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An impedance head combines a force transducer and an accelerometer in one housing for a driving-point vibration test. It measures applied force and acceleration at nearly the same location and along the intended excitation direction. The raw channels yield mechanical accelerance or its reciprocal mechanical dynamic mass; deriving mechanical driving-point mobility or mechanical driving-point impedance additionally requires the acceleration-to-velocity conversion.^[ref]

In the manufacturer's Types8000/8001 example, a piezoelectric acceleration sensor sits above a force gauge in a stiff housing, with separate electrical outputs. The arrangement aims to preserve the phase relation needed for a complex response ratio. This solves a practical apparatus problem: a separate exciter and response sensor can otherwise occupy different points, particularly on a small structure.

An impedance-head mass correction accounts for moving material between the force-sensing plane and the specimen. In a rigid lumped approximation, choose a common positive axis and let that material have effective mass $m_h$ and acceleration complex amplitude $A$. Then Newton's second law givesFor an original in-phase example, let $m_h=0.020\mathrm{kg}$, $A=10m/{\mathrm{s}}^2$ and gauge force $1\mathrm{N}$. Specimen force is $0.80\mathrm{N}$, so specimen accelerance is $12.5\mathrm{k}{\mathrm{g}}^{-1}$ rather than the uncorrected value10. With general signals the correction is a complex subtraction, not subtraction of unsigned amplitudes.

This force-plane correction does not undo every effect of attaching the instrument: added mass and mounting stiffness can change the specimen's actual vibration. The test must also state mounting, calibration, usable frequency range and the motion of any second terminal. Manufacturer guidance distinguishes measurements requiring mass compensation and cases where separate force/acceleration sensors are preferable.^[ref]

Mechanical accelerance is the acceleration-to-force frequency response, $G(s)=A(s)/F(s)$ under zero initial conditions. It has units $\mathrm{k}{\mathrm{g}}^{-1}$. It is related to mechanical receptance and mechanical driving-point mobility byThus an acceleration response cannot be labeled mobility without a frequency-dependent conversion. Its inverse is mechanical dynamic mass, which is generally complex and frequency dependent.^[ref]

For $m\ddot{x}+c\dot{x}+kx=F$,For a pure mass, $G=1/m$, independent of frequency in the ideal rigid model. A supported mass is different: spring and damping forces also enter its response. If $k>0$, accelerance tends to zero as $\omega$ tends to zero, while its high-frequency limit is $1/m$ within the model's useful range.

With $m=2\mathrm{kg}$, $c=4\mathrm{N}s/m$, $k=200N/m$ and $\omega =10rad/s$, the result is $G=2.5i\mathrm{k}{\mathrm{g}}^{-1}$. A peak force $0.8\mathrm{N}$ gives acceleration amplitude $2m/{\mathrm{s}}^2$, leading force by a quarter cycle. The reciprocal response is $-0.4i\mathrm{kg}$: it describes a dynamic ratio, rather than a negative physical amount of inertial material.

An impedance head measures force and acceleration together, making this ratio directly accessible after measurement calibration and signal processing. Driving-point accelerance uses coincident force and response directions. A response at another point is a transfer measurement; a small or vanishing transfer response can arise from the mode shapes and does not establish that the driven structure has infinite physical mass.

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.

Mechanical driving-point mobility is the frequency response relating motion to excitation at the same mechanical port. Using relative velocity $v$ and signed entering force $F$, its zero-initial-state Laplace transform convention iswhere $Z$ is the force/velocity mechanical driving-point impedance. Mobility has units $m/(\mathrm{N}\mathrm{s})$. The measured direction, port and boundary conditions are part of its specification.^[ref]

For an inertial element $b$, dashpot coefficient $c$ and spring stiffness $k$ sharing a port,For $e^{i\omega t}$ complex amplitudes, mobility tends to $i\omega /k$ at low frequency when $k>0$ and to $1/(i\omega b)$ at high frequency when $b>0$. A spring therefore prevents a nonzero steady drift response to constant force. This differs from the steady mechanical mobility of a body under pure linear drag, which has no elastic restoring term.

