On the solarity waves in arteries

Solitary waves are coincided with separatrices, which surround an equilibrium point with characteristics like a center, a sink, or a source. The existence of closed orbits in phase plane predicts the existence of such an equilibrium point. If there exists another saddle point near that equilibrium point, separatrix orbit appears. In order to prove the existence of solution for any kind of boundary value problem, we need to apply the fixed-point theorems. The Schauder's fixed-point theorem was used to show that there exists at least one nontrivial solution for equation of wave motion in arteries. The equation of wave motion in arteries has a nonlinear character, and the amplitude of the wave depends on the wave velocity. There is no general analytical or straightforward method for prediction of the amplitude of solitary waves. Therefore, the solution must be found by numerical or nonstraightforward methods. The methods of saddle point trajectory, escapetime, and escape-energy are introduced and shown that they are applicable methods with enough accuracy. Application of any of these approximate methods depends on the equation of motion, and the user preference. Applying a phase plane analysis, it was shown that the domain of periodic solution is surrounded by a separatrix. The separatrix is coincident with the desired solitary wave. The amplitude of the solitary wave is the most important characteristic of the wave, and will be predicted with each of the above methods

Linear static analysis and finite element modeling for laminated composite plates using third order shear deformation theory

In this paper, deformations of a laminated composite plate due to mechanical loads are presented. Third order shear deformation theory of plates, which is categorized in equivalent single layer theories, is used to derive linear dynamic equations of a rectangular multi-layered composite plate. Moreover, derivation of equations for FEM and numerical solutions for displacements and stress distributions of different points of the plate with a sinusoidal distributed mechanical load for Navier type boundary conditions are presented

CHAOTIC BEHAVIOR OF HYDRAULIC ENGINE MOUNT

ABSTRACT The constitutive relationships of the rubber materials that act as the main spring of a hydraulic mount are nonlinear. In addition to material induced nonlinearity, further nonlinearities may be introduced by mount geometry, turbulent fluid behavior, boundary conditions, temperature, decoupler action, and hysteretic behavior. While all influence the behavior of the system only certain aspects are realistically considered using the lumped parameter approach employed in this research. The nonlinearities that are readily modeled by the lumped parameter approach constitute the geometry and constitutive relationship induced nonlinearity, including hysteretic behavior, noting that these properties all make an appearance in the load-deflection relationship for the mount and may be readily determined via experiment or finite element analysis. In this paper we will shoe that under certain conditions, the nonlinearities involved in the hydraulic engine mounts can show a chaotic response

Suspended Decoupler: A New Design of Hydraulic Engine Mount

Because of the density mismatch between the decoupler and surrounding fluid, the decoupler of all hydraulic engine mounts (HEM) might float, sink, or stick to the cage bounds, assuming static conditions. The problem appears in the transient response of a bottomed-up floating decoupler hydraulic engine mount. To overcome the bottomed-up problem, a suspended decoupler design for improved decoupler control is introduced. The new design does not noticeably affect the mechanism's steady-state behavior, but improves start-up and transient response. Additionally, the decoupler mechanism is incorporated into a smaller, lighter, yet more tunable and hence more effective hydraulic mount design. The steady-state response of a dimensionless model of the mount is examined utilizing the averaging perturbation method applied to a set of second-order nonlinear ordinary differential equations. It is shown that the frequency responses of the floating and suspended decoupled designs are similar and functional. To have a more realistic modeling, utilizing nonlinear finite elements in conjunction with a lumped parameter modeling approach, we evaluate the nonlinear resorting characteristics of the components and implement them in the equations of motion

