<content> Linkage Bars </content>

419e1410-1cc5-4bc2-80f3-472223c5ac5a 250 4 5 false false 0 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Linkage bars animations

x y z pAxis 000 x 000000 y 000000 z 00073 mat :

pR(a,b,n) retunrs a vector of equally spaced n+1 points between a and b. a b n pR a b a - n trunc / 0 n trunc range * + eval 11 line :

pR(a,b,n) retunrs a vector of equally spaced n+1 points between a and b. a# b# n# x# pR U# x# Unknowns : F#(U# 1 el num2str): x# num2str concat str2num xo# 01 n# pR : f# xo# F# vectorize : 12 mat a# b# a# - 1 / f# f# min - f# max f# min - / * + eval 41 line :

a b N pRR p a b 0.5 N * pR : p p reverse stack eval 21 line :

View point at az angle from -y axis and el from +x axis. 3x3 Matrix az el pView a az : l el : 12 mat l l 180 ° * + 360 ° * mod 360 ° * + 360 ° * mod 180 ° * - : l 90 ° * > l 180 ° * l - : a a 180 ° * + : 12 mat l 90 ° * - < l 180 ° * - l - : a a 180 ° * + : 12 mat if a a 360 ° * mod 360 ° * + 360 ° * mod : a cos l sin a sin * - l cos a sin * a sin l sin a cos * l cos a cos * - 0 l cos l sin 33 mat eval 51 line :

View point for proyect into 2D plots. az el pView2 az el pView 13 range 12 range el 11 line :

θ E Po Pg J BarSol ρ 1 Pg length range : V Clear PV 0 : PV ρ el V ρ el : P Pg : 15 mat x y z 13 mat Po MCols : po Po rows : 12 mat P e# k 1 Pg rows range x k po + el y k po + el z k po + el 13 mat P 3 k 1 - (* 1 + el P 3 k 1 - (* 2 + el P 3 k 1 - (* 3 + el 13 mat : for S θ x y z BarE : so S length : 12 mat k 1 E cols range m n 12 mat E 1 k el E 2 k el 12 mat : E 3 k el 0 > S k so + el x m el x n el - (2^y m el y n el - (2^+ z m el z n el - (2^+ E 3 k el (2^- : if 21 line for S 41 line : J J ≡ Je# PV e# PV Jacobian : V J# Je# eval : V eq# V e# eval : 13 mat S Pg ρ el 0 TOL eq# J# al_nleqsolve : 21 line S Pg ρ el 0 TOL e# al_nleqsolve : if P ρ el S ρ el S ρ el abs 0.001 TOL * > (* eval : 51 line :

θ D Po Pg BarSol θ D Po Pg 0 BarSol 11 line :

D# N# BarWireFrame f 1 : A 0 : n D# cols : E D# 12 range 1 n range el : m E max eval : 15 mat μ 1010^: s m m matrix eval : d μ 1 s + (m identity - (* eval : 13 mat N N# transpose eval : E E reverse E augment eval : 12 mat f i 12 n * (range j 1 m range d j E 1 i el el d j E 2 i el el ≤ d j E 2 i el el d j E 1 i el el 1 + : E 1 i el j ≡ s j E 2 i el el E 2 i el : s j E 2 i el el s j E 1 i el el : if 21 line f 0 : if for for while r 1 : G E 11 el 11 mat : E 31 el 0 : 13 mat counter 1 n range δ μ : i 12 n * (range d G r el E 1 i el el δ < (E 3 i el 0 ≡ (& δ d G r el E 1 i el el : ι i : 12 mat if for G r el E 1 ι el ≠ (δ μ < (& G r r 1 + : el s G r el E 1 ι el el : while G r r 1 + : el E 2 ι el : E 3 ι el 1 : E 3 ι n + 1 - 2 n * mod 1 + el 1 : 13 mat 41 line for A 03 matrix eval : i 1 r range A A N G i el col transpose stack eval : for A 91 line :

θ D Po Pg J BarPW P 0 : W 0 : 12 mat t 1 θ rows range t 1 ≡ P t el θ t el D Po Pg J BarSol eval : P t el θ t el D Po P t 1 - el J BarSol eval : if W t el D Po P t el stack BarWireFrame eval : 21 line for P W 12 mat 31 line :

