Large Amplitude Limit Cycle Research Articles

Elimination of limit cycles in floating-point implementations of direct-form recursive digital filters

This paper focuses on the limit cycle analysis of floating-point implementations of direct form recursive digital filters. A sufficient criterion for the absence of zero-input limit cycles is derived for a direct-form implementation with a single quantizer in the recursive loop. Both the unlimited exponent range (UER) limit cycles and those due to exponent underflow are considered. The main result of the paper is that the limit cycle behaviour of the filter is directly related to the maximum gain of the recursive loop. The higher the recursive loop gain, the longer mantissa wordlength is required to eliminate the possible large-amplitude UER limit cycles. Also the root-mean-square (RMS) bound for the underflow limit cycles is directly proportional to the recursive loop gain. The error feedback technique can be used to reduce the peak gain and thus to save bits in the mantissa wordlength in critical applications, as shown by examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonlinear Large Amplitude Aeroelastic Behavior of Composite Rotor Blades

An investigation is made of the nonlinear, large amplitude aeroelastic behavior of hingeless composite rotor blades during hover. The aeroelastic model is capable of dealing with large vibration amplitudes as well as large static deflections including transverse shear and warping deformation and is based on an Euler angle formulation of the basic large deflection equations together with a harmonic balance, finite difference, and NewtonRaphson technique. Nonlinear stall aerodynamics is included by use of the ONERA airforces model. Analysis of large amplitude limit cycles that evolve from linear flutter solutions is performed on a [0/90]35 graphite/ epoxy composite blade model. Numerical results indicate that the nonlinear stall is dominant at moderate amplitudes, but that nonlinear static-dynamic couplings in the structure, which bring a softening effect into the general aeroelastic behavior, could be equally important at large amplitudes.

Small Amplitude Limit Cycles for Cubic Systems

AbstractIn this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systemsWith a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.

Finite element nonlinear panel flutter with arbitrary temperatures in supersonic flow

A finite element frequency domain method for predicting nonlinear flutter response of panels with temperature effects is presented. By using the principle of virtual work, the element nonlinear stiffness formulation for a panel under a combined thermal and aerodynamic loads is derived on the bases of von Karman's large deflection plate theory, the first-order piston theory aerodynamics and the quasi-steady thermal stress theory. The system equations of motion can be mathematically separated into two sets of equations and then solved in sequence. The first set of equations yields the panel thermal-aerodynamic equilibrium and the second set of equations of motion leads to the flutter limit-cycle oscillations. Stability and flutter boundaries can also be obtained from the two sets of system equations. Finite element large amplitude limit-cycle flutter results at different uniform temperatures are obtained for a simply supported square panel and are compared with existing Galerkin/time integration and other finite element solutions. Effects of nonuniform temperature distributions, panel length-to-width ratios, and boundary conditions on flutter responses of rectangular and triangular panels are presented.

Dynamic Control of Centrifugal Compressor Surge Using Tailored Structures

A new method for dynamic control of centrifugal compressor surge is presented. The approach taken is to suppress surge by modifying the compression system dynamic behavior using structural feedback. More specifically, one wall of a downstream volume, or plenum, is constructed so as to move in response to small perturbations in pressure. This structural motion provides a means for absorbing the unsteady energy perturbations produced by the compressor, thus extending the stable operating range of the compression system. In the paper, a lumped parameter analysis is carried out to define the coupled aerodynamic and structural system behavior and the potential for stabilization. First-of-a-kind experiments are then conducted to examine the conclusions of the analysis. As predicted by the model and demonstrated by experiment, a movable plenum wall lowered the mass flow at which surge occurred in a centrifugal compression system by roughly 25 percent for a range of operating conditions. In addition, because the tailored dynamics of the structure acts to suppress instabilities in their initial stages, this control was achievable with relatively little power being dissipated by the movable wall system, and with no noticeable decrease in steady-state performance. Although designed on the basis of linear system considerations, the structural control is shown to be capable of suppressing existing large-amplitude limit cycle surge oscillations.

A New and Interesting Class of Limit Cycles in Recursive Digital Filters

Limit cycles oscillations often occur in recursive digital filters due to the quantization of products in the feedback section, A new and interesting class of limit cycles has been discovered and categorized for second-order sections that round either sign-magnitude or twoscomplement products. These limit cycles are named rolling-pin limit cycles. They are completely defined by three integers and a simple construction rule and exist for B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> − B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> pairs lying within small rectangular regions in the B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> − B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> (coefficient) plane. Each set of integers completely defines the peak amplitude, the length, and the region of existence. The amplitude of these limit cycles can be made close to but does not exceed three times the Jackson peak estimate. Rolling-pin limit cycles often occur in filters with high Q poles located near dc or half the sampling frequency. When these large amplitude limit cycles occur, the idle channel performance of a filter is often unacceptable. Specialized techniques, requiring extra circuitry, can be used to suppress them. Alternatively, it may be less costly and more efficient to avoid the small rectangular regions within which the rolling-pin limit cycles exist in the B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> − B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> coefficient plane.

Large amplitude, self-sustained oscillations in difference equations which describe digital filter sections using saturation arithmetic

The possibility of oscillations due to adder overflow in digital filter sections of order exceeding two is investigated. Specifically, the following problem is posed: is the property that large amplitude limit cycles cannot be self-sustaining in saturation arithmetic special to second-order sections, or is it also true for some or all higher order sections? This issue is resolved and, beyond that, some new and rather interesting results are presented on the stability implications of approximating a nonlinear element in standard filter structures by a range of linear gains. For every order of the section beyond two we are able to identify a set of coefficients corresponding to an absolutely stable underlying linear system, and a set of initial conditions for which the solution (to the nonlinear recursion) is a limit cycle. As an aid to understanding this behavior, we show that a particular natural class of associated linear recursions is always stable if and only if their order does not exceed two. Finally, motivated by the above result and by the viewpoint of describing functions, we make a conjecture according to which these oscillations do not exist if every member of the above stated class of linear systems is absolutely stable. We show that even this conjecture is false for recursions of order three. Thus, to ensure that overflow oscillations do not occur in high-order sections in which saturation arithmetic is used particular care has to be exercised on a case by case basis.

A quantized data attitude-control system.

This analysis investigates a space-vehicle attitude-control system utilizing a quantized attitude-err or signal, wherein the quantized attitude information is passed through a lead compensation network comprised of passive electronic elements. The lead-network response to an input consisting of a series of quantization steps is a series of exponentially decaying pulses that are fed into a biased Schmitt-trigger pulse-shaping circuit, which activates an on/ off torque-producing device that applies corrective torques to the vehicle. Analysis of the operation of the quantized data attitude-control system (QDACS) includes the general solution for the lead-network response to a uniform staircase input. This staircase-response solution is used to derive expressions for the single-pulse on and off switching lines in the phase plane. Boundaries separating regions of constant torque and pulsing torque application are identified, and large-amplitude response and limit cycle modes of operation are analyzed to determine the nature of the response trajectories. Despite its comparatively simple system philosophy and mechanization and also the absence of a mode-switching requirement, the QDACS is shown to have desirable response characteristics over its entire dynamic operational range.