A bifurcation analysis is used to investigate the complex dynamics of a heavily loaded turbine-generator system connected to an infinite busbar through a series capacitor-compensated transmission line. It reveals the existence of self-excited subsynchronous torsional oscillations of a 5-mass rotor, which may eventually lead to the destruction of the shaft or the loss of synchronism of the generator. Specifically, we show that, as the capacitor-compensation value increases and reaches a critical value, called supercritical Hopf bifurcation, the system around the operating point undergoes small single-period oscillations with constant amplitude. This in turn results in a small limit-cycle attractor. As the compensation level increases further, the amplitude of oscillation grows until a secondary Hopf bifurcation is reached. There, the oscillations characterize themselves by two incommensurate periods and bounded amplitudes, signifying the transformation of the limit cycle into two-period quasiperiodic motion called a two-torus attractor. When the capacitor-compensation level passes a third critical value, the amplitude of oscillations becomes unbounded following the destruction of the two-torus attractor and its basin of attraction in a so-called bluesky catastrophe. Interestingly, this scenario repeats itself in the vicinity of three supercritical Hopf bifurcations. The bifurcation analysis is validated with numerical solutions of the differential equations that govern the power system.

A uniformly valid asymptotic expansion is obtained for the regular solution of a class of second-order linear differential equations with three transition points-a turning point and two regular singular points. The solution is found by matching three different solutions obtained using the Langer Transformation. The matching yields the eigenvalues and the eigenfunctions.

Since the most severe roll motion occurs at resonance (known as synchronous rolling), the best way of reducing it is to increase the damping. The most common means of doing so is by the installation of bilge keels. If more control is required, both anti-roll tanks and fins are used. Tanks have the advantage of being able to function when the ship is not underway. The use of tanks with liquid free surfaces for reducing roll motion of ships is an old idea. Many researchers have studied the design of anti-roll tanks. However, most of the past effort has concentrated on studying the performance of anti-roll tanks in damping the roll motion of the ship. Little attention has been paid to the fluid motion inside the tank itself. Another important issue is the tank tuning. Proper tuning of the anti-roll tank, to match the ship's natural frequency, is very important in reducing the roll motion. This paper concentrates on the most familiar type, which is the U-tube passive tank as a mechanical absorber of roll motion. A detailed study, covering tank damping, mass, location relative to the ship CG, and tuning, is presented. New suggestions and observations are stated concerning tank damping and tuning.

The best way of reducing roll motion is by increasing roll damping. Bilge keels are the most common devices for increasing roll damping. If more control is required, anti-roll tanks and fins are used. Tanks have the advantage of being able to function when the ship is not underway. Our objective is to develop design procedures for passive tanks for roll reduction in rough seas. This paper focuses on the design of passive U-tube tanks. The tank-liquid equation of motion is integrated simultaneously with the six-degree-of-freedom (6DOF) equations of the ship motion. The coupled set of equations is solved by using the Large Amplitude Motion Program ‘LAMP’, which is a three-dimensional time-domain simulation of the motion of ships in waves. The unstabilized and stabilized roll motions of a S60-70 ship with forward speed and beam waves have been analyzed. For high-amplitude waves, the unstabilized roll angle exhibits typical nonlinear phenomena: a shift in the resonance frequency, multi-valued responses, and jumps. The performance of a S60-70 ship with a passive tank is investigated in various sea states with different encounter wave directions. It is found that passive anti-roll tanks tuned in the linear or nonlinear ranges are very effective in reducing the roll motion in the nonlinear range. The effect of the tank damping, frequency, and mass on the tank performance is studied. Also, it is found that passive anti-roll tanks are very effective in reducing the roll motion for ships having a pitch frequency that is nearly twice the roll frequency in sea states 5 and 6.