θ D Po Pg BarPW θ D Po Pg 0 BarPW 11 line :

t Po P W ko γ BarPlot Q P vectorize eval : Qo Q max vectorize : Qm Q min vectorize : Λ 0 : 14 mat ko 0 ≡ kn 1 Q rows range : kn ko stack Po rows - eval : if k 1 kn length range : Pv Po P t el stack : p 1 Pv rows range : N P rows 1 - : 13 mat 1.001 Qo 1 col max Qo 2 col max Qo 3 col max pAxis * γ * W t el γ * k length 1 ≡ Λ k el Q kn k el 1 el Q kn k el 2 el Q kn k el 3 el augment γ * : Λ k el Q kn k el 1 el Q kn k el 2 el Q kn k el 3 el augment γ * : mat2sys.1 if P t el γ * x augment Po γ * o augment Pv γ * p num2str vectorize 8 darkgreen augment 61 sys 51 line :

t Po P W ko BarPlot t Po P W ko γ 13 range 12 range el BarPlot 11 line :

t Po P W BarPlot t Po P W 0 BarPlot 11 line :

M# MCols C# 0 : c# 1 M# cols range C# 1 c# el M# c# col : for C# 31 line :

Returns the classification and the angles. a b f g Bar4 T g f + a b + (- b g + a f + (- b f + a g + (- 13 mat : A a 2^g 2^+ b f - (2^- 2 a * g * / a 2^g 2^+ b f + (2^- 2 a * g * / a f + (2^b 2^g 2^+ (- 2 b * g * / a f - (2^b 2^g 2^+ (- 2 b * g * / 14 mat : A AC A acos 1 deg / * A abs 1 ≤ Ɇ cases 11 line : α A 1 el AC A 2 el AC A 3 el AC A 4 el AC 14 mat : M 1 crank T 1 el T 2 el * 0 ≥ (T 3 el 0 ≥ (& T 1 el T 3 el * 0 ≤ (T 2 el 0 ≤ (& 2 rocker T 1 el T 2 el * 0 < (T 3 el 0 < (& T 1 el T 3 el * 0 > (T 2 el 0 > (& 3 0 rocker T 1 el T 2 el * 0 ≥ (T 3 el 0 < (& T 1 el T 3 el * 0 > (T 2 el 0 ≤ (& 4 π rocker T 1 el T 2 el * 0 < (T 3 el 0 ≥ (& T 1 el T 3 el * 0 ≤ (T 2 el 0 > (& 44 mat : ans M 13 findrows 2 el α 1 el α 2 el M 14 findrows 2 el α 3 el α 4 el augment : Inp link a θmin θmax Out link b ψmin ψmax 16 mat ans stack 71 line :

Preserve the angle α for the points hkj in the xy variables α x y h k j BarA y h el y k el - (x j el x k el - (* y j el y k el - (x h el x k el - (* - (α cos * y h el y k el - (y j el y k el - (* x j el x k el - (x h el x k el - (* + (α sin * - :

Rotating point θ x y BarR x θ sin * y θ cos * - :

TOL 105 -^:

γ 36 deg * - 153 deg * pView2 :

<content> Examples </content>

N t τ x y z θ ω Clear 1

Four Bar Mechanism

Reference

https://www.researchgate.net/publication/325172892_
Classification_geometrical_and_kinematic_analysis_of_four-bar_linkages/
link/5bce3cdc299bf1a43d9a38ff/download

There are 3^3 = 27 different configuration types considerating movement performed by input link a and
output link b. If a link performs a complete rotation it is called a crank and a rocker if not, with both limits αmin and αmax for the angle.

The case 0-rocker is when not exist αmin and the link oscilates about 0 with amplitude ± αmax.

The case π-rocker is when not exist αmax and the link oscilates about π with amplitude π ± αmax.

a b f g Bar4

returns the classification and the angles.

