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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 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 dynamic stiffness is the complex force required per unit displacement in a specified zero-initial-state frequency response. If mechanical receptance is $H(s)=X(s)/F(s)$, its scalar inverse is $K_{\mathrm{dyn}}(s)=1/H(s)$, wherever that inverse exists. Units are $N/m$, but this dynamic quantity includes inertia and damping as well as the static spring stiffness. The measured coordinate, applied-force direction and boundary conditions are part of the definition.^[ref]

For $m\ddot{x}+c\dot{x}+kx=F$, zero initial conditions giveThus displacement and force generally differ in phase. With $m=2\mathrm{kg}$, $c=4\mathrm{N}s/m$ and $k=200N/m$, at angular frequency $\omega =10rad/s$ the result is $40iN/m$. A real displacement complex amplitude $X=0.020\mathrm{m}$ requires $F=0.80i\mathrm{N}$, rather than the static estimate $kX=4\mathrm{N}$. Elastic and inertial contributions cancel at this frequency; damping still requires an applied force.

For several coupled coordinates, a mechanical dynamic-stiffness matrix maps displacement to force:A matrix inverse cannot be replaced by reciprocating individual entries. For example, two masses connected by a spring of stiffness $k_c$, each also grounded by stiffness $k_0$, have $D_{11}=D_{22}=a=m{s}^2+k_0+k_c$ and $D_{12}=D_{21}=-k_c$. If coordinate2 is free of external force, $H_{11}=a/(a^2-k_c^2)$, so the driving-point dynamic stiffness is $a-k_c^2/a$ where defined. If coordinate2 is clamped, the required ratio is instead $a$. The measurement constraints determine which inverse is physically relevant.

A mechanical dynamic-stiffness matrix maps displacement coordinates to applied forces in a linear structural dynamics model at a specified transform variable or frequency. Starting with constant mass matrix $\mathbf{M}$, damping matrix $\mathbf{C}$ and mechanical stiffness matrix $\mathbf{K}$,zero initial conditions give $\mathbf{F}(s)=\mathbf{D}(s)\mathbf{X}(s)$ with $\mathbf{D}=s^2\mathbf{M}+s\mathbf{C}+\mathbf{K}$. For the $e^{i\omega t}$ convention, $\mathbf{D}(i\omega )=\mathbf{K}-{\omega }^2\mathbf{M}+i\omega \mathbf{C}$. Its matrix inverse, when it exists, is the complete mechanical receptance matrix. Models and measured responses must use matching coordinates and boundary conditions.^[ref]

Consider two $1\mathrm{kg}$ masses, each grounded by a $100N/m$ spring and joined by another $100N/m$ spring, with no damping. At angular frequency $5rad/s$,A force amplitude $F_1=1\mathrm{N}$ with $F_2=0$ gives $X_1=175/20625\mathrm{m}\approx 8.485\mathrm{mm}$ and $X_2=100/20625\mathrm{m}\approx 4.848\mathrm{mm}$. Clamping coordinate2 instead gives $X_1=1/175\mathrm{m}\approx 5.714\mathrm{mm}$, because the clamp supplies a reaction force.

Partitioning retained and internal coordinates gives mechanical dynamic condensation. With ${\mathbf{F}}_i=0$ and invertible ${\mathbf{D}}_{ii}$,The effective matrix is a Schur complement, generally frequency dependent even when the original mass matrix, damping and mechanical stiffness matrix are constant. In the example its scalar value is $175-100^2/175\approx 117.857N/m$, rather than175. A reduced force-free internal coordinate and a clamped coordinate describe different experiments. At a singular block, that particular elimination is unavailable; it does not establish that the full mechanical system lacks a valid solution.

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.