A method to evaluate the use of actively controlled moving weights on board ships to reduce roll motion is developed. The weights can simulate in principle anti-roll-tank systems, or they can be considered a possible anti-roll device in their own right. The ship, the moving weight, and the control device are considered components of a single dynamic system. The full eight-degree-of-freedom set of coupled governing equations for the complete dynamic system is derived. And a three-degree-of-freedom non-linear approximation for the roll motion only (MOTSIM) is derived from these eight equations. The reduced set of equations is used to determine the influence of various parameters and to evaluate control strategies. A PID controller is developed to command the position of the weight and a servomechanism model is used to predict its actual position. Then, the moving-weight system is incorporated into LAMP (Large–Amplitude–Motion Program), a computer code that integrates the governing equations of the sea and the motion of the ship interactively and simultaneously and predicts the motion of the ship in the time domain. A comparison of the results from the two simulations shows that there is fairly good correlation between the simple and complex models, but the simple model is a little optimistic in predicting the effectiveness of the moving-weight system. The results predict that the moving-weight system can be an excellent roll-suppressing device when the moving weight is as small as 1% of the displacement of the ship and the maximum distance the weight moves is as small as 15% of the half-beam.

We numerically studied the full (six degrees of freedom) motion of a cargo ship without roll stabilizers in rough (sea state 5) conditions for multiple heading angles ranging from 0° (follower seas) to 180° (head seas). We found that the ship exhibits excessive roll motion in quartering (45° off the stern), beam, and head seas. Therefore, roll damping is critical in these conditions. We then investigated the performance of passive and active anti-roll tank (ART) systems and compared their performance in each of the three sea conditions. Each ART consists of three identical tanks, distributed along the centerline of the ship, each of which consists of two vertical ducts connected at the bottom with a horizontal duct. A pump is located at the middle of the horizontal duct of each tank. The pumps are switched on for active ARTs but switched off for passive ones. The loads (forces and moments) exerted on the ship by the ARTs are added to the hydrodynamic loads (e.g., due to pressure and viscous effects) and the thrust in the governing equations of motion of the ship. Whereas both passive and active ARTs are able to reduce the excessive roll motion, active ARTs outperform the passive ones from three perspectives. First, they are more effective in reducing the roll motion. Second, they require much less working liquid. Third, their performance is insensitive to their natural frequencies and, hence, to their geometric design. In addition, we found that head seas are most responsive to ARTs, which suggests that they are effective in mitigating parametric roll.

A considerable fraction of commercial sensors are electrostatically actuated. Many sensor diaphragms are operated in different thermal environments that affect their performance. Because the interplay between the thermal and electrostatic loadings is of interest to designers, in this work we investigate such an interplay. We start with the coupled heat conduction equation and the Saint–Venant plate model. We use nondimensional analysis to show that the dissipation and the elastic coupling vary on a slow scale and hence they can be neglected. Consequently, the heat equation is uncoupled from the plate equation. We consider the case in which the temperature at the boundary is kept at a constant value above the ambient temperature. Substituting the resulting temperature distribution into the plate equation yields an equation with an equivalent compressive load and an electrostatic load due to a DC voltage. Then, a reduced-order model is used to investigate the influence of the dual loading on the plate deflection and their interplay.

Numerical simulations of the flow past a stationary circular cylinder at different Reynolds numbers (Re) have been performed using a computational fluid dynamics (CFD) solver that is based on the unsteady Reynolds-averaged Navier—Stokes equations (RANS). The results obtained are used to develop reduced-order models for the lift and drag coefficients. The models not only match the numerical simulation results in the time domain, but also in the spectral domain. They capture the steady-state region with excellent accuracy. Further, the models are verified by comparing their results in the transient region with their counterparts from the CFD simulations and very good agreement is found. The work performed here is a step towards building models for vortex-induced vibrations (VIV) encountered in risers, spars, and other offshore structures.

