- Email: [email protected]
Mechanisms with discontinuous transmission ratios
Mechanism and Mecnine TheorY, 1973. Vot. 8, pp. 365-369.
Pergamon Press.
Printed In Great Britain
Mechanisms with Discontinuous Transmission Ratios S. Molian*
Received 4 November 1972 Abstract A 6-bar linkage with certain dimensions can provide an instantaneous reversal in the sign of the transmission ratio between input and output. By adding further components, it is possible to construct mechanisms that provide discontinuous transmission ratios without reversal, and without using edge-contacts or deformable members. Introduction MECHANISMS that provide discontinuous transmission ratios, such as the cam mechanism shown in Fig. 1, are frequently used. The operation of any mechanism with a discontinuous transmission ratio will set up impulsive forces in that mechanism since the acceleration at the discontinuity is (theoretically) infinite. The application of such forces at an edge-contact, as in the figure, makes these mechanisms prone to wear and limits their use to low speeds. F r e n c h [ l ] has examined the possibility of producing a discontinuous transmission ratio by discontinuity of action of a single kinematic pair, or exchange of function between two pairs or sets of pairs. He concludes that a sudden increase of transmission ratio can be obtained, but that a sudden decrease is not possible without edge-contact, active components or energy storage. It is, however, the latter condition, which is equivalent to a sudden increase in mechanical advantage, which is usually required. A fast approach to a load, followed by a slow action when the load is engaged, is a common requirement, for example in machine tools and presses. Suppose part of a plane mechanism consists of two links AB and BC connected together by a revolute pair at B (Fig. 2). If point A is connected to the frame by a revolute pair and a force applied to point C than, when A, B and C are collinear, the motion is kinematically indeterminate. Point B can move to either of two positions, shown by broken lines in the figure, as C approaches A. What happens depends on the dynamics of the mechanism. In particular, it is possible for B to approach the limiting position of collinearity from one side of the straight line AC and then to reverse back onto the same side. In what follows, a pair of links capable of acting in this way will be called a change-over dyad. If the dyad is incorporated in a mechanism and the change-over occurs instantaneously, the characteristics of the mechanism will also undergo an instantaneous change. All kinematic pairs will remain in action and such a device will therefore be exempt from the limitations mentioned above. A mechanism using this phenomenon will now be described. *Mechanisms and Machine Design Group, Cranfield Institute of Technology, Bedford, England
365
366
Figure 1. Cam mechanism giving an instantaneous change in transmission ratio, but prone to wear and limited to low-speed operation with moderate loads. C
Figure 2. C h a n g e - o v e r dyad. Instantaneous Reversal The 6-bar linkage shown in Fig. 3a, in which A B = A C and B D = C D . incorporates the change-over dyad as the sub-chain A C D . The input member A B is driven in a clockwise direction. The output member A C rotates in a counter-clockwise direction until the mechanism reaches the configuration shown in Fig. 3b, where B and C coincide. In this configuration the output motion is kinematically indeterminate. Depending on the dynamics of the mechanism, A C will either continue to rotate counter-clockwise or will reverse instantaneously and rotate clockwise with A B , as shown in Fig. 3c. In the latter case the transmission ratio ~)/0' will change instantaneously from - 1 to + 1. This can be proved as follows. A vector-loop analysis (or reasoning from a velocity diagram) applied to the 4-bar chain A B D E in Fig. 3a shows that (b/O = A B sin(/? -- O ) / D E sin (/3-- ~).
(1)
Similarly, considering the 4-bar chain A C D E (010' = A B sin (/3' - 0 ' ) / D E sin (/3' -- 0).
(2)
Combining these results we have 0/0' = sin (13 - 0) sin (B' -- 0')/sin (/3' -- qJ) sin (/3 -- 0). But, because A B D C is symmetrical about the line A D .
(3)
367 /3' - O' = - (/3 - O) + 360 ~
(4)
and therefore 0i0' = - sin (13 - e)/sin (/3' - qJ).
