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Jan Høgsberg, listed as Jan Becker Høgsberg in the DTU research directory, researches structural dynamics, damping and structural control. His primary university profile describes work on modeling structures and improving reduced models used for analysis and design.^[ref]

A 2025 study treats a viscous damper consisting of a dashpot in series with an inerter. Its design analysis shows how interaction with a building's normal modes can produce a resonant damping effect even though the isolated device lacks its own internal resonance. The study emphasizes retaining inertia and flexibility effects from modes that a simple reduced model would discard. Its demonstration is a numerical model of a 20-storey building, not a reported physical building test.^[ref]

An original check for the ideal viscous part alone uses the force magnitude $F=cv$, with relative speed $v$ across the dashpot and positive damping coefficient $c$. If $c=4000\mathrm{N}\mathrm{s}{\mathrm{m}}^{-1}$ and $v=0.05\mathrm{m}{\mathrm{s}}^{-1}$, then $F=200\mathrm{N}$ and the dissipated power isThese illustrative values describe a constitutive element, not the complete composite damper or its tuned building response. Locating the element within the structure and accounting for its coupled motion are essential additional steps in structural dynamics.

DTU's research profile and portrait page documents the researcher.

Malcolm C. Smith is a Cambridge researcher in control theory whose work connects robust control and mechanical network synthesis with physical system design. The Cambridge research profile identifies uncertainty, robustness and mechanical applications as themes of his work; the IEEE Control Systems Society independently identifies his Cambridge affiliation.^[ref]

His 2002 paper Synthesis of Mechanical Networks: The Inerter by Malcolm C. Smith introduced the inerter as a two-terminal mechanical element. Its ideal terminal force is proportional to relative acceleration,where $b$ is mechanical inertance. Both terminals can move, so equal terminal accelerations produce zero force in this ideal constitutive relation. This makes the element useful for synthesizing mechanical responses through networks, alongside springs and dampers.^[ref]

For an original ideal transmission example, relative terminal displacement drives a flywheel with angle $\theta =(x_1-x_2)/r$. With moment of inertia $J$, its kinetic energy isso equivalent mechanical inertance is $b=J/r^2$. Choosing $J=0.002\mathrm{kg}{\mathrm{m}}^2$ and $r=0.01\mathrm{m}$ gives $b=20\mathrm{kg}$. A relative acceleration $5\mathrm{m}{\mathrm{s}}^{-2}$ then gives $100\mathrm{N}$. The coefficient has mass units, but it is not a statement that the flywheel's physical mass is20kilograms.

The calculation neglects friction, compliance and transmission inertia beyond the chosen flywheel. It illustrates the design idea, not measured performance of a particular device. The IEEE Control Systems Society profile includes a labeled headshot.

The mechanical dissipation inequality states that a mechanical energy storage function can increase no faster than energy supplied through the mechanical terminals. With one port and entering power $w=Fv$, its differentiable form and interval form areThe second form remains meaningful when a pointwise time derivative is unavailable. The mechanical supply rate $w$ records the external input; the difference $Fv-\dot{E}\ge 0$ records irreversible loss in the model. This is the mechanical specialization of dissipativity, not a demand that entering power be positive at every instant.^[ref]

For a spring, inerter and dashpot sharing a port,and multiplication by $v$ gives the exact balanceA spring may return energy with $Fv<0$ while $E$ decreases; that does not violate the inequality. Conversely, a negative viscous coefficient fails this storage balance by supplying energy to motion.

For an original free-decay example, omit the spring and take grounded inertial coefficient $b=2\mathrm{kg}$, $c=1\mathrm{N}s/m$ and initial velocity $v_0=1m/s$. With zero applied port force, $b\dot{v}+cv=0$, soAt $t=(b/(2c))\ln 2\approx 0.6931\mathrm{s}$, stored energy has fallen from $1\mathrm{J}$ to $0.5\mathrm{J}$; the missing $0.5\mathrm{J}$ equals ${\int }_0^{t}c{v}^{2}dt$. No external work entered during that interval.

For several ports, use ${\sum }_j{F}_j{v}_j$ as the total supply. When passive elements are connected with power-conserving internal constraints, their internal supply terms cancel, so the sum of their storage functions satisfies the same external inequality. This explains why passive interconnections can be checked element by element, provided every power-carrying terminal is accounted for.