Design of a piecewise linear vibration isolator for jump avoidance

Piecewise linear isolators are smart passive vibration isolators that provide effective isolation for high frequency/low amplitude excitation. This can be done by introducing a soft primary suspension and a relatively damped secondary suspension. Such a piecewise isolator prevents the system from a high relative displacement in low frequency/high amplitude excitation. By employed an averaging method it is possible to obtain an implicit function for frequency response of a symmetric bilinear vibration isolator system under steady-state harmonic excitation. This function is examined for jump avoidance. A condition is derived which when met ensures that the undesirable phenomenon of 'jump' does not occur and the system response is functional. The jump avoidance and sensitivity of the condition are examined and investigated as the dynamic parameters vary. The results of this investigation can be directly employed in design of effective piecewise linear vibration isolators. A linear vibration system is defined as one in which the quantities of mass (or inertia), stiffness, and damping are linear in behaviour and do not vary with time [1]. Although mathematical models employing a linear ordinary differential equation with constant coefficients portray a simple and manageable system for analytical study, in most cases they are an incomplete representation simplified for the sake of analysis. Most real physical vibration systems are more accurately depicted by non-linear governing equations, in which the non-linearity may stem from structural constraints causing a change in stiffness and damping characteristics, or from inherent non-linear behaviour of internal springs and dampers. This paper focuses on a general form of such a non-linear system. This study of piecewise-linear systems will allow hazardous system behaviour over operating frequency ranges to be gauged and controlled in order to avoid premature fatigue damage, and prolong the life of the system

Effects of nonlinearities on the steady state dynamic behavior of electric actuated microcantilever-based resonators

This paper presents the dynamic behavior of microcantilever-based microresonators and compares their steady state behavior for polarized and nonpolarized systems at different levels of nonlinearities. A microcantilever, equipped with a time-varying capacitor, makes the microresonator system. The capacitor is activated by a constant polarization voltage, and an alternative actuating voltage. The partial differential equation of motion of the vibrating electrode can be reduced to a highly nonlinear parametric second order ordinary differential equation. The steady state behavior of the microresonator has been analyzed with and without polarization voltage. The main characteristic of the non-polarized model is explained by the stability of the system in parameter plane. A set of stability chart is provided to predict the boundary between the stable and unstable domains. Furthermore, the main characteristic of the polarized model is determination by the period-amplitude relationship of the system. Applying perturbation methods, analytical equations are derived to describe the frequency response of the system, which are suitable to be utilized in parameter study and design

Third order systems: periodicity condition

Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation, x'''+f(t,x,x',x")=0. In this paper we prove a new theorem, and establish a new sufficient condition for periodicity of a more restricted and better classified third-order system obeying the following third-order ordinary differential equation: xt'''+g1(x')x"+g2(x)x'+g(x,x',t)=e(t). In order to obtain conditions that guarantee the existence of periodic solutions and stable responses, the Schauder's fixed-point theorem has been implemented to prove the third-order periodic theorem for the differential equation. We show the applicability of the new third-order existence theorem by analyzing an independent suspension for conventional vehicles has been modeled as a non-linear vibration absorber with a non-linear third-order ordinary differential equation. Furthermore a numerical method has been developed for rapid convergence, and applied for a sample model. The correctness of sufficient conditions and solution algorithm has been shown with appropriate figures

Improved mathematical modeling of thermal effects in flexural microcantilever resonators dynamics

In a recent research the thermal dependency of material characteristics in dynamic response of microresonator systems is modeled using Lorentzian function and employing perturbation analysis. Thermal phenomena introduce two main effects: damping due to internal friction, and softening due to Young modulus-temperature relationship. The presented mathematical model provided effective equations to study the electrically actuated microbeam resonators. The mathematical model of thermal phenomena in microbeam vibration was introduced by Jazar (2009). In that analysis, using the Zener model, a positive frequency dependent damping and a negative frequency dependent stiffness terms were introduced to mode the effects of warming at resonance (Jazar 2009). In this investigation, the problem will be analyzed from a practical point of view. We introduce a better mathematical model by improving the presented model. The main difference would be including the strain distribution in the damping and stiffness model

Periodicity condition of fourth order ordinary differential equations with application in dynamical systems

An existence theorem is proven to investigate sufficient conditions of a nontrivial solution, for a general nonlinear fourth order ordinary differential equation provided that and are bounded, and satisfy certain sufficient conditions. Then, there exists a bounded space for state variables, in which the equation has an answer. For this purpose we use the existence of the Green's function and Schauder's fixed point theorem. The condition is applied to a nonlinear two degrees of freedom vibrating system, and Bouasse-Sarda regulator mechanical system