The equations of an inverted pendulum with an arbitrary number of links and an asymmetric follower force

Abstract

This paper formulates a problem for and gives the differential equations of plane-parallel motion of an inverted n-link pendulum subject to an asymmetric follower force applied at the upper end via a spring. The physical nonlinearities of springs are taken into account. The possible mechanisms of energy dissipation are described Keywords: inverted multilink pendulum, asymmetric follower force, physical nonlinearity Introduction. Unusual and even dangerous oscillatory phenomena in oil pipelines were discovered in the first half of the 20th century. They were believed to be due to the loss of stability of the stationary solution after the oil velocity exceeds a certain (critical) level. To comprehend and explain the phenomenon, Pflüger [17] used Euler's buckling concept developed to determine the critical compressive force on a vertical elastic column. This concept, formulated more than two centuries ago, worked flawlessly for quite a while. However, Pflüger's problem included a follower force, a factor not previously encountered. It is a force that tracks the varying configuration of the system (the compressive force in Pflüger's problem remains tangent to the deformed elastic column).For this very reason the use of Euler's concept produced a paradoxical result: no critical load in Pflüger's problem in spite of intuitive considerations. Ziegler [18] explained the paradox by using a simple double-link pendulum to model the column under a follower force. He showed that besides static instability there may occur dynamic instability. This is how types of instability of elastic systems are now called [1,5,19,20]. There are also other terms: divergent and flutter (oscillatory) instability.The compressive force in the Ziegler pendulum is directed along the last link. It can be resolved into two components: vertical (present in Euler's problem) and horizontal (absent in Euler's problem). The magnitudes of both components depend on the angle of deviation of the last link from the vertical. The horizontal component is responsible for the follower force's being nonpotentional [2]. Japanese authors [7,8,15] have introduced asymmetric follower forces, the asymmetry caused by angular eccentricity (parameters d and k). The effect of this asymmetry on the bifurcations, stability, and catastrophes of a two-link pendulum are studied in [4,9,[11][12][13][14].Besides modeling an elastic column by a double-link pendulum, it is possible to directly study the column as a continuum system [3,6].Though the two-link approximation allowed Ziegler to explain the contradictions in Pflüger's problem, it raised new questions. One is the number of links. It is clear that approximating a curve with a chord (one-link pendulum) is not the best choice. However, the double-link approximation is insufficient too.Thus, we need to study the plane-parallel motion of an n-link pendulum for n = 3, 4, ... and then to compare the results obtained. It would be ideal to solve the problem for an arbitrary n and then to pass to the limit as n ® + ¥. This is how ...

Theory of inverted pendulum with follower force revisited

“…It is not unreasonable to carry out similar studies for double-link, three-link, etc. pendulums to accomplish the research plan outlined in [1,9]. …”

“…Considering a material point À 1 (õ 1 , y 1 ) of mass m 1 , an imponderable rod OA 1 of length l 1 , a viscoelastic joint O and keeping the notation adopted in [1], we have the following differential equation (it follows from [1] when n = 1) for the angle j 1 between the pendulum and the vertical (Fig. 1a where dis the angular eccentricity of the follower force r P; e is the linear eccentricity; k is the orientation parameter of the follower force;…”

“…It would be more natural to write À, j, Ð, Ì, l, õ, ó, m instead of À 1 , j 1 , Ð 1 , Ì 1 , l 1 , õ 1 , ó 1 , m 1 . However, accomplishing the research program outlined in [1], we will compare the dynamic behavior of a one-link pendulum with that of two-three-, ..., n-link pendulums as discrete models of an elastic rod in order to choose the optimal number of links. Therefore, we will use the subscript 1 to refer to the lower link (which is the first one in the general case and the only link in the present paper) of the pendulum.…”

“…Interpreting the parameters of asymmetric follower force as imperfection factors means that they should be assumed small. However, considering Lurie's interpretation of the follower force as a control factor [4], we recognize the necessity to examine the influence of all values (including large) of these parameters.The effect of the angular eccentricity and orientation of the follower force was studied in [1,2,[9][10][11][12], and the effect of the linear eccentricity in [3]. In [2,3], one of the eccentricities (angular [2] or linear [3]) was set at zero.…”

Influence of concurrent use of springs with characteristics of different types on the equilibrium of an inverted pendulum

“…We will use the subscript "1" in m 1 , j 1 , r Ì 1 , l 1 , r Ð 1 , and À 1 to preserve the notational and numbering consistency and convention for series-connected multilink simple pendulums, which go back to the study [3], where the differential equations of plane-parallel motion of such chain systems are also presented. The abscissa and ordinate of the upper end of the pendulum are defined by expressions (1.1) in [3], which have the following form for the point A 1 of a single-link pendulum:…”