Question

a can be skipped

Consider the following second-order ODE representing a spring-mass-damper system for zero initial conditions (forced response): 2x + 2x + x=u, x(0) = 0, *(0) = 0 where u is the Unit Step Function (of magnitude 1). a. Use MATLAB to obtain an analytical solution x(t) for the differential equation, using the Laplace Transforms approach (do not use DSOLVE). Obtain the analytical expression for x(t). Also obtain a plot of .x(t) (for a simulation of 14 seconds) b. Solve the above differential equation numerically by using the ODE45 function in MATLAB, for a duration of 14 seconds. Plot the displacement x(t). Include the function code, and the code that invokes ODE45 & plots the response. c. Model the above Ordinary Differential equation using Simulink, and obtain a plot of the solution x(t) for a simulation of 14 seconds (the Scope output). Also submit a snapshot of the Simulink model created.

Answer 1

% SOLVE d2x/dt2 + dx/dt + 0.5* x = unit_step
% initial conditions: x(0) = 0, x'(0)=0

t = 0:0.001:14; % time scale

starting_dxdt = 0;
starting_x = 0;

[t,x]=ode45( @rhs, t, [starting_x starting_dxdt] );

plot(t,x(:,1));
xlabel('t'); ylabel('x');

function dxdt=rhs(t,x)
dxdt_1 = x(2);
dxdt_2 = -x(2) - 0.5*x(1) + heaviside(t);
dxdt=[dxdt_1; dxdt_2];
end

Output