1  Introduction

“All models are wrong, but some are useful” - Unknown

1.1 What is Mechanics?

In mechanical engineering, mechanics can be defined as the science that deals with the behaviour of physical bodies when subjected to forces or displacements. It involves the study of motion, deformation and equilibrium under external influences.

Although mechanics has a long history of experimental research, it plays a crucial role today in creating mathematical models that represent reality for further analysis. These models are, by necessity, simplifications of the real world, with the level of abstraction determined by their intended purpose. Such simplifications, known as idealizations, are an integral part of mechanics and will be introduced progressively as needed.

Mechanics is broadly classified into three main areas:

  1. Statics – The study of bodies at rest or in equilibrium under forces.
  2. Dynamics – The study of bodies in motion, further divided into:
  • Kinematics – The study of motion without considering forces.
  • Kinetics – The study of motion with forces taken into account.
  1. Strength of Materials (Solid Mechanics) – The study of deformable bodies under various loads and what deformations and stresses arise and lead to failure.

Another area of mechanics is the mechanics of fluids, also known as fluid dynamics and deals with compressible and incompressible fluids.

Mechanics is a branch of physical science aimed at predicting the accurate behavior of a system under study. Mechanics is a discipline deeply rooted in mathematics, making it deductive in nature. However, since these models are based on observations, they also have an empirical foundation. The study of mechanics is divided into verification and validation: verification focuses on mathematical analysis, while validation involves physical testing. Both are essential in engineering, as models have limitations, but relying solely on testing would be too expensive and time-consuming.

The purpose of studying mechanics is to evaluate the structural engineering requirements of a design. We analyze the angle at which an arm experiences the highest moments to properly size a motor. We determine the forces needed to open or close a lid, the required spring force for assistance, and its optimal placement. For a mountain bike suspension, we assess the necessary damping and stiffness for a 1-meter drop. In roller coaster design, we ensure that gravitational forces on riders remain within safe limits.

The advancement of modern computational tools has enhanced the role of mathematics, shifting the focus away from hand-calculations of overly simplified problems toward conceptual understanding, modeling, and interpretation. In today’s competitive product development landscape, it is not enough to find one design that fulfills the requirements, but rather to optimize factors such as weight, energy efficiency, and cost within ever-shorter timeframes, simulation has become essential. By replicating reality through computational models, engineering education must adapt to this paradigm shift.

Bridging the gap between academia and industry is critical. This goes beyond simply using computers as calculators; the entire process of mathematical modeling must be streamlined for digital computation — a concept known as “Computational Thinking”. Naturally, we embrace this approach.

1.2 Areas of mechanics

Mechanics is the study of physical systems at rest or in motion, along with the forces that influence them. Broadly speaking, it encompasses everything from particles and rigid bodies to deformable materials, fluids, and gases. However, in most academic courses, mechanics primarily focuses on particles and rigid bodies, while other areas are covered under solid mechanics and fluid dynamics. The boundaries between these fields are not always clear-cut.

Mechanics is traditionally divided into two main branches:

  • Statics – The study of forces in systems that remain at rest. This area provides foundational knowledge for further studies in mechanics and related subjects. Key mathematical tools include vector algebra and solving equations.

  • Dynamics – The study of systems in motion, which is further divided into:

    • Kinematics, which describes motion without considering the forces that cause it. For instance, when analyzing 3D printer head, one might track the nozzle’s position over time. Here, differentiation plays a crucial role.

    • Kinetics, which explores the relationship between forces and motion. A classic example is the double pendulum, influenced only by gravity. In this case, the challenge is to determine its position over time and reaction forces, requiring the solution of (non linear) differential equations.

Modern mechanics relies heavily on mathematical modeling and computational methods to analyze and predict physical behavior. This approach helps bridge the gap between theoretical principles and real-world applications.

1.3 Principles of mechanics

Although the principles are few, the complexity of analysis in mechanics arises due o the unlimited applications. For most common applications, we tend to have courses in machine elements to do deep analysis.