Dynamical Tasks with MatLab

This book is for everyone who wants to enrich their knowledge in the field of Theoretical Mechanics and Computational Mathematics using the MATLAB program. One-mass and multi-mass, linear, and non-linear mechanical systems are considered. Solved tasks are in the field of dynamics but can also be used by specialists working in other areas of knowledge. Appropriate examples illustrate the practical use of the numerical integration program for ordinary linear and non-linear differential equations.

The modern modeling of complex mechanical systems is done in two ways. The first way is based on the use of ready-made programs. Modeling with such of these programs is up-to-date and fast, but they are limited within the possibility of the program. The second way of modeling is related to the derivation of differential equations describing the state of the mechanical system and then their solution. Here, at the numerical integration of complex linear and especially non-linear differential equations, the researchers, mostly engineers, experienced difficulties for many years. Now this problem no longer exists. Powerful computer mathematical programs have emerged, one of which is MATLAB – an undisputed leader in the field of modern computational engineering mathematics.

The MATLAB program is difficult to learn. On the one hand, it is quite large, and on the other hand, although it has its own “Help”, the user must have a solid knowledge of mathematics. Many engineers are making methodical mistakes and deviating at one extreme or another. Using the SIMULINK Toolbox, they build simulation models with a complex and large network of links and elements, even for linear tasks. With these models, they solve a small number of equations by exhausting the resources of the program (no matter how perfect it is). They are often bypassing the solution of the basic question in the study of such a class of mechanical systems, namely, the determination of their own frequencies and their own forms. This is because they are missing the matrices and matrix calculus which are implanted at the core of the MATLAB program (shortened by MATrix LABoratory).

That is why the author has built simple, clear, logical, and small simulation models with the main core that does not depend on the number of equations. This is especially true for linear systems.

The presented book does not treat the way of receiving the differential equations describing the motion of mechanical systems. Only numerical solution is shown. The full text of all programs and all structural block schemas is provided. All tasks, except for two non-linear ones, are solved in two ways – through direct integration, using the MATLAB library procedure, and indirectly integrating by using the SIMULINK Toolbox environment. The author considers that such an approach is extremely important for verifying the solution’s veracity.

The book is divided into two parts.

The first part treats linear mechanical systems. It consists also of two parts:

Chapter 1 – one-mass systems;

Chapter 2 – multi-mass systems.

Four types of vibrations are examined – free damping vibrations and forced vibrations due to force, kinematical, and inertia excitations. The Eigen frequencies are obtained for all dynamical models. The Eigen forms are also obtained for all multi-mass systems.

The second part treats non-linear mechanical systems.

It consists also of two parts:

Chapter 3 – one-mass systems;

Chapter 4 – multi-mass systems.

All programs for linear mechanical systems and some programs for non-linear systems are standard. These programs can be used with light preparing and updating for solving specific practical tasks for mechanical systems with many degrees of freedom. Version 6.1 is used, but the programs can also run on the next versions of MATLAB.

The author hopes the book will contribute to the development and improvement of young scientists. He will treat with care and gratitude all readers who, with their useful remarks, recommendations, and suggestions, would help him to improve the book in the next edition.