Design of the crank-rocker mechanism with unit time ratio

Jnl. MechanismsVolume 5. lap.l-4/Pergamon Press1970/Printed in Great Britain

Design of the Crank-Rocker Mechanism with Unit Time Ratio Graduate Student R, Joe Brodell and Assist. Prof. A. H. Soni* Received 31 July 1969 Abstract A planar crank-rocker mechanism is designed for unit time ratio using the design criterion of minimum transmission angle. Explicit relationships providing the dimensions of crank-rocker mechanism are derived and a design chart for a practical range of the the transmission angle is presented. Zusammenfassung--Entwurf eines Kurbelschwing-getriebes mit Einheits-zeit Verh~iltniss: R. J. Brodell und A. H. Soni, Ein ebenes Kurbelschwing-Getriebe wird entworfen for ein Einheits-Zeit Verh/iltniss unter Benutzung des Kriteriums eines minimalen Ubertragungswinkels. Ausdr/Jckliche Verh~iltnisse, die die Abmessungen des Getriebes liefern, sind abgeleitet und Entwurfs-Tabellen fiJr praktische Grenzen des 0bertragungswinkels aufgestellt. Pe3mMe~HpocrrHpoea~He xpmJomm~o-xopo~mlcnoearo MexamaaMa~ ornomerma e ~ m m m a BpeMetm: P. I;polxenux A. X. Corm. 1-IHocKHt~ r ,mOmmmO-KOpOMMCHOBM~R MeXaHH3M cnpocrrHpoeaH ~r~ orrmmerm~te ~ epeMem,~ Hcnom=3y~ ..p~-reptLttMmmMambHaro yrna nepe~a~m. BMpaxemuq oTaomenm~, ZOTOpr-le ]laIoT paaMepM MeXaHIi3Ma, B/a, mc/XeHMH IIpOeKTHpOSOqHble Ta6HHI~,I COCTaBHeH~[~ nparrw~cczax n ~ e ~ o s

yrna nepeaa~. THE PRESENT PAPER is aimed to provide design information for designing a planar crankrocker mechanism with unit time ratio and minimum transmission angle. Such design procedures are utilized by Hain [1], Soni and Harrisberger [2], Volmer [3] and Chen [4]. A planar four-link crank-rocker mechanism, MABQ, is shown in Fig. 1, with link MA as the input link and QB as the output link. As shown in Fig. 2, note MAIB1Q and MA2B2Q are the limit positions of the crank-rocker mechanism. As the input link, MA, moves from the position MAt to MA 2, the output link rotates forward through an angle q~. Further rotation of the input link from the position MA2 to MA~ causes the output link to return from position QB2 to its initial position QBI. The time ratio, as described by Hall [5] is given by

TR-Imput crank rotation for forward stroke Imput crank rotation for reverse stroke 180+= 180-ct"

(1)

When the time ratio becomes unity, ~ becomes zero and the forward and the reverse strokes require 180° rotation of the input crank. The crank rocker mechanism with unit time ratio is designed using the design criterion of minimum transmission angle. The transmis* Oklahoma State University, Stillwater, Okla. 74074, U.S.A.

A

M~ ~''~ - d Figure 1. Four-link

o crank-rocker mechanism. FOti.W~D.

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Figure 2. Limit positions of crank-rocker mechanism. sion angle defined by the angle between the coupler link AB and the output link QB becomes minimum when the input crank is in line with the fixed link MQ and A lies toward Q. The transmission angle becomes maximum when the input crank is in line with the fixed link and A lies away from Q. These conditions are described in Fig. 3. The crank-rocker mechanism is designed to satisfy the condition. /~m~,= 180-/Zm~~.

(2)

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Figure 3. Crank-rocker mechanism in its minimum and maximum transmission angle positions. Application of the Cosine-Law to triangles MA3B3Q and MA,,B,,Q in Fig. 3 yields b 2 + c 2 --(d - a) 2 Cos/.z,~ia -

2bc

(3)

Cos/~ma~-

b2 +c2-(d+a) 2 2bc

(4)

where MA=a, AB=b, BQ= c, and QM=d. Application of the Cosine-Law to the triangles M,42B 2 {2 and MA tBt {2 in Fig. 2 yields Cos 0z - c2 + d 2 - (a + b) 2

2cd

(5)

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(6)

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~0 O1- 02.

(7)

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Utilizing the conditions given by equations (2), (5) and (6), equations (3), (4) and (7) are solved simultaneously to yield

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Design chart of crank-rocker mechanism with optimum transmission angle. Note that for crank-rocker mechanism obtained from the above design chart, the output-crank completes the forward stroke in 180 ° rotation of the input-crank.

Figure 4 presents the design chart to design crank-rocker mechanism for a given set of design parameters gmi, and cp. For a good "quality" of motion, it is recommended that the transmission angle/~m~, be greater than 30 °. For higher speed, a larger value of the transmission angle is desired. Conclusion

The design criterion of minimum transmission angle is utilized to design a planar crankrocker mechanism with unit time ratio. Such design criterion of minimum transmission angle has proved to be efficient for the other types of planar four-link mechanisms. It is interesting to note that satisfying the condition given by equation (2) satisfies the condition that the time ratio for the designed crank-rocker mechanism is unity. Acknowledgement--Th© present paper was presented at the Applied Mechanism Conference, held July 31

and August I, 1969, and sponsored by Oklahoma State University. This paper is released for publication by the directors of the conference, Dr. A. H. Soni and Dr. Lee Harrisberger. R e f e r e nces

[1] H~dNK. How to apply ¢hag-link mehcanisms in the synthesis of mechanisms. Transactions of the Fourth Conference on Mechanisms, pp. 66-75 (1967). [2] So~l A. H. and H~m~,ISBEV, OER LEE. The design of the spherical drag-link mechanisms. Trans. ASME, J. Engng Ind., Series B, 89, 177-181 (1967). [3] Vot.~R. Dic Konstruktion yon Gclenkvierecken mit Hilfe yon Kurvcntafcln. Maschinenbautechnik 5, H.6.S.319.128. (1956). [4] C,¢rENF. Y. An analytical method for synthesizing the four-bar crank-rocker mechanism. Trans. ASME, J. Engng Ind., Series/391, 45-54 (1969).