A Unique Formulation of Piecewise Exact Method to Analyze a Nonlinear Spring System under Harmonic Excitation

2014 ◽  
Vol 31 (3) ◽  
pp. 337-344
Author(s):  
P.-S. Xie ◽  
P.-J. Shih
Keyword(s):  
Exact Solution ◽  
Numerical Approximation ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Harmonic Excitation ◽  
Exact Method ◽  
Nonlinear Stiffness ◽  
Time Error ◽  
Single Degree Of Freedom ◽  
Spring System

AbstractThis paper introduces a unique, efficient, and exact formulation for solving a single-degree-of-freedom system with nonlinear stiffness under a harmonic loading. This formulation is one kind of the piecewise exact method, and its benefit lies in providing the closed-form exact solution in each displacement segment. Since the exact solution is given in each segment, the continuity between two segments can be confirmed. Consequently, no instability errors affect the analysis. To determine the exact solutions in these segments, this research develops a technique that shifts the equilibrium points of the piecewise linear segments, which are discretized from a nonlinear stiffness curve, to new equilibrium points in order to satisfy the typical linear exact solution. Thus, positive- and negative-stiffness linear segments can be solved with this technique. This formulation saves roughly 60% of the calculation time (error < 10−10) as compared to the numerical approximation.

The Dynamic Response of a System With Preloaded Compliance

1988 ◽  
Vol 110 (3) ◽  
pp. 278-283 ◽  
Author(s):  
S. W. Shaw ◽  
P. C. Tung
Keyword(s):  
Dynamic Response ◽  
Piecewise Linear ◽  
Harmonic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Linear Features ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stable Periodic ◽  
Model Equations

We consider the dynamic response of a single degree of freedom system with preloaded, or “setup,” springs. This is a simple model for systems where preload is used to suppress vibrations. The springs are taken to be linear and harmonic excitation is applied; damping is assumed to be of linear viscous type. Using the piecewise linear features of the model equations we determine the amplitude and stability of the periodic responses and carry out a bifurcation analysis for these motions. Some parameter regions which contain no simple stable periodic motions are shown to possess chaotic motions.

The Dynamics of a Harmonically Excited System Having Rigid Amplitude Constraints, Part 1: Subharmonic Motions and Local Bifurcations

1985 ◽  
Vol 52 (2) ◽  
pp. 453-458 ◽  
Author(s):  
S. W. Shaw
Keyword(s):  
Simple Model ◽  
Piecewise Linear ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Coefficient Of Restitution ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Local Bifurcations ◽  
Excited System ◽  
Double Impact

A simple model for the response of mechanical systems having two-sided amplitude constraints is considered. The model consists of a piecewise-linear single degree-of-freedom oscillator subjected to harmonic excitation. Encounters with the constraints are modeled using a simple impact rule employing a coefficient of restitution, and excursions between the constraints are assumed to be governed by a linear equation of motion. Symmetric double-impact motions, both harmonic and subharmonic, are studied by means of a mapping that relates conditions at subsequent impacts. Stability and bifurcation analyses are carried out for these motions and regions are found in which no stable symmetric motions exist. The possible motions that can occur in such regions are discussed in the following paper, Part 2.

Dynamic Stability of an Impact System Connected With Rock Drilling

1969 ◽  
Vol 36 (4) ◽  
pp. 743-749 ◽  
Author(s):  
C. C. Fu
Keyword(s):  
Computer Simulation ◽  
Exact Solution ◽  
Steady State ◽  
Asymptotic Stability ◽  
Piecewise Linear ◽  
Steady State Solution ◽  
External Excitation ◽  
Rock Drilling ◽  
Impact System ◽  
Number Of Cycles

This paper deals with asymptotic stability of an analytically derived, synchronous as well as nonsynchronous, steady-state solution of an impact system which exhibits piecewise linear characteristics connected with rock drilling. The exact solution, which assumes one impact for a given number of cycles of the external excitation, is derived, its asymptotic stability is examined, and ranges of parameters are determined for which asymptotic stability is assured. The theoretically predicted stability or instability is verified by a digital computer simulation.

The Dynamics of a Nonharmonically Excited System Having Rigid Amplitude Constraints

1992 ◽  
Vol 59 (3) ◽  
pp. 693-695 ◽  
Author(s):  
Pi-Cheng Tung
Keyword(s):  
Dynamic Response ◽  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Coefficient Of Restitution ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Excited System

We consider the dynamic response of a single-degree-of-freedom system having two-sided amplitude constraints. The model consists of a piecewise-linear oscillator subjected to nonharmonic excitation. A simple impact rule employing a coefficient of restitution is used to characterize the almost instantaneous behavior of impact at the constraints. In this paper periodic and chaotic motions are found. The amplitude and stability of the periodic responses are determined and bifurcation analysis for these motions is carried out. Chaotic motions are found to exist over ranges of forcing periods.

