Subharmonic Resonance Conditions Research Articles

Parametrically Excited Nonlinear Pneumatic Artificial Muscle Under Hard Excitation: A Theoretical and Experimental Investigation

In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle subjected to time-varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales. The time and frequency response, phase portraits, and basin of attraction are plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters are evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters is illustrated.

Nonlinear dynamics of the rotating beam with time-varying speed under aerodynamic loads

In this research, the bifurcation analysis is inspected for a rotatory pre-twisted beam which is subjected to the time varying angular velocity and aerodynamic forces. Aerodynamic loads are obtained by the Piston and Whitehead theories. It is assumed that a time-dependent periodic component is superposed on the angular velocity. The extended Hamilton's principle is utilized to derive the dynamic equations of motion in flapwise and chordwise directions. Then a combination of the Galerkin and multiple scales methods is established to obtain the modulation equations describing amplitude and phases of the interacting modes. Eigen value analysis is performed on the linear part of the equations and the possibility of 1:1 internal resonance condition is examined. The evolution of amplitudes is obtained versus the amplitude and frequency of the periodic angular velocity function. Primary and subharmonic resonance conditions are tuned between the angular velocity frequency and natural frequencies of the beam. Numerical simulations indicate that the nonlinear phenomena such as jump, saturation, and double jumping appear in nonlinear vibration of rotating beam with varying speed.

Bifurcations in a Pre-Stressed, Harmonically Excited, Vibro-Impact Oscillator at Subharmonic Resonances

This study investigates the behavior of a damped, inelastic, sinusoidally forced impact oscillator which has its barrier placed such that the oscillator always vibrates under compression about its subharmonic resonant frequencies. The Poincaré sections at near subharmonic resonance conditions exhibit finger-shaped chaotic attractors, similar to the strange attractor mapping of Hénon and the ones found by Holmes in his study of chaotic resonances of a buckled beam. The number of such fingers are observed to increase as the barrier distance from the equilibrium is decreased. These chaotic states are interspersed with regimes of periodic behavior, with the periodicity being in accordance with well defined period adding laws. This study also focuses on the ordered behavior of the one-impact period-[Formula: see text] orbits around the higher subharmonics of the oscillator.

Numerical Investigation of Nonlinear Dynamics of a Pneumatic Artificial Muscle With Hard Excitation

Abstract In this work, a numerical analysis has been carried out to study the nonlinear dynamics of a system with pneumatic artificial muscle (PAM). The system is modeled as a single degree-of-freedom system and the governing nonlinear equation of motion has been derived to study the various responses of the system. The system is subjected to hard excitation and hence the subharmonic and superharmonic resonance conditions have been studied. The second-order method of multiple scales (MMS) has been used to find the response, stability, and bifurcations of the system. The effect of various system parameters on the system response has been studied using time response, phase portraits, and basin of attraction. In these responses, while the saddle node bifurcation is found in both super and subharmonic resonance conditions, the Hopf bifurcation is found only in superharmonic resonance condition. By changing different system parameters, it has been shown that the response with three periods leads to chaotic response for superharmonic resonance condition. This study will find applications in the design of PAM actuators.

Modal analysis and dynamic responses of a rotating Cartesian manipulator with generic payload and asymmetric load

Here, investigation to explore the effect of generic payload and externally applied asymmetric load on the calculation of modal parameters and dynamic performance of a rotating flexible manipulator under prismatic motion has been established. We thus have developed a dynamic model of a rotating Cartesian manipulator with a payload whose center of gravity doesn’t coincide with the point of attachment, to determine the modal parameters i.e., natural frequency and corresponding mode-shape. These modal parameters are then illustrated graphically upon varying parameters like offset parameters (i.e., offset mass, offset inertia, offset length), mass and stiffness of rotary actuator, and amplitude and frequency of asymmetric load. An investigation into the nonlinear dynamics of the system accounting of geometric nonlinearity has been executed while obtained results have been validated numerically within the permissible error at the assorted critical points in frequency characteristic curves. Current research further investigates the influences of offset parameters, mass and stiffness of the actuator, frequency and amplitude of axial force on the steady state responses for the primary and sub-harmonic resonance conditions to reveal the built-in saddle-node and pitchfork bifurcation due to which the system losses its structural stability. This work enables an insight into the modal characteristics and nonlinear behavior of a rotating-Cartesian manipulator with a generic payload under asymmetric axial force and prismatic motion.

