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

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It is shown that there is a magnitude of the follower force at which two limit cycles, stable and unstable, are born in the phase space of a double simple pendulum Keywords: double simple pendulum, stable limit cycle, unstable limit cycle, Hopf bifurcation Introduction. Pflüger was the first in the 20th century to try to assess the effect of follower force on the dynamic behavior of an elastic bar [27]. To find the magnitude of the follower force that causes instability of the rectilinear equilibrium mode, he used Euler's concept, which had been reliably applied in science since the 19th century. However, Pflüger arrived at the unexpected conclusion that there is no critical load, which contradicted experimental observations. Ziegler [28] resolved Pflüger's paradox by modeling an elastic cylindrical pipe with a moving fluid inside by a double simple pendulum subject to a follower force. It is the follower force that appeared to be responsible for the disagreement between Pflüger's theoretical finding and experimental observations: the corresponding term of the linearized perturbed equations of motion includes a skew-symmetric matrix, whereas this matrix for a dead compressive load is symmetric.Ziegler's work pioneered a new area in the theory of elastic stability [16]: dynamic stability, unlike static stability. The emergence of the concepts of bifurcation, catastrophe, and limit cycle gave rise to different terminology in the theory of dynamic systems: divergent and flutter instability instead of static and dynamic instability.Off-center compression in the theory of elastic bars is usually attributed to the linear eccentricity of the compressive force. Japanese scientists also introduced linear eccentricity [17,18,26]. The recent studies have demonstrated that except linear elastic systems governed by Hooke's law, there are mechanical systems with physical nonlinearity. Of the nonlinear characteristics of deformable systems, hard and soft characteristics are most popular [6]. They are of experimental origin. We will use the analytic approximations of the corresponding empirical formulas from [25]. These approximations were used in [19] to set up the theory of plane-parallel motion of an n-link simple pendulum under the action of a follower force and in [20][21][22][23][24] to study the motion of single-and double-link simple pendulums.Some features of the dynamic behavior of double pendulums under the action of a compressive follower force associated with limit cycles in the phase space are described in [5], which studies the behavior of dynamic systems near the stability boundaries and equilibrium states in the critical cases described by Lyapunov [13]. Further progress is associated with specific dynamic systems. Dynamic systems with so-called central (or elementary) symmetry are introduced in [7-9]. Methods for describing periodic motion in such systems are described in [10][11][12]. They are based on Lyapunov's, Poincaré's, and Malkin's methods [13,14].1. Problem Formulation. Let m 1 and m 2 be the masses of...

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confidence: 99%

Limit cycles of a double pendulum subject to a follower force

Lobas

Ichanskii

2009

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“…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].…”

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

Lobas

Kovalchuk

2008

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The stability boundaries for the vertical equilibrium position of a triple simple pendulum subject to asymmetric follower force are analyzed. The effect of the upper spring and the magnitude of the follower force on the roots of the characteristic equation is studied Keywords: triple simple pendulum, follower force, stability domain, vertical position Introduction. The dynamic behavior of inverted multilink pendulums with a spring at the upper end subject to a follower force have recently attracted considerable interest due to numerous problems of mechanical and instrument engineering and the necessity of modeling continuum elastic systems by discrete systems. Pflüger's and Ziegler's works [22,24] gave an impetus to relevant studies in many countries. The pioneering study [22] was initiated after encountering problems with oil pipelines. Subsequently, the dynamic stability of a fluid-conveying cantilever pipe was analyzed in continuum [1, 2] and discrete formulations. Tending to put Ziegler's academic problem formulation into practice, researchers modified it in [10-12, 20, 23]. The Austrian mechanicians [23] "inverted" the Ziegler pendulum, while the Japanese scientists [10,11,20] assumed asymmetry of the follower force, which may be due to the off-center application of the compressive load, manufacturing errors, imperfections, etc. The combination of these factors determines the asymmetry of the follower force described as linear and angular eccentricities in the paper [12], which sets up a generalized mathematical model of an inverted simple pendulum with an arbitrary number of links. Modeling a one-dimensional continuum elastic system by an n-link pendulum in a potential force field, the paper [21] explains the Indian rope trick. 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].The nature and physical feasibility of follower forces are discussed in the review [9], which was motivated by studies with polar views on whether follower forces are real. A distinctive feature of follower forces is that they are positional, yet nonpotential: they are represented by a skew-symmetric matrix in the linearized equations of perturbed motion (like radial-correction forces in gyroscopy [7] and cornering forces on tires [3]).A triple pendulum is a better representative of multilink pendulums than a double pendulum because it has a link (middle) that is affected by the neighboring links. However, this typical feature of the triple pendulum complicates the problem by increasing the order of the dynamic system from fourth to sixth. It is ...

“…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.…”

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“…A specific composition is determined by the values of the so-called influence coefficients q 1 , q 2 , q 3 , q 11 , q 12 , q 13 , q 21 , q 22 , and q 23 , each taking the value 1 or 0: If all the three springs are nonlinear hard, then q q q 1 11 = . For a one-link pendulum [4][5][6][7], there are nine such combinations (m = 2): hh, hs, hl, sh, ss, sl, lh, hs, ll. A two-link pendulum (m = 3) provides 27 combinations: hhh, hhs, hhl, hsh, hss, hsl, hlh, hls, hll, shh, shs, shl, ssh, sss, ssl, slh, sls, sll, lhh, lhs, lhl, lsh, lss, lsl, llh, 11s, 111.…”

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

Lobas¹,

Kovalchuk²,

Bambura³

2007

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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...