04-11-2016, 04:32 PM
1464841022-MMT2009PennockIsrarVariableSpeedTransmission.pdf (Size: 800 KB / Downloads: 5)
abstract
This paper investigates the kinematics of an adjustable six-bar linkage where the rotation
of the input crank is converted into the oscillation of the output link. This single-degree-of
freedom planar linkage will be used as a variable-speed transmission mechanism where
the input crank rotates at a constant speed and the output link consists of an overrunning
clutch mounted on the output shaft. The analysis uses a novel technique in which kinematic
coefficients are obtained with respect to an independent variable. Then kinematic
inversion is used to express the kinematic coefficients with respect to the input variable
of the linkage. This technique decouples the position equations and provides additional
insight into the geometry of the adjustable linkage. The angle through which the output
link oscillates, for each revolution of the input crank, can be adjusted by a control arm. This
arm allows a fixed pivot to be temporarily released and moved along a circular arc about a
permanent ground pivot. The paper shows how to determine the angle of oscillation of the
output link for a specified position of the fixed pivot and investigates the extreme positions
of the output link corresponding to the extreme positions of a point on the coupler link. For
this reason, the paper includes a study of the geometry of the path traced by a coupler
point and determines the location of the ground pivot of the control arm which will cause
the output link to remain stationary during a complete rotation of the input crank. Finally,
the paper shows how the kinematic analysis results can be used, in a straightforward manner,
to redesign the control arm.
. Introduction
This paper focuses on the kinematics and design of an adjustable six-bar linkage which is proposed here as a variablespeed
transmission mechanism. Such mechanisms are used to change the ratio of the input motor speed to the output shaft
speed. Many variable-speed transmission mechanisms incorporate cams, planetary gear arrangements, multiple drive belt
and pulley systems to power a lawn mower or tractor [1], to improve the wheel drive of garden equipment [2], to connect
an air-conditioning compressor to the prime mover of the automobile [3], to improve an exercise driving mechanism [4], and
to control the movement of valves in an engine [5]. The disadvantages of such mechanisms include high manufacturing
costs, high maintenance costs, large housing requirements, shaking vibrations, and slip. Variable-speed transmission mechanisms
based on only linkages can be used to reduce design complexity and can also be superior to alternative designs especially
at high speeds. Horton [6], for example, designed a linkage-type variable-speed transmission mechanism to replace the
cone-and-belt type of drive that was used in some textile machines. Gasoline-driven railroad section cars have employed
adjustable linkages where the continuous rotation of the input crank is converted into an intermittent rotation of the output
shaft. Several of these mechanisms can be used in combination to approximate a continuous rotation to the output shaft. One approach to obtain a variable-speed output is to utilize an adjustable crank–rocker mechanism in which the length of the
crank, or the rocker, is adjusted to obtain the required oscillation range, or timing, of the stroke [7,8]. Another approach
is to relocate the coupler, or follower, joints in order to change the orientation of the mechanism to obtain the range of output
positions for a single rotation of the crank [9,10]. In the latter case, a control arm is commonly designed that adjusts one
of the links to rotate along a circular path. The design of such a control arm is discussed in the present paper.
Basically, the adjustable six-bar linkage presented in this paper and shown in Fig. 1 is a Stephenson III six-bar linkage [11–
13] with an additional link, referred to as the control arm. The input crank is denoted as link 2, the clutch is denoted as link 6,
the coupler link is denoted as link 4, and the control arm is denoted as link 7. The continuous rotation of the input crank is
converted into an oscillation of the clutch mounted on the output shaft. The input crank rotates at a constant speed and the
angle through which the clutch oscillates, for each revolution of the input crank, will be changed manually by moving the
location of the fixed ground pin (or pivot) P of link 5. The position of this pin is changed by unlocking the pin and moving
the control arm to a new location on the circular arc (denoted by positions 1–6 in Fig. 1).
The linkage has two modes of operation; namely: (i) the working mode in which the control arm is locked at an arbitrary
position (i.e., the position of pin P is fixed on the circular arc) and the clutch can oscillate. Since the control arm is locked then
it is a part of the ground link and the resulting linkage is a Stephenson III six-bar linkage, see Fig. 2a; and (ii) the adjustment
mode in which the clutch is locked and the control arm can move along a circular arc about a permanent ground pivot. Since
the clutch is locked then it is a part of the ground link and the resulting linkage is a Watt II six-bar linkage, see Fig. 2b. The
paper begins with a kinematic analysis of the linkage in the working mode (the adjustment mode is the focus of the redesign
or synthesis of the linkage, see Section 4). The analysis is performed using the method of kinematic coefficients [14,15] which
provides a concise description of the geometric properties of the linkage and provides insight into the point trajectory problem.
