University of Florida
EML 4220
MECH2620 Vibration and Control (Unit 1) Free Response of Mechanical Systems Unit 1, Part 2: Free Response of Spring/Damper & Mass/Spring/Damper Systems References Mechanical Vibrations Sections 2.6.1 and 2.6.2 We will now look at the free response of a spring and damper acting in parallel. In this example we are assuming the mass of the system
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MECH2620 Vibration and Control (Unit 1) Free Response of Mechanical Systems Unit 1, Part 2: Free Response of Spring/Damper & Mass/Spring/Damper Systems References Mechanical Vibrations Sections 2.6.1 and 2.6.2 We will now look at the free response of a spring and damper acting in parallel. In this example we are assuming the mass of the system is negligible compared to the damping and stiffness. As examples, this system is typical of a suspension unit from a car or bike. We will use the same steps as before only this time we will show that there is only a single solution to the characteristic equation and hence the system is known as a first order system. 1) Choose axis system and apply +ve deflection. Draw Free Body Diagram 2) Apply Newton’s 2nd Law ( Force = md 2 x/dt2 Torque = Id 2/dt2 ) 3) Write out Equation of Motion 4) Apply trial solution Ae t 5) Find Characteristic Equation and solve to find root S1 6) Determine General Solution 1) Free Body Diagram 2) Newtons 2nd Law 3) Equation of Motion 4) Trial solution x(t) = Ae t k c + ve This study source was downloaded by 100000853497421 from CourseHero.com on 05-22-2023 13:12:48 GMT -05:00 https://www.coursehero.com/file/116965776/MECH2620-Unit1-Handout-2-2docx/ 5) Characteristic Equation, Solve to find root S1 6) General Solution Hence the general solution for free motion of a first order system is of the form x(t) = Ae-t/ Where is the time constant (in this case =c/k) A is the initial displacement

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