We analyze the stability of a boost power-factor-correction (PFC) circuit using a hybrid model. We consider two multi-loop controllers to control the power stage. For each closed-loop system, we treat two separate cases: one for which the switching frequency is approaching infinity and the other for which it is finite but large. Unlike all previous analyses, the analysis in this paper investigates the stability of the converter in the saturated and unsaturated regions of operation. Using concepts of discontinuous systems, we show that the global existence of a smooth hypersurface for the boost PFC circuit is not possible. Subsequently, we develop conditions for the local existence of each of the closed-loop systems using a Lyapunov function. In other words, we derive the conditions for which a trajectory will reach a smooth hypersurface. If the trajectories do not reach the sliding surface, then the system saturates. As such, the stability of the period-one orbit is lost. Using the conditions for existence and the concept of equivalent control, we show why, for the second closed-loop system, the onset of the fast-scale instability occurs when the inductor current approaches zero. For this system, we show that the onset of the fast-scale instability near zero-inductor current occurs for a lower line voltage. Besides, when the peak of the line voltage approaches the bus voltage, the fast-scale instability may occur not only at the peak but also when the inductor current approaches zero. We develop a condition which ensures that the saturated region does not have any stable orbits. As such, a solution that leaves the sliding surface (if existence fails) cannot stabilize in the saturated region. Finally, we extend the analysis to the case in which the converter operates with a finite but large switching frequency. As such, the system has two fundamental frequencies: the switching and line frequencies. Hence, the dynamics of the system evolve on a torus. We show two different approaches to obtaining a solution for the closed-loop system. For the second closed-loop system, using the controller gain for the current loop as a bifurcation parameter, we show (using a Poincaré map) the mechanism of the torus breakdown. If the mechanism of the torus breakdown is known, then, depending on the post-instability dynamics, a designer can optimize the design of the closed-loop converter.

We develop analytic methodologies for stability analysis of a parallel dc-dc converter using its switching model, discrete model (based on nonlinear map) and averaged model. We use these methodologies to investigate the dynamics of an interleaved and a synchronized parallel dc-dc buck converter. Using the switching and discrete models we show the mechanism and boundaries of fast-scale and slow-scale instabilities. We show the difference in mechanism of fast-scale instability between the interleaved and a synchronized converter. The averaged model does not show this difference. It also differs considerably (from the other two models) in prediction of instability boundaries.

We investigate the feasibility of a variable-geometry truss (VGT) based architecture for suppressing payload pendulations in ship-mounted cranes. The VGT assembly is conceived to be retrofitted onto the boom tip of ship-mounted cranes. A simplified planar model is developed. A control point along the cable hoisting the payload is constrained to move along a straight path with a given control input (acceleration) imparted via the actuators embedded in the VGT assembly. Control laws based on either linear quadratic or fuzzy control methodologies are developed in order to minimize an assigned cost functional. Their effectiveness is compared through extensive numerical simulations. The performance of the VGT architecture and associated control laws is analyzed when the crane is subject to the most severe combination of resonant excitations: a primary resonant roll excitation at the natural frequency of the controlled system, and a principal-parametric resonant heave excitation, both corresponding to sea state three and higher. The proposed strategy exhibits enough control authority over the system dynamics, greatly reducing the severe and undesirable resonant pendulations caused by the ship motions in a broad-band frequency range. Moreover, its disturbance-rejection capabilities are exerted with feasible control efforts, which are localized in the segment of the crane where they are needed.

Roll motion is an undesirable feature of the behavior of a ship in rough seas, and so it is natural to consider ways of reducing it. The most common devices for increasing roll damping are bilge keels. However, the effectiveness of keels is limited, and anti-roll tanks and fins are used when more control is required. Moreover, unlike keels, anti-roll tanks can be used when the ship is not underway. Our objective is to develop design procedures for passive tanks for roll reduction in rough seas. To this end, we develop an improved model of the passive tank-liquid motion in this paper. This tank consists of U-shaped tubes placed side by side along the length of the ship. The equations of six-degrees-of-motion (6DOF) that govern the tank-liquid are coupled with those that govern the 6DOF motion of the ship, and all of the equations are integrated simultaneously in the time domain using the Large Amplitude Motion Program (“LAMP”). LAMP is a three-dimensional time-domain simulation of the motion of ships in waves. The unstabilized and stabilized roll motions of a S60-70 ship with forward speed and beam waves have been analyzed. For high-amplitude waves, the variation of the roll angle with the encounter-wave frequency exhibits typical nonlinear phenomena: a shift in the resonance frequency, multi-valued responses, and jumps. The performance of passive tanks on a S60-70 ship with forward speed is investigated in an irregular sea with different encounter-wave directions. It is found that passive anti-roll tanks tuned in the nonlinear range are very effective in reducing the roll motion. The effect of the tank mass and distribution of tank tubes on the performance of the tank system is studied. Also, it is found that passive anti-roll tanks are very effective in reducing the roll motion in sea state five of a ship whose pitch frequency is nearly twice its roll frequency.

