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ENGG952 - Engineering Computing

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University Of Wollongong

   

Engineering Computing (ENGG952)

   

Added on  2020-04-29

ENGG952 - Engineering Computing

   

University Of Wollongong

   

Engineering Computing (ENGG952)

   Added on 2020-04-29

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FACULTY OF ENGINEERING AND INFORMATION SCIENCESUNIVERSITY OF WOLLONGONGENGG952 ENGINEERING COMPUTINGSPRING SESSION – 2017GROUP OR INDIVIDUAL DETAILSNAMES OF MEMBERSREGISTRATION NUMBERSDATE OF SUBMISSION
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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. The mechanical system in the free vibration will oscillate with its natural frequency and eventually settle 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 in the 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 final solution 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
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tspan=[0:0.1:50];y0=[0.02;0];[t,y]=ode45('unforced2',tspan,y0);plot(t,y(:,1));grid onxlabel('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));grid onxlabel('time')ylabel('Displacement')title('Under-damped Modelling of MSD system (C=5)')hold on;%% order 647236function yp=unforced2(t,y)c=40;m=20;k=20;yp= [y(2); (-((c/m)*y(2))-((k/m)*y(1)))];%% order 647236function yp=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 and critically-damped system are shown in the same axes. The critically damped reduces to zero faster than the under-damped. The damping coefficient determines the rate at which the signal reduces its amplitude or magnitude as it travels.
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