Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.     

The Drazin inverse in multibody system dynamics

Summary The numerical analysis of multibody system dynamics is based on the equations of motion as differential-algebraic systems. A thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches. The main tool is the Drazin inverse, a generalized matrix inverse, which preserves the eigenvalues. The results are illustrated by a realistic truck model. Finally, the approach is extended to the nonlinear index 2 formulation.     

Non-linearity of multibody dynamic equations with respect to Lagrange multipliers: application to railway dynamics

In this paper, the dynamics of multibody systems with closed kinematical chains of bodies is considered. The main focus is set on non-linearity of the multibody equations with respect to the Lagrange multipliers. When closed chains are considered, loop cutting procedure is a solution to express the constraint equations associated with the loops. Dynamic equations of the multibody tree-like structure are thus completed with the constraint forces via the Lagrange multipliers. In the considered case of railway vehicles, constraints arise from the contact between the rigid wheels and the rails. Corresponding contact forces applied to the wheels appears via the Lagrange multipliers λ and the tangent creep forces as well. Resulting differential-algebraic equations can be transformed into an ODE system and then time-integrated using the coordinate partitioning method [3], when the system is linear with respect to λ. This paper presents an algorithm allowing us to solve this system in case of nonlinearities with respect to λ, which is typical of wheel/rail contact force models. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Using Multibody Systems for the Investigation of Dynamic Aberrations in High Precision Optics

The coupling of multibody system dynamics and optical simulations by using ray tracing through moving lens systems will be summarized for both rigid and flexible lens systems. Furthermore, a method will be introduced for efficient simulations in the time domain. This method provides a direct integration of the optical simulation into the equations of motion of a multibody system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computer system for kinematic and dynamic analysis synthesis of rigid and flexible multibody systems

In the paper an overview of a general numerical algorithm and program system library for deriving the kinematic constraint equations and dynamic equations of motion, as well as, computation of their first and second order partial derivatives with respect to kinematic parameters of motion, design parameters and mass and inertia characteristics for rigid and flexible multibody systems is presented. These are the main basic computational modules for implementation of kinematic and dynamic synthesis, optimization and design. The main theoretical basis consists in matrix methods for deriving the kinematic constraints and dynamic equations, as well as, the generalized Newton – Euler dynamic equations for rigid and flexible bodies, and finite element discretization in relative coordinates. Block–scheme of the computational procedures and problem oriented program compilation is presented. An example of kinematic synthesis of six–link path generating mechanism with singular points is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized inertia forces of rigid and flexible bodies

In the paper a method for deriving the dynamic equations of motion of rigid and flexible multibody systems is presented. Three dimensional matrices are applied for working out the configuration space dynamic equations. Finite elements are used for modeling of flexible systems. Novel generalized Newton‐Euler equations and relations for inertia forces in nodes of flexible elements are proposed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification and higher order analysis of manipulator singularities

A purely algebraic approach to higher order analysis of (singular) configurations of rigid multibody systems with kinematic loops (CMS) is presented. Rigid body con.gurations are described by elements of the Lie group SE(3) and so the rigid body kinematics is determined by an analytical map f : V → SE(3), where V is the configuration space, an analytic variety. Around regular configurations V has manifold structure but this is lost in singular points. In such points the concept of a tangent vector space does not makes sense but the tangent space C_qV (a cone) to V can still be defined. This tangent cone can be determined algebraically using the special structure of the Lie algebra se (3), the generating algebra of the special Euclidean group SE (3), and the fact that the push forward map f_*, the tangential mapping C_qV → se (3), is given in terms of the mechanisms screw system. Moreover the differentials of f of arbitrary order can be expressed algebraically. The tangent space to the configuration space can be shown to be a hypersurface of maximum degree 4, a vector space for regular points. It is the structure of the tangent cone to V that gives the complete geometric picture of the configuration space around a (singular) point. Identification of the screw system and its matrix representation with the kinematic basic functions of the CMS allows an automatic algebraic analysis of mechanisms.     

Mathematical modeling of complex mechanical systems

Mathematical modeling of mechanical systems based on multibody system models is a well tested approach. Generating the equations of motion for complex multibody systems with a large number of degrees of freedom is difficult with paper and pencil. For this reason methods for automatic equation generation have been developed. Most methods result in numerical equations of motion without explicit information about the parameters. In this paper a method is described resulting in symbolic equations of motion. The method allows also the determination of the constraint forces which are important for design purposes. The inverse problem of dynamics is also easily solved.     

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.     

滚装船在波浪中横摇时船上滑动重载荷的同步效应

由于波浪和内部滑动车辆共同作用,使滚装船的横摇加剧,这是许多滚装船发生倾覆的重要原因之一．对由滚装船和多辆滑动车辆组成的浮基多体系统,取滚装船的横摇角和多辆自由滑动车辆在甲板上的横向位移为此系统的自由度,通过对船舶在波浪中所受表观重力和表观浮力的分析,导出波浪作用力矩,运用多体系统动力学方法,建立了系统的动力学方程．以某型海峡滚装渡轮为例,对在多辆车自由滑动和波浪共同作用下的滚装船浮基多体系统的横摇响应和车辆位移响应进行了数值计算,得出了多个自由滑动的重载荷因相互碰撞在舷侧舱壁的约束下随着时间的延长其运动将趋于同步的结论．     

Parametric control of the motions of non-linear oscillatory systems

A new class of control and optimization problems for the motions of oscillatory systems, utilizing adjustable variation of the parameters, is considered. Forms of the equations of motion are proposed suitable for the use of asymptotic methods of non-linear mechanics and optimal control. The basic control modes are investigated: by bang-bang-type variation of the acceleration, velocity or the value of a parameter within relatively narrow limits. Approximate time-optimal and time-quasi-optimal control laws are constructed for non-linear oscillatory systems with regulated relative equilibrium position and stiffness. Some interesting features of the motions are observed and discussed.     

Modelling of three-dimensional mechanical systems using point coordinates with a recursive approach

In this paper, the dynamic simulation of constrained mechanical systems that are interconnected of rigid bodies is studied using projection recursive algorithm. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. Closed-chain system is transformed to open-chain by cutting suitable kinematical joints and introducing cut-joint constraints. For the resulting open-chain system, the equations of motion are generated recursively along the serial chains. An example is chosen to demonstrate the generality and simplicity of the developed formulation.     

Numerical calculation of the tangent stiffness matrix in materials modeling

The Finite Element Method in the field of materials modeling is closely connected to the tangent stiffness matrix of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness matrix can be derived analytically only with great effort, if at all. Numerical methods are therefore widely used for its calculation. We present our method to calculate the tangent stiffness matrix for the logarithmic strain measure. The approach is compared with other commonly used procedures. A significant increase in accuracy can be achieved with the proposed method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)