Classical transmission-angle problem for slider–crank mechanisms

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Abstract

The classical transmission-angle problem for slider–crank mechanisms is the determination of the dimensions of planar slider–crank mechanisms with optimum transmission angle for given values of the slider stroke and corresponding crank rotation. In this study complex algebra is used to solve this classical problem. The solution is obtained as the root of a cubic equation within a defined range, which can be easily implemented on a computer.

Introduction

Transmission-angle optimization of planar four-bar linkages was one of the synthesis problems solved by Alt [1], improved by Meyer zur Capellen [2] and Volmer [9] and published by the VDI. (Verein Deutscher Ingenieure) as guideline No. 2130 [3]. Freudenstein and Primrose [4] obtained a closed form analytical solution which was determined as the root of a cubic equation. Meyer zur Capellen [2] and Volmer [5] treated a similar problem for planar slider–crank mechanisms graphically and the result are given in nomogram forms in VDI guideline No. 2132 [6]. In [2], [5] the link lengths are expressed using the initial crank angle as the parameter.

For spatial four-bar and slider–crank mechanisms, Söylemez and Freudenstein [7] used algebraic methods to determine the loci of the moving pivots and a computer-aided optimization algorithm for the transmission ratio optimization. Nomograms similar to Alt charts have been prepared for simply skewed four-bar mechanisms. In case of right angled four-bar mechanisms (in which the input and output links are at right angles) similar transmission ratio optimization has been performed using a single design parameter and for symmetric right angled four-bar, constant transmission ratio can be obtained [8].

In the present study, for planar slider–crank mechanisms the loci of the fixed and moving pivots of the crank and the link lengths are expressed in terms of a single parameter. For the full rotatability of the crank, the ranges for the parameter are determined. The transmission-angle optimization is analytically treated and the optimum is found as the root of a cubic equation within a specified range.

Section snippets

Dead centers of slider–crank mechanisms

In Fig. 1 a planar slider–crank mechanism is shown. The link lengths are a=A0A (crank); b=AB (rocker) and c is the eccentricity (c>0). For a certain input crank angle θ the slider is at a position x and the transmission angle is μ.

The dead centers of the slider–crank mechanism are when the crank and the coupler links are collinear as extended (A0AeBe) or folded (A0AfBf) forms (Fig. 2). The stroke s=BeBf is the total displacement of the slide while the crank rotates by an angle φ between dead

Ranges of φ and λ

According to Grashof's rule, a slider–crank mechanism with a full rotatable crank must satisfy the following two inequalities:b⩾aandb−a⩾c.Using , , these conditions yield the ranges for φ and λ as:π2⩽φ⩽tan−1−1c,1tan2φ/2⩽λ⩽1.

Transmission-angle optimization

The transmission angle μ at any input crank angle θ is given by:μ=cos−1c+asinθb.

The minimum transmission angle is when θ=π/2:μmin=cos−1c+ab.Expressing μmin in terms of λ and φ:cosμmin=λ+12(1−λ2)sin(φ)[(1+λ2)+(1−λ2)cos(φ)]1/2since λ is a free design parameter, the necessary condition for the minimum transmission angle to be a maximum is dμmin/dλ=0. If the value of λ which makes the derivative equal to zero is λ=λopt, differentiating Eq. (20) and setting dμmin/dλ=0 yields:2[λopt2(1−cosφ)+(1+cosφ)]

Discussion and conclusion

Instead of charts, diagrams and graphical methods for the synthesis of mechanisms, use of exact analytical methods is believed to be an important tool for the present design approach. The method presented for the slider–crank mechanism is hoped to give a faster and more accurate result for today's mechanism designers.

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