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Find the shape of a chain of springs

Description
Find the shape of a hanging chain where each chain link is a spring.


Source Files





springs.mos

(!*********************************************************************
   Mosel NL examples
   =================
   file springs.mos
   ````````````````
   Find the shape of a hanging chain by minimising its potential energy
   where each chain link is a spring that buckles under compression and 
   each node (connection between links) has a weight hanging from it.  
   The springs are assumed weightless.
   SOCP problem (convex quadratic objective, convex second-order cone 
   constraints)

   Based on AMPL model springs.mod
   Source: http://www.orfe.princeton.edu/~rvdb/ampl/nlmodels/noncute/springs/  
   Reference: "Applications of Second-Order Cone Programming",
   M.S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, 1998
 
   (c) 2013 Fair Issac Corporation
       author: S. Heipcke, Sep. 2013
*********************************************************************!)

model "springs"
 uses "mmxnlp"

 parameters
  N = 15	                    ! Number of chainlinks
 ! A = (AX, AY) and B = (BX, BY) are the positions of end nodes
  AX = 0 
  AY = 0
  BX = 2
  BY = -1
  G = 9.8                       ! Acceleration due to gravity
  K = 100                       ! Stiffness of springs
 end-parameters

 declarations
  RN = 0..N
  RN1 = 1..N
  x: array(RN) of mpvar         ! x-coordinates of endpoints of chainlinks
  y: array(RN) of mpvar         ! y-coordinates of endpoints of chainlinks
  MASS: array(RN) of real       ! Mass of each hanging node
  t: array(RN1) of mpvar        ! Extension of each spring
  LENGTH0: real                 ! Rest length of each spring
 end-declarations

! Initializing data
 LENGTH0 := 2*sqrt((AX-BX)^2+(AY-BY)^2)/N
 
! Try different settings for node weights:
 forall(j in RN | j>0 and j<N) MASS(j):=1     
! forall(j in RN | j>0 and j<N) MASS(j):=2*(N-j)
! forall(j in RN | j>0 and j<N) MASS(j):=5*j
 MASS(0):=0; MASS(N):=0  

! Decision variables
 forall(j in RN) x(j) is_free
 forall(j in RN) y(j) is_free
 forall(j in RN | j>0) t(j)>=0

! Objective: minimise the potential energy
 potential_energy:= sum(j in RN) MASS(j)*G*y(j) + (K/2)*sum(j in RN1) t(j)^2

! Bounds: positions of endpoints
! Left anchor
 x(0) = AX; y(0) = AY
 ! Right anchor
 x(N) = BX; y(N) =BY

! Constraints: positions of chainlinks
 forall(j in RN| j>0) 
  Link(j):= sqrt((x(j)-x(j-1))^2+(y(j)-y(j-1))^2) <= LENGTH0+t(j);

! Setting start values
 forall(j in RN) setinitval(x(j), (j/N)*BX+(1-j/N)*AX);
 forall(j in RN) setinitval(y(j), (j/N)*BY+(1-j/N)*AY);

! Solving
 setparam("XPRS_verbose", true)
 minimise(potential_energy)

! Solution reporting
 writeln("Solution: ", getobjval)
 forall(j in RN) 
  writeln(j, ": (", strfmt(getsol(x(j)),10,5), ", ", 
          strfmt(getsol(y(j)),10,5), ")")

end-model

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