Influence of material and geometrical nonlinearities on the bifurcations of equilibrium states of a two-link pendulum

Abstract

The equilibrium states of an inverted two-link simple pendulum with an asymmetric follower force are classified depending on the characteristics of the springs (hard, soft, or linear) at the upper end and at the hinges. Phase portraits are plotted. The bifurcation points on the equilibrium curves are identified. Emphasis is on fold and cusp catastrophes Keywords: two-link inverted simple pendulum, asymmetric follower force, spring characteristics of different typesIntroduction. In the mid-20th century, increased speeds of oil piping occasioned dangerous oscillatory phenomena in the pipe-fluid system. In particular, the unexpected vibrations of the Transarabian oil pipeline discovered in 1950 were described by Paidoussis and Issid in 1974 [23, 26]. Experts associated them with the buckling of a compressed slender column. Pflüger [22] tried to find the critical compressive force based on Euler's concept of buckling for a slender column compressed at the ends by longitudinal forces (active one and subgrade reaction). The result appeared paradoxical: no critical force existed in spite of what intuition obviously suggested. Ziegler [27] was able to explain the paradox. He modeled a slender column by a discrete system-a two-link pendulum. Certainly, the disagreement between Pflüger's and Ziegler's results is not due to the discrete approximation of a continuous system. This, in particular, follows from stability analyses of elastic pipelines where their continuous nature is taken into account [3,10]. It is after Ziegler [9, 28] that static and dynamic stability began to be distinguished in the theory of stability of elastic systems [1,2,9].A distinctive feature of the problem solved by Pflüger [22] and Ziegler [27] is the presence of a so-called follower force that remains tangential to the curved (curvilinear) axis of the column. The specific behavior of mechanical systems with follower forces is due to the fact that these forces (like the forces of radial correction in gyroscopy) are represented by a skew-symmetric matrix in the linearized equations of perturbed motion. In arbitrary elastic systems, such forces follow any changes in their configuration. By this definition, the cornering forces acting on tires can also be attributed to follower forces.Koiter [15] doubted whether follower forces are realistic. Sugiyama, Langthjem, and Ryu [25] held the opposite opinion. The philosophical sense and physical feasibility of follower forces were discussed by Elishakoff in the review [11]. Japanese scientists [13,14,20] introduced angular eccentricity to model manufacturing imperfections of pipes and inaccuracy in the application of compressive loads. Linear eccentricity is possible too [4][5][6][7]. It was shown in [8] that as the angular eccentricity of the follower force is smoothly varied, a two-link pendulum may jump from one stable equilibrium state to another, even if its springs are linear. The statement of dynamic problems for chain systems consisting of series-connected simple pendulums was extended in [19] to in...

Bifurcations of a limit cycle in nonlinear dynamic systems

“…An approximate solution of the limit cycle equations is found through a qualitative analysis Keywords: domain of periodic solutions, bifurcation, drift, skew-symmetry principle Introduction. Many recent publications pay much attention to the bifurcations of the equilibrium states of mechanical systems [7,8]. The study [13] is concerned with modeling the interaction between a rigid body and a medium and with the symmetry and bifurcations of the vector field of dynamic systems.…”

Stability domains of the vertical equilibrium state of a triple simple pendulum

“…The problem formulation for a chain of pendulums was extended in [19] to include physical nonlinearity: in addition to linear springs connecting links, springs with hard (tangensoid) and soft (arctangensoid) characteristics were introduced with analytic description of relevant experimental results. The results of [12,19] were used in [13][14][15][16][17]. When the so-called significant parameters are varied, qualitative changes such as bifurcations and catastrophes occur in the manifold of equilibrium states of a double pendulum [4][5][6]18].…”

“…(2.2) is zero remains in the domain of practically feasible values of c and P. If c is kept constant and P is smoothly increased, the representative point goes from the asymptotic-stability domain D(6, 0) to the domain D(5, 1) and then to D(4, 2).4. Decomposition of System (1.1) in Degenerate Cases.If we set m 3 = 0, l 3 = 0, c 3 = 0, m 3 = 0, and j 3 = j 2 in the differential equations of motion of a triple pendulum when e = 0, then the third equation in (1.1) will become an identity and the first two equations will go over into the equation of perturbed motion of a double pendulum given in[17]. If we set j 2 = j 3 = j 1 in the specific case of m 2 = m 3 = 0, l 2 = l 3 = 0, c 2 = c 3 = 0, m 2 = m 3 = 0, then the first equation in (1.1) will become the equation of motion of an inverted single pendulum[13] and the last two equations in (1.1) become identities.…”

Asymptotic stability boundaries for the equilibrium positions of a double pendulum with vibrating point of suspension

“…Studying the oscillations of multilink pendulums is a task of current importance in theoretical mechanics [6]. Significant results on the theory of oscillations of pendulum systems subject to various forces were obtained in [2,3,6,[9][10][11][12][13]. The energy of a double pendulum on the boundary of complex and simple motions was calculated in [12].…”

“…Significant results on the theory of oscillations of pendulum systems subject to various forces were obtained in [2,3,6,[9][10][11][12][13]. The energy of a double pendulum on the boundary of complex and simple motions was calculated in [12].A classical study on the dynamics of a pendulum with vibrating point of suspension is [2], which established the fundamental possibility of stabilizing the upper equilibrium position by subjecting the point of suspension to high-frequency vibrations. This result was applied in engineering.…”