Dynamic Stability Analysis of Size-Dependent Viscoelastic/Piezoelectric Nano-beam

2022 ◽  
Author(s):  
Zahra Tadi Beni ◽  
Yaghoub Tadi Beni
Keyword(s):  
Stability Analysis ◽  
Dynamic Stability ◽  
Piezoelectric Materials ◽  
Equilibrium Points ◽  
Geometrical Nonlinearity ◽  
Saddle Points ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Bifurcation Points ◽  
Size Dependent

This paper analyzes the dynamic stability of an isotropic viscoelastic Euler–Bernoulli nano-beam using piezoelectric materials. For this purpose, the size-dependent theory was used in the framework of the modified couple stress theory (MCST) for piezoelectric materials. In order to capture the geometrical nonlinearity, the von Karman strain displacement relation was applied. Hamilton’s principle was also employed to obtain the governing equations. Furthermore, the Galerkin method was used in order to convert the governing partial differential equations (PDEs) to a nonlinear second-order ordinary differential one. Dynamic stability analysis was performed and the effects of such parameters as viscoelastic coefficients, size effect, and piezoelectric coefficient were investigated. The results showed that in this system, saddle points, central points, Hopf bifurcation points, and fork bifurcation points could be created, and the phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, and homoclinic orbits.

Coupled effects of surface interaction and damping on electromechanical stability of functionally graded nanotubes reinforced torsional micromirror actuator

2021 ◽  
pp. 030932472110368
Author(s):  
Weidong Yang ◽  
Menglong Liu ◽  
Linwei Ying ◽  
Xi Wang
Keyword(s):  
Casimir Force ◽  
Volume Fraction ◽  
Nonlocal Parameter ◽  
Saddle Points ◽  
Functionally Graded ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Squeeze Film ◽  
Squeeze Film Damping ◽  
Tilting Angle

This paper demonstrated the coupled surface effects of thermal Casimir force and squeeze film damping (SFD) on size-dependent electromechanical stability and bifurcation of torsion micromirror actuator. The governing equations of micromirror system are derived, and the pull-in voltage and critical tilting angle are obtained. Also, the twisting deformation of torsion nanobeam can be tuned by functionally graded carbon nanotubes reinforced composites (FG-CNTRC). A finite element analysis (FEA) model is established on the COMSOL Multiphysics platform, and the simulation of the effect of thermal Casimir force on pull-in instability is utilized to verify the present analytical model. The results indicate that the numerical results well agree with the theoretical results in this work and experimental data in the literature. Further, the influences of volume fraction and geometrical distribution of CNTs, thermal Casimir force, nonlocal parameter, and squeeze film damping on electrically actuated instability and free-standing behavior are detailedly discussed. Besides, the evolution of equilibrium states of micromirror system is investigated, and bifurcation diagrams and phase portraits including the periodic, homoclinic, and heteroclinic orbits are described as well. The results demonstrated that the amplitude of the tilting angle for FGX-CNTRC type micromirror attenuates slower than for FGO-CNTRC type, and the increment of CNTs volume ratio slows down the attenuation due to the stiffening effect. When considering squeeze film damping, the stable center point evolves into one focus point with homoclinic orbits, and the dynamic system maintains two unstable saddle points with the heteroclinic orbits due to the effect of thermal Casimir force.

scholarly journals Local Stability Analysis On Lotka-Volterra Predator-Prey Models Functional Responses With Constant prey Refuge

2021 ◽  
Author(s):  
Chongming Li
Keyword(s):  
Stability Analysis ◽  
Functional Response ◽  
Local Stability ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Stable Node ◽  
Type I ◽  
Type Ii ◽  
Functional Responses ◽  
Type Ii Functional Response

The dynamical behaviours of the predators and prey can be described by studying the local stability of the planar systems. Type I functional response shows that the rate of consumption per predator is proportional to prey’s density while type II functional response is related to the situation that predators would reach satiation as they consumed sufficient amount of prey. We seek out a method of using transformation to reduce the number of parameters of original models and then study the stability analysis of equilibrium points. Under suitable restrictions on the new parameters, we prove that the positive interior equilibrium is a stable node for the system of type I and type II functional responses. Moreover, in the case of type II functional response, the boundary equilibria can have more types of stability other than saddle points.

scholarly journals Global Portraits of Nonminimal Teleparallel Inflation

Universe ◽  
2021 ◽  
Vol 7 (6) ◽  
pp. 179
Author(s):  
Laur Järv ◽  
Joosep Lember
Keyword(s):  
Saddle Point ◽  
Fixed Points ◽  
Saddle Points ◽  
Heuristic Method ◽  
Gravity Models ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Late Time ◽  
Nonminimal Coupling ◽  
De Sitter

