A.H. von Flotow

Damping of structural vibrations with piezoelectric materials and passive electrical networks

Abstract The possibility of dissipating mechanical energy with piezoelectric material shunted with passive electrical circuits is investigated. The effective mechanical impedance for the piezoelectric element shunted by an arbitrary circuit is derived. The shunted piezoelectric is shown to possess frequency dependent stiffness and loss factor which are also dependent on the shunting circuit. The generally shunted model is specialized for two shunting circuits: the case of a resistor alone and that of a resistor and inductor. For resistive shunting, the material properties exhibit frequency dependence similar to viscoelastic materials, but are much stiffer and more independent of temperature. Shunting with a resistor and inductor introduces an electrical resonance, which can be optimally tuned to structural resonances in a manner analogous to a mechanical vibration absorber. Techniques for analyzing systems which incorporate these shunting cases are presented and applied to a cantilevered beam experiment. The experimental results for both the resistive and resonant shunting circuits validate the shunted piezoelectric damping models.

Comparison and extensions of control methods for narrow-band disturbance rejection

Techniques for designing and implementing algorithms for the control of periodic narrowband disturbances are discussed, and the strong similarities between the different methodologies are shown. The analysis of linear time invariant feedback systems results in a suggestion of how to extend two of the LQ-based multiple-input, multiple-output methodologies to achieve improved performance and stability robustness properties. In the analysis of the adaptive feedforward methods, it is concluded that the popular filtered-x LMS algorithm is useful for implementation, but is best analyzed from a classical linear time invariant feedback perspective. This perspective results in a suggestion of how to extend the multiple error LMS algorithm to achieve improved performance and stability robustness. Stability bounds of the adaptive feedforward approaches are derived in terms of allowable model error. >

Dynamics and nonlinear adaptive control of an autonomous unicycle: theory and experiment

Equations of motion for an autonomous unicycle are given as derived using Kanes formalism, and the unstable nonminimum-phase behavior of the linearized system is illustrated. Lateral and longitudinal dynamics nominally decouple for low yaw rates but couple nonlinearly for large yaw rates and yaw accelerations. Time scale separation is employed to reduce the model order for inner loop controller design, where an unusual LQG (linear quadratic Gaussian) structure is proposed. Strong dependence of the lateral dynamics on wheel speed necessitates continuous gain scheduling of the lateral controller. A significant nonlinearity exists due to dry friction in yaw between the wheel and floor. Bang-bang control is employed to overcome this. The proposed unicycle has been built and the implemented controller structures and test results are indicated in the paper.<<ETX>>

A travelling wave approach to power flow in structural networks

Abstract A structural network is an assemblage of slender one-dimensional members. Each member is demarcated, by definition, by two junctions. A junction may involve only one member (a termination) or may be an interconnection of many members. In this paper, computationally and theoretically, the dynamics of such networks are investigated. The focus of the analysis is on elastic disturbance propagation and power flow in the networks. Spatially local models are assembled into a global, frequency domain description. This global description of the response is then investigated for local and global power flow. The procedures serves to identify disturbance transmission paths and to choose and evaluate control procedures, both active and passive. The techniques are demonstrated by application to several examples.

Optimal control of power flow at structural junctions

This paper describes several techniques for deriving optimal feedback compensators for structural waveguides at junctions. A frequency dependent cost functional, composed of power flow and control effort, is minimized. Control of power flow, by modifying junction reflection and transmission properties, enables incoming vibrational power to be selectively absorbed. Matched termination, non-causal, causal fixed-form and Wiener-Hopf feedback solutions are derived. These solutions, including a positive real approximation to the Wiener-Hopf solution, are illustrated through an extensive example for the free end of a dispersive Bernoulli-Euler beam. Several interesting results arise from this research. A matched termination, absorbing all impinging energy, is a subset of the optimal, non-causal solution. As a result, performance ranging from that achieved with simple rate feedback to that exceeding the matched termination is possible with increasingly complex configurations of control hardware. In the formulation of the control, information about the spectral content of the incoming waves can be used to frequency-tailor the control performance. The wave mode control formulation reveals that performance can be improved by using more than one distinct actuator and sensor at the active junction, a process not generally practiced in structural control. Finally, given the appropriate control hardware and particular structural geometries, optimal wave control can be used to eliminate resonant behavior or dynamically isolate one structural region from another.