Mechanical driving-point impedance describes the relation between a driving force and motion at the same mechanical port. Here the convention iswhere $F,V$ are Laplace transforms with zero initial conditions, and $v$ is the signed relative velocity of the two terminals. Choose signs so $Fv$ is entering power. The units of this mechanical impedance are $\mathrm{N}s/m$. Its inverse is the mechanical driving-point mobility. Conventions must be checked explicitly: in The Inerter: A Retrospective by Malcolm C. Smith, the force–current analogy calls $V/F$ the mechanical impedance and $F/V$ the admittance.^[ref]

For a spring, dashpot and inerter sharing the same terminals, relative velocities agree and forces add. HenceAt a sinusoidal angular frequency $\omega$, using complex amplitudes proportional to $e^{i\omega t}$,The real part determines mean absorbed power, while the imaginary part records the out-of-phase reactive response. With peak velocity amplitude $|V|$, the mean entering power is $\frac{1}{2}\mathrm{Re}(Z)|V{|}^2$.

For an original example, choose $b=10\mathrm{kg}$, $c=30\mathrm{N}s/m$ and $k=1000N/m$. At $\omega =10rad/s$, the inertial and elastic terms cancel, giving $Z=30\mathrm{N}s/m$. A relative velocity amplitude $0.020m/s$ requires force amplitude $0.60\mathrm{N}$, and mean absorbed power is $0.006\mathrm{W}$. The individual spring and inerter still store and exchange energy.

A mechanical impedance measurement drives the chosen port and records synchronized force and motion there. A load cell measures force; displacement or an accelerometer can supply motion. For an acceleration complex amplitude $A$, $V=A/(i\omega )$, so the phase as well as magnitude matters. Fixture inertia, sensor measurement calibration and the second terminal's motion belong in the measurement model. Using a response at a different point instead defines a transfer relation, not this driving-point relation. Initial stored motion also prevents interpreting an arbitrary transient quotient as a zero-initial-state transfer function.

A mechanical energy storage function assigns a nonnegative stored-energy quantity $E$ to the state of a mechanical model so that its increase never exceeds entering work. For a one-port passive mechanical network with entering power $Fv$, the mechanical dissipation inequality isThis state-based accounting includes energy present before the measurement begins. A temporary outward power is compatible with release of stored energy.^[ref]

For an ideal linear spring, $E_k=\frac{1}{2}k{x}^2$ is elastic potential energy. For an ideal inerter, $E_b=\frac{1}{2}b{v}^2$ is kinetic energy associated with terminal relative motion. An ideal dashpot need not store mechanical energy: its entering power $c{v}^2$ is dissipated. If these elements share terminals, their state variables obey $v=\dot{x}$ andThis equality proves the inequality for $c\ge 0$. In a larger network, summing element balances cancels internal transfers, leaving power through the external terminals and total dissipation.

For an original numerical check, let $k=800N/m$, $b=5\mathrm{kg}$ and $c=10\mathrm{N}s/m$. Start at $x=0.050\mathrm{m}$ and $v=0.20m/s$, so $E=1.10\mathrm{J}$. Maintain that relative velocity for $0.10\mathrm{s}$: displacement reaches $0.070\mathrm{m}$ and stored energy becomes $2.06\mathrm{J}$. Dissipation is $c{v}^2\Delta t=0.040\mathrm{J}$, so input work is $0.96+0.04=1.00\mathrm{J}$. Counting all entering work as newly stored energy would overestimate storage.

The mechanical available storage is the maximum net work that could be recovered under allowed future terminal motions; it is a related operational quantity, rather than automatically every form of internal physical energy. A model's storage function is a certificate that must respect its accessible states and port variables. With zero external power, a positive-definite storage function can supply a Lyapunov function, but nonincreasing energy alone does not establish that every motion decays to rest: a lossless spring–inerter oscillation keeps a constant nonzero stored energy.

