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Learning Objectives. Learn to analyze a general second order system and to obtain the general solution. Identify the over-damped, under-damped, and critically damped solutions. Convert complex solution to real solution. Suspended “mass-spring-damper” equivalent system.
1.1 Drag and general Damping Forces. To achieve our objective of finding a more accurate model for oscillatory phenomena, we need to first find the correct Newton’s second law equation for such systems. Thus we need to better determine the forces acting on our oscillating object.
Damping is a frictional force, so it generates heat and dissipates energy. When the damping constant b is small we would expect the system to still oscillate, but with decreasing amplitude as its energy is converted to heat.
Free Response of Critically Damped 2nd Order System For a critically damped system, ζ = 1, the roots are real negative and identical, i.e. ss12==−ζωn (15) The solution form X(t) = A e st is no longer valid. For repeated roots, the theory of ODE’s dictates that the family of solutions satisfying the differential equation is () n (12)
Lecture 7 - Step Response & System Behaviour to print. In this lecture, I will continue to consider Laplace transform, particularly for a 1st order and a 2nd order system. I will develop some insights into how these systems behave both in the time domain in response to a step input, and in the frequency domain (that is, in response to sinusoids ...
This document discusses the response of a second-order system, such as the mass-spring-dashpot shown in Fig. 1, to a step function. The modeling of a step response in MATLAB and SIMULINK will also be discussed. Fig. 1. Single-degree-of-freedom mass-spring-dashpot system.
Figure 2 shows the response of the series RLC circuit with L=47mH, C=47nF and for three different values of R corresponding to the under damped, critically damped and over damped case. We will construct this circuit in the laboratory and examine its behavior in more detail. (a) Under Damped. R=500Ω (b) Critically Damped. R=2000 Ω (c) Over Damped.