On the secondary resonance of a MEMS resonator: A conceptual study based on shooting and perturbation methods

Nonlinear dynamics of a clamped–clamped capacitive micro-beam resonator subjected to subharmonic excitation of order one-half is studied. The micro-beam resonator is sandwiched with two piezoelectric layers throughout the length, and as a result of piezoelectric actuation a tensile/compressive axial load is induced along the length which is used as a frequency tuning tool. The resonator is subjected to a combination of a bias DC and harmonic AC electrostatic actuations. In order to determine the frequency response subharmonic resonance condition, both perturbation and shooting methods are applied. The stability of the periodic solutions and the bifurcations types are also studied. It is shown that the application of perturbation method imposes some limitations on the order of magnitudes of the terms in the differential equation of the motion; as a result out of the domain where the ordering assumption of the perturbation solution does not hold, some periodic solutions as well as some vital bifurcation points are missed. It is shown that on the frequency domain, the resonator exhibits both softening and hardening behaviors whereas this is not predicted by the perturbation scheme. The effect of DC and AC actuation voltages on the qualitative response of the system is determined. It is shown that based on the polarity of the piezoelectric actuation, the frequency response curves can be shifted both in forward and backward directions which can be used in the design of novel RF MEMS filters/sensors.

Nonlinear Dynamics of Axially Loaded Piezoelectric Energy Harvester

In the present work nonlinear dynamics of axially loaded energy harvester which is modeled as a fixed-guided beam with tip mass having piezoelectric patch has been investigated. The system is subjected to a base excitation and periodic axial load. The governing equation of motion of the system is developed using Lagrange principle. A closed form solution has been developed to find out the response and generated voltage using method of multiple scales (MMS). Frequency response of the system is obtained for superharmonic and subharmonic resonance conditions and parametric study has been carried out in order to investigate the variation of output voltage with different system parameters.

Bifurcation analysis of nonlinear milling process with tool wear and process damping: Sub-harmonic resonance under regenerative chatter

Abstract In this paper, bifurcation analysis is performed for the nonlinear milling process under sub-harmonic resonance and regenerative chatter, with tool wear and process damping effects. Multiple-scales approach is used to construct analytical approximate solutions for non-autonomous parametrically excited equations of the system with time delay terms. The new bifurcation parameters are the detuning parameter (deviation of the tooth passing frequency from three times of the chatter frequency), damping ratio (affected by process damping) and tool wear width. Jump phenomenon and multi-values responses are observed in the first order solution under sub-harmonic resonance condition. Periodic, quasi-periodic and chaotic behaviour of the limit cycles are predicted in the presence of regenerative chatter. Change of the detuning parameter leads to the cyclic-fold (tangent) and secondary Hopf (Neimark) bifurcations. It is observed that as damping (affected by process damping) varies, cyclic-fold, secondary Hopf and supercritical symmetry-breaking (pitchfork) bifurcations occur. For the variation of tool wear width, period-doubling (flip), symmetry-breaking and secondary Hopf bifurcations occur. Moreover, for slight values of damping or large values of tool wear width, chaotic behaviour is dominant.

Vibration control of a transversely excited cantilever beam with tip mass

The problem of controlling the vibration of a transversely excited cantilever beam with tip mass is analyzed within the framework of the Euler–Bernoulli beam theory. A sinusoidally varying transverse excitation is applied at the left end of the cantilever beam, while a payload is attached to the free end of the beam. An active control of the transverse vibration based on cubic velocity is studied. Here, cubic velocity feedback law is proposed as a devise to suppress the vibration of the system subjected to primary and subharmonic resonance conditions. Method of multiple scales as one of the perturbation technique is used to reduce the second-order temporal equation into a set of two first-order differential equations that govern the time variation of the amplitude and phase of the response. Then the stability and bifurcation of the system is investigated. Frequency–response curves are obtained numerically for primary and subharmonic resonance conditions for different values of controller gain. The numerical results portrayed that a significant amount of vibration reduction can be obtained actively by using a suitable value of controller gain. The response obtained using method of multiple scales is compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. Numerical simulation for amplitude is also obtained by integrating the equation of motion in the frequency range between 1 and 3. The developed results can be extensively used to suppress the vibration of a transversely excited cantilever beam with tip mass or similar systems actively.

