Question

Consider the Spring-Mass-Damper system: 17 →xt) linn M A Falt) Ffl v) Consider the following system parameter values: Case 1: m= 7 kg; b = 1 Nsec/m; Fa = 3N; k = 2 N/m; x(0) = 4 m;x_dot(0)=0 m/s Case2:m=30kg;b=1Nsec/m; Fa=3N; k=2N/m;x(0)=2m;x_dot(0)=0m/s Use MATLAB in order to do the following: 1. Solve the system equations numerically (using the ODE45 function). 2. For the two cases, plot the Position x(t) of the mass, on the same graph (include proper titles, axis labels, units and legends), for a simulation of 20 seconds. 3. From the graph obtained in part (a), for which of the two cases does it take a longer time for the position response to settle down (that is, to reach steady-state)? Explain physically, why this is so, using the parameter values provided for the two cases. 4. For the two cases, plot the Velocity xdot(t) of the mass, on the same graph (include proper titles, axis labels, units and legends), for a simulation of 20 seconds.

Answer 1

The following equations are solved in matlab:

mx + bxtkn=F HT let, 4,= * 9 = my + b => ý = = = Uz = ý, + ky, = F - - t-ky, - byz m

MATLAB code:

%Given time span

tspan = [0 20];

%Given Inputs - case 1

m1 = 7;

b1 = 1;

Fa1= 3;

k1 = 2;

y01 = [4 0];

%Given Inputs - case 1

m2= 30;

b2 = 1;

Fa2= 3;

k2 = 2;

y02 = [2 0];

%Numerical solution of differential equations

[T1,Y1] = ode45(@(t,y) coupled_diff(t,y,m1,b1,Fa1,k1),tspan,y01);

[T2,Y2] = ode45(@(t,y) coupled_diff(t,y,m2,b2,Fa2,k2),tspan,y02);

tiledlayout(2,1)

nexttile

plot(T1,Y1(:,1),'LineWidth',2);

hold on

plot(T2,Y2(:,1),'LineWidth',2);

hold off

legend('case-1','case-2','Location','best')

title('Displacement')

xlabel('time,s')

ylabel('x,m')

nexttile

plot(T1,Y1(:,2),'LineWidth',2);

hold on

plot(T2,Y2(:,2),'LineWidth',2);

hold off

legend('case-1','case-2','Location','best')

title('Velocity')

xlabel('time,s')

ylabel('velocity,m/s')

%Function definition

function f = coupled_diff(t,y,m,b,F,k)

f = zeros(2,1);

f(1) = y(2);

f(2) = (F-k*y(1)-b*y(2))/m;

end

Output:

Displacemen . Eman case-1 -case-2 X,m 15 10 time, s Velocity velocity,mis case-1 case 2 10 time, s

It takes longer for case-1 to stabilize because of higher initial displacement and relatively lower mass. Higher masses are damped faster.