We construct global phase portraits of inflationary dynamics in teleparallel gravity models with a scalar field nonminimally coupled to torsion scalar. The adopted set of variables can clearly distinguish between different asymptotic states as fixed points, including the kinetic and inflationary regimes. The key role in the description of inflation is played by the heteroclinic orbits that run from the asymptotic saddle points to the late time attractor point and are approximated by nonminimal slow roll conditions. To seek the asymptotic fixed points, we outline a heuristic method in terms of the “effective potential” and “effective mass”, which can be applied for any nonminimally coupled theories. As particular examples, we study positive quadratic nonminimal couplings with quadratic and quartic potentials and note how the portraits differ qualitatively from the known scalar-curvature counterparts. For quadratic models, inflation can only occur at small nonminimal coupling to torsion, as for larger coupling, the asymptotic de Sitter saddle point disappears from the physical phase space. Teleparallel models with quartic potentials are not viable for inflation at all, since for small nonminimal coupling, the asymptotic saddle point exhibits weaker than exponential expansion, and for larger coupling, it also disappears.

Stability and Chaos Analysis of Nonlinear Roll Motion of Trimaran Ship With Variable Lay-Out Under Wind and Waves

2020 ◽  
Author(s):  
Yihan Zhang
Keyword(s):  
Dynamic Stability ◽  
Great Influence ◽  
Wave Force ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Roll Motion ◽  
Asymmetric System ◽  
Stability Performance ◽  
Combined Action ◽  
Nonlinear Roll Motion

Abstract The combined action of wind and waves has a great influence on the dynamic stability of roll motion of a trimaran ship, which may get into chaotic situation even capsizing. The lay-out of the trimaran is the main factor influencing the roll performance and its dynamic stability. In order to study the stability performance of the roll motion, firstly, the nonlinear roll motion equations under transverse wind and beam waves are established, in which the main coefficients are obtained by CFD method combined with model test. Then, the Hamilton system is used to analyze the phase portraits of the homoclinic and heteroclinic orbits under different transverse spacing. Finally, the Melnikov function is used to calculate the critical wave threshold of the asymmetric system under the combined action of wind load and wave force, and the Lyapunov exponent based on RHR algorithm was used to verify it. A series of significant conclusions are obtained by comparing the calculation models of different transverse spacing, which can provide references for the design of the trimaran ship.

scholarly journals Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

2018 ◽  
Vol 12 (7) ◽  
pp. 37
Author(s):  
Ning Han ◽  
Mingjuan Liu
Keyword(s):  
Periodic Solution ◽  
Saddle Points ◽  
Hamiltonian Function ◽  
Original System ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Restoring Force ◽  
Remarkable Feature ◽  
Approximate System ◽  
Irrational Nonlinearity

This paper focuses on a novel rotating mechanical model which provides a cylindrical example of transition from smooth to discontinuous dynamics. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting both smooth and discontinuous characteristics due to the geometry configuration. By using nonlinear dynamical technique, the unperturbed dynamics of the proposed system are studied including the irrational restoring force, stability of equilibria, Hamiltonian function and phase portraits. Note that a pair of double heteroclinic-like orbits connecting two non-standard saddle points are proposed in discontinuous case. For the perturbed system, we introduce a cylindrical approximate system for which the analytical solutions can be obtained successfully to reflect the nature of the original system without barrier of the irrationalities. Melnikov method is employed to detect the chaotic thresholds for the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing in smooth regime. Finally, numerical simulations show the efficiency of the proposed method and demonstrate the predicated periodic solution and chaotic attractors. It is found that a good degree of correlation is demonstrated in the bifurcation diagram, the phase portraits of periodic solution, the chaotic attractor’ structures and the Lyapunov characteristics between the original system and approximate system.

scholarly journals Local Stability Analysis On Lotka-Volterra Predator-Prey Models Functional Responses With Constant prey Refuge

2021 ◽  
Author(s):  
Chongming Li
Keyword(s):  
Stability Analysis ◽  
Functional Response ◽  
Local Stability ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Stable Node ◽  
Type I ◽  
Type Ii ◽  
Functional Responses ◽  
Type Ii Functional Response

The dynamical behaviours of the predators and prey can be described by studying the local stability of the planar systems. Type I functional response shows that the rate of consumption per predator is proportional to prey’s density while type II functional response is related to the situation that predators would reach satiation as they consumed sufficient amount of prey. We seek out a method of using transformation to reduce the number of parameters of original models and then study the stability analysis of equilibrium points. Under suitable restrictions on the new parameters, we prove that the positive interior equilibrium is a stable node for the system of type I and type II functional responses. Moreover, in the case of type II functional response, the boundary equilibria can have more types of stability other than saddle points.

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