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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.
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system, ζ = 1, the roots are real negative and identical, i.e. ss12==−ζωn (15) The solution form X(t) = A e st is no longer valid.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)
Critically-Damped RLC Circuit – 𝜁𝜁= 1 𝑠𝑠 1 = −𝛼𝛼+ 𝛼𝛼 2 −𝜔𝜔 0 2, 𝑠𝑠 1 = −𝛼𝛼+ 𝛼𝛼 2 −𝜔𝜔 0 2 If 𝜁𝜁= 1, then 𝛼𝛼= 𝜔𝜔 0 𝛼𝛼 2 −𝜔𝜔 0 2 = 0– i.e., the discriminant is zero 𝑠𝑠 1 and 𝑠𝑠 2 are real and identical Complimentary solution has the ...
INTRODUCTION. This tutorial discusses the response of a second-order system to initial conditions, including initial displacement and initial velocity. The mass-spring-dashpot system shown in Fig. 1 is an example of a second-order system.
critically damped ( = 1) and overdamped ( >1) systems 26/39 Process Control Second Order Models and Response
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.
Effect of Damping Ratio on System Response . Depending on whether the quantity (ζ 2 −1) is negative, zero, or positive, the system is underdamped, critically damped, or overdamped, respectively. Underdamped: When this quantity is negative (ζ<1), the system is said to be underdamped. This is, by far, the most common case for structural systems.
