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ENGG952 Engineering Computing

   

Added on  2020-05-01

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UNIVERSITY OF WOLLONGONGFACULTY OF ENGINEERING AND INFORMATION SCIENCESENGG592 ENGINEERING COMPUTINGSPRING SESSION -2017ASSIGNMENT 2STDUENT NAMESTUDENT REGISTRATION NUMBERDATE OF SUBMISSION
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QUESTION 1MOTION OF A SPRING MASS SYSTEM (50%)The motion of a damped spring mass system is described by the following ordinary differential equationWhere x is displacement from equilibrium position (meter), t is time (second), m is the mass and equal 20 kg, c is the damping coefficient (N.sec/meter). The damping coefficient, c, takes on twovalues of 5 (under damped), 40 (critically damped). The spring constant k=20N/meter. The initialvelocity is zero, and the initial displacement x=1 meter.Solution(i)Transform the problem to a system of two first order initial value ODES.From Newton’s second law,∑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.
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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)t(ii)Second order RK Heun method over a period of timeThe Heun’s method evaluates the slope at the beginning and at the end of the step. The generalized idea embodied in the Heun’s method is the Runge-Kutta method which uses a weighted average of the slope evaluated at multiple in the step. %% order 647236tspan=[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)))];
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