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Geometric design of slider-crank mechanisms for desirable slider positions and velocities

Geometrische Konstruktion von Schubkurbelmechanismen zur Realisierung von gewünschten Stellungen und Geschwindigkeiten des Gleitsteines

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Abstract

In problems requiring uniform cutting speeds uniform flow rates etc. slider-crank mechanisms provide economical solutions. Thus, the problem of designing slider-crank mechanisms for desirable slider positions and velocities is handled within a geometric framework in this paper. Here, the rotation of the crank has been related to the translation of the slider through a linear function. A unified methodology comprising the so-called Subdomain, Galerkin methods together with the classical collocation method is implemented to find the values of the parameters involved. The effectiveness of the methodology has been demonstrated on numerical examples. Since more than one solution results from the solution process, it is always possible to form multi-loop mechanisms. Moreover, designs with low velocity errors in the forward stroke exhibit similar feature in the backward stroke too.

Zusammenfassung

Schubkurbelmechanismen liefern ökonomische Lösungen für Probleme die gleichbleibende Schneidengeschwindigkeiten. Fliessgeschwindigkeiten usw. in bestimmten Stellungen benötigen. Deshalb wird das Problem der Konstruktion von Schubkurbelmechanismen zur Realisierung von gewünschten Stellungen und Geschwindigkeiten in dieser Arbeit im Rahmen der geometrischen Möglichkeiten behandelt. Hier wird die Drehung von der Kurbel auf die Schiebung des Gleitsteines durch eine lineare Funktion bezogen. Um die Werte von existierenden Parametern zu finden, wird eine vereinte Methodologie, die aus der sogenannten Subdomain und Galerkin Methode zusammen mit der klassischen Kollokation Methode bestehen, implementiert. Die Wirksamkeit der Methodologie wird an numerischen Beispielen demonstriert. Am Ende des Lösungsprozesses wird immer mehr als eine Lösung erzielt, so dass sich immer mehrere, mögliche Schubkurbelmechanismen ergeben. Konstruktionen mit geringen Abweichungen von den gewünschten Geschwindigkeiten bei der vorwärtsgerichteten Schiebung zeigen ähnliche Eigenschaften, wie die rückwärtsgerichtete Schiebung.

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Abbreviations

A i ; B i ; B si :

Coefficients

B ci ; C ci ; C i :

Coefficients

C si ; P i ; Q i , i=1−5:

Coefficients

A pL ; A ssL ; As0L:

Intermediate coefficients

A ooL ; A soL , L=1,2:

Intermediate coefficients

a1L ; a00L ; b3L :

Computable constants

b2L ; b1L ; b oL :

Computable constants

c3L ; c2L ; c1L :

Computable constants

c0L , L=1,2:

Computable constants

DEL:

Increment (rad)

f(x):

A function of x variable

G(s,ψ):

Displacement function

R x :

Input scale (rad/m)

R y :

Output scale (m/m)

s :

Position of the slider (m)

s 0 :

Initial position of the slider (m)

w :

Angular speed of crank (rad/s)

w i , i=1−5:

Weighting functions

\(\widehat{xOA_{0}}';\widehat{xOA_{n}}'\) :

Position of crank at the start and end of uniform motion in the backward stroke

x 1 :

Crank length (m)

x 2 :

Connecting rod length (m)

x 3 :

Eccentricity (m)

[x n ,x 0]:

Function interval (m)

[x i−1,x i ], i=1−5:

Subinterval (m)

x i :

Accuracy point (m)

y i :

Function value at accuracy point (m)

y0;y n :

Initial and final values of dependent variable (m)

V :

Linear speed of slider (m/s)

z1;z2;z3:

Design parameters

Δs :

Uniform–velocity region (m)

Δx :

Independent variable range (m)

Δψ :

Amount of crank rotation (rad)

δ :

Connecting rod angle (rad)

ψ :

Crank angle (rad)

ψ0;ψ n ;:

Initial and final crank angle (rad)

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Authors and Affiliations

  1. MACTIMARUM Research and Application Center, Çukurova University, Adana, Turkey

    İ. D. Akçalı

  2. Mechanical Engineering Department, Çukurova University, Adana, Turkey

    M. A. Arıoğlu

Authors
  1. İ. D. Akçalı
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  2. M. A. Arıoğlu
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Correspondence to İ. D. Akçalı.

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Akçalı, İ.D., Arıoğlu, M.A. Geometric design of slider-crank mechanisms for desirable slider positions and velocities. Forsch Ingenieurwes 75, 61–71 (2011). https://doi.org/10.1007/s10010-011-0134-7

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