Faculty of Engineering and Information Sciences
Added on - 29 Apr 2020
FACULTY OF ENGINEERING AND INFORMATION SCIENCESUNIVERSITY OF WOLLONGONGENGG952 ENGINEERING COMPUTINGSPRING SESSION – 2017GROUP OR INDIVIDUAL DETAILSNAMES OF MEMBERSREGISTRATION NUMBERSDATE OF SUBMISSION
ASSIGNMENT 2QUESTION 1 (50%)Mass-Spring-Damper system for an Ordinary Differential EquationFree vibrations in a mechanical system are caused by initial conditions of the parameters:displacement, velocity, or acceleration where no external force is allowed to interact. Themechanical system in the free vibration will oscillate with its natural frequency and eventuallysettle down to zero as a result of damping effects.∑f=mamd2xdt2+c(dxdt)+kx=0Dividing through by mass, m,d2xdt2+cm(dxdt)+kmx=0The natural frequency of the system and the damping ratio are replaced as shown inthe following equation,d2xdt2+2ζw0(dxdt)+w02x=0Where,w0=√km,ζ=c2√kmThe solution to the differential equation.λ1,2=−2ζω0±√4ζ2ω02−4ω022Therefore,λ1,2=−ζω0±ω0√ζ2−1The transient solution of the system is given as,xt=Ae(−ζω0+ω0√ζ2−1)t+Be(−ζω0−ω0√ζ2−1)tThe constants A and B are obtained when the initial conditions are inserted. The finalsolution is obtained as,From the initial conditions,w0=1,ζ=0.125xt=Ae(−0.125+ω0√ζ2−1)t+Be(−ζω0−ω0√ζ2−1)tThe Runge-Kutta Heun Method considering a certain range or period of time
tspan=[0:0.1:50];y0=[0.02;0];[t,y]=ode45('unforced2',tspan,y0);plot(t,y(:,1));gridonxlabel('time')ylabel('Displacement')title('Critically-damped Modelling of MSD system (C=40)')%% order 647236tspan=[0:0.1:50];y0=[0.02;0];[t,y]=ode45('unforced1',tspan,y0);plot(t,y(:,1));gridonxlabel('time')ylabel('Displacement')title('Under-damped Modelling of MSD system (C=5)')holdon;%% order 647236functionyp=unforced2(t,y)c=40;m=20;k=20;yp= [y(2); (-((c/m)*y(2))-((k/m)*y(1)))];%% order 647236functionyp=unforced1(t,y)c=5;m=20;k=20;yp= [y(2); (-((c/m)*y(2))-((k/m)*y(1)))];The graphical illustration of the behavior of the mass-spring-damper for an under-damped andcritically-damped system are shown in the same axes. The critically damped reduces to zerofaster than the under-damped. The damping coefficient determines the rate at which the signalreduces its amplitude or magnitude as it travels.