Multivalued Response Research Articles

Dynamical properties of a forced vibration isolation system with real-power nonlinearities in restoring and damping forces

In this paper, the time-delayed cubic velocity feedback and the real-power form of restoring and damping forces are combined to improve the performance of vibration isolation. The primary resonance, dynamic stability and transmissibility are studied for a forced vibration isolation system with real-power exponents in restoring and damping force terms under base excitation. Based on the method of multiple scales, the equivalent damping is first proposed to interpret the effect of feedback control on the dynamic behavior of the real-power isolation system, such as the frequency island phenomenon. In this regard, the isolation system indicates the softening behavior for under-linear restoring force and hardening behavior for over-linear restoring force. And multi-valued responses, especially five-valued responses, are found under under-linear restoring force. To verify this result, the stability boundaries are characterized and presented excellent agreement with the responses. Then, for avoiding the jump phenomenon, an analytical criterion is derived and confirmed by the numerical simulation. Furthermore, with the purpose of suppressing the amplitude peak and governing the resonance stability, appropriate feedback gains and time delays are derived. Finally, the effects of the system parameters on the energy transmissibility is investigated. Results show that the strategy proposed in this paper is practicable and feedback parameters are significant factors to improve the isolation effectiveness for the real-power isolation system.

Nonlinear Dynamics of Shrouded Turbine Blade System with Impact and Friction

Dry friction dampers are passive devices used to reduce the resonant vibration amplitudes in turbine bladed systems. In shrouded turbine blade systems, in addition to the stick- slip motion induced by dry friction during the contact state in the tangential direction, the interface also undergoes intermittent separation in the normal direction. The problem can thus be treated as a combination of impact and friction. In this work, the dynamics of dry friction damped oscillators which are representative models of dry friction damped bladed system is investigated. A one dimensional contact model which is capable of modeling the interface under constant and variable normal load is used. The steady state periodic solutions are obtained by multi - harmonic balance method (MHBM). Frequency response plots are generated for different values of normal load using the arc length continuation procedure. The MHBM solutions are validated using numerical integration. A single degree of freedom (dof) model under constant normal load with constant and variable friction coefficients, a dry friction damped two dof system under constant normal load and a two dof system under variable normal load are investigated. In the presence of variable normal load, the system shows multivalued frequency response and jump phenomenon. The optimal value of the normal load which gives minimum resonant response is also obtained.

Stability of the Synchronous Response of a System of Multiple Translational Vibration Absorbers

In this paper, a system of perfectly tuned multiple nonlinear vibration absorbers attached in parallel to a main mass under harmonic excitation is considered. The absorbers are identical and of translational type with similar cubic nonlinearities in their springs. The averaging method is used to transform the non-autonomous system of equations that describe the absorbers’ dynamics to an averaged autonomous system. The averaged system is then studied to check the stability of absorbers’ synchronous response and existence of other non-synchronous responses. The present work shows that the absorbers’ synchronous response is stable and no other responses exist when the springs in the absorbers are of hardening type. When the springs are of softening type, at certain excitation force amplitude, multi-valued synchronous responses exist. Only one is the desired response where absorbers are functioning well in suppressing vibration of the main mass. The current research work also shows jump instabilities in the synchronous response of softening springs absorbers. These findings are checked against numerical simulations of the nonlinear equations that describe the dynamics of the system considered. The effect of high nonlinearities in absorbers’ springs on their synchronous response is investigated by numerically simulating the systems’ nonlinear equations. In this situation, numerical simulations did observe some non-synchronous responses of the absorbers. This suggests that when designers of vibration absorbers decide to use multiple identical highly nonlinear absorbers, they need to be aware of the absorbers’ non-synchronous responses that may exist and have negative effects on absorbers’ task of suppressing unwanted vibrations.

Multi-valued responses of a nonlinear vibro-impact system excited by random narrow-band noise

We investigate the multi-valued responses of a non-linear vibro-impact oscillator with a one-sided barrier subject to random narrow-band excitation. The frequency response of the system is obtained using the Krylov-Bogoliubov averaging method. Meanwhile, the backbone curve and the critical equation of unstable region are also derived for the deterministic case. Then, the method of moment is applied to obtain the iterative calculation equation for the mean-square response amplitude under the stochastic case. Excitation frequency, nonlinearity intensity, damping parameters, especially the distance between the system’s static equilibrium position and the barrier can lead to triple-valued response under certain case. In some conditions the impact system may have two or four steady-state solutions, which is an interesting phenomenon for impact system. The unstable region is one uniform part while under smaller nonlinearity intensity it is divided into two parts. Moreover, we also find that as random noise intensity increases, the pervasion of the phase trajectories strengthens, and then destroys the topological property of the phase trajectories.

