Question

find the solution in working model.

5). A mass m moves over a horizontal surface under external harmonic excitation. The mass is attached to one end of a nonlinear spring and to one end of a linear damper whose other ends are anchored in an immovable wall (see Fig.). Plot x(t) and v(x) for different initial conditions. Parameters: m = l, b = 0.1. h1-0.8, ω-38, spring parameters are shown in Fig. h1coswt F(x) F(x) C1 = 8

Answer 1

for positive x direction

Displacement x(t) 0.1r 0.05 -0.05 -0.1 0.15 0 12345 6 78910 time (s)

v(x) 0.3 0.2 0.1H o -0.1 0.2 0.3 0.4 0 12345 6 78910 time (s)

similary for other direction

% To test the mathematical model for a spring, mass and damper system

global M K R Amp freq mg

%Interval of Integration
tspan = [0 10];
% tspan = [0 10];

%initial conditions, at t = 0,
x0 = [ 0; ...%p1(t = 0) = 0 => initial momentum = 0, mass at rest
0];%qk(t = 0) = 1 => Initial position

%Input Parameters
M = 1;%40;%Mass = 1Kg.
K = 8;%0.5;%50;%spring stiffness, N/m
zeta = 1;%0.25;%4;%0.005;%-0.1;%
R = 0.1;%2*zeta*sqrt(K*M);%10;%damping, N-s/m
Amp = 0.8;%0.001;%10;%0;%0;
freq = 3.8;%0;%0.5;%0;%0;

g = 9.81;
mg = M*g;

%calling the solver
options = odeset('OutputFcn', 'odeplot', 'OutputSel', [1 2]);
[t,x] = ode45('simpleode', tspan, x0, options);
[v] = velocity(t, x);
plotsimple% plot results

%ODE file for simple.m

function out1 = simpleode(t, x)
global M K R Amp freq mg

%The state vector = x, received after intergration from the solver, at this instant.
p1 = x(1);   %momentum of mass
qk = x(2);   %displacement of mass

%Derivation of System Equations from the Bond graph.
%I. What do the elements give to the system?
% F = mg;

F = Amp*cos(freq*t);   %unit sine wave input
e4 = F;

f1 = p1/M;
%e2 = (qk/(1-qk^2));%K*qk;

e2 = K*qk;

f3 = f1;
e3 = R*f3;

%II. What does the system give to the elements with integral causality?
e1 = e4 - e3 - e2;
dp1 = e1;

f2 = f1;
dqk = f2;
% end

%Time derivative of the state vector

dx(1) = dp1;
dx(2) = dqk;

out1 = dx';               %Output of the ODE file simpleode.m, function simpleode

plot(t,x(:,2));
title('Displacement x(t)');
xlabel('time (s)');
ylabel('Displacement (m)');
pause;

plot(t,v(:,1));
title('v(x)');
xlabel('time (s)');
ylabel('Displacement (m)');
pause;