<content> <p>Unit-proof integrator, Jacobian and Newton Solver with line search</p> </content>

5de53ab5-bae7-43bc-bc8c-7538a076e25f 50 Martin Kraska 3 5 false true 0 7 decimal

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Chain with pendant. Version without Maxima, full unit support

Simple integrator (trapezoidal rule)

f# x# a# b# I b# a# - IsDefined n# 20 : dx# b# a# - n# / : eval p# 1 n# range dx# * dx# 2 / - a# + eval : dx# f# x# p# ≡ at vectorize sum (* eval 41 line f# x# a# b# I if :

x sin x 0 π I 2

Estimate of Jacobian by central differential quotient with step h

R# v# x# h# J J# x# rows 0 matrix : j# 1 x# rows range dx# x# 0 * : dx# j# el h# 2 / x# j# el UnitsOf * : x#.1 x# dx# - : x#.2 x# dx# + : J# J# R# v# x#.2 ≡ vectorize at R# v# x#.1 ≡ vectorize at - h# x# j# el UnitsOf * / augment eval : 51 line for J# 31 line :

Newton-Raphson procedure with line search (seems to be not really needed)

X# NR ls# 0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1110 mat : J# R Vars X# h J : R# R Vars X# ≡ vectorize at eval : dX# J# J# UnitsOf / vectorize (1 -^R# R# UnitsOf / vectorize * eval : dX# dX# X# UnitsOf * vectorize : f ls# 1 el : rmin 109^: j# 1 ls# length range dxj# dX# ls# j# el * : R# R Vars X# dxj# - (≡ vectorize at eval : r j# el R# R# UnitsOf / vectorize norme : r j# el rmin < rmin r j# el : f ls# j# el : 21 line if 41 line for X# dX# f * - r rmin f 14 mat 91 line :

Scale factors to try in the line search

Units are removed from Jacobian and residual before computing the correction of the unknowns

Units of unknowns are applied to correction

Line search. The correction is scaled with a set of factors and the one with the least residual is used.

ropes and their slope

x y.1 h.1 b.1 x x.h.1 - b.1 / cosh 1 - (* + :

x y.2 h.2 b.2 x x.h.2 - b.2 / cosh 1 - (* + :

x y'.1 x y.1 x diff : x x.h.1 - b.1 / sinh

x y'.2 x y.2 x diff : x x.h.2 - b.2 / sinh

Moments of the ropes due to dead weight about left support (just abbreviations)

M.1 G.l x 1 x y.1 x diff (2^+ sqrt * x 0 x.0 I * :

M.2 G.l x 1 x y.2 x diff (2^+ sqrt * x x.0 L I * :

Equations

R H.1 0 y.1 - H.2 L y.2 - x.0 y.1 x.0 y.2 - p S * 1 x y.1 x diff (2^+ sqrt x 0 x.0 I - 1 p - (S * 1 x y.2 x diff (2^+ sqrt x x.0 L I - A.v B.v + G S G.l * + (- A.v x y.1 x diff x 0 ≡ at F.x * - (- B.v x y.2 x diff x L ≡ at F.x * - G F.x x y.2 x diff x x.0 ≡ at x y.1 x diff x x.0 ≡ at - (* - B.v L * x.0 G * M.1 + M.2 + (- 101 sys :

Position of left support

Position of right support

Position of the load

Length of the left part of the cable

Length of the right part of the cable

Global vertical load balance

Reaction of left support tangential to cable

Reaction of right support tangential to cable

Vertical force balance at load point

Global moment balance about left support

Apply the context and convert to vector

R R sys2mat :

H.1 20 cm * :

Elevation of left support

S 40 cm * :

Length of the cable

H.2 20 cm * :

Elevation of right support

G.l 0.7 gf cm / * :

Weight per length of the cable

L 28 cm * :

Horizontal distance of supports

G 10 gf * :

Suspended weight

p init

init 0

1. Activate init=1 and recalculate to reset the solver
2. Set init=0 and press F9 (Recalculate) to perform individual Newton iterations

Variables

Perturbation for Jacobian

Initial guess

Vars h.1 b.1 x.h.1 h.2 b.2 x.h.2 x.0 A.v B.v F.x 101 mat :

h 0.000000001 :

init ~X.0 1 cm * 10 cm * 15 cm * 1 cm * 10 cm * 10 cm * 14.3 cm * 20 N * 20 N * 5 N * 101 mat : if

N ~X.0 NR :

Newton iteration

~X.0 N 1 el :

Modified solution

N 3 el 7.97 109 -^*

Norm of residuals

N 4 el 1

Line search parameter

~X.0 UnitsOf 1 m * 1 m * 1 m * 1 m * 1 m * 1 m * 1 m * 1 kg m * s 2^/ * 1 kg m * s 2^/ * 1 kg m * s 2^/ * 101 mat

Apply solution

h.1 b.1 x.h.1 h.2 b.2 x.h.2 x.0 A.v B.v F.x 101 mat Vars Vars ~X.0 ≡ vectorize at :

x.1 01 0.1 range x.0 * :

x.2 01 0.1 range L x.0 - (* x.0 + :

x.1 x.1 y.1 vectorize augment 1 cm / * x.2 x.2 y.2 vectorize augment 1 cm / * 0 H.1 L H.2 x.0 x.0 y.1 2 cm * - 3 2 mat 1 cm / * o augment x.0 x.0 y.1 x.0 x.0 y.1 2 cm * - 2 2 mat 1 cm / * 4 1 sys

x.0 cm 24.1

N 3 el 7.97 109 -^*

Norm of residuals

A.v gf 14.7

B.v gf 23.3

F.x gf 8.25

Geometric residuals

R 15 range el mm 0.0000 0.0000 0.0000 0.0000 0.0000 51 mat