Visualization in mechanics: the dynamics of an unbalanced roller

It is well known that mechanical engineering students often find mechanics a difficult area to grasp. This article describes a system of equations describing the motion of a balanced and an unbalanced roller constrained by a pivot arm. A wide range of dynamics can be simulated with the model. The equations of motion are embedded in a graphical user interface for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the system dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration the simulation tool has been applied to a range of roller–pivot arm configurations. In addition, approximations to the equations of motion are explored and a second-order model is shown to be accurate for a limited range of parameters.     

Classroom note: Maple® version of the ‘Indian rope trick’

If the point of suspension of a multiple pendulum is suitably oscillated then the pendulum can remain in motion in an upside-down position. Since such pendulums can model flexible materials, this inverted motion is sometimes referred to as an ‘Indian rope trick’. Despite the complexity of the governing differential equations, this rope trick can be successfully demonstrated using a popular computer algebra system to formulate and solve the equations, and display an animation of the motion.     

A damped bipedal inverted pendulum for human–structure interaction analysis

The bipedal inverted pendulum with damping has been adopted to simulate human–structure interaction recently. However, the lack of analysis and verification has provided motivation for further investigation. Leg damping and energy compensation strategy are required for the bipedal inverted pendulum to regulate gait patterns on vibrating structures. In this paper, the Hunt–Crossley model is adopted to get zeros contact force at touch down, while energy compensation is achieved by adjusting the stiffness and rest length of the legs. The damped bipedal inverted pendulum can achieve stable periodic gait with a lower energy input and flatter attack angle so that more gaits are available, compared to the template, referred to as spring-load inverted pendulum. The measured and simulated vertical ground reaction force-time histories are in good agreement. In addition, the dynamic load factors are also within a reasonable range. Parametric analysis shows that the damped bipedal inverted pendulum can achieve stable gaits of 1.6 to 2.4 Hz with a reasonable first harmonic dynamic load factor, which covers the normal walking step frequency. The proposed model in this paper can be applied to human–structure interaction analysis.     

Chaos in a four-dimensional system consisting of fundamental lag elements and the relation to the system eigenvalues

A simple system consisting of a second-order lag element (a damped linear pendulum) and two first-order lag elements with piecewise-linear static feedback that has been derived from a power system model is presented. It exhibits chaotic behavior for a wide range of parameter values. The analysis of the bifurcations and the chaotic behavior are presented with qualitative estimation of the parameter values for which the chaotic behavior is observed. Several characteristics like scalability of the attractor and globality of the attractor-basin are also discussed.     

The multiple pendulum problem via Maple®

The way in which computer algebra systems, such as Maple, have made the study of physical problems of some considerable complexity accessible to mathematicians and scientists with modest computational skills is illustrated by solving the multiple pendulum problem. A solution is obtained for four pendulums with no restriction on the size of the angles through which they move, and for as many as a hundred pendulums if the angles remain sufficiently small. An attractive and informative graphical display of the motion in animated form can be obtained from the solution, and the effect of varying the initial conditions can be investigated visually.     

无阻尼振动方程解的稳定性

在无干扰力的环境中,定性分析无阻尼振动方程解的稳定性,可归结为下述几个问题:牛顿第二运动定律应用于振动建模描述力与运动的关系,线性化方程是求解微分方程的有效方法;单摆振动的等时性与非等时性特征表明,微分方程的解不仅决定于方程本身,而且也决定于解的初值;微分方程定性理论,特别是李雅普诺夫第二方法,是研究非线性微分方程解的稳定性的有效手段;如何构造李雅普诺夫函数,至今仍是一个吸引人的研究课题.     

