scholarly journals A Transformation Method for Solving Conservative Nonlinear Two-Degree-of-Freedom Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Daniel Olvera Trejo ◽  
Inés Ferrer Real ◽  
Oscar Martínez-Romero
Keyword(s):  
Numerical Integration ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Transformation Method ◽  
Degree Of Freedom ◽  
Nonlinear Transformation ◽  
Equivalent Representation ◽  
Representation Form ◽  
Two Degree Of Freedom

A nonlinear transformation approach based on a cubication method is developed to obtain the equivalent representation form of conservative two-degree-of-freedom nonlinear oscillators. It is shown that this procedure leads to equivalent nonlinear equations that describe well the numerical integration solutions of the original equations of motion.

scholarly journals Investigation of the Equivalent Representation Form of Strongly Damped Nonlinear Oscillators by a Nonlinear Transformation Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Transformation Method ◽  
Nonlinear Transformation ◽  
Equivalent Representation ◽  
Oscillatory Systems ◽  
Representation Form ◽  
Engineering Problems ◽  
Linear And Nonlinear ◽  
Strongly Damped

We use a nonlinear transformation method to develop equivalent equations of motion of nonlinear homogeneous oscillatory systems with linear and nonlinear odd damping terms. We illustrate the applicability of our approach by using the equations of motion that arise in many engineering problems and compare their amplitude-time curves with those obtained by the numerical integration solutions of the original equations of motion.

scholarly journals Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Alex Elías-Zúñiga ◽  
Daniel Olvera ◽  
Inés Ferrer Real ◽  
Oscar Martínez-Romero
Keyword(s):  
Numerical Integration ◽  
Structural Engineering ◽  
Duffing Oscillator ◽  
Equations Of Motion ◽  
Fractional Power ◽  
Polymeric Materials ◽  
Nonlinear Damping ◽  
Equivalent Representation ◽  
Representation Form ◽  
Dynamics Response

In this work we consider the nonlinear equivalent representation form of oscillators that exhibit nonlinearities in both the elastic and the damping terms. The nonlinear damping effects are considered to be described by fractional power velocity terms which provide better predictions of the dissipative effects observed in some physical systems. It is shown that their effects on the system dynamics response are equivalent to a shift in the coefficient of the linear damping term of a Duffing oscillator. Then, its numerical integration predictions, based on its equivalent representation form given by the well-known forced, damped Duffing equation, are compared to the numerical integration values of its original equations of motion. The applicability of the proposed procedure is evaluated by studying the dynamics response of four nonlinear oscillators that arise in some engineering applications such as nanoresonators, microresonators, human wrist movements, structural engineering design, and chain dynamics of polymeric materials at high extensibility, among others.

scholarly journals Energy Method to Obtain Approximate Solutions of Strongly Nonlinear Oscillators

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Potential Energy ◽  
Analytical Solutions ◽  
Energy Method ◽  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Equivalent Representation ◽  
Strongly Nonlinear ◽  
Representation Form ◽  
Strongly Nonlinear Oscillators

We introduce a nonlinearization procedure that replaces the system potential energy by an equivalent representation form that is used to derive analytical solutions of strongly nonlinear conservative oscillators. We illustrate the applicability of this method by finding the approximate solutions of two strongly nonlinear oscillators and show that this procedure provides solutions that follow well the numerical integration solutions of the corresponding equations of motion.

scholarly journals Lyapunov Equivalent Representation Form of Forced, Damped, Nonlinear, Two Degree-of-Freedom Systems

2018 ◽  
Vol 8 (4) ◽  
pp. 649 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Luis Palacios-Pineda ◽  
Daniel Olvera-Trejo ◽  
Oscar Martínez-Romero
Keyword(s):  
Degree Of Freedom ◽  
Equivalent Representation ◽  
Representation Form ◽  
Two Degree Of Freedom

Nonlinear Dynamic of Cantilever Functionally Graded Plates Under the Thermalmechanical Loads

2009 ◽  
Author(s):  
Yu-xin Hao ◽  
Wei Zhang ◽  
Jian-hua Wang
Keyword(s):  
Nonlinear System ◽  
Functionally Graded Materials ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Governing Equations ◽  
Graded Materials ◽  
Two Degree Of Freedom

An analysis on nonlinear dynamic of a cantilevered functionally graded materials (FGM) plate which subjected to the transverse excitation in the uniform thermal environment is presented for the first time. Materials properties of the constituents are graded in the thickness direction according to a power-law distribution and assumed to be temperature dependent. In the framework of the Third-order shear deformation plate theory, the nonlinear governing equations of motion for the functionally graded materials plate are derived by using the Hamilton’s principle. For cantilever rectangular plate, the first two vibration mode shapes that satisfy the boundary conditions is given. The Galerkin’s method is utilized to discretize the governing equations of motion to a two-degree-of-freedom nonlinear system under combined thermal and external excitations. By using the numerical method, the two-degree-of-freedom nonlinear system is analyzed to find the nonlinear responses of the cantilever FGMs plate. The influences of the thermal environments on the nonlinear dynamic response of the cantilevered FGM plate are discussed in detail through a parametric study.