Harmonic response of a class of finite extensibility nonlinear oscillators

Finite extensibility oscillators are widely used to simulate those systems that cannot be extended to infinity. For example, they are used when modelling the bonds between molecules in a polymer or DNA molecule or when simulating filaments of non-Newtonian liquids. In this paper, the dynamic behavior of a harmonically driven finite extensibility oscillator is presented and studied. To this end, the harmonic balance method is applied to determine the amplitude–frequency and amplitude–phase equations. The distinguishable feature in this case is the bending of the amplitude–frequency curve to the frequency axis, making it asymptotically approach the limit of maximum elongation of the oscillator, which physically represents the impossibility of the system reaching this limit. Also, the stability condition that defines stable and unstable steady-state solutions is derived. The study of the effect of the system parameters on the response reveals that a decreasing value of the damping coefficient or an increasing value of the excitation amplitude leads to the appearance of a multi-valued response and to the existence of a jump phenomenon. In this sense, the critical amplitude of the excitation, which means here a certain value of external excitation that results in the occurrence of jump phenomena, is also derived. Numerical experiments to observe the effects of system parameters on the frequency–amplitude response are performed and compared with analytical calculations. At a low value of the damping coefficient or at a high value of excitation amplitude, the agreement is poor for low frequencies but good for high frequencies. It is demonstrated that the disagreement is caused by the neglect of higher-order harmonics in the analytical formulation. These higher-order harmonics, which appear as distinguishable peaks at certain values in the frequency response curves, are possible to calculate considering not the linearized frequency of the oscillator but its actual frequency, which is strongly amplitude dependent. On the other hand, for a high value of the damping coefficient or a low value of excitation amplitude, the agreement between numerical and analytical calculations is excellent. For these cases, the system is prevented from exploring large amplitudes of vibration and therefore the nonlinearity is not much evidenced.

Modeling and analysis of electrostatic MEMS filters

We present an analytical model and closed-form expressions describing the response of a tunable MEMS filter made of two electrostatic resonators coupled by a weak microbeam. The model accounts for the filter geometric and electric nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system to produce a reduced-order model. We predict the filter deflection and static pull-in voltage by solving a boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the filter poles (the natural frequencies delineating the filter bandwidth). We found a good agreement between the results obtained using our model and published experimental results. We found that, when the input and output resonators are mismatched, the first mode is localized in the softer resonator whereas the second mode is localized in the stiffer resonator. We demonstrated that mismatch between the resonators can be countermanded by applying different DC voltages to the resonators. As the effective nonlinearities of the filter grow, multi-valued responses appear and distort the filter performance. Once again, we found that the filter can be tuned to operate linearly by choosing a DC voltage that makes the effective nonlinearities vanish.

Experimental results and numerical simulations on strongly non-linear rolling of multihulls in moderate beam seas

A trimaran and a pentamaran ship have been investigated to study the effect of transversal hull separation on roll motion in waves. Four different configurations for the trimaran have been tested in beam waves and two for the pentamaran, all using the same main hull. Typical outriggers with small beam and draught have been used. The experiments in regular waves revealed the extreme sensitivity of these hull typologies to roll motion in beam waves, even in mild sea conditions (wave steepness=1/60). Two configurations of the trimaran and one of the pentamaran exhibited a multivalued roll response curve in the low frequency range. This is, to the best of the authors knowledge, the first experimental evidence of bifurcations in roll motion of multihull ships, which are usually designed and selected for their superior stability characteristics and low rolling. Three mathematical models have been used with the aim of reproducing the experimental roll response curve. Two models are based on a partially semi-empirical extension of the standard linear seakeeping equations in the non-linear range, while one model is based on a non-linear one-degree-of-freedom modelling of roll motion. Two models are based on a relative angle approach for the non-linear restoring, while one model is based on an absolute angle approach for the computation of the non-linear restoring.

A Review of the Systems Approach to the Analysis of Dynamic-Mode Atomic Force Microscopy

The atomic force microscope (AFM) is one of the foremost tools for imaging, measuring and manipulating matter at the nanoscale. This brief presents a review of the systems and control approach to analyzing the challenging dynamic-mode operation of the AFM. A Lure system perspective of the AFM dynamics facilitates the application of powerful tools from systems theory for the analysis. The harmonic balance method provides significant insights into the steady-state behavior as well as a framework for identifying the tip-sample interaction force. A simple piecewise-linear tip-sample interaction model and its identification using the harmonic balance method is presented. The dominant first harmonic is analyzed using multivalued frequency responses and the corresponding stability conditions. The ability of the simple tip-sample interaction model to capture the intricate nonlinear behavior of the first harmonic is demonstrated. This also points to the importance of studying the higher harmonics to obtain finer details of the tip-sample interaction. The suitability of the Lure system perspective for the analysis of the higher harmonics is demonstrated.

