 tuning values and 18 damping values, with 101 frequencies across the displayed band. It returns the smallest feasible value on that grid, not a continuous or all-frequency optimum. With zero damping, the bare structure has an unbounded reference amplitude at resonance. The run stops if the motion leaves the five-metre display range; positions are never clipped. In the drawing the two masses rest one behind the other so that both absolute positions can be shown; the relative stroke is the difference of their displacements, not the gap between the blocks. The block sizes, chosen spring rest lengths and rigid drive linkage are schematic. Each record uses one fixed drawing scale that covers both the absolute and the relative travel. The force actuator acts only on the primary mass, and the absorber has no spring or dashpot to the ground. Ideal rails balance gravity.

Physics tutorial

Protect the structure, budget the stroke

BackgroundA small second mass is attached to the main structure by a spring and a dashpot. As it moves, it pushes back on the structure, changes how the structure responds and creates two resonances.

Why it mattersLecture 26 of MIT Engineering Dynamics derives the harmonic response of a two-degree-of-freedom system and demonstrates a dynamic absorber. This Lab adds a measurement over a finite time window and a finite grid search that keeps the absorber stroke within a limit.

Start with the essentials

Focus question: Can taming one resonance create a worse peak nearby, or make the absorber travel too far?
One-sentence intuition: Judge a damper by the response across the whole displayed frequency band and by the relative stroke, not by the structure amplitude at one frequency.

Core mathematical model

Two coupled force balances

M\ddot x+C\dot x+Kx+c(\dot x-\dot y)+k(x-y)=F_0\cos\omega t,\quad m\ddot y+c(\dot y-\dot x)+k(y-x)=0

Both positions are measured from the fixed ground; only the connecting spring and dashpot depend on the relative motion.

Tuning and mass ratio

\omega_0=\sqrt{K/M},\quad \omega_a=\sqrt{k/m},\quad \mu=m/M

The absorber frequency is the frequency of the absorber on its own; once it is coupled to the structure, the actual modes are different.

Complex dynamic stiffness

\begin{bmatrix}K+k-M\omega^2+i\omega(C+c)&-k-i\omega c\\-k-i\omega c&k-m\omega^2+i\omega c\end{bmatrix}\begin{bmatrix}X\\Y\end{bmatrix}=\begin{bmatrix}F_0\\0\end{bmatrix}

Solving this system gives the steady harmonic response after infinite time, which serves as an independent reference.

Independent energy balance

E=\tfrac12M\dot x^2+\tfrac12m\dot y^2+\tfrac12Kx^2+\tfrac12k(y-x)^2,\quad E-E_0=W-Q

The dissipated energy is integrated from the power lost in both dashpots; it is not found by subtracting energies.

Design search on a grid with a stroke limit

\min_{r_a,\zeta_a}\max_{\omega\in\mathcal G}|X(\omega)|,\quad \max_{\omega\in\mathcal G}|Y(\omega)-X(\omega)|\le s_{\max}

The search checks only a finite grid of frequencies and parameters. The true maximum between grid points can be larger.

Common difficulties

Tuning does not remove the two resonances

Typical misconceptionIf the notch is perfect, the structure is safe at every frequency.

Better mental modelThe notch sits between the two split resonance peaks. Check both peaks and the stroke curve.

Position is not stroke

Typical misconceptionThe absorber position tells you how far it travels on its mount.

Better mental modelTravel is measured relative to the main structure, which is itself moving.

The grid search has limits

Typical misconceptionThe search finds the best possible damper.

Better mental modelIt only tests one fixed frequency band, one finite grid and one convention for harmonic loading.

Run the experiment

01
Compare tuning
Use Acquire full record with Tuned absorber and again with Mistuned. Orbit the two carts and look at the equal and opposite forces where they are joined; compare the absolute displacement with the relative stroke.
What to observe: The motion of both masses and the resonance peaks change together.
02
Find the antiresonance
Choose Ideal antiresonance and use Acquire full record.
What to observe: The exact steady response of the structure has a notch, yet the finite record can still show a startup transient that dies away.
03
Limit the absorber stroke
Lower Allowed steady relative stroke and use Search feasible grid minimum.
What to observe: The best grid point that meets the limit may change, or no grid point may meet it at all.
04
Check the calculation
Choose Conservative exchange and open Energy & power without resetting the cursor. Compare the separate energy terms, then halve the integration step.
What to observe: Energy keeps moving between the masses, while the energy-balance error shrinks.