Mechanical inertance $b$ is the coefficient relating the terminal force of an ideal inerter to relative terminal acceleration:The forces here are applied to the device. Its mechanical inertance has SI units $\mathrm{N}{\mathrm{s}}^2{\mathrm{m}}^{-1}=\mathrm{kg}$. The constitutive law was introduced by Malcolm C. Smith as a two-terminal mechanical-network element.^[ref]

Mechanical inertance is a response coefficient and need not equal the assembly's translating mass. For an ideal rack and pinion driving a flywheel, let relative terminal speed $v$ produce angular velocity $\omega =v/r$. Comparing stored energies giveswhere $J$ is the flywheel's moment of inertia and $r$ is the pinion's pitch radius. In an original example, $J=0.0125\mathrm{kg}{\mathrm{m}}^2$ and $r=0.005\mathrm{m}$ give $b=500\mathrm{kg}$. A relative acceleration of $0.30\mathrm{m}{\mathrm{s}}^{-2}$ then requires150newtons. Halving $r$ with $J$ fixed multiplies $b$ by four; it does not multiply the physical mass by four.

At relative speed $0.20\mathrm{m}{\mathrm{s}}^{-1}$, the same ideal element stores10joules. If both terminals instead have identical acceleration, the ideal inertial force is zero regardless of their common acceleration. This two-terminal behavior distinguishes the coefficient from simply attaching a mass to a fixed reference.

A measurement can estimate mechanical inertance from the slope of force against relative acceleration over a range where the ideal model is adequate. Friction, backlash and mechanical compliance can produce additional force components, so one force-to-acceleration ratio at an arbitrary operating point need not identify a constant inertance.

Mechanical network synthesis constructs an arrangement of mechanical elements to realize a specified relation between applied force and motion. In the linear two-terminal setting, a natural specification is the mechanical impedancewhere $F(s)$ and $V(s)$ are Laplace transforms of terminal force and relative velocity, with zero initial conditions. Choosing elements and their connections to obtain a desired $Z(s)$ is the synthesis problem; calculating $Z(s)$ for an already chosen arrangement is the corresponding analysis problem. Malcolm C. Smith's 2002 work introduced the inerter as a genuinely two-terminal inertial element, allowing a broader class of passive mechanical networks than a construction restricted to grounded masses.^[ref]

A linear spring with stiffness $k$, a dashpot with coefficient $c$, and an inerter with mechanical inertance $b$ have respective mechanical impedancesFor elements attached between the same two moving terminals, the relative velocity is common and their forces add, so their mechanical impedances add. Consequently the targetcan be realized by a $20\mathrm{kg}$ inerter, a $100\mathrm{N}s/m$ dashpot and a $2000N/m$ spring sharing those terminals. The units of the resulting mechanical impedance are $\mathrm{N}s/m$. At angular frequency $\omega =10rad/s$, the sinusoidal mechanical impedance is $Z(i\omega )=100\mathrm{N}s/m$: the spring and inerter contributions are equal and opposite imaginary terms. A sinusoidal relative velocity of amplitude $1m/s$ therefore needs a terminal force amplitude of $100\mathrm{N}$ in this ideal example.

The apparent cancellation does not remove stored energy. With relative displacement $x$ and $v=\dot{x}$, the terminal power satisfiesThe inerter and spring exchange stored energy, while the dashpot dissipates it. This balance verifies that the example is a passive mechanical network for nonnegative $b,c,k$. A practical inerter can use a rack and pinion to drive a flywheel; friction, backlash and finite travel limit the frequency and displacement range over which the synthesized response matches the ideal mechanical impedance. Thus synthesis gives a testable target for an apparatus, rather than a guarantee that a real mechanism has no parasitic effects.^[ref]

A passive mechanical network can store and dissipate energy, but cannot supply more net work than was previously stored or delivered through its terminals. For a two-terminal mechanical element, choose signed force $F$ and relative velocity $v$ so that $p=Fv$ is power entering the network. With a nonnegative mechanical energy storage function $E$, passivity requiresIn particular, the net extracted energy over that interval cannot exceed $E(t_0)$. This permits temporary outward power: a stretched spring can release stored elastic potential energy. A requirement that $Fv$ be nonnegative at every instant would incorrectly exclude such a spring.^[ref]