The method has been applied to the kinematic analysis of a wide variety of single-degree-of-freedom planar mechanisms;
for example, a variable-stroke engine [16], a planar eight-bar linkage [17], and cooperating robots manipulating a
payload [18]. The technique has also been used to study the path curvature of a geared seven-bar mechanism [19] and
the double flier eight-bar linkage [20].
The paper introduces a novel technique to determine the kinematic coefficients of the linkage with respect to an independent
variable. This technique decouples the position equations and gives additional insight into the geometry of the linkage.
In order to obtain kinematic coefficients with respect to the input variable, the paper presents equations for a kinematic
inversion. The paper also investigates the extreme positions of the clutch; i.e., the clutch stops and changes direction of rotation.
The extreme positions of the clutch correspond to stationary positions of a point fixed in the coupler link (i.e., the velocity
of the coupler point is zero). For this reason, the paper includes a kinematic analysis of the coupler point (which for
convenience is chosen here as point B). Again the method of kinematic coefficients is used to provide geometric insight into
the analysis.
The paper also addresses the problem of how to determine the angle through which the clutch will oscillate for a complete
rotation of the input crank. The range of angular displacement of the clutch varies with the position of the fixed ground
pivot (and the control arm) on the circular arc. In order for the linkage to be changed from the working mode to the adjustment
mode, the clutch must dwell for a complete rotation of the crank. This paper investigates the location of the fixed pivot
for the clutch to remain stationary in the working mode. The kinematics literature contains six-link dwell mechanisms that
convert the uniform motion of the input link into periodic, or intermittent, motion of the output link. For example, Sandgren
[21] applied nonlinear programming techniques to design multiple-dwell six-link mechanisms. Kota et al. [22,23] developed
rules of thumb for a systematic analysis and classification of straight-line, circular-arc, symmetrical curves, parallel motion, and dwell mechanisms. The rules were based on path curvature theory and were used to develop computer-aided-design
packages [24,25]. Drawbacks to this approach include: (i) a four-bar linkage was selected and a dyad was designed iteratively
creating computational problems; and (ii) the dwell-link did not, in fact, dwell for a complete rotation of the input crank.
A constant radius of curvature and a constant center of curvature of the path traced by the coupler point of a planar linkage,
during a complete rotation of input crank, indicate that the coupler point is moving on a circular arc. If the location of the
center of curvature of coupler point B, see Fig. 1, is coincident with point D on the clutch then the clutch will remain stationary
for a complete rotation of the input crank. The novel technique presented in this paper will determine the geometric conditions
for the clutch to dwell for a complete rotation of the input crank. The geometry of the path traced by coupler point B
(i.e., the radius of curvature and the center of curvature of the path) is obtained from the method of kinematic coefficients.
Mechanisms with an exact dwell are not possible in practical engineering due to manufacturing tolerances, vibrations, and
backlash. The linkage, obtained here in the redesign, will reduce the change in the angular displacement of the clutch to a
tolerance within 0.01.
The kinematic analysis is performed by differentiating the vector loop equations for the linkage with respect to an independent
variable that will decouple the 4 4 matrix into two 2 2 matrices. Without loss in generality, the independent
variable is chosen to be the angular position of the clutch. The determinants of the two coefficient matrices provide insight
into the conditions for the clutch to remain stationary. A study of the first determinant yields the conditions for a toggle position;
i.e., the configurations of the input crank for the clutch to reach an extreme position. While a study of the second
determinant yields the conditions for the clutch to dwell, for a complete rotation of the input crank. Finally, the paper presents
a redesign of the linkage; i.e., the length of the control arm and the location of the permanent ground pivot are determined,
such that the linkage can operate in the working mode and the adjustment mode by rotating the control arm about
the permanent ground pivot along a circular arc.
