Weak discontinuities in one-dimensional compressible nonideal gas dynamics

This article concerns the study of various parameter effects on the propagation of weak discontinuities by using the method of characteristics. Analytical solutions of the quasi-linear system of hyperbolic partial differential equations (PDEs) are obtained and examined the evolutionary behavior of shock in the characteristic plane. The general behavior of solutions to the Bernoulli equation, which determines the evolution of weak discontinuity in a nonlinear system, is studied in detail. Also, we discuss the formation and distortion of compressive and expansive discontinuities under the van der Waals parameter effect and small particles for planar and cylindrical symmetric flow. The comparison between planar flow and cylindrical symmetric flow is studied under the influence of nonidealness and mass fraction of dust particles. It is found that the compressive waves become shock after a certain lapse of time. The medium considered here is the mixture of van der Waals gas with small dust particles.

Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design

Thick ellipsoids were recently introduced by the authors to represent uncertainty in state variables of dynamic systems, not only in terms of guaranteed outer bounds but also in terms of an inner enclosure that belongs to the true solution set with certainty. Because previous work has focused on the definition and computationally efficient implementation of arithmetic operations and extensions of nonlinear standard functions, where all arguments are replaced by thick ellipsoids, this paper introduces novel operators for specifically evaluating quasi-linear system models with bounded parameters as well as for the union and intersection of thick ellipsoids. These techniques are combined in such a way that a discrete-time state observer can be designed in a predictor-corrector framework. Estimation results are presented for a combined observer-based estimation of state variables as well as disturbance forces and torques in the sense of an unknown input estimator for a hovercraft.

Treating the Solid Pendulum Motion by the Large Parameter Procedure

In this paper, we consider the dynamical description of a pendulum model consists of a heavy solid connection to a nonelastic string which suspended on an elliptic path in a vertical plane. We suppose that the dimensions of the solid are large enough to the length of the suspended string, in contrast to previous works which considered that the dimensions of the body are sufficiently small to the length of the string. According to this new assumption, we define a large parameter ε and apply Lagrange’s equation to construct the equations of motion for this case in terms of this large parameter. These equations give a quasi-linear system of second order with two degrees of freedom. The obtained system will be solved in terms of the generalized coordinates θ and φ using the large parameter procedure. This procedure has an advantage over the other methods because it solves the problem in a new domain when fails all other methods for solving the problem in such a domain under these conditions. It is one of the most important applications, when we study the slow spin motion of a rigid body in a Newtonian field of force under an external moment or the rotational motion of a heavy solid in a uniform gravity field or the gyroscopic motions with a sufficiently small angular velocity component about the major or the minor axis of the ellipsoid of inertia. There are many applications of this technique in aerospace science, satellites, navigations, antennas, and solar collectors. This technique is also useful in all perturbed problems in physics and mechanics, for example, the perturbed pendulum motions and the perturbed mechanical systems. The results of this paper also are useful in moving bridges and the swings. For satisfying the validation of the obtained solutions, we consider numerical considerations by one of the numerical methods and compare the obtained analytical and numerical solutions.

Solution of the Riemann problem for an ideal polytropic dusty gas in magnetogasdynamics

The main aim of this paper is, to obtain the analytical solution of the Riemann problem for a quasi-linear system of equations, which describe the one-dimensional unsteady flow of an ideal polytropic dusty gas in magnetogasdynamics without any restriction on the initial data. By using the Rankine-Hugoniot (R-H) and Lax conditions, the explicit expressions of elementary wave solutions (i. e., shock waves, simple waves and contact discontinuities) are derived. In the flow field, the velocity and density distributions for the compressive and rarefaction waves are discussed and shown graphically. It is also shown how the presence of small solid particles and magnetic field affect the velocity and density across the elementary waves. It is an interesting fact about this study that the results obtained for the Riemann problem are in closed form.

Reconstruction of Governing Equations from Vibration Measurements for Geometrically Nonlinear Systems

Data-driven system identification procedures have recently enabled the reconstruction of governing differential equations from vibration signal recordings. In this contribution, the sparse identification of nonlinear dynamics is applied to structural dynamics of a geometrically nonlinear system. First, the methodology is validated against the forced Duffing oscillator to evaluate its robustness against noise and limited data. Then, differential equations governing the dynamics of two weakly coupled cantilever beams with base excitation are reconstructed from experimental data. Results indicate the appealing abilities of data-driven system identification: underlying equations are successfully reconstructed and (non-)linear dynamic terms are identified for two experimental setups which are comprised of a quasi-linear system and a system with impacts to replicate a piecewise hardening behavior, as commonly observed in contacts.

A survey on well-balanced and asymptotic preserving schemes for hyperbolic models for chemotaxis

Chemotaxis is a biological phenomenon widely studied these last years, at the biological level but also at the mathematical level. Various models have been proposed, from microscopic to macroscopic scales. In this article, we consider in particular two hyperbolic models for the density of organisms, a semi-linear system based on the hyperbolic heat equation (or dissipative waves equation) and a quasi-linear system based on incompressible Euler equation. These models possess relatively stiff solutions and well-balanced and asymptotic-preserving schemes are necessary to approximate them accurately. The aim of this article is to present various techniques of well-balanced and asymptotic-preserving schemes for the two hyperbolic models for chemotaxis.

Refined response axis decoupling axiom for a coupled vibrating system with spectrally-varying mount properties

Real-life engine mounts inherently exhibit considerable frequency- and amplitude-dependent characteristics, and the base flexibility has a significant effect on engine vibration and forces transmitted to the vehicle body. A new analytical formulation is proposed that incorporates spectrally-varying stiffness and damping properties of multi-dimensional mounts in the presence of a compliant base (with many vibration modes over the applicable lower frequency regime). A refined analytical axiom for the response axis decoupling of coupled system is also mathematically formulated using spectral response axis decoupling indices. Two examples are chosen to prove the refined axiom. Firstly, a powertrain mounting system with two hydraulic mounts is redesigned in terms of their stiffness and damping properties, and mount locations for both powertrain and sub-frame systems used the refined axiom in torque roll axis (TRA) direction. Frequency and time domain results demonstrate that the TRA of the redesigned powertrain mounting system is indeed decoupled from other powertrain motions. The effects of parameter uncertainties on the response axis decoupling indices are also examined. Then, a laboratory experiment consisting of a powertrain, three powertrain mounts including two hydraulic mounts, a sub-frame, and four bushings is then used to mathematically validate the refined axiom in vertical axis direction. The quasi-linear system formulation of the coupled system is also verified by comparing the frequency responses with the results obtained by the direct (matrix) inversion method and measurements.

The Biot Model for Anisotropic Poro-Elastic Media: The Viscoelastic Fluid Case

We show that an existence theorem for the completely anisotropic, time-harmonic poro-elastic boundary value problem can be established for the linear anisotropic Biot equations. Using the existence of these solutions, we present a scheme for solving the quasi-linear system for a nonlinear fluid–fluid viscosity such as the Carreau type.