Take $b=2\mathrm{kg}$, $c=4\mathrm{N}s/m$ and $k=200N/m$. At $\omega =10rad/s$ the reactive terms cancel and $Y=0.25m/(\mathrm{N}\mathrm{s})$. A peak force amplitude $0.8\mathrm{N}$ gives peak velocity $0.20m/s$ and mean entering power $0.08\mathrm{W}$. These statements concern the same port; a response measured elsewhere is a transfer mobility.

A response/force ratio using acceleration instead is mechanical accelerance, $A/F=sY$; using displacement gives mechanical receptance, $X/F=Y/s$. Confusing these ratios introduces a frequency-dependent magnitude and phase error. With several coupled ports, matrix inversion rather than elementwise reciprocation is needed to exchange complete impedance and mobility descriptions.

Mechanical receptance is a displacement-to-force frequency response. With zero initial conditions, the Laplace transform convention is $H(s)=X(s)/F(s)$, with units $m/N$. Mechanical driving-point mobility is $Y=sH$ and mechanical accelerance is $s^{2}H$. The port, direction and boundary conditions must accompany the ratio.^[ref]

For a single-coordinate model $m\ddot{x}+c\dot{x}+kx=F$,Its inverse is mechanical dynamic stiffness. If $k>0$, the zero-frequency limit $H(0)=1/k$ agrees with static mechanical compliance. At finite frequency, inertia and damping change both magnitude and phase; replacing the denominator by $k$ would discard them.

For an original example, use $m=2\mathrm{kg}$, $c=4\mathrm{N}s/m$ and $k=200N/m$. At $\omega =10rad/s$, stiffness and inertia cancel, giving $H=-0.025im/N$. A peak force amplitude $0.8\mathrm{N}$ therefore gives displacement amplitude $0.020\mathrm{m}$, lagging force by a quarter cycle. Static displacement under the same force magnitude would be only $0.004\mathrm{m}$.

For sufficiently high frequency, this model has $H(i\omega )\sim -1/(m{\omega }^2)$, revealing an inertia-dominated response with a phase reversal. A mechanical impedance measurement can infer receptance from acceleration by dividing the acceleration/force response by $-{\omega }^2$ for nonzero $\omega$. This conversion becomes sensitive to low-frequency measurement errors and is not a substitute for specifying the actual displacement measurement or integration model. With multiple coupled coordinates, the receptance matrix is the inverse of the complete dynamic-stiffness matrix, rather than a collection of elementwise reciprocal entries.

Ole Døssing is the credited author of Brüel & Kjær's Structural Testing Part I: Mechanical Mobility Measurements by Ole Døssing, whose title page identifies the author and an April1988revision. The handbook develops experimental ways to relate applied force to structural motion, with emphasis on measured frequency responses rather than vibration amplitude alone.^[ref]

The distinction is important when diagnosing a vibrating structure. A large response may reflect a large applied force, a strong structural response at that frequency, or both. The handbook describes measuring the exciting force and the motion simultaneously, using a force transducer, an accelerometer and a dual-channel analyzer. The complex ratio retains both magnitude and phase. It distinguishes mechanical accelerance, acceleration divided by force, from mechanical driving-point mobility, velocity divided by force at the same point.^[ref]

An original known-mass check makes the apparatus interpretable. For a freely translating mass $m=0.50\mathrm{kg}$ in a regime where suspension forces are negligible, Newton's second law predictsA10newton sinusoidal force amplitude should therefore produce20metres per second squared of acceleration amplitude in the same phase. With the convention $e^{i\omega t}$, velocity amplitude is $a/(i\omega )$, so at $\omega =100\mathrm{rad}{\mathrm{s}}^{-1}$ the mechanical driving-point mobility is $-0.020i\mathrm{m}{\mathrm{N}}^{-1}{\mathrm{s}}^{-1}$. Confusing the two response ratios loses both a factor of frequency and the phase shift.

Døssing's handbook describes this known-mass measurement calibration as a check of the complete response-to-force measurement chain. The mass must move in the intended direction and its effective moving mass must be understood. It checks a ratio of transducer responses, rather than separately determining each transducer's absolute sensitivity.^[ref]