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Reduction of structural vibrations is of major interest in mechanical engineering for lowering sound emission of vibrating structures, improving accuracy of machines, and increasing structure durability. Besides optimization of the mechanical design or various types of passive damping treatments, active structural vibration control concepts are efficient means to reduce unwanted vibrations. In this contribution, two different semiactive control concepts for vibration reduction are proposed that adapt to the normal force of attached friction dampers. Thereby, semiactive control concepts generally possess the advantage over active control in that the closed loop is intrinsically stable and that less energy is required for the actuation than in active control. In the chosen experimental implementation, a piezoelectric stack actuator is used to apply adjustable normal forces between a structure and an attached friction damper. Simulation and experimental results of a benchmark structure with passive and semiactively controlled friction dampers are compared for stationary narrowband excitation. For simulations of the control performance, transient simulations must be employed to predict the achieved vibration damping. It is well known that transient simulation of systems with friction and normal contact requires excessive computational power due to the nonlinear constitutive laws and the high contact stiffnesses involved. However, commercial finite-element codes do not allow simulating feedback control in a general way. As a remedy, a special simulation framework is developed which allows efficiently modeling interfaces with friction and normal contact by appropriate constitutive laws which are implemented by contact elements in a finite-element model. Furthermore, special model reduction techniques using a substructuring approach are employed for faster simulation.

Semiactive control strategies for vibration reduction offer interesting alternatives to passive means of damping enhancement or fully active vibration control (AVC). Hereby, the term semiactive means that passive system properties, such as friction, material damping, or fluid viscosity, are actively controlled. This intrinsically eliminates the problem of system destabilization due to spillover effects encountered in AVC applied to flexible structures [

A beam-like friction damper element is attached to a beam-like metal benchmark structure by a normal screw and an adaptive screw. The principle is depicted schematically in Figure

Sketch of the benchmark structure (steel, 775 mm length, 40 mm width, and 3 mm thickness) with adaptive friction damper beam (steel, 160 mm length, 40 mm width, and 3 mm thickness).

Photograph of the experimental setup with structure, damper beam, piezoelectric stack, force cell, accelerometers, and attached shaker stinger (cf. Figure

The structure is discretized by the finite-element (FE) method using ANSYS (Figure

Finite-element model (≈30000 DOFs and 632 nodes in contact) in typical bending deformation with the two screws that impose the external clamping force pairs

In a generic way, the discretized structural dynamics of the two substructures, namely, the main structure and the attached friction damper beam, are given by

Constitutive equations are implemented for the normal contact and the tangential contact in the interface by node-to-node contact elements. The former is a bilinear stiffness relationship (Figure

(a) Normal force stiffness relation. (b) 1D Jenkins friction model. (c) Jenkins model hysteresis loop (thick) in comparison to the Coulomb friction hysteresis (thin). Note, that for increasing tangential stiffness, the hysteresis approximates the Coulomb model hysteresis.

The nonlinear system of (

Measured and simulated FRFs for impulse excitation and

Two controllers each consisting of an appropriate nonlinear control law plus an observer to estimate nonmeasurable variables required by the control are introduced in the following. The first control is denoted by hysteresis-optimal control and is motivated by experimental investigations. They show that relatively simple dynamical friction models are often capable of modeling the most dominant friction effects in structures with local joints [

For that, it is assumed that the dominant damping effects are located in the contact area close to the adaptive screw and can be modeled by a discrete friction model. Then, the dissipated work

In general, the required tangential displacement

This model is derived by rigid connection of the damper beam at one end

Closed control loop with hysteresis-optimal control law.

For the second proposed control law, Lyapunov’s direct method is applied by choosing the mechanical system energy as Lyapunov function

The proposed controls are investigated for the benchmark structure with a damper beam at

A

For the Lyapunov-type control, a piezoelectric force cell of high sensitivity and bandwidth is added (cf. Figure

For nonlinear mechanical structures, comparing FRFs requires special care because the obtained FRFs are nonlinear. More specifically, their resonance frequencies, peak amplitudes, and peak forms depend on the excitation signal as well as amplitudes. Consequently, the amplitude is controlled during sine sweep measurements to make the excitation independent of the structural impedance for consistent comparisons. Very low sweep velocities (0.1 Hz/s) are employed to obtain steady-state conditions which approximate step-sine testing and to avoid interaction between the interesting effects of the semiactive structural control and the shaker control. Due to the very small relative displacements outside resonances, the control is only effective close to resonances which allows restricting the evaluation around the resonance frequencies to save simulation and measurement time.

In Figures

Hysteresis-optimal control: measured (top) and simulated (bottom) accelerance FRFs for controlled sine-sweep excitation with (solid) and without (dashed) control

Mode 2,

Mode 3,

Mode 4,

Mode 2,

Mode 3,

Mode 4,

Lyapunov-type control: measured (top) and simulated (bottom) accelerance FRFs for controlled sine-sweep excitation with (solid) and without (dashed) control (

Mode 2,

Mode 3,

Mode 4,

Mode 2,

Mode 3,

Mode 4,

In the passive case, the minimal possible force

Multimodal, semiactive vibration controllers that adapt the normal force applied to friction damper beams by piezoelectric stack actuators are investigated for a generic benchmark structure in experiments and simulations. They are shown to efficiently damp structural resonances for different excitation amplitudes and vibration modes. Which of the investigated controller concepts suits best for a certain application depends mainly on the actuator principle, the power considerations, and whether the excitation being rather broadband or narrowband. Based on the results of the beam experiment, the proposed friction damper is used to reduce the vibrations of machine tools [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The support of the DFG (German Research Foundation) with Project SPP 1156 is gratefully acknowledged.