扁锥面网壳非线性动力分岔与混沌运动

对曲面为正三角形网格的3向扁锥面单层网壳,用拟壳法建立了轴对称非线性动力学方程．在几何非线性范围内给出了协调方程．网壳在周边固定条件下,通过Galerkin作用得到一个含2次、3次的非线性微分方程,通过求Floquet指数讨论了分岔问题．为了研究混沌运动,对一类非线性动力系统的自由振动方程进行了求解,继之给出了单层扁锥面网壳非线性自由振动微分方程的准确解,通过求Melnikov函数,给出了发生混沌的临界条件,通过数值仿真也证实了混沌运动的存在．     

The motion characteristics of the double‐pendulum system with skew walls

Many components of machines and other technological devices (chains and bodies hanging on the ropes) can be modeled as multiple pendulums situated in tubes or holes of limited space. The investigation of motion of such systems represents the substance of many real technological problems; therefore, it stirred up motivation to perform research on vibration of a double pendulum situated between two skew rigid walls. The analyzed system was set into motion by the horizontal movement of its suspending. The results of the simulations show that the system can exhibit both the regular and chaotic movements depending on the excitation frequency. The development of the computational model, which is applicable for more complex investigations, and learning more on the motion character of a double pendulum, the movement of which is limited by skew walls, are the principal contributions of the presented article.     

The dynamics of a spherical pendulum with a vibrating suspension

The motion of a spherical pendulum whose point of suspension performs high-frequency vertical harmonic oscillations of small amplitude is investigated. It is shown that two types of motion of the pendulum exist when it performs high-frequency oscillations close to conical motions, for which the pendulum makes a constant angle with the vertical and rotates around it with constant angular velocity. For the motions of the first and second types the centre of gravity of the pendulum is situated below and above the point of suspension, respectively. A bifurcation curve is obtained, which divides the plane of the parameters of the problem into two regions. In one of these only the first type of motion can exist, while in the other, in addition to the first type of motion, there are two motions of the second type. The problem of the stability of these motion of the pendulum, close to conical, is solved. It is shown that the first type of motion is stable, while of the second type of motion, only the motion with the higher position of the centre of gravity is stable.     

A New approach to finding the control that transports a system from one phase state to another

In their previous papers, the authors have considered the possibility of applying the theory of motion for nonholonomic systems with high-order constraints to solving one of the main problems of the control theory. This is a problem of transporting a mechanical system with a finite number of degrees of freedom from a given phase state to another given phase state during a fixed time. It was shown that, when solving such a problem using the Pontryagin maximum principle with minimization of the integral of the control force squared, a nonholonomic high-order constraint is realized continuously during the motion of the system. However, in this case, one can also apply a generalized Gauss principle, which is commonly used in the motion of nonholonomic systems with high-order constraints. It is essential that the latter principle makes it possible to find the control as a polynomial, while the use of the Pontryagin maximum principle yields the control containing harmonics with natural frequencies of the system. The latter fact determines increasing the amplitude of oscillation of the system if the time of motion is long. Besides this, a generalized Gauss principle allows us to formulate and solve extended boundary problems in which along with the conditions for generalized coordinates and velocities at the beginning and at the end of motion, the values of any-order derivatives of the coordinates are introduced at the same time instants. This makes it possible to find the control without jumps at the beginning and at the end of motion. The theory presented has been demonstrated when solving the problem of the control of horizontal motion of a trolley with pendulums. A similar problem can be considered as a model, since when the parameters are chosen correspondingly it becomes equivalent to the problem of suppression of oscillations of a given elastic body some cross-section of which should move by a given distance in a fixed time. The equivalence of these problems significantly widens the range of possible applications of the problem of a trolley with pendulums. The previous solution of the problem has been reduced to the selection of a horizontal force that is a solution to the formulated problem. In the present paper, it is offered to seek an acceleration of a trolley with which it moves by a given distance in a fixed time, as a time function but not a force applied to the trolley, while the velocities and accelerations are equal to zero at the beginning and end of motion. In this new problem, the rotation angles of pendulums are the principal coordinates. This makes it possible to find a sought acceleration of a trolley on the basis of a generalized Gauss principle according to the technique developed before. Knowing the motion of a trolley and pendulums it is easy to determine the required control force. The results of numerical calculations are presented.     

Chaotic behavior in differential equations driven by a Brownian motion

In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.     

