Nicolae Pandrea

Numerical Analysis with Applications in Mechanics and Engineering: Teodorescu/Numerical Analysis with Applications in Mechanics and Engineering

A much-needed guide on how to use numerical methods to solve practical engineering problemsBridging the gap between mathematics and engineering, Numerical Analysis with Applications in Mechanics and Engineering arms readers with powerful tools for solving real-world problems in mechanics, physics, and civil and mechanical engineering. Unlike most books on numerical analysis, this outstanding work links theory and application, explains the mathematics in simple engineering terms, and clearly demonstrates how to use numerical methods to obtain solutions and interpret results.Each chapter is devoted to a unique analytical methodology, including a detailed theoretical presentation and emphasis on practical computation. Ample numerical examples and applications round out the discussion, illustrating how to work out specific problems of mechanics, physics, or engineering. Readers will learn the core purpose of each technique, develop hands-on problem-solving skills, and get a complete picture of the studied phenomenon. Coverage includes:How to deal with errors in numerical analysisApproaches for solving problems in linear and nonlinear systemsMethods of interpolation and approximation of functionsFormulas and calculations for numerical differentiation and integrationIntegration of ordinary and partial differential equationsOptimization methods and solutions for programming problemsNumerical Analysis with Applications in Mechanics and Engineering is a one-of-a-kind guide for engineers using mathematical models and methods, as well as for physicists and mathematicians interested in engineering problems.

Numerical (analytical-based) model for the study of vehicle frontal collision

This article describes and solves a mathematical model useful to study a vehicles kinematics before and after collision, and the structure behaviour during an impact. The algorithm is based on solving differential equations using Runge–Kutta numerical method. To extract input data, a full-scale numerical model is solved using LS-Dyna. The friction coefficient between vehicle and barrier is evaluated and compared with the input value. The motion of the vehicle is solved and then passengers are added. The acceleration and displacement of the passengers centre of gravity are computed for different cases of impact angle and safety restrain systems stiffness. To extend the capabilities for accident-reconstruction applications, a procedure to construct deformation force pulse is presented. Also, a method for the evaluation of vehicles kinematic parameters using force–deformation curve is presented. The performances of the open architecture of the analytical model are presented and applied for the study of passengers head motion.

Aspects in the Synthesis of a Variable Compression Ratio Mechanism

The mechanism considered in this paper is a VCR one consisting in a crank, a shaft, an intermediate triangular element and a control lever and it was described in previous papers of the authors. The authors start from a classical crank-shaft mechanism for which the extreme positions are known. The first stage of the synthesis consists in determination of the constraint function which has to be fulfilled by the new mechanism so that the extreme position remain unchanged. The new step consists in imposing new conditions for the mechanism so that some new positions have to be obtained. The main hypothesis is that the positions of different characteristic points of the mechanism may be written as continuous functions of the certain input data. Due to aspect, the synthesis of the mechanism implies continuous variations of the output data and, consequently, there exists at least one solution for the synthesis process. In each case the authors determine the extreme positions of the piston. These extreme positions are also continuous functions of the input data. Two main approaches are discussed in the paper. One approach consists in the exact determination of the solution using a numerical procedure. The second one is an approximate one and consists in the determination of an approximate solution of the synthesis and verifying the deviation of this solution. For the new mechanism obtained by synthesis the authors determined the reduced velocities and accelerations of different characteristic elements. Some aspects of the wear are discussed with the aid of the reduced relative velocities.

A new approach in the study of frictionless collisions using inertances

This paper presents a complete study on the collision without friction of two rigid bodies with and without bilateral constraints. Our goal is to obtain the same formulae for the impulse, the energy of the lost velocities, and the loss of kinetic energy as in the case of the collision of two particles. The study is performed with the aid of the notion of inertance. The dependence on the constraints is given by the inertances. The calculations are realized using the results from the theory of screws (plückerian coordinates). The obtained formulae are general, written in matrix form, and they may be easily used in any practical problem. We give the general algorithms for the collision of two rigid bodies with and without bilateral constraints. The equality of the coefficients of restitution in the Newton, Poisson, and energetic models is also proved. The numerical examples highlight the theory.