For a spring, dashpot and inerter connected between the same two terminals, let $x$ be relative displacement and $v=\dot{x}$. Their constitutive equations giveFor $k,b,c\ge 0$, integration proves the passivity inequality: the extra term $\int c{v}^{2}dt$ is dissipated energy. The inerter stores kinetic energy associated with relative motion, so a passive mechanical network need not be a device that merely resists displacement or converts all input work into heat. This is central to mechanical network synthesis using springs, dashpots and inerters.^[ref]

For example, a spring with $k=1000N/m$ initially extended by $0.10\mathrm{m}$ stores $5\mathrm{J}$. It can deliver up to $5\mathrm{J}$ while relaxing, despite being passive. A purely viscous element with $c=-10\mathrm{N}s/m$ would instead have input power $-10v^2$: maintaining $v=1m/s$ for one second would extract $10\mathrm{J}$ even from an initially unenergized model. An apparatus producing that sustained behavior needs another energy source.

An experimental check records terminal force and relative velocity simultaneously and integrates their product, accounting for initial energy and every power-carrying connection. A brief interval of negative measured power is compatible with stored-energy release. For a linear zero-initial-state network, the mechanical driving-point impedance provides a complementary frequency description; a nonnegative real part at measured sinusoidal frequencies checks mean absorbed power there, but a finite frequency sweep alone does not establish the full time-domain passivity inequality.

A rack and pinion is a meshing gear pair that converts between rotation and straight-line motion. The pinion is the circular gear and the rack has teeth along a straight path. The ideal motion is described using the gear pitch circle, rather than the outer tips of the teeth. Manufacturer diagrams compare rack travel over one full pinion revolution and show how tooth spacing affects that travel.^[ref]

For a fixed pinion axis and a guided rack, let $r$ be the pitch-circle radius and choose consistent positive directions. With angular displacement $\theta$ in radians,where $\omega$ and $\alpha$ are angular velocity and angular acceleration. For an original example, $r=20\mathrm{mm}$ and a quarter-turn give $x=10\pi \mathrm{mm}\simeq 31.4\mathrm{mm}$. A rotation rate of2revolutions per second gives $v=80\pi \mathrm{mm}{\mathrm{s}}^{-1}\simeq 251\mathrm{mm}{\mathrm{s}}^{-1}$.

Ideal lossless work transfer also givesThus a1newton-metre torque at the same20millimetre pitch radius corresponds to50newtons of tangential rack force. This is a kinematic and energetic model; it omits tooth deformation and friction. Backlash makes reversal especially important: a small rotation can take up clearance before the rack moves. The manufacturer's catalogue explicitly describes backlash adjustment through mounting geometry.^[ref]

In a flywheel-based inerter, a rack and pinion makes relative terminal translation rotate the flywheel. The motion conversion then changes the apparent mechanical inertance without requiring the translating assembly to have that same physical mass.

Structural dynamics studies the time-dependent motion of structures under applied forces and their response after a disturbance. Inertia, stiffness and damping jointly determine that motion. A common linear small-displacement model iswhere $u$ collects displacement coordinates, $M$ is the mass matrix, $C$ the damping matrix, $K$ the stiffness matrix and $f(t)$ the applied-force vector. Boundary conditions and the chosen coordinates are part of the model; a static force balance omits the velocity and acceleration terms.

A scalar version makes the consequences quantitative. Choose $m=1000\mathrm{kg}$, $k=40000\mathrm{N}{\mathrm{m}}^{-1}$ and $c=400\mathrm{N}\mathrm{s}{\mathrm{m}}^{-1}$. Its undamped natural frequency isFor a constant 100-newton force, static displacement is $100/k=2.5\mathrm{mm}$. For a sinusoidal force $100\cos ({\omega }_{0}t)$, the steady-state displacement amplitude of this linear forced harmonic oscillator is insteadThese are original model values, not measurements of a building. They show why the force amplitude alone does not determine displacement: forcing frequency and damping matter.

Real structures have multiple normal modes. A 2025 study of a viscous inerter damper uses a numerical 20-storey building example and shows that inertia and flexibility associated with discarded modes affect the reduced model's tuning. Matching one nominal mode while ignoring those contributions can therefore misrepresent the coupled device–structure behavior.^[ref]

Measured motion can test a chosen model, while its predictions remain limited by assumptions about linearity, boundary conditions and loading. A large predicted response is a reason to examine those assumptions, not proof that an actual structure will follow the linear response indefinitely.