Steven R. Hall

Frequency response of linear time periodic systems

A frequency response for linear time periodic (LTP) systems that exploits the one-to-one map induced by geometrically periodic signals is developed. This map is described by an integral operator, based on the GP (geometrically periodic) test input, and a generalized harmonic balance approach, based on an EMP (exponentially modulated periodic) input. The singular values or principal gains of the LTP operator are discussed, and the LTP principal gain diagram is described. Directional properties of the LTP operator are discussed, and notions of the domain and range spaces are presented. The framework of linear operators described has lead to the development of a comprehensive open-loop analysis theory for LTP systems, including a characterization of poles, transmission zeros, and their directional properties.<<ETX>>

Extensions of mixed-µ bounds to monotonic and odd monotonic nonlinearities using absolute stability theory†

In this paper we make explicit connections between classical absolute stability theory and modern mixed-μ analysis and synthesis. Specifically, using the parameter-dependent Lyapunov function framework of Haddad and Bernstein and the frequency dependent off-axis circle interpretation of How and Hall, we extend previous work on absolute stability theory for monotonic and odd monotonic nonlinearities to provide tight approximations for constant real parameter uncertainty. An immediate application of this framework is the generalization and reformulation of mixed-μ analysis and synthesis in terms of Lyapunov functions and Riccati equations. This observation is exploited to provide robust, reduced-order controller synthesis while avoiding the standard D, N - K iteration and curve-fitting procedures.

Minimum induced power requirements for flapping flight

(Received 21 October 1994 and in revised form 3 May 1996) The Betz criterion for minimum induced loss is used to compute the optimal circulation distribution along the span of flapping wings in fast forward flight. In particular, we consider the case where flapping motion is used to generate both lift (weight support) and thrust. The Betz criterion is used to develop two different numerical models of flapping. In the first model, which applies to small-amplitude harmonic flapping motions, the optimality condition is reduced to a one-dimensional integral equation which we solve numerically. In the second model, which applies to largeamplitude periodic flapping motions, the optimal circulation problem is reduced to solving for the flow over an infinitely long wavy sheet translating through an inviscid fluid at rest at infinity. This three-dimensional flow problem is solved using a vortex-lattice technique. Both methods predict that the induced power required to produce thrust decreases with increasing flapping amplitude and frequency. Using the large-amplitude theory, we find that the induced power required to produce lift increases with flapping amplitude and frequency. Therefore, an optimum flapping amplitude exists when the flapping motion of wings must simultaneously produce lift and thrust.

Design of a high efficiency, large stroke, electromechanical actuator

Large stroke, electromechanical actuator designs are considered. Special emphasis is placed on actuators designed to power a trailing edge servo-flap system for feedback control of helicopter rotor vibration, acoustics and aerodynamic performance. A survey was conducted comparing the advantages and disadvantages of a number of actuator designs. The major conclusions from this survey indicate that any successful actuator design will utilize a high bandwidth active material, produce large amplification of the active material stroke and incorporate a simple compressive pre-stress mechanism, while remaining efficient in a mass normalized sense. The mass efficiency, defined as the ratio of the specific work performed by the actuator to the specific energy available in the active material element, was used as a metric to rate the actuators considered in the survey. This metric is appropriate in aerospace applications where weight is critical. The most feasible discrete actuators are those where the active material reacts against an inert support frame housing. An upper bound on the mass efficiency of this type of actuator is shown to be a function of the ratio of active material to frame specific modulus. A new high efficiency discrete actuator, the X-frame actuator, is described. A prototype of this actuator was built and tested to confirm the predicted performance. The prototype demonstrates an output energy density of 14.6 ft lb/slug. It has a bandwidth of about 540 Hz when driving a nearly impedance-matched load.

Adoption of active learning in a lecture-based engineering class

Three years ago, the Department of Aeronautics and Astronautics at MIT expanded its repertoire of active learning strategies and assessment tools with the introduction of muddiest-point-in-the-lecture cards, electronic response systems, concept tests, peer coaching, course web pages, and web-based course evaluations. This paper focuses on the change process of integrating these active learning strategies into a traditional lecture-based multidisciplinary course, called unified engineering. The description of the evolution of active learning in unified engineering is intended to underscore the motivation and incentives required for bringing about the change, and the support needed for sustaining and disseminating active learning approaches among the instructors.