Wave propagation, power flow, and resonance in a truss beam

Abstract Wave propagation in a periodic truss-work beam is investigated computationally. The analysis is based upon the transfer matrix of a single bay of the structure. The results, with members modeled as rods with pinned joints, agree well with results obtained from an equivalent continuum model of the same structure. The inclusion of bending in member models shows that both the pinned rod model and the equivalent continuum models lose fidelity above the first resonant frequency of lateral motion of the members. Modeled with beam members, the truss exhibits complicated mechanical filtering properties, which are illuminated by investigation of wave-mode power flow. Boundary conditions are applied in wave-mode co-ordinates by reformulation in terms of reflection matrices. The phase closure principle is invoked to predict natural frequencies of a fixed-free portion of the truss. It is found that closely spaced resonant frequencies are not identified by this method. Computed results show other subtle erroneous characteristics which are attributed to computational inaccuracy.

Active control of bending wave propagation at acoustic frequencies

Abstract An analytical and experimental investigation of the possibility of actively blocking the propagation of bending waves along a uniform beam is described. The investigation has the form of a case study; a thin brass plate-beam is used, it is excited with a short duration impulse, and the resulting disturbance spreads dispersively as it travels along the beam. A short portion of the beam is used as an active block. Strain gage sensors are used to drive thin piezoceramic bending moment actuators through a dynamic compensator. The compensation is designed in the frequency domain with reference to the beam equation, but independent of boundary conditions. The analytical work described includes the nominal design and its performance, the performance degradation due to modeling errors, and the performance degradation due to approximate implementation of the dynamic compensator. The laboratory implementation, in which analog electronics were used for the compensator, essentially verifies some of the sensitivity predictions.

Linear control design for active vibration isolation of narrow band disturbances

When an engine is mounted on a structurally flexible vehicle, it is important to inhibit the transmission of vibrations of the engine into the structure. In the past, vibration isolation has been achieved using passive schemes. Active control at the mount should permit a marked improvement in its vibration isolation characteristics. Three control schemes for actively controlling engine mounts are presented. The first method uses classical control ideas and physical insights for designing a compensator that provides broadband control. Stability issues are considered and provide motivation for pursuing other control methodologies. The second and third control schemes are variations on linear quadratic Gaussian (LQG) theory with augmented states added to the system matrix. Knowledge of the disturbance spectrum dictates the choice of the augmented states for both of the LQG schemes. The resulting compensators are a sequence of notch filters where the notches occur at the disturbance frequencies.<<ETX>>

Comparison of two LQG-based methods for disturbance rejection

When the disturbances acting on a system reflect the dynamics of the process generating them (i.e. disturbances other than white noise), this information can be exploited for the design of disturbance rejection controllers. Two modern control approaches for designing linear disturbance rejection controllers are discussed: disturbance modeling and frequency shaping of cost functionals. Both methods are extensions of linear-quadratic-Gaussian theory. It is shown that although the philosophies used to develop the two control schemes are very different, there is a distinct relationship between them. This is formalized by showing that the two methods are equivalent for the single-input-single-output case. The duality of the two methods is discussed for the multi-input-multi-output case.<<ETX>>

The Acoustic Limit of Control of Structural Dynamics

The acoustic limit of active control of structural dynamics is investigated; the limit as the control bandwidth includes a very large number of natural modes of the structure. The point is made that, in this limit, modal analysis cannot provide reasonably accurate models of the structural dynamics and that control design with respect to modal models is then of questionable value. Alternative modeling approaches are reviewed. A particular wave propagation formalism, applicable to modeling the acoustic response of networks of slender structural members, is described in some detail. Control options designed with reference to this formalism are reviewed, and speculations as to future developments of such control are offered.