The remainder of the paper is arranged as follows: Section 2 presents the kinematic analysis of the adjustable six-bar linkage
in the working mode. A novel technique, that ensures the output link will dwell, is explained in some detail. The section
also presents relationships that convert the kinematic coefficients with respect to an independent variable to kinematic coef-
ficients with respect to the input variable. Section 3 presents a kinematic analysis of the path traced by a point fixed in the
coupler link of the mechanism. The velocity of the coupler point, the radius of curvature and the center of curvature of the
path traced by this point are obtained in terms of kinematic coefficients. The section also shows that two toggle positions
Numerical example
(I) Kinematic analysis. The dimensions of a variable-speed transmission mechanism, with the notation shown in Fig. 3, are
presented in Table 1.
The angle \CDB = b = 15 and the ground links O2O5, O5O6 and O2O6 make angles of 178, 251, and 210, respectively,
with the horizontal X-axis. The location of the fixed ground pivot O5 (or pin P) is O2 O5 = R15 = 55 cm, and the ground pivot
OP is located 40 cm at an angle of 239 from the ground pivot O2. The length of the control arm is 50 cm and the angle the arm
makes with the X-axis is 133.
Three problems are addressed here, namely: (i) the variation in the angular position of clutch link for a complete revolution
of the input crank and the angular variations of links 4 and 5 for a complete revolution of the input crank; (ii) the
radius of curvature and the center of curvature of the path traced by point B; and (iii) for the given control arm and location
of the control arm: (a) the angle of the control arm such that input is a crank; and (b) the angular variation of the clutch for
the fixed ground pivot P positioned along the arc. The procedure to solve problem (i) is to determine the angular position of
links 3, 4, 5 and 6, see Eqs. (3), from the Newton–Raphson iterative technique. The procedure to solve problem (ii) is to determine
the radius of curvature and the center of curvature of path traced by point B from Eqs. (28) and (30). Finally, the procedure
to solve problem (iii) is to repeat problem (i) for different positions of the fixed pivot P. If the geometry of the
mechanism does not permit the input link to be a crank then Eqs. (3) will not converge and a solution will not exist.
Solution to problem (i). A plot of the angular position of the clutch against the position of the input crank is shown in
Fig. 6a. The plot shows that the clutch oscillates from h6 = 236.5 to h6 = 293; i.e., a total angular variation D h6 = 56.5,
for one rotation of the input crank.
Plots of the angular positions of links 4 and 5 against the position of the input crank are shown in Fig. 6b. Since the angular
positions of links 4 and 5 are not the same then the determinant in Eq. (33a) cannot be zero.
Solution to problem (ii). Substituting the given data into Eqs. (10) and (13) and the results into Eqs. (14)–(17) gives the
kinematic coefficients. Then substituting the known values into Eqs. (19)–(28) gives the radius of curvature of the path
traced by point B, see Fig. 7a. Note that the radius of curvature varies from 12.8 cm at the crank position h2 = 228 to
45.2 cm at crank position h2 = 137. The discontinuities, which occur at the crank positions h2 = 57 and h2 = 228, are due
to a change in the direction of rotation of the coupler link 4 at the two toggle positions given by Eqs. (31b). Finally, substituting
the known values into Eqs. (30) gives the center of curvature of the path traced by point B, see Fig. 7b.
Solution to problem (iii). The procedure is to position the control arm from 0 6 h7 6 360 with differentials of Dh7 = 5 and
repeat problem (i). The position of the control arm where the input is the crank and the corresponding angular variation of
clutch is presented in Table 2. The input link is a crank for the control arm positions 130 6 h7 6 160.
(II) Kinematic synthesis. The problem is to redesign the control arm so as to maintain the angular change of the clutch to a
tolerance within 0.01. The dimensions of links 2, 3, 4, 5 and 6 and the location of the ground pivots of links 2 and 6 are as
given in Table 1.
Recall that Section 4 showed that the clutch will dwell when the angle of links 4 and 5 are the same. The first condition
given by Eq. (33b) is used here to satisfy the objective that the clutch will remain stationary for a complete rotation of the
input crank. This is possible when the fixed pivot P coincides with point D as shown in Fig. 5b. The redesigned mechanism is
shown in Fig. 8. In this case, the length of the control arm is 16.78 cm and the location of the ground pivot OP is 48.612 cm at
an angle of 195.3 from the fixed X-axis.
Fig. 8. The redesi