A NOTE ON THE EQUIVALENCE OF RATIONALIZABILITY CONCEPTS IN GENERALIZED NICE GAMES

Moulin (1984) describes the class of nice games for which the solution concept of point-rationalizability coincides with iterated elimination of strictly dominated strategies. As a consequence nice games have the desirable property that all rationalizability concepts determine the same strategic solution. However, nice games are characterized by rather strong assumptions. For example, only single-valued best responses are admitted and the individual strategy sets have to be convex and compact subsets of the real line ℝ. This note shows that equivalence of all rationalizability concepts can be extended to multi-valued best response correspondences. The surprising finding is that equivalence does not hold for individual strategy sets that are compact and convex subsets of ℝn with n ≥ 2.

Uniqueness conditions for strongly point-rationalizable solutions to games with metrizable strategy sets

This paper introduces the iterative solution concept of strong point-rationalizability as a strengthening of standard point-rationalizability by stipulating that any two players have identical beliefs about the strategy choices of their common opponents. By adopting and generalizing the contraction-property approach of Moulin [Moulin, H., 1984. Dominance solvability and cournot stability. Mathematical Social Sciences 7, 83–102] and Bernheim [Bernheim, B.D., 1984. Rationalizable strategic behavior. Econometrica 52, 1007–1028] we derive sufficient conditions for the existence of unique strongly point-rationalizable strategies in strategic games with best response functions. These uniqueness results are derived under fairly general assumptions and are especially useful for complete and bounded, as well as finite strategy sets. For games with monotonic individual best response functions, equivalence between a unique strongly point-rationalizable and a unique point-rationalizable solution in the standard sense is established. Furthermore, the existence of unique fixed points is proven under conditions that generalize for bounded metric spaces an established fixed point theorem by Bonsall [Bonsall, F.F., 1962. Lectures on Some Fixed Point Theorems of Functional Analysis. Tata Institute of Fundamental Research, Bombay] and Smart [Smart, D.R., 1974. Fixed Point Theorems. Cambridge University Press, Cambridge, London]. Possible extensions of our findings to games with multi-valued best response correspondences are discussed.

Super- and sub-harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balance method

This article presents a new multi-term harmonic balance method (HBM) for nonlinear frequency response calculations of a torsional sub-system containing a clearance type nonlinearity. The ability of the simplified subsystem to capture the salient behavior of the larger system is verified by the comparison of results to experimental data. Unlike previous analytical and numerical methods, the proposed HBM includes adaptive arc-length continuation and stability calculation capabilities to find periodic solutions in multi-valued nonlinear frequency response regimes as well as to improve convergence. Essential steps of the proposed HBM calculations are introduced, and it is validated by comparing time and frequency domain predictions with those yielded by numerical solutions, experimental studies, or analog simulations for several examples. Then, nonlinear frequency response characteristics of an oscillator with clearance nonlinearity are examined with focus on super- and sub-harmonics. We also explore some issues that are not fully resolved in the literature. For instance, the effect of mean operating point is examined for α = 0 and α = 0.18 - 0.25 cases where α is the stiffness ratio of the piecewise-linear elastic function. In addition, the number of harmonic terms that must be included in the HBM response calculations, given sinusoidal excitation, has been investigated. Finally, some simple analytical predictions for super and sub-harmonic resonances are presented.

The method of programmed iterations in abstract control problems

A direct version (“direct” in the sense of constructing control procedures - quasistrategies) of the method of programmed iterationsis considered for the abstract problem of control of bundles of trajectories, information about which is realized by means of a certain signal. Conditions for the guaranteed solvability of the problem of encounter with a functional set, as defined by an iterative procedure in the space of multivalued responses to signals reaching the system, are investigated.

An ab initio approach to the calculation of current-voltage characteristics of programmable molecular devices

Molecular electronics can be developed if we are able to program a random arrangement of molecules or a field-programmable molecular random array. The ansatz that small molecules can be programmed needs to be demonstrated; this means characterizing the smallest molecular system with programmable features. We demonstrate that even two molecules in a series conformation can have multivalued responses and, thus, is able to be programmed; we also indicate how to extend this programmability to other molecular circuit conformations. Current programs for the calculation of current-voltage characteristics of electronic circuits, needed for such demonstrations, are only capable of predicting single-valued characteristics. We present results from our ab initio procedures that couple the molecular approaches with a practical analysis of molecular circuits having strong nonlinearities.

Thermally Induced Principal Parametric Resonance in Circular Plates

We consider the problem of large‐amplitude vibrations of a simply supported circular flat plate subjected to harmonically varying temperature fields arising from an external heat flux (aeroheating for example). The plate is modeled using the von Karman equations. We used the method of multiple scales to determine an approximate solution for the case in which the frequency of the thermal variations is approximately twice the fundamental natural frequency of the plate; that is, the case of principal parametric resonance. The results show that such thermal loads produce large‐amplitude vibrations, with associated multi‐valued responses and subcritical instabilities.

