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Second order differential equation for spring-mass systems

Let's look at modeling the motion of a spring-mass system (a harmonic oscillator) using a second-order differential equation. From Newton's Second Law, we arrive at mx'' + cx' + kx = 0 (or a forcing function), where x(t) is the position of the spring-mass over time, m is the mass, c is the damping coefficient, and k is the constant from Hooke's Law. We focus on the effects of damping and how to detect what kind of damping a spring-mass system has based on the roots of the characteristic equation. Four types of spring motion are discussed based on the roots: undamped (no damping force), underdamped (small damping), critically damped (damping force just prevents oscillation), and overdamped (large damping). Different damping leads to different behaviors, which we can illustrate with MATLAB simulations. #mathematics #math #differentialequations #ordinarydifferentialequations #stemeducation #harmonicoscillator #hookeslaw #physics #matlab #matlabsimulation #iitjammathematics