We review crane models available in the literature, classify them, and discuss their applications and limitations. A generalized formulation of the most widely used crane model is analyzed using the method of multiple scales. We also review crane control strategies in the literature, classify them, and discuss their applications and limitations. In conclusion, we recommend appropriate models and control criteria for various crane applications and suggest directions for further work.

Traditionally, a container crane is modeled as a simple pendulum with either a flexible or a rigid hoisting cable, and a lumped mass at the end of the cable. However, in the case of quay-side container cranes, the actual configuration of the hoisting mechanism is significantly different; it consists typically of a set of four hoisting cables. The cables are hoisted from four different points on a trolley and are attached on the load side to four points on a spreader bar used to lift containers. A controller design based on the actual model will most likely result in a response superior to those based on simple pendulum models. In this paper, we develop a mathematical model of the actual quay-side container crane. A simplified model is then used to obtain the gain and time delay for a delayed feedback controller, which will be used for the control of payload sway oscillation. Performance of the controller is simulated on a 1/10th scale computer model of a 65 ton container crane using the full model. Simulation results are verified experimentally on a 1/10th scale model of the same container crane.

An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse deformations of a composite laminated beam accounting for the midplane stretching are derived. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single nonlinear fourth-order partial–integral–differential equation governing the transverse deformations. We find out that the governing equation for the postbuckling of symmetric or asymmetric composite beams has the same form as that of beams made of an isotropic material. Composite beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are considered. A closed-form solution for the postbuckling deformation is obtained as a function of the applied axial load, which is beyond the critical buckling load. To study the vibrations that take place in the vicinity of a buckled equilibrium position, we exactly solved the linear vibration problem around the first buckled configuration. Solving the resulting eigen-value problem results in the natural frequencies and their associated mode shapes. Both the static response represented by the postbuckling analysis and the dynamic response represented by the free vibration analysis in the postbuckling domain strongly depend on the lay-up of the laminate. Variations of the beam’s midspan rise and the fundamental natural frequency of the postbuckling domain vibrations with the applied axial load are presented for a variety of lay-up laminates. The ratio of the axial stiffness to the bending stiffness was found to be a crucial parameter in the analysis. This control parameter, through the selection of the appropriate lay-up, can be manipulated to help design and optimize the static and dynamic behavior of composite beams.

The hydrodynamic cavitation over axisymmetric projectiles is computed using the unsteady incompressible Navier–Stokes equations for multi-fluid elements. The governing equations are discretized on a structured grid using an upwind difference scheme with flux limits. A preconditioning dual-time stepping method is used for the unsteady computations. The eigensystem is derived for the Jacobian matrices. This eigensystem is suitable for high-density ratio multi-fluid flows and it provides high numerical stability and fast convergence. This method can be used to compute single- as well as multi-phase flows. Cavitating flows over projectiles with different geometries are computed and the results are in good agreement with available experimental data and other published computations.

We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.

We investigate theoretically the nonlinear normal modes of a vertical cantilever beam excited by a principal parametric resonance. We apply directly the method of multiple scales to the governing nonlinear nonautonomous integral-partial-differential equation and associated boundary conditions. In the absence of damping, it is shown that the system has nonlinear normal modes, as defined by Rosenberg, even in the presence of the parametric excitation. We calculate the spatial correction to the linear mode shape due to the effects of the inertia and curvature nonlinearities and the parametric excitation. We compare the result obtained with the direct approach with that obtained using a single-mode Galerkin discretization. The deviation between the two predictions increases as the oscillation amplitude increases.

