LPV Model of Bouncing Ball

This example shows how to construct a linear parameter-varying (LPV) representation of a system that exhibits multi-mode dynamics using lpvss.

The function BouncingBallDF defines the dynamics of this system. For more information on this model, see Using LTI Arrays for Simulating Multi-Mode Dynamics.

Define uncompressed spring lengths a1 and a2 and initial mass heights h1 and h2.

a1 = 12;    % uncompressed length of spring 1 (mm)
a2 = 20;    % uncompressed length of spring 2 (mm)
h1 = 100;   % initial height of mass m1 (mm)
h2 = a2;    % initial height of mass m2 (mm)

Moderate Floor Stiffness

First simulate the response for a soft surface by setting the spring constant for the second mass k2 to 300 g/s^2. The state is $$\mathit{x}=\left[{\mathit{y}}_1,{\dot{\mathit{y}}}_1,{\mathit{y}}_2,{\dot{\mathit{y}}}_2\right]$$.

The input is the constant $\mathit{u}$ = 1 (the forcing term is $\mathit{F}=\mathit{mgu}$).

k2 = 300;   % spring constant for second mass (g/s^2)
sys = lpvss("ygap",@(t,p) BouncingBallDF(t,p,k2)); % p = ygap = y1-y2

t = 0:0.05:40;
pFcn = @(t,x,u) x(1)-x(3);
xinit = [h1;0;h2;0];
[y,~,~,~,p] = step(sys,t,pFcn,RespConfig('InitialState',xinit));

figure(1), plot(t,y(:,1),t,y(:,2),t,abs(p-a1))
legend('y1','y2','|p-a1|')
title('k2 = 300')

The curves show the position of the masses. At the start of simulation, mass 1 undergoes free fall until it hits mass 2. The collision causes the mass 2 to be displaced, but it recoils quickly and bounces mass 1 back. The two masses are in contact for the time duration where ${\mathit{y}}_{\mathrm{gap}}<\mathrm{a1}$. When the masses settle down, their equilibrium values are determined by the static settling due to gravity.

High Floor Stiffness

Now simulate the response for a hard surface by setting k2 to 30,000 g/s^2.

k2 = 3e4;   % spring constant for second mass (g/s^2)
sys = lpvss("ygap",@(t,p) BouncingBallDF(t,p,k2));

t = 0:0.05:80;
[y,~,~,~,p] = step(sys,t,pFcn,RespConfig('InitialState',xinit));

figure(2), plot(t,y(:,1),t,y(:,2),t,abs(p-a1))
legend('y1','y2','|p-a1|')
title('k2 = 3e4')

For high floor stiffness, the collision does not cause the mass 2 to be displaced. It instead bounces the mass 1 back to a higher position; also, it takes more bounces for mass 1 to settle.

Data Function Code

type BouncingBallDF.m

function [A,B,C,D,E,dx0,x0,u0,y0,Delays] = BouncingBallDF(~,p,k2)
% p = gap y1-y2
m1 = 7;     % first mass (g)
k1 = 100;   % spring constant for first mass (g/s^2)
c1 = 2;     % damping coefficient associated with first mass (g/s)

m2 = 20;    % second mass (g)
%k2 = 300;   % spring constant for second mass (g/s^2)
c2 = 5;     % damping coefficient associated with second mass (g/s)

g = 9.81;   % gravitational acceleration (m/s^2)

a1 = 12;    % uncompressed lengths of spring 1 (mm)
a2 = 20;    % uncompressed lengths of spring 2 (mm)

if p>a1
   % Uncoupled
   A11 = [0 1; 0 0];
   B11 = [0; -g];
   C11 = [1 0];
   D11 = 0;

   A12 = [0 1; -k2/m2, -c2/m2];
   B12 = [0; -g+(k2*a2/m2)];
   C12 = [1 0];
   D12 = 0;

   A = blkdiag(A11, A12);
   B = [B11; B12];
   C = blkdiag(C11, C12);
   D = [D11; D12];
else
   A = [ 0        1,         0,             0; ...
      -k1/m1,   -c1/m1,    k1/m1,         c1/m1;...
      0,        0,         0,             1; ...
      k1/m2,    c1/m2,     -(k1+k2)/m2,   -(c1+c2)/m2];

   B = [0; -g+k1*a1/m1; 0; -g+(k2/m2*a2)-(k1/m2*a1)];
   C = [1 0 0 0; 0 0 1 0];
   D = [0;0];
end
E = [];
dx0 = [];
x0 = [];
u0 = [];
y0 = [];
Delays = [];