Power Requirements for Large-Amplitude Flapping Flight

In this paper, a method is presented for computing the circulation distribution along the span of a e apping wing that minimizes the power required to generate a prescribed lift and thrust. The power is composed of three parts: useful thrust power, induced power, and proe le power. Here, the thrust and induced power are expressed in terms of the Kelvin impulse and kinetic energy associated with the sheet of trailing and shed vorticity left behind the e apping wing. The proe le power is computed using a quasisteady approximation of the two-dimensional viscous drag polar at each spanwise station of the wing. A variational principle is then formed to determine the necessary conditions for the circulation distribution to be optimal. Included in the variational principle is a constraint that the wing not stall. This variational principle, which is essentially the viscous extension of the well-known Betz criterion for optimal propellers, is discretized using a vortex-lattice model of the wake, and the optimum solution is computed numerically. The present method is used to analyze a conventional propeller as well as a rigid wing in forward-e ight e apping about a hinge point on the longitudinal axis.

Development of a piezoelectric servoflap for helicopter rotor control

A servoflap that uses a piezoelectric bender to deflect a trailing edge flap for use on a helicopter rotor blade was designed, built, and tested. This servoflap design is an improvement over a design developed previously at MIT. The design utilizes a new flexure mechanism to connect the piezoelectric bender to the control surface. The efficiency of the bender was improved by tapering its thickness with length. Also, the authority of the actuator was increased by implementing a nonlinear circuit to control the applied electric field, allowing a greater range of actuator voltages. Experiments were carried out on a bench test article to determine the frequency response of the actuator, as well as hinge moment and displacement capabilities. Flap deflections of or more were demonstrated while operating under no-load conditions at frequencies up to 100 Hz. The data indicate that, if properly scaled, the actuator will produce flap deflections greater than at the 90% span location on a full-scale helicopter. In addition, the first mode of the actuator was at frequency of the target model rotor. Proper inertial scaling of this actuator could raise this modal frequency to greater than on an operational helicopter, which is adequate for most rotor control purposes. A linear state space model of the actuator was derived. Comparisons of this model with the experimental data highlighted a number of mild nonlinearities in the actuators response. However, the agreement between the experiment and analysis indicate that the model is a valid tool for predicting actuator performance.

Control of uncertain structures using an H(infinity) power flow approach

A technique is described for generating guaranteed stable control laws for uncertain, modally dense structures with collocated sensors and actuators. By ignoring the reverberant response created by reflections from other parts of the structure, a dereverberated mobility model can be developed that accurately models the local dynamics of the structure. This is similar in many respects to a wave-based model, but can treat more general structures, not only those that can be represented as a collection of waveguides. This model can be determined directly from transfer function data using an analysis technique based on the complex cepstrum. In order to minimize the effect of disturbances propagating through the structure, the power dissipated by the controller is maximized in an //<» sense. This guarantees that the controller is positive real and, thus, that the system will remain stable for any uncertainty, provided that the power flow is correctly modeled. The approach is demonstrated for two examples. The resulting controllers are much more effective than simple collocated rate feedback.

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.

Design and testing of a double X-frame piezoelectric actuator

The development and testing of a new actuator for helicopter rotor control, the Double X-Frame, is described. The actuator is being developed for wind tunnel and flight testing on an MD-900 helicopter. The double X-frame actuator has a number of design innovations to improve its performance over the original X-frame design of Prechtl and Hall. First, the double X-frame design uses two X-frames operating in opposition, which allows the actuator stack preloads to be applied internally to the actuator, rather than through the actuation path. Second, the frames of the actuator have been modified to improve the actuator form factor, and increase the volume of active material in the actuator. Testing of the double X-frame piezoelectric actuator was conducted in order to determine its performance (stroke and stiffness) and robustness. In general, stiffness test data compared well with the analytical predictions. The actuator stroke was about 15% less than expected, probably due to the stack output being less in the actuator than as measured in single stack segment testing in the lab. The actuator was also tested dynamically, to determine its frequency response. Actuator robustness was evaluated by measuring its performance when subjected to the effects of blade bending, vibration, and centrifugal loading. Blade elastic bending and torsion deformations were simulated by shimming of the actuator mounts. To assess the impact of the blade vibrations, the actuator and bench test rig were mounted on a hydraulic shaker and subjected to flapwise or chordwise accelerations up to 30 g. To assess the impact of centrifugal force loading, the actuator and bench test rig were spun in the University of Maryland vacuum chamber, so that the actuator was subjected to realistic accelerations, up to 115% of nominal. Results showed that actuator output (force times stroke) was largely unaffected by dynamic and steady accelerations or elastic blade deformations.