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G Agnes, Beavercreek, OH, USA Copyright # 2001 Academic Press doi:10.1006/rwvb.2001.0190

Active vibration absorbers combine the benefits of mechanical vibration absorbers with the flexibility of active control systems. Mechanical vibration absorbers use a small mass coupled to the structure via a flexure (in some form) to add a resonant mode to the structure. By tuning this resonance, the vibration of the structure is reduced. Two limitations on the performance of mechanical vibration absorbers are the mechanism’s strokelength and the added mass.

The active vibration absorber replaces the flexure and mass with electronic analogs. Active vibration absorbers can achieve larger effective strokelengths with less mass added to the system. In addition, they can be implemented via strain actuators located in regions of high strain energy instead of large displacement as required by mechanical vibration absorbers – a benefit for some applications.

The drawback to active vibration absorbers is the need for power and electronics. Higher cost and complexity results from the custom analog circuits which must be built or the digital controllers which must be implemented. Finally, unlike mechanical vibration absorbers, active vibration absorbers can lead to spillover, destabilizing the system.

In the following sections, the equations of motion for three common implementations are discussed: the piezo-electric vibration absorber, positive position feedback control and the active vibration absorber. Other combinations of position, velocity, and acceleration feedback are possible (depending on sensor availability) but will not be discussed herein.

Piezo-electric Vibration Absorber

Piezo-electric materials act as a transformer between mechanical and electrical energy. Common forms include a ceramic (PZT) and a polymer (PVDF). When a piezo-electric material undergoes strain, electrical charge is produced on its electrode. By creating a resonant electrical shunt via a resistor and inductor, an electrical resonator is formed. (The piezo-electric material acts as a capacitor in an L–R–C circuit.) By tuning the resonant frequency, the vibration of the structural system is reduced. In practice, inductances on the order of kilohenries are required for lowfrequency modes. The inductor is therefore often implemented as an active circuit, requiring power to operate. Hence the piezo-electric vibration absorber is considered an active vibration absorber.

Equations of Motion

Amodalmodelofastructurecontainingpiezo-electric materials can be idealized as shown in Figure 1. The base system consists of a mass constrained by a struc- tural spring, KS, and a piezo-electric spring, KP, arranged in parallel. The displacement of the struc- ture, X, is to be minimized by a tuned resonant circuit, with charge Q, on the piezo-electric electrodes. The systemthushastwo-degrees-of-freedomorfourstates.

The linear constitutive equations for piezo-electric materials, simplified for one-dimensional transverse actuation are:

ÿh31 cD

Figure 1 The piezo-electric vibration absorber.

The views expressed in this article are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the US Government.

Here, the standard IEEE notation is used (i.e., E is electric field, T is stress, D is electrical displacement, S is strain, "S is electrical permittivity, h is the piezoelectric coupling constant, and cD is the elastic modulus). Assuming a standard patch-like application, these equations may be rewritten in terms of variables more convenient for this study. The equations for piezo-electric spring are thus:

where V is the voltage or the piezo-electric electrode, F is the force of the spring, Q is the charge flowing into the patch electrodes, X is the displacement of the spring, CSP is the capacitance of the patch under constant strain, KDP is the stiffness of the piezo-electric spring under constant charge, and H is the electro- mechanical coupling parameter. Note that coefficients in these equations can be modified for more complicated geometries, but would assume a similar form.

Placing an inductive–resistive (LR) shunt across the electrodes of the piezo-electric spring, the equations of motion for the mass in Figure 1 are:

Here, L is the shunt inductance; R, the shunt resistance; M, the structural mass; and F t is an external disturbance. These equations are next nondimensionalized:

with the nondimensional quantities used in eqns [4a] and [4b] defined as:

Time has been nondimensionalized such that t oDT. Note that eqns [4a], [4b] and [5] differ from those in the literature since the constant charge

(or shorted) stiffness, KD, of the piezo-electric spring is used in place of the usual constant voltage stiffness, KE. This simplifies the tuning. Also note, that the coupling term is the generalized electro- mechanical coupling coefficient, ~K31, which can be determined experimentally – as a modal quantity.