We investigate the nonlinear response of an infinitely long, circular cylindrical shell to a primary-resonance excitation of one of its flexural modes, which is involved in a one-to-two internal resonance with the breathing mode. The excited flexural mode is involved in a one-to-one internal resonance with its orthogonal flexural mode. There are two simultaneous internal (autoparametric) resonances: two-to-one and one-to-one. The method of multiple scales is directly applied to the partial-differential equations to obtain a system of six first-order nonlinear ordinary-differential equations governing modulation of the amplitudes and phases of the three interacting modes. In the absence of damping, the modulation equations are derivable from a Lagrangian, reflecting the conservative nature of the system. The modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence their bifurcations. The response may be either a two-mode or a three-mode solution. For certain excitation parameters, the equilibrium three-mode solutions undergo Hopf bifurcations. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability, and hence their bifurcations.

The transverse vibrations of a circular disk of uniform thickness rotating about its axis with constant angular velocity are analyzed. The results specialized to the linear case of disks clamped at the center and free at the periphery are in good agreement with those reported in the literature. The natural frequencies of spinning hard and floppy disks are obtained for various nodal diameters and nodal circles. Primary resonance is shown to occur at the critical rotational speed at which, in the linear analysis, the spinning disk is unable to support arbitrary spatially fixed transverse loads. Using the method of multiple scales, we determine a set of four nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of two interacting modes. The symmetry of the system and the loading conditions are reflected in the symmetry of the modulation equations. They are reduced to an equivalent set of two first-order equations whose equilibrium solutions are determined analytically. The stability characteristics of these solutions is studied; the qualitative behavior of the response is independent of the mode being considered.

We experimentally investigated nonlinear combination resonances in two graphite-epoxy cantilever plates having the configurations (90/30/-30/-30/30/90)_s and (-75/75/75/-75/75/-75)_s. As a first step, we compared the natural frequencies and modes shapes obtained from the finite-element and experimental-modal analyses. The largest difference in the obtained frequencies for both plates was 6%. Then, we transversely excited the plates and obtained force-response and frequency-response curves, which were used to characterize the plate dynamics. We acquired time-domain data for specific input conditions using an A/D card and used them to generate time traces, power spectra, pseudo-state portraits, and Poincaré maps. The data were obtained with an accelerometer monitoring the excitation and a laser vibrometer monitoring the plate response. We observed the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaIYaaabeaakiab% gUcaRiabeM8a3naaBaaaleaacaaI3aaabeaaaaa!45C9!\[\Omega \approx \omega _2 + \omega _7 \] in the quasi-isotropic plate and the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGa% aiikaiabeM8a3naaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3n% aaBaaaleaacaaI1aaabeaakiaacMcaaaa!4AAD!\[\Omega \approx (1/2)(\omega _2 + \omega _5 )\] and the internal combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaI4aaabeaakiab% gIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGaaiikaiabeM8a3n% aaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3naaBaaaleaacaaI% XaGaaG4maaqabaGccaGGPaaaaa!4FDC!\[\Omega \approx \omega _8 \approx (1/2)(\omega _2 + \omega _{13} )\] in the ±75 plate, where the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeM8a3naaBaaaleaacaWGPbaabeaaaaa!3F16!\[\omega _i \] are the natural frequencies of the plate and Ω is the excitation frequency. The results show that a low-amplitude high-frequency excitation can produce a high-amplitude low-frequency motion.

The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.

A novel approach for implementing an active nonlinear vibration absorber is presented. The absorber, which is built in electronic circuitry, takes advantage of the saturation phenomenon that occurs when two natural frequencies of a system with quadratic nonlinearities are in the ratio of two-to-one. When the system is excited at a frequency near the higher natural frequency, there is a small ceiling for the system response at the higher frequency and the rest of the input energy is channeled to the low-frequency mode.

