Description

Coil compression spring design problem, finds minimum volume
of wire for the production of a coil compression spring.

Small Model of Type : MINLP

Category : GAMS Model library

Main file : spring.gms

$title Coil Compression Spring Design (SPRING,SEQ=206)

$onText
Coil compression spring design problem, finds minimum volume
of wire for the production of a coil compression spring.


Sangren, E, Nonlinear Integer and Discrete Programming in Mechanical
Design Optimization. Journal of Mechanical Design, Trans.
ASME 112 (1990), 223-229.

Keywords: mixed integer nonlinear programming, coil compression spring designment,
          engineering
$offText

Set i      'possible wire types' / g-1*g-11 /;

Parameter
   b(i)    'allowable wire diameters  (in)' / g-1    0.207
                                              g-2    0.225
                                              g-3    0.244
                                              g-4    0.263
                                              g-5    0.283
                                              g-6    0.307
                                              g-7    0.331
                                              g-8    0.362
                                              g-9    0.394
                                              g-10   0.4375
                                              g-11   0.500  /
   Pload   'preload                   (lb)' /   300    /
   Pmax    'max working load          (lb)' /  1000    /
   delm    'max deflection            (in)' /     6    /
   delw    'deflection from preload   (in)' /     1.25 /
   lmax    'max free length of spring (in)' /    14    /
   coilmax 'max coil diameter         (in)' /     3    /
   S       'max shear stress'               / 189e3    /
   G       'shear modulus of material'      / 11.5e6   /;

Variable
   coil    'coil diameter'
   wire    'wire diameter'
   del     'deflection'
   N       'number of coils'
   C       'constant'
   K       'constant'
   y(i)    'wire choice'
   volume;

Binary  Variable y;
Integer Variable n;

Equation
   voldef  'definition of total volume of coil spring'
   defC
   defK
   shear
   defdel
   freel
   CoilWidth
   defWire
   selectWire;

voldef..     volume =e= (N + 2)/2*PI*coil*wire**2;

DefC..       C =e= coil/wire;

DefK..       K =e= (4*C - 1)/(4*C - 4) + 0.615/C;

shear..      8*Pmax/PI*K*C/wire**2 =l= S;

Defdel..     del =e= 8*N/G*power(C,3)/wire;

freel..      Pmax*del + 1.05*(N + 2)*wire =l= lmax;

CoilWidth..  coil + wire =l= coilmax;

DefWire..    wire =e= sum(i, b(i)*y(i));

SelectWire.. sum(i, y(i)) =e= 1;

del.up  = delm/Pload;
del.lo  = delw/(Pmax - Pload);
wire.lo = smin(i, b(i));
c.lo    = 1.1;
coil.lo = 2*wire.lo;
n.lo    = 1;

Model spring / all /;

* Starting point
y.l(i) = 1/card(i);
wire.l = sum(i, b(i)*y.l(i));
coil.l = coilmax - wire.l;
c.l    = coil.l/wire.l;

solve spring using minlp minimizing volume;