Double crank

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In this example N + 1 is the numbers of frames in the animation, n the variable for it,
The two first rows in E are the node numbers and the third the lenght for each bar.
Po are the x,y,z coordinates of the fixes nodes and Pg the guess values for the coordinates of the movible nodes.
θo is a vector with N + 1 values for the rotating angle.

N 40 : n 1 N 1 + (range : θo 0 ° * 360 ° * N pR : 13 mat a 1.4 : b 1.2 : f 1 : g 0.5 : 14 mat Po 000 g 0023 mat : Pg 0 b 0 f b 023 mat : 12 mat E 234341 b f a 33 mat : θ x y z BarE θ x 4 el y 4 el BarR z 3 el z 4 el 31 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W BarPlot

a b f g Bar4 Inp link a θmin θmax Out link b ψmin ψmax crank Ɇ Ɇ crank Ɇ Ɇ 26 mat

We can change the speed of θo adding a mapping expression in pR. Also can change the z-values, by fixed values or using τ, ranged between 0 and 1

N 40 : n 1 N 1 + (range : θo 02 π * N x 2^pR : 13 mat a 1.4 : b 1 : f 1.2 : g 0.4 : zo 0.5 : 15 mat Po 000 g 0 zo - 23 mat : Pg g b - 0 a 0023 mat : 12 mat E 234341 b f a 33 mat : τ t 1 - N / : 12 mat θ x y z BarE θ x 4 el y 4 el BarR z 3 el z 4 el zo τ * - 31 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W BarPlot

Double rocker

In this example we use Bar4 for get the angle bounds and construct θo for the forward and retrun paths.

N 60 : n 1 N 1 + (range : θo 29 ° * 88 ° * 0.5 N * pR : 13 mat θo θo θo reverse stack eval : a 1.4 : b 1.2 : f 0.5 : g 1 : 14 mat Po 000 g 0023 mat : Pg 0 b 0 f b 023 mat : 12 mat E 234341 b f a 33 mat : θ x y z BarE θ x 4 el y 4 el BarR z 3 el z 4 el 31 mat : P W 12 mat θo E Po Pg BarPW : 71 line

t Po P W BarPlot

a b f g Bar4 Inp link a θmin θmax Out link b ψmin ψmax rocker 28.0981 88.5675 rocker 60.8236 132.7786 26 mat

Crank - rocker

We add an small zo, which not change the mechanism type, and the angles ψ

N 40 : n 1 N 1 + (range : 12 mat θo 02 π * N e 3 x *^1 - pR : a 0.4 : b 1.2 : f 1.4 : g 1 : zo 0.1 : 15 mat Po 000 g 0023 mat : Pg g b zo a 0 zo 23 mat : 12 mat E 234341 b f a 33 mat : θ x y z BarE θ x 4 el y 4 el BarR z 3 el zo - z 4 el zo - 31 mat : P W 12 mat θo E Po Pg BarPW : 71 line

t Po P W BarPlot

a b f g Bar4 Inp link a θmin θmax Out link b ψmin ψmax crank Ɇ Ɇ rocker 70.5288 126.8699 26 mat

<content> Chebyshev Linkages </content>

Chebyshev Linkage

It's a four bar linkage with the interesting point at the center of bar f

N 60 : n 1 N 1 + (range : θo 37 ° * 103 ° * N pRR : 13 mat f 1 : a 5 f * : b 5 f * : g 4 f * : zo 0.1 : 15 mat Po 000 g 0023 mat : Pg g b zo f b 00 b zo - 33 mat : 12 mat E 23453451 b f f a 34 mat : θ x y z BarE θ x 5 el y 5 el BarR 180 ° * x y 354 BarA z 3 el zo - z 4 el z 5 el zo + 51 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W 4 BarPlot

a b 2 f * g Bar4 Inp link a θmin θmax Out link b ψmin ψmax rocker 36.9 101.5 rocker 78.5 143.1 26 mat

Chebyshev Lambda Linkage

It's a 5 bar based in a previous example of 4 bar.
We introduce also some noise in the node 2 and a small height.

