Vibrations and waves

This guide is specifically designed for second-year students in Science and Technology programs at Algerian universities and engineering schools. It forms part of the official curriculum for the "Mechanical Vibrations and Waves" course, taught during the second year (L2-S3) for students majoring in Science and Technology (ST) and Materials Science (SM).

The primary aim of this program is to introduce second-year students to key concepts in physics and mechanics related to vibration and mechanical wave phenomena, while clearly distinguishing between the two. The unit is divided into two main sections: the first section focuses on Vibrational Motion, and the second on Waves. This guide is crafted to effectively meet the recommendations outlined in the official curriculum.

In accordance with the official curriculum, Lagrange's equations and differential equations are emphasized for the study of mechanical vibrations, without relying on the laws of classical mechanics. These concepts are highlighted in alignment with the official curriculum for both Mechanical Engineering and Methods Engineering majors. For Electrical Engineering students, however, the focus is placed on the second section, which covers Mechanical Waves.

In the context of this scientific guide, the first part covers five fundamental topics related to mechanical vibrations, along with a summary of the necessary mechanical and mathematical concepts needed to support students throughout the course.

We have simplified Lagrange’s equations as much as possible for a single-particle system with one degree of freedom. The classification of various types of constraints has been omitted, as the primary focus is on the concept of generalized forces and, in the case of dissipative systems, the introduction of the dissipation function. The generalization to systems with multiple degrees of freedom is presented without formal proof. The analysis of oscillations is intentionally limited to low-amplitude oscillations and is covered in the chapter titled "Free Oscillations of Systems with One Degree of Freedom." Only viscous friction, proportional to velocity, is considered in this guide.

The subsequent chapter, dedicated to forced oscillations in systems with one degree of freedom, explores the concepts of resonance and mechanical impedance. The study of oscillatory systems with multiple degrees of freedom is confined to systems with two degrees of freedom, approached through the superposition of specific sinusoidal solutions. This section facilitates the integration of resonance and impedance concepts and introduces the phenomenon of antiresonance.

The formal approach also recommends representing the propagation of mechanical waves using a linear string model, ensuring a smooth transition from a discrete medium to a continuous medium. This transition will be explored in detail in the second part of the guide.

Finally, we trust that this guide will provide valuable insights and be a useful resource for students and the broader academic community.

Dr : DJOUDI Tarek