A new approach in the study of the collisions with friction using inertances

This paper presents a complete study on the collision with friction of one or two rigid bodies without constraints. The differential formula between the velocities and impulse uses the notion of inertance resulting from the theory of screws (Plückerian coordinates). One thus may calculate the kinematic and dynamic parameters, the velocities and the kinetic energies of the two rigid solids after the collision, and the variation of the kinetic energy. The calculation is detailed for the Newton, Poisson, and energetic variants of the coefficient of restitution. The variation of the kinematic and dynamic parameters in relation to the coefficient of restitution and coefficient of friction for all the three variants are presented and discussed. A numerical example highlights the theory.

Numerical Results Obtained on Models of Dynamical Systems Used in the Study of Active Suspensions

The semi-active and active suspensions are more and more often used in the construction of medium price automotive. The models used by the specialists in the study of the dynamical response of such suspensions are diverse and start from simple models arriving to complex ones. In this paper we perform an analysis of them and of the numerical results obtained by simulation on many models of semi-active and active suspensions. The results are presented as diagrams obtained by using some performance criteria.

Elements of Cam’s Synthesis for a Distribution Mechanism for the Miller-Atkinson Cycle

In this paper we present the synthesis of the cam for a distribution mechanism used for Miller-Atkinson cycle and described in previous works. The input data contain the angles at which the valve opens and closes, respectively, the dimensions of the main parts, the maximum displacement of the valve. and the law of motion of the valve. The authors consider that the law of motion of the valve is described by numerical values, even if in the simulation process they use analytical formulae for this displacement. The head of the valve is considered to be circular. A particular case of flat head of the valve is also described. The relative displacement of the lever and the valve is discussed in two hypotheses: symmetric position and asymmetric position of the contact point with respect to the axis of the valve. The contact between the head of the valve is assured by a cylindrical roller. The contact between the cam and the lever is considered in two main cases: roller tappet and flat tappet. In all cases the cam is obtained by numerical synthesis using a convenient angular degree. A great attention is paid to the convexity of the cam. The resulted cam is transformed in a convex one using the Jarvis March. The authors also perform a comparison between the theoretical profile of the cam and the profile of the convex cam, and between the imposed law of motion of the valve and the real law of motion of the valve using the convex cam. The reduced velocity and acceleration are obtained in all cases. Some aspects of the wear are discussed using the relative linear and angular velocities. The inertia of the mechanism is discussed with the aid of linear and angular accelerations of the elements. The most important resulted tables and diagrams are also presented. The paper closes with a paragraph of conclusions.

Model for the study of active suspensions

In the beginning of the paper are briefly presented various types of semi-active and active suspensions. The advantages and disadvantages of the solutions analyzed are highlighted. There is a mathematical model for the study of an active suspension. In the next chapter we present a calculation model of active suspension controls based on elements of linear dynamic systems theory. The optimal synthesis of the dynamic system is based on a set performance criterion. At the end of the paper are presented the numerical results obtained for a two-degree system model used in several configurations.

Numerical Differentiation and Integration

This chapter gives introduction to numerical differentiation by means of an expansion into a Taylor series and interpolation polynomials, and numerical integration. The numerical integration formulas include the Newton-C??Tes quadrature formulae, the trapezoid formula, Simpsons formula, Eulers and Gregorys formulae, Rombergs formula, and Chebyshevs quadrature formulae. In addition, the chapter considers quadrature formulae of Gauss type obtained by orthogonal polynomials, calculation of improper integrals, Kantorovichs method, and the Monte Carlo method for calculation of definite integrals. These are followed by applications.

Solution of Algebraic Equations

This chapter deals with the determination of limits of the roots of polynomials, including their separation. Three methods are considered, namely, Lagranges method, the Lobachevski-Graeffe method, and Bernoullis method. In addition, the chapter talks about Bierge-Viete method and Lin methods. These are followed by applications.