A common format is developed for a mass and an inerter-based resonant vibration absorber device, operating on the absolute motion and the relative motion at the location of the device, respectively. When using a resonant absorber a specific mode is targeted, but in the calibration of the device it may be important to include the effect of other non-resonant modes. The classic concept of a quasi-static correction term is here generalized to a quasi-dynamic correction with a background inertia term as well as a flexibility term. An explicit design procedure is developed, in which the background effects are included via a flexibility and an inertia coefficient, accounting for the effect of the non-resonant modes. The design procedure starts from a selected level of dynamic amplification and then determines the device parameters for an equivalent dynamic system, in which the background flexibility and inertia effects are introduced subsequently. The inclusion of background effect of the non-resonant modes leads to larger mass, stiffness and damping parameter of the device. Examples illustrate the relation between resonant absorbers based on a tuned mass or a tuned inerter element, and demonstrate the ability to attain balanced calibration of resonant absorbers also for higher modes.

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Shunting of piezoelectric transducers and suitable electric circuits constitutes an effective passive approach to resonant vibration damping of structures. Most common design concepts for resonant resistor-inductor (RL) shunt circuits rely on either maximization of the attainable modal damping or minimization of the frequency response amplitude. However, the former is suboptimal near resonance due to constructive interference of the two modes with identical frequency, and the latter results in reduced implemented damping. This article proposes an explicit pole placement–based design procedure for both series and parallel RL circuits. The procedure relies on equal modal damping and sufficient separation of the complex poles to avoid constructive interference of the two modes. By comparison with existing design procedures, it is demonstrated that the present calibration leads to a balanced compromise between large modal damping and effective response reduction with limited damping effort.

SUMMARYIntroduction of algorithmic damping by increasing the parameter values in the Newmark algorithm leads to undesirable low-frequency damping and reduced order of accuracy. It is demonstrated, how these effects can be removed by introducing an extra damping term in the form of a first order linear filter. When the linear filter is discretized in time, the state variable associated with the filter can be eliminated, leading to a weighted average of the equations of motion at two consecutive times. The filter procedure contains the known versions of alpha weighted Newmark methods as special cases, but gives a different and improved weighting of the excitation terms. A complete analysis of the properties of the algorithm when used on linear systems is given, including the frequency response properties. It is demonstrated that the effect of 'overshoot' is the consequence of a conservation relation that operates on a modified form of the mechanical energy of the system, and analytic results are presented for the magnitude of the effect.

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A resonant, piezoelectric shunt tuning procedure is based on the limiting eigenvalue problems associated with short and open circuiting of the piezoelectric domains attached to a vibrating structure. Whereas short circuit and open circuit frequencies are directly obtained from the eigenvalues, the associated electrical equation further determines a modal charge as an short circuit reaction force and a new modal voltage from the electric deflection in the open circuit limit. By a modal representation with short circuit mode shapes the structural equation fully decomposes, while the inherent contributions from non-resonant vibration modes in the corresponding electrical equation consistently define an effective modal capacitance, conveniently estimated by the modal charge to voltage ratio. The shunt tuning is obtained from the governing characteristic equation for the targeted vibration mode, in which the residual mode correction is explicitly represented by the effective modal capacitance. The tuning procedure with an effective modal capacitance is generalized to multiple piezoelectric domains with independent shunts for simultaneous damping of multiple vibration modes. The accuracy of the proposed shunt tuning methods is demonstrated by a numerical beam example with three piezoceramic patch pairs and independent resistive–inductive shunts. The analysis is carried out in a commercial finite element program, in which the required modal frequencies, charge and voltage are readily available as output to the short circuit and open circuit eigenvalue problems.