M067 · Vibration design

Tuned-Mass Damper Design

Attach a sliding mass, spring and dashpot to a primary structure. Release the structure directly, inspect the two motions and compare exact force-response curves with an acquired three-cycle measurement. Search tuning and damping under a relative-stroke 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 ledger 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 $\text{Ready}$

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 $0$

DESIGN / ACQUIRE / AUDIT

Can a small secondary mass suppress both resonance peaks?

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\,\%

Drag the mass or use the keyboard ranges. Acquire the full record before comparing late-cycle measurements. Changing the experiment cancels incompatible scans.

Two linear horizontal degrees of freedom under harmonic force excitation; this is not an earthquake or base-motion model. The absorber coordinate is absolute and its stroke is relative to the primary. Exact curves are infinite-time references; finite windows may retain startup transients. The search uses 21 tuning values and 18 damping values, with 101 frequencies in the displayed band; it is a feasible grid minimum, not a continuous or all-frequency optimum. The bare system can have an unbounded resonant reference when its damping is zero. Motion stops beyond the five-metre display domain without clipping.

Start with the essentials

Focus question: Can reducing one resonance create a worse nearby peak or an excessive stroke?
One-sentence intuition: Compare the full displayed frequency band and the relative stroke, rather than the primary amplitude at a single 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 absolute; only the connecting spring and dashpot use relative motion.

Tuning and mass ratio

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

The auxiliary frequency belongs to the absorber alone; coupling changes the actual modes.

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 matrix gives independent infinite-time harmonic references.

Independent energy ledger

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

Dissipation integrates both dashpots; it is not defined by subtracting energy.

Sampled constrained design

\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 finite frequency and parameter grids define the search; interpolated extrema can be larger.

Common difficulties

Tuning does not remove both modes

Typical misconceptionA perfect notch guarantees safe vibration everywhere.

Better mental modelThe notch lies between split resonances; inspect both peaks and the stroke curve.

Coordinate and stroke differ

Typical misconceptionThe absorber position equals its mechanical travel.

Better mental modelTravel is relative to the moving primary structure.

Grid design has a boundary

Typical misconceptionThe search finds the globally optimal damper.

Better mental modelIt tests a fixed finite band, grid and harmonic loading convention.

Run the experiment

01
Compare tuning
Acquire tuned and mistuned records.
What to observe: The two motions and resonance peaks move together.
02
Locate antiresonance
Choose ideal antiresonance and acquire the complete record.
What to observe: The exact primary notch can coexist with a decaying finite-time transient.
03
Constrain the absorber
Reduce allowable stroke and run the search.
What to observe: A feasible grid solution may change or disappear.
04
Audit the calculation
Use conservative exchange, then halve the integration step.
What to observe: Energy transfer persists while the ledger defect decreases.