Purpose of the Article

This article provides a basic understanding of dynamics, focusing on the motion of objects and the forces that affect this motion. It is intended for readers who are new to the subject or need a refresher on fundamental concepts.

Static vs. Dynamic Systems

Dynamics is the study of forces and their effects on motion. It contrasts with statics, which deals with systems in equilibrium where forces are balanced and there is no motion. In dynamics, we analyze how forces cause changes in the motion of objects. For example, a high school example of the mass-spring system and Hook's law illustrates how much deflection occurs in a spring when a mass is attached to it. This relationship is written as $F = kx$, where $F$ is the force applied, $k$ is the spring constant, and $x$ is the displacement from the equilibrium position. So we all know that any elastic body showing a deflection when a force is applied to it, is following Hook's law. In this example, we talked about the deflection or displacement as an eventual quantity in result of the force applied. This study of a system is called statics, and this displacement, do you wonder, is called the static response of the system, or the steady state response of the system. In this case, the system is in equilibrium, and the forces are balanced. On the other hand, one could How much time did it take for the spring to reach this maximum deflection? This is a question that can be answered by the study of dynamics. In this case, we are interested in the time it takes for the system to reach its maximum deflection, which is called the dynamic response of the system. The dynamic response is the behavior of the system over time as it responds to an applied force.

More Formal Definitions

Statics

Statics is the branch of mechanics that study the systems in equilibrium (not moving) or at rest. In the case of above example, the study of the system is called statics when following is ensured:

$$ \begin{equation} \dot{x} = 0 \quad \text{and} \quad \ddot{x} = 0 \label{eq:statics} \end{equation} $$ Where $\dot{x}$ is the velocity and $\ddot{x}$ is the acceleration of the system. In this case, the system is not moving, and the forces are balanced.

Dynamics

What happens when the system is not in equilibrium? In this case, the system is in motion, and essentially, we can study the motion (how system is moving with respect to time). Hence, Dynamics is the branch of mechanics that study the systems in motion. As the per the Newton's second law, all the forces acting on a system has to be nulled $$ \begin{align} m \ddot{x} + c \dot{x} + k x = F \label{eq:dynamics} \end{align} $$

Where $F$ is the external force applied to the system, $m$ is the mass of the system, $c$ is the damping coefficient, and $k$ is the spring constant. In this case, the system is in motion, and the forces are not balanced. Eq. (\ref{eq:dynamics}) is a second-order ordinary differential equation (ODE) that describes the motion of the system. The solution to this ODE gives us the position of the system as a function of time, which is called the dynamic response of the system. As $F \ne 0$, this response is called Forced Response of the system. In the case of a system with no external force applied, the system is said to be in free motion, and the response is called Free Response of the system, and the equation becomes $$ \begin{equation} m \ddot{x} + c \dot{x} + k x = 0 \label{eq:free_response} \end{equation} $$ Where the solution to this ODE gives us the natural frequency of the system, which is the frequency at which the system oscillates when it is not subjected to any external force.