Chaos and Period-Adding; Experimental and Numerical Verification of the Grazing Bifurcation

Summary Experimental results are presented for a single-degree-of-freedom horizontally excited pendulum that is allowed to impact with a rigid stop at a fixed angle to the vertical. By inclining the apparatus, the pendulum is allowed to swing in an effectively reduced gravity, so that for each fixed less than a critical value, a forcing frequency is found such that a period-one limit cycle motion just grazes with the stop. Experimental measurements show the immediate onset of chaotic dynamics and a period-adding cascade for slightly higher frequencies. These results are compared with a numerical simulation and continuation of solutions to a mathematical model of the system, which shows the same qualitative effects. From the model, the theory of discontinuity mappings due to Nordmark is applied to derive the coefficients of the square-root normal form map of the grazing bifurcation for this system. The grazing periodic orbit and its linearisation are found using a numerical continuation method for hybrid systems. From this, the normal-form coefficients are computed, which in this case imply that a jump to chaos and period-adding cascade occurs. Excellent quantitative agreement is found between the model simulation and the map, even over wide parameter ranges. Qualitatively, both accurately predict the experimental results, and after a slight change in the effective damping value, a striking quantitative agreement is found too.     

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells北大核心CSCD

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

An optimization technique for verified location of trajectories with prescribed geometrical behaviour in the chaotic forced damped pendulum

The present paper studies the forced damped pendulum equation, equipped with Hubbard’s parameters (Hubbard in Am Math Mon 8:741–758, 1999). With the aid of rigorous computations, his 1999 conjecture on the existence of chaos was proved in Bánhelyi et al. (SIAM J Appl Dyn Syst 7:843–867, 2008) but the problem of finding chaotic trajectories remained entirely open. In order to approximate a wide range of chaotic trajectories with arbitrary precision, the present paper establishes an optimization method capable to locate finite trajectory segments with prescribed geometrical behavior.     

Bifurcation of slow motions in a stiff spring pendulum

We consider free oscillations of a double pendulum, where one of the pendula is modelled as a very stiff spring. Contrary to a single spring pendulum numerical simulations show an unexpected large influence of the fast longitudinal oscillations on the slow pendulum oscillations even for extremely large values of the stiffness. The transition from the regular motion, which is governed by the dynamics of a rigid double pendulum close to a periodic orbit, to the irregular motion with large contributions from the longitudinal oscillations occurs due to a subcritical symmetry breaking bifurcation of the periodic solution. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction and quatitative characterization of a chaotic saddle from a pendulum experiment

We experimentally study the behaviour of a parametrically driven damped pendulum in a parameter region where a transient chaotic motion is observed. We reconstruct the chaotic saddle and a chaotic attractor near an interior crisis in a stroboscopic phase representation and give an estimation of the corresponding f() spectra.     

曲率的形状梯度和经典梯度:微纳米曲面上的驱动力

近期的实验和分子动力学模拟均表明:圆锥面上粘附液滴能自发地定向运动,且自发定向运动的方向与粘附面的亲水、疏水性质无关．针对这一重要现象,拟从曲面微纳米力学几何化的角度,提供一般性的理论解释．借助于粒子对势,研究了孤立粒子与微纳米硬曲面之间的相互作用,分析了粒子/硬曲面相互作用的几何学基础．可以证实:（a） 粒子/硬曲面的作用势均具有统一的曲率化形式,均可以统一地表达成曲面平均曲率和Gauss曲率的函数;（b） 基于曲率化的作用势,能够实现曲面微纳米力学的几何化;（c） 曲率与曲率的内蕴梯度构成卷曲空间上的驱动力;（d） 驱动力方向与曲面的亲水、疏水性质无关,解释了自发定向运动实验．     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Optimal decision policy for real options under general Markovian dynamics

The Least-Squares Monte Carlo Method (LSM) has become the standard tool to solve real options modeled as an optimal switching problem. The method has been shown to deliver accurate valuation results under complex and high dimensional stochastic processes; however, the accuracy of the underlying decision policy is not guaranteed. For instance, an inappropriate choice of regression functions can lead to noisy estimates of the optimal switching boundaries or even continuation/switching regions that are not clearly separated. As an alternative to estimate these boundaries, we formulate a simulation-based method that starts from an initial guess of them and then iterates until reaching optimality. The algorithm is applied to a classical mine under a wide variety of underlying dynamics for the commodity price process. The method is first validated under a one-dimensional geometric Brownian motion and then extended to general Markovian processes. We consider two general specifications: a two-factor model with stochastic variance and a rich jump structure, and a four-factor model with stochastic cost-of-carry and stochastic volatility. The method is shown to be robust, stable, and easy-to-implement, converging to a more profitable strategy than the one obtained with LSM.