https://en.wikipedia.org/wiki/Chebyshev_lambda_linkage

N 40 : n 1 N 1 + (range : θo 02 π * N pR : 13 mat a 1 : zo 0.3 : τ t 1 - N / : 13 mat Po 00 zo 2 a * 0.2 2 π * τ * sin * 023 mat : Pg a - 00 0.5 a * 2 a * 02 a * 4 a * 033 mat : 12 mat E 12343445 a 2.5 a * 2.5 a * 2.5 a * 34 mat : θ x y z BarE θ x 3 el y 3 el BarR 180 ° * x y 345 BarA z 3 el 0.5 - z 4 el z 5 el 0.5 + 51 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W BarPlot

a 2.5 a * 2.5 a * 2 a * Bar4 Inp link a θmin θmax Out link b ψmin ψmax crank Ɇ Ɇ rocker 78.5 143.1 26 mat

Chebyshev Table Linkage

N 40 : n 1 N 1 + (range : θo 105 ° * 35 ° * N pRR : 13 mat a 1 : b 5 a * : g 4 a * : 13 mat Po 0004 a * 0023 mat : Pg 2 a * a 02 a * 4 a * 04 a * 4 a * 04 a * 3 a * 04 a * 5 a * 053 mat : 12 mat E 12345533745676 2.5 a * 5 a * 2.5 a * 2 a * a a 2.5 a * 37 mat : θ x y z BarE θ x 3 el y 3 el BarR 180 ° * x y 136 BarA 180 ° * x y 657 BarA 31 mat : P W 12 mat θo E Po Pg J BarPW : 61 line

t Po P W 3 5 1 2 mat BarPlot <content> Hoecken linkage - sliding joint </content>

Hoecken linkage

This example shows how to implement an sliding joint, using 0 as distance between nodes in the E matrix.

N 60 : n 1 N 1 + (range : θo 0 ° * 360 ° * N pR : 13 mat a 1 : ν 6 : 12 mat Po 0002 a * 0023 mat : Pg a 00 ν a * 0023 mat : 12 mat E 132344 a 2 a * ν a * + 033 mat : θ x y z BarE θ x 3 el y 3 el BarR 180 ° * x y 324 BarA 21 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W BarPlot <content> Roberts linkage </content>

Roberts linkage

It is a four-bar linkage which converts a rotational motion to approximate straight-line motion

N 60 : n 1 N 1 + (range : θo 30 ° * 97 ° * N pRR : 13 mat a 2 : b 1 : zo 0.2 : 13 mat Po 0002 b * 0023 mat : Pg 3 b * a 0 b 0 zo 0.5 b * a 033 mat : 12 mat E 2345334515 a a a a b 35 mat : θ x y z BarE θ x 5 el y 5 el BarR z 3 el z 4 el zo - z 5 el 41 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W 4 BarPlot

a a b 2 b * Bar4 Inp link a θmin θmax Out link b ψmin ψmax rocker 29.0 97.2 rocker 82.8 151.0 26 mat

<content> Parallel motion linkage </content>

Parallel motion linkage

It is a six-bar mechanical linkage described by James Watt

N 60 : n 1 N 1 + (range : θo 5 ° * - 30 ° * N pRR : 13 mat a 2 : b 1 : zo 0.0 : 13 mat Po 0002 a * 2 b * - 023 mat : Pg 2 a * 002 a * b - 0 a b - 0 a 2 b * - 0 a 0053 mat : 12 mat E 17375467345662 a a 2 b * b b a a 37 mat : θ x y z BarE θ x 7 el y 7 el BarR 180 ° * x y 173 BarA 180 ° * x y 756 BarA 31 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W 4 5 1 2 mat BarPlot <content> Peaucellier–Lipkin linkage </content>

Peaucellier–Lipkin linkage

It is the first true planar straight line mechanism.

