We choose mechanical systems as the second group of dynamic systems to study. We first consider a simple case of the motion of a spring attached with a rigid body

of mass. Newton's law of motion governs the motion. With assumptions about the spring and the mass, we derive a differential equation model for the motion. A system which connects multiple springs and masses expands the number of dynamic variables. A thorough understanding of the modeling of a singe mass-on-spring will allow is to derive system equations for complex cases. They give us useful examples for studying system issues and the effect of feedback. Many mechanical systems produce oscillations. A simple case is one mass on one spring. The motion follows a linear differential equation with constant coefficients. Since there is oscillation, it cannot be modeled with a first-order equation. We know that Newton's law of motion governs the system. So, it is a second-order differential equation.A figure is shown below to describe the spring-mass systems Begin with a simple case of a single spring attached with a mass. The variable y(t) is the position of the mass m. It is measured from where the spring-mass is at rest,where y = 0. That position is an equilibrium. Away from that point, the spring is stretched and it has a restoring force to pulls the spring back toward the equilibrium. We assume that the restoring force is linear proportional to the stretch, that is it is -ky. The negative sign is important, for the spring force is against the stretch. Now suppose the mass is not at y = 0, the spring force will cause the mass to move. The motion follows Newton's law of motion: m\frac{d^2y}{\differentialD t^2}=-ky We follow the same approach of system analysis as we did with water-tank systems. We first investigate a system, reasoning about the nature of the system and applicable laws or rules. From that we establish a mathematical model which the dynamic behavior of the system follows. Then we analyze the system using the mathematical models to gain definitive and quantitative understanding of the system. The quantitative analysis is aimed for system design ideas or compensation strategies.The model of the spring-mass system, Equation (1), is a second-order linear differential equation. From the knowledge of differential equations, the solution of equation (1) describes pure oscillations. Task 1 : 1. Suppose at t= 0, y(0) = C1 and y(0) = C2. Find the solution of Equation(1). 2. Suppose the motion of m induces a friction force. The friction is against the motion and is assumed linearly proportional to the velocity: -c(dy/dt).Include this term in the equation and find the solution.