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Mechanical dynamic mass is the complex force-to-acceleration ratio $M_{\mathrm{dyn}}(s)=F(s)/A(s)$ for a stated mechanical excitation and response. It is the scalar reciprocal of mechanical accelerance where that response is nonzero. Its units are kilograms, but a supported structure need not have a real, positive or constant dynamic ratio. An impedance head supplies separate force and acceleration channels from which this response can be estimated.^[ref]

For $m\ddot{x}+c\dot{x}+kx=F$ with zero initial conditions,Equivalently, mechanical dynamic stiffness equals $s^2{M}_{\mathrm{dyn}}$. A freely translating ideal mass gives $M_{\mathrm{dyn}}=m$; a spring-supported mass includes the elastic and dissipative terms. With $m=2\mathrm{kg}$, $k=200N/m$ and $c=4\mathrm{N}s/m$, at $\omega =5rad/s$ the result is $(-6-0.8i)\mathrm{kg}$. At $10rad/s$ it is $-0.4i\mathrm{kg}$. The negative real part in the first case describes a restoring-force contribution relative to acceleration; it is not a negative quantity of material.

An impedance-head mass correction subtracts the effective moving mass below the force-sensing plane from the measured scalar dynamic ratio, provided that material shares the measured acceleration. This is a force-plane correction, not removal of the sensor's physical influence on the structure. At an antiresonance, a small response can produce a very large reciprocal ratio; dividing by a channel near its noise floor makes that estimate unreliable. With coupled measurement coordinates, exchanging complete force/acceleration descriptions requires a matrix inverse, rather than the reciprocal of each transfer entry.

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.

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.