01-10-2012, 11:15 AM
Elementary Kinematics from an Advanced Standpoint
Elementary Kinematics.pdf (Size: 113.08 KB / Downloads: 30)
INTRODUCTION
At the beginning of the 20th century the celebrated German
mathematical physicist Felix Klein (1849-1925)
presented a series of lectures to high-school math and
science teachers during the Christmas and Easter periods
when there were no classes. The intent was to make
the teachers aware of the necessity for them to be familiar
with their subjects at the highest possible level of
authority and rigor so as to be able to explain the material
in the clearest, most effective way. His lecture notes
were later published as books and translated to English
[1, 2]. The two volumes are called “Elementary Mathematics
from an Advanced Standpoint” and subtitled
“Geometry” and “Arithmetic, Algebra and Analysis”,
respectively. These continue to provide inspiration. It
is in this vein that the following simple example in planar
kinematics is treated using the projective geometry
of point, line and conic and a common substitution to
convert trigonometric to algebraic functions.
INCLINED “SCOTCH YOKE”
Figure 1 shows a “Scotch yoke” mechanism wherein a
slotted slider is driven by crank-pin P which rotates
about centre O. The block which slides in the slot is attached
to the crank-pin with a revolute joint. Distance
OP is r. The crank angle is µ measured counterclockwise
positive from a line o on O and in the direction
of slider displacement. The motion of P relative to
the slot is along line b inclined at angle ¯ to o. This
mechanism can be described as an RRPP closed, single
loop chain; a four-bar mechanism variant. Consider the
following aspects of the kinematics of this device.
POLARITY AND DISPLACEMENT LIMITS
It is shown in Figure 1 that the limits of slider displacement
b1 are established where line b is tangent to the
circle traced by the motion of crank-pin P. One could
simply intersect a line n, on O and normal to b, with
the circle and define where the two parallel lines b0 and
b00 intersect o. However if the locus of P were a general
conic, instead of an origin centred circle of given radius,
then the method outlined below would be quite useful
to obtain n. This makes use of the so-called conic polarity
relationship which defines the planar line polar
to any point in the plane and with respect to a conic
at hand.
CONCLUSION
The three topics introduced here in the context of a
simple kinematic analysis, viz., projective geometry,
trigonometric to algebraic conversion and the theory
of conics, were not the easiest way to solve the problem
posed. Nevertheless by introducing these methods
along with an easy-to-follow example it is hoped
that the reader sees how to use them to formulate robust
constraint equations which lead to clear solutions
amenable to efficient computation and unambiguous results
in other, more complicated situations. Although
not treated herein, design problems lurk in the background.
I.e., how to choose r and ¯ to fulfill specified
kinematic performance? This mechanism admits two
design variables. What opportunities does this present
to the designer?