(5)
Thus, when the mechanism approaches the change-over configuration and /3' ~ / 3 ; the limiting value of the transmission ratio is - I. As soon as the change-over occurs, however, the transmission ratio b e c o m e s + 1.
Z~~,-~i
...... ~/,HI~/I~E-~~L (a)
(b)
Figure 3.
Cc) Six-bar mechanism incorporating a change-over dyad and giving instantaneous reversal of output motion.
Instantaneous Change of Transmission Ratio without Reversal The m e c h a n i s m shown in Fig. 3 provides two angular velocities, that of link A B which is rotated continuously in the same sense, and that of link A C which undergoes an instantaneous reversal. By using some form of differential mechanism to combine these two angular velocities we can obtain a uni-directional motion with an instantaneous change in velocity or angular velocity. T w o such m e c h a n i s m s are shown in Fig. 4: A mechanism combining the 6-bar linkage with a differential pulley (Fig. 4a)
368
I
(b)
Figure 4. Mechanisms giving instantaneous change in transmission ratio without the use of edge-contacts. and another using additional links to make up a 10-bar linkage (Fig. 4b). The latter, since it uses only lower pairs, is perhaps of greater interest and a model (Fig. 5) was constructed. In both these mechanisms there is an instantaneous decrease in output velocity, and therefore an instantaneous increase in mechanical advantage, as the 6-bar chain passes through the change-over configuration. The model (Fig. 5), when placed in a vertical position and operated manually at slow and moderate speeds, consistently exhibited the desired change-over. When operated quickly, or placed in an inverted position, the change-over did not occur. A qualitative explanation can easily be given. T h e input link AB drives link ED and the latter in turn drives A C by attempting to pull A C and CD into a straight line. At slow speeds, when only static forces need be considered, this alignment is impossible because it would set up infinite forces in the links. In practive, bearing clearances and slight elastic effects enable point C to remain slightly below the line AD until B coincides with C and the desired configuration (Fig. 3c) occurs. At high speeds the inertia of links A C and CD, and of the attached differential linkage, will overcome the gravitational forces and enable A C to rotate continuously through the change-over configuration. When the mechanism is inverted the gravitational forces will themselves cause this to occur. Apart from this inconsistent operation, the mechanism is unsatisfactory in this basic form because large forces occur near the changeover. These difficulties can probably be o v e r c o m e by providing a counter-weight or spring acting on A C to bring this link into line with CD, together with a striking plate attached to BD and acting on CD to ensure the change-over. Experimental work on these lines is now being undertaken.
369
Figure 5. Model of a mechanism giving instantaneous change of transmission ratio and using only revolute pairs. Ackm~wledgements-The work described here is part of a general programme of research into the theory and design of mechanisms financed by a grant from the Science Research Council. The model shown in figure 5 ~vas designed and constructed by Mr. J. Heathorn.
Reference [11 FRENCH M. J., Force and corner-power relationships in two regime and many degree of freedom mechanisms. I. Mech. E. Conference on Mechanisms, Paper C82/72, pp. 64-67, September 1972.
Mecanismes a Rapport de Transmission Discontinus
S. Molian R e s u m e - Un changement soudain du rapport de transmission est generalement effectue par un mecanisme utilisant le contact par glissement (Fig. 1), mais de tels mecanismes subissent une certaine usure qui interdit leur utilisation & veitsses elev6es et couples forts. L'auteur propose I'alternative suivante, utilisant la propriet6 instable des deux tiges que montre la Fig. 2. Ces deux organes font partie d'un assemblage & 6 barres que montre la Fig. 3; la tige AB est entrafne & vitesse constant et la tige AC se renverse instantanement. En combinant les mouvements de ces deux tiges on peut obtenir un mecanisme qui fournit un changement de velocite instantan& Des exemples de tels mecanismes sont donnes en Fig. 4 et une photographie de la maquette de I'un d'eux apparait en Fig. 5. M M . T . Vol. 8 No. -t.-F
Pergamon Press.
