A first-order mathematical model has been developed in a theoretical study of the dynamics of a load suspended by a cable from a crane that moves along a straight track. The model is the basis of a proposed method of computer control of the velocity as well as the position of the crane to minimize or prevent swinging of the load. (Traditionally, only the position of the crane is controlled; there is no engineering provision against swinging of the load.) Velocity and position control to prevent swinging would be highly desirable, especially in situations in which there are requirements for precise placement of large loads and/or the delays incurred in waiting for damping of pendulum oscillations of loads are unacceptable. Because most modern cranes are already controlled by computers and include position-indicating control subsystems, the implementation of this method of control would entail little or no additional equipment and thus should be relatively inexpensive.
The mathematical model describes the coupled motions of the crane head, cable, and load, with simplifying assumptions that (1) friction on the cable and load is negligible, (2) the mass of the cable is negligible compared to that of the load, (3) the load acts like a point mass, (4) the length of the cable remains constant during the controlled motion, and (5) the angle between the cable and the vertical is never more than a few degrees, so that the first-order pendulum approximation is justified. The basic equation of the model is
where Vp is the velocity of the crane head (which can be controlled), Vm(t) is the velocity of the load mass, l is the distance (approximately equal to the length of the cable) between the center of mass of the load and the point of attachment of the cable to the crane head, g is the gravitational acceleration, and t is time. Mathematically, the control problem for accelerating (or, equivalently, decelerating) the load without inducing pendulum oscillations is to choose a crane-head velocity function Vp(t) such that the load velocity will be a desired function Vm(t) , subject to the boundary conditions that, at the end of the acceleration and or deceleration interval, both the crane and the load must be at the same horizontal position and velocity and must not be accelerating.
Suppose, for example, that the crane head and the load mass are initially moving together at a constant velocity with the load mass directly under the crane head, and it is desired that the load mass be stopped smoothly at a given location. In principle, there are infinitely many velocity functions Vp(t) that can produce this effect. Essentially all of them start by decelerating the crane head first. This causes the load mass to move in front of the crane head, so that the cable pulls back on the load mass, decelerating it.
Then, after substantial deceleration by the load mass, the crane head is made to move forward faster than the load mass, moving back over the load mass and stopping the deceleration of the load mass just as Vm(t) reaches zero (see figure).
There are many variations on this theme; in all of them, the crane head is made to move back and forth in order to make the cable exert acceleration and deceleration forces on the load mass and the crane head is repositioned to prevent the load mass from swinging or starting any other motion not desired. Of course, one should choose a desired Vm(t) with acceleration and/or deceleration gentle enough that the Vp(t) needed to obtain it is not so jerky and/or does not require so much precision as to lie outside the operational range of the crane motor and its control system.
Some aspects of the theory remain to be addressed. In particular, it would be desirable to generalize the mathematical model to the cases in which the length of the cable changes during the controlled motion, the angle is large enough that the model must include the full nonlinearity of the dynamics, and/or the crane can be moved along either or both of two orthogonal horizontal axes. A further problem to be considered is that of the degree to which the position and velocity of the crane can be measured and controlled and the effect to which limitations in control degrade the desired motion.
This work was done by Robert C. Youngquist, James P. Strobel, and Stan Starr of I-NET for Kennedy Space Center. For further information, access the Technical Support Package (TSP) free on-line at www.nasatech.com/tsp under the Mechanics category. KSC-11942