A working model of using saturation to suppress the vibrations of a rigid beam connected to a DC motor has been built. An electronic oscillator is built, and its frequency is set at one-half the frequency of the beam. The output from a sensor on the beam is multiplied by the output from the electronic oscillator and a suitable gain, and the result is used as the forcing term for the oscillator. At the same time, the output from the oscillator is squared and multiplied by a suitable gain, and that result is used as the input to the motor. The oscillator/actuator and the beam act as the two modes of a two-degree-of-freedom quadratically coupled system with a 2:1 autoparametric resonance. When the beam is excited by a harmonic force, its motion quickly becomes saturated, and most of the energy imparted to the beam by the harmonic force is transferred to the electronic circuit and from there to the actuator. Thus, the harmonic force is made to work against itself. As a result, the motion of the beam always remains small.

Approximate methods for analyzing the vibrations of an Euler – Bernoulli beam resting on a nonlinear elastic foundation are discussed. The cases of primary resonance (Ω ≈ Ω_n) and subharmonic resonance of order one-half (Ω ≈ 2 Ω_n), where Ω is the excitation frequency and Ω_n is the natural frequency of the nth mode of the beam, are investigated. Approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equation and boundary conditions. The amplitude and phase modulation equations show that single-mode discretization leads to erroneous qualitative as well as quantitative predictions. Regions of softening (hardening) behavior of the system, the spatial dependence of the response drift, and frequency-response curves are numerically evaluated and compared using both approaches.

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.

The nonlinear, single-mode responses of a fixed-fixed, buckled beam are investigated under the case of a uniform, transverse, harmonic excitation. In order to avoid axial slipping and to obtain meaningful data, a clamping apparatus was designed to maximize the clamping force applied to the beam. To fully characterize the single-mode responses, data were obtained at various levels of buckling up to 3.3 times the thickness of the beam. The data demonstrate that at a low level of buckling, supercritical period doubling occurs during an amplitude sweep in which the first mode is directly excited. However, as the buckling level increases, the period-doubling bifurcation becomes subcritical during such amplitude sweeps. In addition, a period-five motion, broadband responses, and responses with an unexplained sideband structure were observed.

An approach for implementing an active nonlinear vibration absorber for flexible structures is presented. The technique exploits the saturation phenomenon exhibited by multidegree-of-freedom systems with quadratic nonlinearities possessing two-to-one autoparametric resonances. The strategy consists of introducing second-order controllers and coupling each of them with the plant through a sensor and an actuator, where both the feedback and control signals are quadratic. Once the structure is forced near its resonances, the oscillatory response is suppressed through the saturation phenomenon. We present theoretical and experimental results of the application of the proposed vibration absorber. The structure consists of a cantilever beam, the feedback signal is generated by a strain gage, and the actuation is achieved through piezoceramic patches. The equations of motion are developed and analyzed through perturbation techniques and numerical simulation. Then, the strategy is tested by assembling the controllers in electronic components and suppressing the vibrations of the first and second modes of two beams.

Methods for the study of weakly nonlinear continuous (distributed-parameter) systems are discussed. Approximate solution procedures based on reduced-order models via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equations and boundary conditions. By means of several examples and an experiment, Nayfeh and co-worker had shown that reduced-order models of nonlinear continuous systems obtained via the Galerkin procedure can lead to erroneous results. A method is developed for producing reduced-order models that overcomes the shortcomings of the Galerkin procedure. Treatment of these models yields results in agreement with those obtained experimentally and those obtained by directly attacking the continuous system.

An experimental validation of the suitability of reduction methods for studying nonlinear vibrations of distributed-parameter systems is attempted. Nonlinear planar vibrations of a clamped-clamped buckled beam about its first post-buckling configuration are analyzed. The case of primary resonance of the nth mode of the beam, when no internal resonances involving this mode are active, is investigated. Approximate solutions are obtained by applying the method of multiple scales to a single-mode model discretized via the Galerkin procedure and by directly attacking the governing integro-partial-differential equation and boundary conditions with the method of multiple scales. Frequency-response curves for the case of primary resonance of the first mode are generated using both approaches for several buckling levels and are contrasted with experimentally obtained frequency-response curves for two test beams. For high buckling levels above the first crossover point of the beam, the computed frequency-response curves are qualitatively as well as quantitatively different. The experimentally obtained frequency-response curves for the directly excited first mode are in agreement with those obtained with the direct approach and in disagreement with those obtained with the single-mode discretization approach.