Printed In Great Britain
Mechanisms with Discontinuous Transmission Ratios S. Molian*
Received 4 November 1972 Abstract A 6-bar linkage with certain dimensions can provide an instantaneous reversal in the sign of the transmission ratio between input and output. By adding further components, it is possible to construct mechanisms that provide discontinuous transmission ratios without reversal, and without using edge-contacts or deformable members. Introduction MECHANISMS that provide discontinuous transmission ratios, such as the cam mechanism shown in Fig. 1, are frequently used. The operation of any mechanism with a discontinuous transmission ratio will set up impulsive forces in that mechanism since the acceleration at the discontinuity is (theoretically) infinite. The application of such forces at an edge-contact, as in the figure, makes these mechanisms prone to wear and limits their use to low speeds. F r e n c h [ l ] has examined the possibility of producing a discontinuous transmission ratio by discontinuity of action of a single kinematic pair, or exchange of function between two pairs or sets of pairs. He concludes that a sudden increase of transmission ratio can be obtained, but that a sudden decrease is not possible without edge-contact, active components or energy storage. It is, however, the latter condition, which is equivalent to a sudden increase in mechanical advantage, which is usually required. A fast approach to a load, followed by a slow action when the load is engaged, is a common requirement, for example in machine tools and presses. Suppose part of a plane mechanism consists of two links AB and BC connected together by a revolute pair at B (Fig. 2). If point A is connected to the frame by a revolute pair and a force applied to point C than, when A, B and C are collinear, the motion is kinematically indeterminate. Point B can move to either of two positions, shown by broken lines in the figure, as C approaches A. What happens depends on the dynamics of the mechanism. In particular, it is possible for B to approach the limiting position of collinearity from one side of the straight line AC and then to reverse back onto the same side. In what follows, a pair of links capable of acting in this way will be called a change-over dyad. If the dyad is incorporated in a mechanism and the change-over occurs instantaneously, the characteristics of the mechanism will also undergo an instantaneous change. All kinematic pairs will remain in action and such a device will therefore be exempt from the limitations mentioned above. A mechanism using this phenomenon will now be described. *Mechanisms and Machine Design Group, Cranfield Institute of Technology, Bedford, England
365
366
Figure 1. Cam mechanism giving an instantaneous change in transmission ratio, but prone to wear and limited to low-speed operation with moderate loads. C
Figure 2. C h a n g e - o v e r dyad. Instantaneous Reversal The 6-bar linkage shown in Fig. 3a, in which A B = A C and B D = C D . incorporates the change-over dyad as the sub-chain A C D . The input member A B is driven in a clockwise direction. The output member A C rotates in a counter-clockwise direction until the mechanism reaches the configuration shown in Fig. 3b, where B and C coincide. In this configuration the output motion is kinematically indeterminate. Depending on the dynamics of the mechanism, A C will either continue to rotate counter-clockwise or will reverse instantaneously and rotate clockwise with A B , as shown in Fig. 3c. In the latter case the transmission ratio ~)/0' will change instantaneously from - 1 to + 1. This can be proved as follows. A vector-loop analysis (or reasoning from a velocity diagram) applied to the 4-bar chain A B D E in Fig. 3a shows that (b/O = A B sin(/? -- O ) / D E sin (/3-- ~).
(1)
Similarly, considering the 4-bar chain A C D E (010' = A B sin (/3' - 0 ' ) / D E sin (/3' -- 0).
(2)
Combining these results we have 0/0' = sin (13 - 0) sin (B' -- 0')/sin (/3' -- qJ) sin (/3 -- 0). But, because A B D C is symmetrical about the line A D .
(3)
367 /3' - O' = - (/3 - O) + 360 ~
(4)
and therefore 0i0' = - sin (13 - e)/sin (/3' - qJ).
