3.3.2. Equation of Motion

In such a system, the force applied to the mass causes it to move, and this movement can be modeled by Newton’s Second Law of Motion:

F(t)=ma(t)

where a(t) is the acceleration of the mass and F(t) is the applied force.

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. This means that if you apply a force to an object (like a mass), it will accelerate in the direction of the force. In mathematical terms, if you know the force applied to the mass, the mass itself, and the resulting acceleration, you can express this relationship using the formula: F(t) = m * a(t). Here, F(t) is the force at time t, m is the mass, and a(t) is the acceleration. If force is applied, the mass will respond by moving and its velocity will change.

Consider pushing a shopping cart. When you push it with a certain force, it begins to move and speeds up. The more force you apply, the faster it will accelerate. This everyday experience relates directly to Newton's Second Law, where your push is equivalent to the applied force.

For the spring and damper, the forces are given by:

• Spring force: Fspring=−kx(t)
• Damping force: Fdamper=−bdx(t)/dt

where x(t) is the displacement of the mass.

In a mass-spring-damper system, two main forces act on the mass: the spring force and the damping force. The spring force pulls the mass back towards its equilibrium position and is proportional to the displacement (how far the mass has been stretched or compressed from its resting position). This is represented as F_spring = -kx(t), where k is the spring constant and x(t) is the displacement. The damping force, on the other hand, opposes the motion and is proportional to the velocity of the mass. It is expressed as F_damper = -b(dx(t)/dt), where b is the damping coefficient. Together, these forces determine how the mass will move over time based on how they change with position and velocity.

Imagine a pogo stick. When you bounce on it, the spring pushes you back up, and the damping (like the friction in the pogo stick's mechanics) slows down your bounce. The balance between these forces gives you a smooth ride, just like in our mass-spring-damper system.

The total force on the mass is the sum of the applied force, the spring force, and the damping force. According to Newton's second law:

ma(t)=Fapplied−Fspring−Fdamper

Substituting the expressions for the spring and damping forces:

md²x(t)/dt²=Fapplied(t)−kx(t)−b dx(t)/dt

Rearranging this:

md²x(t)/dt²+bdx(t)/dt+kx(t)=Fapplied(t)

In a mass-spring-damper system, the net force acting on the mass is calculated by summing the applied force, the spring force, and the damping force. Using Newton's second law, we can express this relationship as: ma(t) = F_applied - F_spring - F_damper. Substituting the earlier expressions for spring and damper forces gives us a second-order differential equation: m(d²x(t)/dt²) + b(dx(t)/dt) + kx(t) = F_applied(t). This equation describes how the position of the mass changes over time, influenced by the forces acting on it.

Think of holding a rubber band (spring) while someone tries to pull it (applied force). As you stretch it more (displacement), the rubber band wants to pull back (spring force) and at the same time, if you pull too fast, the resistance from your hand (damping force) increases. The resulting motion of the rubber band as you pull varies based on these opposing effects.

This is the differential equation that governs the motion of the mass. It is a second-order differential equation.

The equation we derived, md²x(t)/dt² + b(dx(t)/dt) + kx(t) = F_applied(t), is called a second-order differential equation. This type of equation involves derivatives of the second degree, which means it can describe systems where both acceleration and velocity play crucial roles. In mechanical systems, second-order differential equations are common because they capture the dynamics of motion, including oscillations and damping behavior.

Consider a swing moving back and forth. The motion of the swing is governed by similar principles: gravity (like spring force) tries to pull it back while air resistance slows it down (like damping force). The swing's path can be modeled with second-order differential equations much like the mass-spring-damper system.

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