Non-linear vibration of a single link viscoelastic Cartesian manipulator

In this present work, the non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator is studied. The temporal equation of motion with complex coefficients of the system is obtained by using D’Alembert's principle and generalized Galarkin method. The temporal equation of motion contains non-linear geometric and inertia terms with forced and non-linear parametric excitations. It may also be found that linear and non-linear damping terms originated from the geometry of the large deformation of the system exist in this equation of motion. Method of multiple scales is used to determine the approximate solution of the complex temporal equation of motion and to study the stability and bifurcation of the system. The response obtained using method of multiple scales are compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. The response curves obtained using viscoelastic beams are compared with those obtained from a linear Kelvin–Voigt model and also with an equivalent elastic beam. The effect of the material loss factor, amplitude of base excitation, and mass ratio on the steady state responses for both simple and subharmonic resonance conditions are investigated.

Non-linear dynamics of a flexible single link Cartesian manipulator

The present work deals with the non-linear vibration of a harmonically excited single link roller-supported flexible Cartesian manipulator with a payload. The governing equation of motion of this system is developed using extended Hamilton's principle, which is reduced to the second-order temporal differential equation of motion, by using generalized Galerkin's method. This equation of motion contains both cubic non-linearities of geometric and inertial type in addition to linear forced and non-linear parametric excitation terms. Method of multiple scales is used to solve this non-linear equation and study the stability and bifurcations of the system. Influence of amplitude of the base excitation and mass ratio on the steady state response of the system is investigated for both simple and subharmonic resonance conditions. Critical bifurcation points are determined from the fixed-point responses and periodic, quasi-periodic responses are also found for different system parameters. The results obtained using the perturbation analysis are compared with the previously published experimental work and are found to be in good agreement. This work will be useful for the designer of a flexible manipulator.

Numerical simulations of chaotic dynamics in a model of an elastic cable

The finite motions of a suspended elastic cable subjected to a planar harmonic excitation can be studied accurately enough through a single ordinary-differential equation with quadratic and cubic nonlinearities. The possible onset of chaotic motion for the cable in the region between the one-half subharmonic resonance condition and the primary one is analysed via numerical simulations. Chaotic charts in the parameter space of the excitation are obtained and the transition from periodic to chaotic regimes is analysed in detail by using phase-plane portraits, Poincare maps, frequency-power spectra, Lyapunov exponents and fractal dimensions as chaotic measures. Period-doubling, sudden changes and intermittency bifurcations are observed.

Planar non-linear oscillations of elastic cables under subharmonic resonance conditions

Planar non-linear oscillations of elastic cables under 1 2 and 1 3 subharmonic resonance conditions are studied. In both cases a perturbation analysis is accomplished, with various orderings considered for the damping and excitation parameters. The regions of existence of steady state non-trivial subharmonics, and the frequency-response curves, are determined, as well as stability and domains of attraction of the solutions. A parametric investigation of the dynamic phenomena for technical cables is performed. Results of numerical integration are also presented, and used to discuss the accuracy of the asymptotic solutions and to evaluate the importance of the free secondary component in the system response.

The influence of an internal resonance on non-linear structural vibrations under subharmonic resonance conditions

The response of structural elements to a simple harmonic, transverse excitation is considered. The effects of both initial curvature and midsurface stretching are included; thus, the governing equations contain both quadratic and cubic terms. A perturbation technique, the method of multiple scales, is used to determine the response. Attention is focused on the subharmonic resonance Ω≅2ω2, where Ω is the forcing frequency and ω2 is a natural frequency. If the system possesses an internal resonance of the form ω2≅2ω1, energy may be transferred from the second mode to the first mode. The structural response is investigated in the presence, and in the absence, of this internal resonance. A comparison of the results reveals that the amplitude of the response can be significantly reduced by the presence of such an internal resonance. This suggests a means of passive vibration control. Also, the internal resonance causes a saturation phenomenon and a role reversal between the directly and indirectly excited modes.