Study of an inclined, segmented cantilever with tip load
The problem of a tip-loaded TV-segment cantilever with torsion springs at the joints is formulated through nonlinear difference equations. The solutions are obtained by both numerical and perturbation methods. As the base is rotated the cantilever shows nonlinear behavior such as nonuniqueness, hysteresis, and catastrophic changes. A stability criterion is proposed. Introduction. Load-bearing segmented rods occur in many situations involving both strength and flexibility. For example, the human limbs and the spine have been modelled as connected rigid rods. Partial rigidity of the system is attained by adequate support at the joints through the tendons. The mechanical analog is represented by the cantilever arm. Of great interest is the configuration of the arm under loading, since exact displacements of the tip are essential in design. The paper is a theoretical study of a very simple model of an "arm" or cantilever. We assume there are N identical rigid segments linked end to end in a plane. In the unloaded state the linked segments, with one end attached to a base, resemble a straight cantilever. There are N linear torsion springs at the joints to produce a certain amount of stiffness to the system. Under a tip weight the cantilever bends as shown in Fig. 1(a). We are interested in the configuration of the cantilever as the base is rotated. The motion is assumed to be slow enough so that dynamics can be ignored. Formulation. Let 6n be the angle of the nth segment with the vertical (Fig. 1(b)). Let the resisting torque of the joints be proportional to the angle differences between adjacent segments with A as the spring constant. A moment balance gives F'esmen + wn+l-dn) = wn-dn_l), (i) where F' is the tip load and £ is the length of the segment. Normalize F' by kf£ and drop primes. Equation (1) yields the nonlinear difference equation 0*+. 2en + en-i + Fsin K = 0. « = 1 t0 N(2) Received September 9, 1992. 1991 Mathematics Subject Classification. Primary 73G99, 73K99, 39A10. ©1995 Brown University 81 82 C. Y. WANG and Q. DU