Abstract:
A mechanism is a constrained kinematic chain composed of
gears, links, cams, or the like. Mechanisms are the building blocks
of all machines, and, as such, their evaluation is of considerable
importance to the mechanical designer. Because the mathematical
analyses of systems with four or more moving members has been
prohibitively complex, the principal method of mechanism evaluation
has been graphical. With the development and widespread distribution
of high-speed digital computer systems, however, mathematical
methods for complex-linkage evaluation have become practicable.
Because of the somewhat universal nature of the digital computer
language called FORTRAN, it is possible for an analyst to develop
a system of analysis which not only he, but any other person as well,
may use. In this paper are developed analytic techniques and computer
programs which evaluate the principal class of mechanisms --
plane motion linkages with a single degree of freedom, employing
turning joints, and having either an angular or translational input
motion. The fundamental premise is that a link may be represented
by a complex vector, and a linkage may be represented by a set of
these vectors in the form of closed polygons. The position vectors,
which are known and are considered to be functions of time, sum to
zero about a closed path. There exist, in a single degree of freedom
linkage, one-half as many independent closed paths as n-Iinks free to
rotate (i.e., links which are neither the crank nor the frame). Therefore,
one-half-n independent vector sums may be written. By separating
the sums into their real and imaginary parts, n independent
equations result. The first and second time-derivatives of the n-set
provides two n-sets of linear algebraic equations. These two sets
are solved, by the computer, for the n-unknown angular rates and
the n-unknown angular accelerations. With these values of angular
kinematic quantities, the computer estimates the angular rotations
over a particular time interval. Through use of a simple iterative
process, these estimated angular values are improved. If the angular
transition is not large, convergence is rapid. After calculation of
angular rates and accelerations in this new position, the process is
repeated, as before, until the desired range of operation of the linkage
has been traversed. Having determined the linkage's angular kinematic values
for a particular position (or instant of time), the computer uses the
values to calculate the translational kinematic data of any specified
point on the linkage. The translational values are placed in an absolute
reference by using the drive-member frame pin as the datum and
summing the translational vector components to the desired point.
The mathematical system of equations and logic are validated
by the successful evaluation of an eleven-bar, ram-drive linkage
system,
It is acknowledged that these methods and the associated
computer programs, although a significant improvement over widely
practiced linkage evaluation methods, are but the first step in the use
of the digital computer in mechanical design. Work must also be done
to develop computer methods to assist the mechanical designers with
dynamic evaluation, stress analysis, bearing loading, space budgeting,
force analysis, and so forth.