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1. 1. Table of ContentsIntroduction ................................................................................... 2The mass-spring-damper ................................................................. 2System behavior ............................................................................. 3 Critical damping (ζ = 1) .................................................................. 4 Over-damping (ζ > 1)..................................................................... 4 Under-damping (0 ≤ ζ < 1) ............................................................. 4The equations used ......................................................................... 5The program created ....................................................................... 6References...................................................................................... 7
2. 2. IntroductionIn this project we aim to determine The (undamped) natural frequency of the system The damped frequencyof the system The damping ratio The rise time The settling time The percentage overshoot By knowing the stiffness of the spring, the damping coefficient and the mass which the spring applied to.The mass-spring-damperAn ideal mass–spring–damper system with mass m, spring constant K and dampingcoefficient C is subject to an oscillatory force.And damping forceTreating the mass as a free body andapplying Newtons second law, the totalforce Ftot on the body isWhere: a is the acceleration of the mass x is the displacement of the mass relative to a fixed point of referenceSince
3. 3. Then the following parameters are then defined:LetWhere:System behaviorThe behavior of the system depends on the relative values of the two fundamentalparameters, the natural frequency ωn and the damping ratio ζ. In particular, thequalitative behavior of the system depends crucially on whether the quadraticequation for γ has one real solution,two real solutions, or two complexconjugate solutions.
4. 4. Critical damping (ζ = 1)When ζ = 1, there is a double root γ (defined above), which is real. The system is saidto be critically damped. A critically damped system converges to zero as fast aspossible without oscillating. An example of critical damping is the door closer seenon many hinged doors in public buildings. The recoil mechanisms in most guns arealso critically damped so that they return to their original position, after the recoildue to firing, in the least possible time.Over-damping (ζ > 1)When ζ > 1, the system is over-damped and there are two different real roots. Anover-damped door-closer will take longer to close than a critically damped doorwould.The solution to the motion equation isUnder-damping (0 ≤ ζ < 1)Finally, when 0 ≤ ζ < 1, γ is complex and the system is under-damped. In thissituation, the system will oscillate at the natural damped frequency ωd, which is afunction of the natural frequency and the damping ratio. To continue the analogy, an
5. 5. underdamped door closer would close quickly, but would hit the door frame withsignificant velocity, or would oscillate in the case of a swinging door.In this case, the solution can be generally written asThe equations usedm=input(the mass=)k=input(the stifness of the spring=)c=input(the damping coefficient=)wn=(k/m)^.5zeta=c/(2*m*wn)wd=wn*(1-zeta^2)^0.5x=atan(sqrt(1-zeta^2)/zeta)settlingtime=4/(zeta*wn)risetime=(pi-x)/wdprcentageovershoot=exp(-zeta*pi/sqrt(1-zeta^2))*100t=[0:0.01:1];for j=1:lengthof tr(j)=1-(((exp(-zeta*wn*t(j))).*sin(wd*t(j)+x))./sqrt(1-zeta^2));endplot(t,r)
6. 6. The program created