Nonlinear Truck Ride Analysis

1974 ◽  
Vol 96 (2) ◽  
pp. 597-602 ◽  
Author(s):  
G. R. Potts ◽  
H. S. Walker
Keyword(s):  
Numerical Integration ◽  
Nonlinear Equations ◽  
Digital Computer ◽  
Dry Friction ◽  
Shock Absorber ◽  
Equations Of Motion ◽  
Deflection Curve ◽  
Force Velocity ◽  
Nonlinear Equations Of Motion ◽  
Dry Friction Damping

The nonlinear vibratory motions of a three-axle semitrailer truck were investigated via the use of a digital computer. The nonlinear equations of motion are presented and a method of numerical integration is discussed. The analysis allows any shape of suspension force-deflection curve (including wheel hop, suspension stops, and dry friction damping) and a similar liberality of shock absorber force-velocity characteristics. An experimental vibration study, performed on a model truck, is described and the results compare favorably with the calculated results of the numerical integration.

Forced Oscillations of Non-Linear Two-Degree-of-Freedom Systems

1966 ◽  
Vol 8 (3) ◽  
pp. 252-258 ◽  
Author(s):  
G. N. Bycroft
Keyword(s):  
Numerical Integration ◽  
Transient Response ◽  
Internal Resonance ◽  
Perturbation Technique ◽  
Degree Of Freedom ◽  
Forced Oscillations ◽  
Oscillatory Systems ◽  
Non Linear ◽  
Two Degree Of Freedom ◽  
Good Agreement

This paper shows how the Lighthill-Poincaré perturbation technique may be used to determine the transient response of ‘lightly coupled’ non-linear multi-degree-of-freedom oscillatory systems subject to arbitrary forcing functions. The results in general are complex but simplify in many important cases. A comparison is made between the analytical results and results obtained by a numerical integration of the equations on a computer. Good agreement is noted. The method fails under conditions of ‘internal resonance’ of the system.

Flow-Induced Vibration of Long-Span Gates

2017 ◽  
pp. 294-386
Keyword(s):  
Vortex Shedding ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Full Scale ◽  
Degree Of Freedom ◽  
Theoretical Development ◽  
Flow Induced Vibration ◽  
Long Span ◽  
Free Discharge ◽  
Two Degree Of Freedom

In this chapter the theoretical equations for fluctuating pressures due to vertical and streamwise gate motions developed in Chapters 4 and 5 are used to derive equations of motion for long-span gates with underflow, overflow and simultaneous over- and underflow. Theoretical development of analysis methods is supported by laboratory and full-scale measurements. Specifically, this chapter considers long-span gate instabilities including one degree-of-freedom vibration of gates with underflow and free discharge, one degree-of-freedom vibration of a gate with submerged discharge and vortex shedding excitation, a two degree-of-freedom vibration of long-span gates with only underflow, and two degrees-of-freedom vibration of long-span gates with simultaneous over and underflow. A method is developed to predict pressure loading on the crest of the gate with overflow.

Vibration Control of Two-Degree-of-Freedom Structures Utilizing Sloshing in Nearly Square Tanks

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Shota Ninomiya
Keyword(s):  
Vibration Control ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Response Curves ◽  
Chaotic Motions ◽  
Installation Angle ◽  
Two Degree Of Freedom ◽  
Theoretical Results

This paper investigates the vibration control of a towerlike structure with degrees of freedom utilizing a square or nearly square tuned liquid damper (TLD) when the structure is subjected to horizontal, harmonic excitation. In the theoretical analysis, when the two natural frequencies of the two-degree-of-freedom (2DOF) structure nearly equal those of the two predominant sloshing modes, the tuning condition, 1:1:1:1, is nearly satisfied. Galerkin's method is used to derive the modal equations of motion for sloshing. The nonlinearity of the hydrodynamic force due to sloshing is considered in the equations of motion for the 2DOF structure. Linear viscous damping terms are incorporated into the modal equations to consider the damping effect of sloshing. Van der Pol's method is employed to determine the expressions for the frequency response curves. The influences of the excitation frequency, the tank installation angle, and the aspect ratio of the tank cross section on the response curves are examined. The theoretical results show that whirling motions and amplitude-modulated motions (AMMs), including chaotic motions, may occur in the structure because swirl motions and Hopf bifurcations, followed by AMMs, appear in the tank. It is also found that a square TLD works more effectively than a conventional rectangular TLD, and its performance is further improved when the tank width is slightly increased and the installation angle is equal to zero. Experiments were conducted in order to confirm the validity of the theoretical results.

CHAOS AND HYPERCHAOS IN SHAPE MEMORY SYSTEMS

2002 ◽  
Vol 12 (03) ◽  
pp. 645-657 ◽  
Author(s):  
M. A. SAVI ◽  
P. M. C. L. PACHECO
Keyword(s):  
Shape Memory ◽  
Intelligent Systems ◽  
Equations Of Motion ◽  
Memory Systems ◽  
Degree Of Freedom ◽  
Dynamical Analysis ◽  
Dimensional System ◽  
Shape Memory Actuators ◽  
Martensitic Phase Transformations ◽  
Two Degree Of Freedom

Shape memory and pseudoelastic effects are thermomechanical phenomena associated with martensitic phase transformations, presented by shape memory alloys. The dynamical analysis of intelligent systems that use shape memory actuators involves a multi-degree of freedom system. This contribution concerns with the chaotic response of shape memory systems. Two different systems are considered: a single and a two-degree of freedom oscillator. Equations of motion are formulated assuming a polynomial constitutive model to describe the restitution force of oscillators. Since equations of motion of the two-degree of freedom oscillator are associated with a five-dimensional system, the analysis is performed considering two oscillators, both with single-degree of freedom, connected by a spring-dashpot system. With this assumption, it is possible to analyze the transmissibility of motion between two oscillators. Results show some relation between the transmissibility of order, chaos and hyperchaos with temperature.

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