These equations are of the same form as those for the mechanical vibration absorber. The coordinates x and q are proportional to the system displacement and shunt charge, respectively, thus maintaining physical significance. In the next section, the equations of motion for a single-degree-of-freedom structure under positive position feedback control will be derived of this same form. First, however, the tuning and response of the PVA will be considered.

Controller Design

The design of a piezo-electric vibration absorber involves three factors: ~Kij; L and R. The value of ~Kij is maximized by locating the piezo-electric material in areas of high strain energy. This constant may be determined as a modal constant by considering either the open- and short-circuit resonant mode:

or analytically by:

where o is the natural frequency with the mass of the piezo-electric device included, but its stiffness neglected.

Given the piezo-electric coupling coefficient, a broadband vibration absorber analagous to Den- Hartog’s equal peak implementation can be formed by setting:

For multimodal applications, numerical optimization must be used to determine the proper electrical network to suppress the vibration of the structure.

System Response

The response of a single-degree-of-freedom system to harmonic excitation is shown in Figure 2. The tuning

2 ABSORBERS, ACTIVE in eqns [8] and [9] was used. As the piezo-electric coupling is increased, the response of the system decreases. In practice, the piezo-electric coupling is generally limited to at most 0.3.

The impulse response of the system with the a 0:2 and the shunt tuned as presented above is shown in Figure 3. Note the system response is damped. The low coupling factors and the fragile nature of piezo-ceramics limit the application of piezo-electric vibration absorbers when higher damping is desired. In the next section, another implementation of an active vibration absorber is discussed: positive position feedback control.

Positive Position Feedback

Modern control design is traditionally performed using first-order dynamical equations. The positive position feedback (PPF) algorithm developed by Goh and Caughey and implemented by Fanson and Caughey uses second-order compensation, allowing physical insight to vibration control by active modal addition. In this algorithm, a position signal is compensated by a second order filter for feedback control. For linear systems, the PPF controller is stable even in the presence of unmodeled actuator dynamics. In addition it is possible to transform the dynamical equations to modal space and design independent second-order feedback compensators for individual modes. Many numerical and experimental implementations of the PPF control scheme may be found in the vibrations literature.

Figure 2 PVA frequency response as the piezo-electric coupling varies.

Figure 3 PVA results for a single-degree-of-freedom system with a 0:2. (A) System; (B) controller.

ABSORBERS, ACTIVE 3

Equations of Motion

A modal model of a structure containing an actuator can be idealized as shown in Figure 4.

A single-degree-of-freedom with mass, M, viscous damping, C, and stiffness, K, is driven by an external force, F. The displacement, X,o f M is controlled by an actuation force, U. The equations of motion are:

Equation [10] can be nondimensionalized. For positive position feedback, U is defined:

The sensor and controller gains, H and G, have been set equal (as is conventional) and defined as

G H go2c. The equations of motion for the combined system are therefore:

Controller Design

Using eqns [13a] and [13b], a controller with pre- scribed closed loop damping ratio xp can be found.

The design requires three factors: g; oc and zc. The value of g is determined such that actuator saturation is avoided. In practice, this may require experimentally setting g for the worst case disturbance.

Again a broadband vibration absorber analogous to the equal peak tuning can be determined. Given the feedback gain, the controller is tuned by setting:

For low values of g this leads to ‘equal peak’ results similar to the piezo-electric vibration absorber. Using this tuning law, the variance in system natural frequency, damping ratio, and pole locations are plotted in Figure 5 as the gain is varied. Note that this law is not an equal peak law at higher values of g.

For multimodal application, numerical optimization must be used to determine the proper parameter

Figure 4 An actuated structure.

Figure 5 Positive position feedback results for a singledegree-of-freedom system as feedback gain is increased from 0 to 0.7. (A) Natural frequencies; (B) damping ratio; (C) pole locations.

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