echo on
clc
% Example 6
% The Bungee Jumpers: Alex, Barry, and Carol
%
% Define the weights
wa = 220; % lbs
wb = 165; % lbs
wc = 102; % lbs
%
% Define the spring constants
c1 = 50; % lbs/ft
c2 = 20; % lbs/ft
c3 = 20; % lbs/ft
%
% Define the lengths of the cords
L1 = 15; % ft
L2 = 15; % ft
L3 = 15; % ft
%
% Calculate the elongations
e1 = (wa + wb + wc)/c1
e2 = (     wb + wc)/c2
e3 = (          wc)/c3
pause

% Determine the actual positions below the bridge
za = -(L1 + e1)
zb = za - (L2 + e2)
zc = zb - (L3 + e3)
pause
clc
% An alternate matrix approach
%
% The vector of elongations is given by
% e = A*z - L
%
% The structural matrix, A, is
A = [1 0 0; -1 1 0; 0 -1 1]

% The vector of cord lengths is
L = [L1 L2 L3]'
pause

% The vector of internal forces is given by
% y = C*e
%
% The matrix of material constants is
C = [c1 0 0; 0 c2 0; 0 0 c3]
pause

% Since we are at equilibrium, the forces
% acting on each jumper must balance
% f + A'*y = 0
%
% The force vector is
f = -[wa wb wc]'
pause

% Compute the inverse of A'
Atinv = inv(A')

% Solve for y
y = Atinv * (-f)

pause
% Compute the inverse of the material constants matrix
Cinv = inv(C)

% Calculate the elongations
e = Cinv * y
pause

% Compute the inverse of A
Ainv = inv(A)

% Calculate the actual positions
z = -Ainv*(e+L)
pause


% A more succinct calculation
% -f = A'*y
%    = A' * C * e
%    = A' * C * (A * z - L)
%    = A' * C * A * z - A' * C * L
AtCA = A'*C*A
z = -AtCA \ (-f + A'*C*L)
echo off