(5)
Thus, when the mechanism approaches the change-over configuration and /3' ~ / 3 ; the limiting value of the transmission ratio is - I. As soon as the change-over occurs, however, the transmission ratio b e c o m e s + 1.
Z~~,-~i
...... ~/,HI~/I~E-~~L (a)
(b)
Figure 3.
Cc) Six-bar mechanism incorporating a change-over dyad and giving instantaneous reversal of output motion.
Instantaneous Change of Transmission Ratio without Reversal The m e c h a n i s m shown in Fig. 3 provides two angular velocities, that of link A B which is rotated continuously in the same sense, and that of link A C which undergoes an instantaneous reversal. By using some form of differential mechanism to combine these two angular velocities we can obtain a uni-directional motion with an instantaneous change in velocity or angular velocity. T w o such m e c h a n i s m s are shown in Fig. 4: A mechanism combining the 6-bar linkage with a differential pulley (Fig. 4a)
368
I
(b)
Figure 4. Mechanisms giving instantaneous change in transmission ratio without the use of edge-contacts. and another using additional links to make up a 10-bar linkage (Fig. 4b). The latter, since it uses only lower pairs, is perhaps of greater interest and a model (Fig. 5) was constructed. In both these mechanisms there is an instantaneous decrease in output velocity, and therefore an instantaneous increase in mechanical advantage, as the 6-bar chain passes through the change-over configuration. The model (Fig. 5), when placed in a vertical position and operated manually at slow and moderate speeds, consistently exhibited the desired change-over. When operated quickly, or placed in an inverted position, the change-over did not occur. A qualitative explanation can easily be given. T h e input link AB drives link ED and the latter in turn drives A C by attempting to pull A C and CD into a straight line. At slow speeds, when only static forces need be considered, this alignment is impossible because it would set up infinite forces in the links. In practive, bearing clearances and slight elastic effects enable point C to remain slightly below the line AD until B coincides with C and the desired configuration (Fig. 3c) occurs. At high speeds the inertia of links A C and CD, and of the attached differential linkage, will overcome the gravitational forces and enable A C to rotate continuously through the change-over configuration. When the mechanism is inverted the gravitational forces will themselves cause this to occur. Apart from this inconsistent operation, the mechanism is unsatisfactory in this basic form because large forces occur near the changeover. These difficulties can probably be o v e r c o m e by providing a counter-weight or spring acting on A C to bring this link into line with CD, together with a striking plate attached to BD and acting on CD to ensure the change-over. Experimental work on these lines is now being undertaken.
369
Figure 5. Model of a mechanism giving instantaneous change of transmission ratio and using only revolute pairs. Ackm~wledgements-The work described here is part of a general programme of research into the theory and design of mechanisms financed by a grant from the Science Research Council. The model shown in figure 5 ~vas designed and constructed by Mr. J. Heathorn.
Reference [11 FRENCH M. J., Force and corner-power relationships in two regime and many degree of freedom mechanisms. I. Mech. E. Conference on Mechanisms, Paper C82/72, pp. 64-67, September 1972.
Mecanismes a Rapport de Transmission Discontinus
S. Molian R e s u m e - Un changement soudain du rapport de transmission est generalement effectue par un mecanisme utilisant le contact par glissement (Fig. 1), mais de tels mecanismes subissent une certaine usure qui interdit leur utilisation & veitsses elev6es et couples forts. L'auteur propose I'alternative suivante, utilisant la propriet6 instable des deux tiges que montre la Fig. 2. Ces deux organes font partie d'un assemblage & 6 barres que montre la Fig. 3; la tige AB est entrafne & vitesse constant et la tige AC se renverse instantanement. En combinant les mouvements de ces deux tiges on peut obtenir un mecanisme qui fournit un changement de velocite instantan& Des exemples de tels mecanismes sont donnes en Fig. 4 et une photographie de la maquette de I'un d'eux apparait en Fig. 5. M M . T . Vol. 8 No. -t.-F