Vibration Isolation From Harmonic Disturbances Through Use of Internally Rotating Masses

2015 ◽  
Author(s):  
Eric Smith ◽  
Al Ferri
Keyword(s):  
Kinetic Energy ◽  
Vibration Isolation ◽  
Harmonic Excitation ◽  
Impulsive Loading ◽  
Qualitative Behavior ◽  
Single Degree Of Freedom ◽  
Transmitted Force ◽  
A Chain ◽  
Isolation Systems ◽  
Parameter Values

This paper considers the use of a chain of translating carts or housings having internally rotating eccentric masses in order to accomplish vibration isolation. First a single degree-of-freedom system is harmonically excited to uncover the qualitative behavior of each rotating mass. The simple model is then expanded into a chain of housings, containing rotating eccentric masses, which are interconnected with springs. The internal rotating eccentric masses are damped along their circular pathway by means of linear viscous damping. Due to the lack of elastic or gravitational constraint on the rotating eccentric masses, they provide a nonlinear inertial coupling to their housings. Previous research has shown that such systems are capable of reducing shock or impulsive loading by converting some of the translational kinetic energy into rotational kinetic energy of the internal masses. This paper examines the potential for vibration isolation of a chain of such systems subjected to persistent, harmonic excitation. It is seen that the dynamics of these systems is very complicated, but that trends are observed which have implications for practical isolation systems. Using simulation studies, tradeoffs are examined between displacement and transmitted force for a range of physical parameter values.

scholarly journals Basic Constraints for Design Optimization of Cubic and Bistable NES

2021 ◽  
pp. 1-51
Author(s):  
Zhenhang Wu ◽  
Sebastien Seguy ◽  
Manuel Paredes
Keyword(s):  
Multiple Scales ◽  
Energy Levels ◽  
Conservative System ◽  
Harmonic Excitation ◽  
Absorption Efficiency ◽  
Complex Variables ◽  
Maximum Efficiency ◽  
Nonlinear Stiffness ◽  
Optimal Efficiency ◽  
Different Response

Abstract This work mainly concentrates on the optimization of cubic and bistable NES to find the maximum efficiency point under harmonic excitation. The conservative system is considered to reveal the inner property of the damping system. With the application of the multiple scales method and the complex variables method, the threshold of excitation and different response regimes are distinguished under the assumption of 1:1 resonance. The maximum efficiency point of cubic and bistable NES occurs when SMR disappears. The factors that affect the optimal efficiency limit are explored. The result indicates that the maximum absorption efficiency level is mainly determined by the damping parameters. Compared with the cubic case, the bistable case involves more complex regimes in terms of chaos oscillation. The influence of damping parameters on the chaos threshold is discussed to adopt different energy levels. With the help of analytical predictions, the proper nonlinear stiffness is determined for certain harmonic excitation. This work offers some fundamental insights into the optimal design of cubic and bistable NES.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

scholarly journals Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 755
Author(s):  
Rebiha Benterki ◽  
Jaume LLibre
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Upper Bounds ◽  
Linear Hamiltonian Systems ◽  
Straight Lines

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

scholarly journals Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Kaibin Chu ◽  
Zhengwei Zhu ◽  
Hui Qian ◽  
Huagan Wu
Keyword(s):  
Control Function ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Stable Node ◽  
The Novel ◽  
Analog Multiplier ◽  
Piecewise Linearity ◽  
Index 2 ◽  
Hardware Circuit ◽  
Zero Index

With new three-segment piecewise-linearity in the classic Chua’s system, two new types of 2-scroll and 3-scroll Chua’s attractors are found in this paper. By changing the outer segment slope of the three-segment piecewise-linearity as positive, the new 2-scroll Chua’s attractor has emerged from one zero index-1 saddle-focus and two symmetric stable nonzero node-foci. In particular, by newly introducing a piecewise-linear control function, an improved Chua’s system only with one zero index-2 saddle-focus and two stable nonzero node-foci is constructed, from which a 3-scroll Chua’s attractor is converged. Some remarks for Chua’s nonlinearities and the generating chaotic attractors are discussed, and the stabilities at the three equilibrium points are then analyzed, upon which the emerging mechanisms of the novel 2-scroll and 3-scroll Chua’s attractors are explored in depth. Furthermore, an analog electronic circuit built with operational amplifier and analog multiplier is designed and hardware circuit experiments are measured to verify the numerical simulations. These novel 2-scroll and 3-scroll Chua’s attractors reported in this paper are completely different from the classic Chua’s attractors, which will enrich the dynamics of the classic Chua’s system.

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