Rigid bodies mounted on compliant supports are encountered in machine foundations, precision platforms, transport systems, rotating equipment, and vibration-isolated technological units [1], [2]. In these systems, passive elastic and viscous elements can provide acceptable attenuation away from resonance, but their performance is limited when the excitation frequency approaches a natural frequency. The limitation is especially important for multi-axis systems because translational and rotational motions are coupled through the geometry of the support locations [3–5].

Active vibration control offers an additional mechanism for modifying the dynamic response. A practical electromechanical actuator is not an ideal force source: The generated mechanical force follows from voltage, current, electrical resistance and inductance, back electromotive force, and the actuator force constant. Therefore, a mechanically complete model should retain the electrical dynamics when the actuator bandwidth or time constant is comparable with the vibration time scale [6].

The contribution of this work is a unified six-degree-of-freedom formulation in which three translations and three rotations of the rigid body are coupled to viscoelastic support forces and electromechanical actuator dynamics [7]. The framework distinguishes two control effects: active damping, which primarily reduces the resonance peak, and active stiffness, which shifts the resonance location. This distinction is used to interpret the numerical frequency-response results and to identify a tradeoff between resonance attenuation and low-frequency amplification [8], [9].

The objective is to: 1) formulate the six-degree-of-freedom mechanical equations, 2) couple them with the actuator electrical equations, 3) obtain a frequency-domain representation suitable for resonance analysis, and 4) quantify the influence of active stiffness, active damping, and actuator time constant using the parameter sets supplied in the source document [10–12].

The numerical study highlights three points relevant to the design of active vibration isolators. First, active damping primarily controls the height of the resonance peak. Increasing c_eff strongly decreases the response at the passive resonance frequency. Second, active stiffness influences both the quasi-static response and the resonance location. A controller with excessively small effective stiffness can produce excellent resonance suppression while allowing a large low-frequency displacement; this behavior is visible for active branch 2. Third, the actuator time constant changes the effective control action. In the supplied parameterization, larger T is associated with stronger equivalent damping for branches 2 and 3, which increases attenuation near but also emphasizes the importance of considering electrical dynamics and phase lag.

The six-degree-of-freedom formulation is important because active forces applied away from the mass center create control moments as well as translations. A one-dimensional model cannot represent cross-axis coupling, products of inertia, or geometric lever arms of the supports. In contrast, the present matrix form can be extended directly to multiple rigid bodies, multiple support groups, non-identical actuators, and base excitation. The same formulation also enables eigenvalue placement, robust-control design, and actuator saturation constraints to be added without changing the underlying mechanical topology.

A limitation of the present draft is that the available source data contain parameter sets and analytical relationships but not experimental measurements or a complete numerical integration record of the coupled electrical–mechanical state equations. Accordingly, the reported curves are model-based calculations. For journal submission, the strongest next step would be experimental or high-fidelity finite-element validation, together with sensitivity analysis for uncertain stiffness, damping, electrical resistance, inductance, and time delay.

I. A coupled electromechanical model was formulated for a rigid body with three translational and three rotational degrees of freedom on viscoelastic supports.

II. The actuator model explicitly retains inductance, resistance, back EMF, current-to-force conversion, and displacement–velocity feedback, making the control force dynamically realizable rather than instantaneous. III. For the normalized passive model, the resonance amplitude at ω = 1.1 rad/s equals 5.00. Active tuning reduces this amplitude by approximately 67–93% for the supplied parameter sets.

IV. Aggressive active damping with low effective stiffness can introduce large low-frequency displacement. The keff = 1.05 branch provides the most balanced broadband behavior among the three supplied active branches.

V. The six generalized coordinates exhibit decaying transient envelopes with 2% envelope times of approximately 18.6–24.5 s for the supplied λ values.

VI. The model provides a suitable basis for optimization of controller gains, actuator bandwidth, and support geometry, but experimental or high-fidelity numerical validation is required before engineering deployment.

In this article, the author contributed equally, and the author jointly developed the ideas and achieved the results through collaborative thinking.

In this article, all authors contributed equally, and the authors jointly developed the ideas and achieved the results through collaborative thinking.

This section indicates any support that is not included in the Author Contribution or Funding sections. Such support may include administrative and technical support or in-kind donations (e.g., materials used for experiments).

Data available on request due to ethical reasons.

The author declares that he has no conflict of interest. "Funders played no role in the design of the study, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results."

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This article does not contain any studies with human participants performed by the author.