Leonard Meirovitch

On the problem of observation spillover in self-adjoint distributed-parameter systems

The problem of observation spillover in self-adjoint distributed-parameter systems is investigated. Observation spillover occurs when the output of a limited number of sensors, located at various points on the distributed domain, cannot synthesize the modal coordinates exactly. To this end, two techniques of state estimation (namely, observers and modal filters) are described. Both techniques can be used to extract modal coordinates from the system output and to implement feedback controls. It is shown that, if the residual modes are included in the observer dynamics, observation spillover cannot lead to instability in the residual modes. The problem of the unmodeled modes does remain, however. It is also shown that the modal filters have some very attractive features. In particular, modal filters can be designed to estimate the modal coordinates with such accuracy that observation spillover can be virtually eliminated. In addition, when modal filters are used, in conjunction with a sufficiently large number of sensors, the entire infinity of the system modes can be regarded as modeled, which implies that actual distributed control of the system is possible. It is also demonstrated that modal filters are quite easy to design and are not plagued by instability problems.

A general substructure synthesis method for the dynamic simulation of complex structures

Abstract In this paper a general substructure synthesis method is developed for the dynamic analysis of complex flexible structures. The motion of each substructure is represented by a given number of substructure admissible functions. Substructure admissible functions are often low-order polynomials, and hence computationally easy to work with. The otherwise disjoint substructures are connected together to form a whole structure by imposing approximate geometric compatibility conditions by means of the method of weighted residuals. The behavior of the estimated eigenvalues obtained by the substructures synthesis method can be ascertained by means of a bracketing theorem. The estimated eigenvalues do converge to the actual eigenvalues of the original structure, although it is necessary to consider two limiting processes, one in which the number of substructure admissible functions is increased and the other in which the number of internal boundary weighting functions is increased.

Convergence of the classical Rayleigh-Ritz method and the finite element method

It is demonstrated in this paper that convergence of the classical Rayleigh-Ritz method can be vastly improved by introducing a new class of admissible functions, called quasi-comparison functions

Response of slightly damped gyroscopic systems

Abstract A second-order perturbation theory is developed for the response of slightly damped gyroscopic systems. The solution is based on the eigensolution for undamped gyroscopic systems and is expressed in terms of real quantities alone.

A perturbation technique for gyroscopic systems with small internal and external damping

Abstract The response of linear damped gyroscopic systems can be obtained by means of techniques of linear systems theory, which involves the computation of the transition matrix. The response is in terms of complex quantities, which is likely to cause computational difficulties as the order of the system increases. In the absence of damping, it is possible to derive the response of a linear gyroscopic system with relative ease by working with real quantities alone. When damping is small, one can use a perturbation approach to produce the response by regarding the undamped gyroscopic system as the unperturbed system. In a previous paper, a perturbation analysis was used to derive the response of a gyroscopic system with small internal damping. This paper extends the approach to the case of external damping, which is characterized not only by symmetric coefficients multiplying velocities but also by skew symmetric coefficients multiplying displacements, where the latter terms are known as circulatory. A numerical example is presented.

Control of spinning flexible spacecraft by modal synthesis

A procedure is presented for the active control of a spinning flexible spacecraft. Such a system exhibits gyroscopic effects. The design of the controller is based on modal decomposition of the gyroscopic system. This modal decoupling procedure leads to a control mechanism implemented in modular form, which represents a distinct computational advantage over the control of the coupled system. Design procedures are demonstrated for two types of control algorithms, linear and nonlinear. The first represents classical linear feedback approach and the second represents an application of on-off control, both types made feasible by the modal decomposition scheme.

A Method for the Liapunov Stability Analysis of Force-Free Dynamical Systems

A new method for the stability analysis of force-free dynamical systems described by simultaneous sets of ordinary and partial differential equations of motion is presented. Such systems, referred to as hybrid, arise naturally in connection with the motion of spinning flexible bodies. The method is based on Liapunovs second method and works directly with the hybrid system of equations. It involves the construction of a Liapunov functional that takes into account automatically motion integrals resulting from the absence of external forces on the system. The general theory is particularly suitable for the stability analysis of torque-free spinning satellites containing distributed elastic members. As an illustration, the case of spin stabilization of a satellite with flexible appendages is solved.

Active vibration suppression of a cantilever wing

Abstract A method for the active vibration suppression of a cantilever wing is presented. The approach is based on modal control, in which a modal feedback control law relating the motion of the control surfaces to the controlled modes is implemented. Modal displacements and velocities required for feedback are extracted from sensor measurements by means of modal filters. A numerical example is presented.

Some problems associated with the control of distributed structures

Control of structures can be carried out conveniently by modal control, whereby the structure is controlled by controlling its modes. Modal control requires the estimation of the modal states for feedback, which can present a problem. One approach that does not require modal state estimation is direct feedback control, which implies collocated sensors and actuators. This paper examines some problems encountered in direct feedback control of distributed structures in conjunction with pole placement. A perturbation technique permits the computation of control gains for multi-input systems. The paper demonstrates that the difficulties experienced in using direct feedback in conjunction with pole placement are endemic to the approach.

A general dynamic synthesis for structures with discrete substructures

Abstract This paper presents a substructure synthesis method for the dynamic simulation of complex structures, where the structures consist of an assemblage of discrete substructures. An analogy between distributed and discrete structures is extensively invoked. To stimulate the motion of discrete substructures, the concept of “admissible vectors” is introduced, where admissible vectors represent the discrete counterpart of admissible functions for distributed substructures. The individual substructures are forced to act as a whole structure by imposing certain geometric compatibility on internal boundaries shared by any two substructures. A numerical example illustrating the method is presented.