N 60 : n 1 N 1 + (range : θo 30 ° * 30 ° * - N pRR : 13 mat a 0.8 : b 2.2 : c 3.2 : zo 0.0 : 14 mat Po 000 a 0023 mat : Pg 2 a * a 0 c 002 c * b 0 c b + 2 b * 043 mat : 12 mat E 11234566434563 c c a b b b b 37 mat : θ x y z BarE θ x 3 el a - y 3 el BarR 11 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W BarPlot <content> Jansen's linkage </content>

Jansen's linkage

It is a planar leg mechanism. Here we use the option "J" for the solver.
Values are from https://github.com/cvigoe/Jansen/blob/master/jansen.pdf

N 40 : n 1 N 1 + (range : θo 0 ° * 360 ° * N pR : 13 mat k.32 4.15 : k.27 3.93 : k.25 4.01 : k.35 5.58 : 14 mat k.56 3.94 : k.67 3.67 : k.68 6.57 : k.78 4.90 : 14 mat k.34 5.00 : k.47 6.19 : k.14 1.50 : λ 0.78 : a 3.80 : 15 mat Po 000 a 0 λ 23 mat : Pg k.14 0 k.34 k.34 - 00 a k.25 + 0 λ a k.25 + 0 λ - a 0 k.27 - a k.25 + 0 k.27 k.78 + (- 63 mat : 12 mat E 2443325667153725767884 k.25 k.34 k.47 k.32 k.35 k.27 k.56 k.67 k.68 k.78 k.14 311 mat : θ x y z BarE θ x 4 el z 4 el BarR 11 mat : P W 12 mat θo E Po Pg J BarPW : 81 line

t Po P W BarPlot

Shigley's Phases of the Foot-Path

<content> Trammel of Archimedes </content>

Trammel of Archimedes

This example shows how to move a 'fixed' node, using the special variable t, which is the variable in the SMath plots for animations (the range is given by n).

Another way to do that is

τ t 1 - N / : Po 02 b * 2 π * τ * sin * 013 mat : 21 line

Usually, the trammel have only one node, but here we add 3.
The last equation in BarE is for force x2 to change the sign.

N 40 : n 1 N 1 + (range : θo 02 π * N pR : 13 mat a 2 : Po 02 a * θo t el sin * 013 mat : Pg 2 a * 00 a 00 0.75 a * 00 1.5 a * 0043 mat : 12 mat E 12333345 a a 0.75 a * 1.5 a * 34 mat eval : θ x y z BarE 0 ° * x y 213 BarA 0 ° * x y 243 BarA 0 ° * x y 253 BarA x 2 el 2 a * θ cos * - 41 mat : P W 12 mat θo E Po Pg BarPW : 61 line

t Po P W 3 4 5 1 3 mat BarPlot

Variation in 3D

N 40 : n 1 N 1 + (range : θo 0 ° * 360 ° * N pR : 13 mat τ t 1 - N / : a 3 : c a : k 1 : κ 145 ° * - cos / abs : b 2 a * κ / : 15 mat b 4 : Po 02 a * θo t el sin * 02 a * θo t el cos * 00002 a * θo t el 45 ° * + cos * - 33 mat : Pg a 00002 b * - 23 mat : E 1233444455 a a 00035 mat : θ x y z BarE z 4 el y 5 el 2 b * θ 0 ° * - sin * - z 5 el 2 b * θ 0 ° * - cos * - x 5 el 41 mat : P W 12 mat θo E Po Pg J BarPW : 91 line

t Po P W BarPlot

d.45 8.54399925677409 :

A P 1 el (: 30000823 mat

B Po t 1 ≡ at eval : 00060000 4.2426 - 33 mat

k n B Po t k ≡ at (: R k el B 3 row A 1 row - norme : 21 line for

R max 6.7082

R min 3

<content> Sewing machine </content>

Sewing machine

Just trial and error for get some stable lenghts.

N 40 : n 1 N 1 + (range : θo 02 π * N pR : 13 mat Po 0001021 - 0533 mat : Pg 0.5 0030 5.5 00 5.5 2 - 0 5.5 10 7.5 2 - 084 - 01073 mat : 12 mat E 1255676894458633109107 0.5 4 3.5 3115 2.5 2.5 5310 mat : θ x y z BarE θ x 4 el z 4 el BarR 170 ° * x z 637 BarA 170 ° * x z 1065 BarA y 4 el 41 mat : P W 12 mat θo E Po Pg BarPW : 51 line

t Po P W BarPlot

Alvaro