Design of passive anti-roll tanks for roll stabilization in the nonlinear range

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.

Exact Solutions for Free Vibration in an Asymmetrical Duffing Equation

An algorithm is proposed for determining the exact solutions for free vibration in an asymmetrical Duffing oscillator. The system is composed of a spring function of the Duffing type (with linear and cubic terms with respect to displacement) accompanied by a constant force. It is reduced from the most comprehensive 3rd-order polynomial having arbitrary terms from a constant to the 3rd power. By employing a bilinear transformation, the system is successfully converted into a regular Duffing equation whose exact solutions already exist and are abstracted here. The whole procedure of obtaining the present solution is summarized in a flow chart. Numerical values of conversion constants, skeleton curves and waveforms are computed and illustrated for the typical cases of asymmetrical (hard, soft and snap-through) spring systems. It is demonstrated that the skeleton curves and waveforms are asymmetrical and that some of them exhibit multivalued responses for a given frequency.

A Study of the Influence of Bearing Clearance on Lateral Coupled Shaft/Disk Rotordynamics

This study examines the influence of bearing clearance on the dynamic behavior of a rotating, flexible disk/shaft system. Most previous work in nonlinear rotordynamics has tended to concentrate separately on shaft vibration or on bladed disk vibration, neglecting the coupling dynamics between them. The current work examines the important rotordynamic behavior of coupled disk/shaft dynamics. A simplified nonlinear model is developed for lateral vibration of a rotor system with a bearing clearance nonlinearity. The steady-state dynamic behavior of this system is explored using numerical simulation and limit cycle analysis. It is demonstrated that bearing clearance effects can produce superharmonic vibration that may serve to excite high-amplitude disk vibration. Such vibration could lead to significantly increased bearing loads and catastrophic failure of blades and disks. In addition, multivalued responses and aperiodic behavior were observed.

On the narrow-band random response distribution function of a non-linear oscillator

The response of a non-linear oscillator excited by random narrow-band noise is discussed. Expressions are found for the probability density function of the response. The approximate expressions involve stochastic averaging followed by partial linearization of the resulting Itô equations. The probability density functions provide a reasonable description of aspects such as multivalued response in the Duffing oscillator and limit cycle response in the van der Pol oscillator.

Thermal runaway in microwave heated ceramics: A one-dimensional model

The heating of a ceramic slab under TM-illumination is modeled and analyzed in the small Biot number regime. The temperature distribution is almost spatially uniform in this limit and its evolution in time is governed by a first-order nonlinear amplitude equation. This equation admits a time independent solution which is a multivalued function of the microwave power. The graph of this steady temperature as a function of the microwave power gives an S-shaped response curve when the electrical conductivity is modeled either as an exponential function of temperature or an Arranius law. The dynamics of the heating process are deduced from the amplitude equation and the multivalued response, and are dependent upon the microwave power and initial conditions. For certain initial conditions and power levels the system evolves to the upper branch of the response curve which corresponds to thermal runaway. Other initial conditions and power levels force the system to evolve to the lower branch and a safe sintering temperature. This heating process can be controlled in some sense by varying the microwave power in time at a rate commensurate with the thermal changes. Specifically, the power is allowed to change in an exponential fashion from a higher to a lower power level. This relationship is turned into a differential equation, which is appended to the amplitude equation to form an autonomous system of the second order. This system is analyzed using a phase-plane method. The analysis shows the existence of a stable manifold which divides the phase plane into two parts: Trajectories in the region above this curve correspond to runaway heating while those below yield stable sintering. Various heating scenarios are presented and discussed.

Thermal Runaway and its Control in Microwave Heated Ceramics

ABSTRACTThe heating of a ceramic slab under TEM illumination is modeled and analyzed in the small Biot number regime. The temperature distribution is almost spatially uniform in this limit and its evolution in time is governed by a first order nonlinear amplitude equation. This equation admits a time independent solution which is a multivalued function of the microwave power. The dynamics of the heating process are deduced, from the amplitude equation and the multivalued response, and are dependent upon the microwave power and initial conditions. The results of this analysis give a plausible explanation of certain difficulties arising in sintering experiments, such as thermal runaway. A simple control process is presented and analyzed which mitigates against these deleterious effects. Abbreviated parameter studies are performed showing trends in the controlled heating process.A quasi-three-dimensional problem modeling the heating process of a thin cylindrical sample in a waveguide applicator is also presented. For certain excitations the physical phenomenon deduced from this model and the required control process are the same as those obtained for the slab. For other excitations there is a strong spatial structure along the axis of the sample. For a certain choice of parameters the middle portion of the sample is at an elevated temperature while the remaining portion is at a much lower temperature. This phenomenon may be useful in joining applications.