I need to find the displacement and velocity characteristics of a suspended mass on a boat. I am struggling to develop a state space expression that can take a sinusoidal input, and then to plot graphs of displacement and velocity from this.

clear
t=0:0.1:10;                                          %time peroid
Y0= input('wave amplitude ') ;                        %Wave amplitude
l= input('length of wave ') ;                        %length of the wave
u=10;                                                %Boat Velocity
w=u/l;                                               %frequency of wave
y=Y0*sin(w*t);                                        %wave height model

Dr= input('damping ratio ');                         %Required Damping Ratio
if (Dr<0 || Dr>=1)                                   % test for acceptable damping ratio.
error('Damping ratio not in acceptable range!')
end
k=17000;                                             %Spring Constant of suspension
m=100;
wn=(k/m)^0.5;
wd=wn*(1-Dr^2)^0.5;


if (wd<6)
    error('Suspension excessively soft')            %test  for suspension softness
end

r=w/wn;                                             %Frequency ratio 
b=2*Dr*(k*m)^(0.5);                                 %Damping coefficient
i=(1-r^2)^2+(2*Dr*r)^2;                             
X0=(r^2)*Y0/(i^0.5);                                %Maximum amplitude of displacement
T=atan((2*Dr*r)/(1-r^2));                           %Spatial Frequency

x=X0*sin(w*t-T);                                    %Displacement of sprung mass


A = [0 1; -k/m -b/m];                            %state space matrices
B = [0 1/m]';
C = [x 0];                                       %I have put y into the input matrix as that is the wave input  
D = [0];

sys=ss(A,B,C,D);

plot(t,x)

Answer 1

[Birdman]

You need to define your outputs(with C matrix) correctly. Since you would like to see both displacement and velocity, then C matrix has to be

C=[1 0;0 1];

Also, to simulate the system under a sine input, you need to use lsim command:

lsim(sys,x,t);

The overall code:

clear
t=0:0.1:10;                                          %time peroid
Y0= input('wave amplitude ') ;                        %Wave amplitude
l= input('length of wave ') ;                        %length of the wave
u=10;                                                %Boat Velocity
w=u/l;                                               %frequency of wave
y=Y0*sin(w*t);                                        %wave height model
Dr= input('damping ratio ');                         %Required Damping Ratio
if (Dr<0 || Dr>=1)                                   % test for acceptable damping ratio.
error('Damping ratio not in acceptable range!')
end
k=17000;                                             %Spring Constant of suspension
m=100;
wn=(k/m)^0.5;
wd=wn*(1-Dr^2)^0.5;
if (wd<6)
    error('Suspension excessively soft')            %test  for suspension softness
end
r=w/wn;                                             %Frequency ratio 
b=2*Dr*(k*m)^(0.5);                                 %Damping coefficient
i=(1-r^2)^2+(2*Dr*r)^2;                             
X0=(r^2)*Y0/(i^0.5);                                %Maximum amplitude of displacement
T=atan((2*Dr*r)/(1-r^2));                           %Spatial Frequency
x=X0*sin(w*t-T);                                    %Displacement of sprung mass
A = [0 1; -k/m -b/m];                            %state space matrices
B = [0 1/m]';
C = [1 0;0 1];                                       %I have put y into the input matrix as that is the wave input  
D = [0];
sys=ss(A,B,C,D);
lsim(sys,x,t)

[Reuben Salisbury]

that's great, thank you!