Dynamic modeling, input-shaped maneuvering and vibration suppression of flexible body using quasi-coordinates and euler parameters

In dealing with the dynamics of a flexible body, the rigid-body motions and elastic vibrations are analyzed separately. However, rigidbody motions cause vibrations, and elastic vibrations affect rigid-body motions, indicating the inherent coupling between rigid-body motions and elastic vibrations. The coupled equations of motion for a flexible body can be derived by means of Lagrange’s equations in terms of quasi-coordinates. The resulting equations of motion are hybrid and nonlinear. This paper proposes a unified approach for the equations of motion for a flexible body based on the perturbation method and the Lagrange’s equations of motion in terms of quasicoordinates and Euler parameters to analyze a more general case maneuvering. The resulting equations consist of zero-order nonlinear equations of motion which depict rigid-body motions and first-order time-varying linear equations of motion which depict perturbed rigid-body motions and elastic vibration. Hence, the input-shaped maneuvering can be applied to the zero-order equation considering the induced vibrations. Since the input-shaped maneuvering alone cannot achieve vibration suppression, the vibration suppression controller combined with the input-shaped maneuvering is proposed in this study. As a numerical example, a hub with elastic appendages is considered. Numerical results show that the unified modeling approach proposed in this paper is effective in numerical simulation and control design.     

Flapwise bending vibration analysis of rotating composite cantilever beams

A modeling method for the modal analysis of a rotating composite cantilever beam is presented in this paper. Linear differential equations of motion are derived using the assumed mode method. For the modeling, hybrid deformation variables are employed and approximated to derive the equations of motion. Symmetrical laminated composite beams are considered to obtain the numerical results. The effects of the dimensionless angular velocity, the hub radius and the fiber orientation angle on the variations of modal characteristics are investigated.     

Vibration analysis of rotating composite cantilever plates

A modeling method for the vibration analysis of rotating composite cantilever plates is presented in this paper. The coupling effects between inplane motions and the bending motion are considered and explicit mass and stiffness matrices are derived for the modal analysis. Numerical results are obtained and some of them are compared to those of a commercial program to confirm the accuracy of the present method. Numerical results show that the coupling effects become important only when laminates are stacked up unsymmetrically. Incidentally, natural frequencies loci veering, loci crossing, and associated mode shape variations are observed.     

Bending and vibration of circular cylindrical beams with arbitrary radial nonhomogeneity

This paper presents a new approach for analyzing transverse bending and vibration of circular cylindrical beams with radial nonhomogeneity. The radial nonhomogeneity may be continuous or piecewise-constant, corresponding a functionally graded circular cylinder or a multi-layered circular cylinder, respectively. Different from the Euler-Bernoulli and Timoshenko theories of beams, our analysis considers shear deformation, but does not need to introduce a shear correction factor. Using the shear-stress-free condition at the surface of the cylinder, coupled governing equations for deflection and rotation angle are derived, and then converted to a single governing equation. The influences of gradient index on the deflection and stress distribution for cantilever and simply-supported beams are studied. Natural frequencies of free vibration of a cylindrical beam with circular cross-section are calculated for different power-law gradients. In particular, a circular cylindrical shell may be taken as a special case of a bi-layered cylinder where the material properties of the inmost cylinder vanish. For this case, the natural frequencies for simply-supported and clamped-clamped cylindrical shells are evaluated and compared with those using three-dimensional theory. Our results coincide well with the previous.     

Utilization of heat quantity to model thermal errors of machine tool spindle

Thermal error modeling of the spindle plays an important role in predicting thermal deformation and improving machining precision. Even though the modeling method using temperature as the input variable is widely applied, it is less effective due to severe loss of thermal information and pseudo-hysteresis between temperature and thermal deformation. This paper presents a novel modeling method considering heat quantity as the input variable with theoretical analysis and experimental validation. Firstly, the change of thermal state of a metal part being heated is discussed to reveal the essence of the relationship between heat, thermal deformation and temperature, and the theoretical basis of the modeling method proposed in this paper is elaborated. Subsequently, the relationship between thermal deformation and heat quantity is further studied through modeling the thermal deformations of stretching bar and bending beam using heat quantity as the independent variable, and the stretching model is verified based on finite element method. Then, the thermal error models of the spindle are developed with the heat elastic mechanics theory and the lumped heat capacity method. In succession, the parameter identification of thermal error models is carried out experimentally using the least square method. The average fitting accuracy of these models is up to 91.3%, which verifies the good accuracy and robustness of the models. In addition, these models are of good prediction capability. The proposed modeling method deepens the research of thermal errors and will help to promote the application of relevant research results in the actual production.     

Classical buckling of a thin-walled tube subjected to bending moment and internal pressure

The classical buckling of a thin-walled circular cylindrical shell which is subjected simultaneously to uniform bending moment and internal pressure is investigated in an approximate fashion by assuming an appropriate eigenmode and obtaining a “best-fit” solution of the resulting equations. In this way, explicit formulae are derived for critical loading combinations and for the characteristics of the modal form. These formulae agree reasonably well with a wide range of numerical data previously obtained by Seide and Weingarten. A feature of the scheme is that it is feasible to interpret the results in physical terms, and to envisage the roles of bending, twisting and stretching in the critical solution.     

Modeling of flexure-hinge type lever mechanisms

A general approach for the design of flexure-hinge type lever mechanisms that are commonly used in the design of translational micro-positioning stages is presented in this paper. The paper focuses on the development of design equations that can accurately predict the behavior of such stages especially the “lost motion” due to hinge stretching. The development of these equations is based on a static analysis of a general configuration of a single-axis, translational, flexure-hinge type, piezo-driven micro-positioning stage using a multi-lever structure. The displacement ratio between the input and output motions of one of the levers, plus the stiffness at either end of this lever are obtained based on the analysis. The overall displacement and stiffness of the micro-positioning stage are then obtained by cascading the individual results from every lever. The developed equations include the effects of the flexure hinge bending and stretching. The developed equations were applied to design of a vertical motion micro-positioning stage. The stiffness and displacement of this stage are predicated by these equations, which are compared to the stiffness and displacement directly obtained from the finite element modeling and actual testing of the same micro-positioning stage. The comparison shows that the analytical approach gave nearly similar results (within 10%), which implies that the developed equations can accurately predict the characteristics of flexure-hinge type lever mechanisms.     

线性分布载荷作用下双模量圆板的计算分析

双模量圆板在线性载荷作用下,会形成各向同性的拉伸区和压缩区.可将双模量圆板看成两种各向同性材料组成的层合板,采用弹性理论建立双模量圆板在外载荷作用下的静力平衡方程,利用此静力平衡方程即可确定双模量圆板的中性面位置.建立双模量圆板在外载荷作用下的弯曲微分方程,求得双模量圆板在线性分布载荷作用下的挠度方程.通过算例讨论分析双模量对圆板弯曲变形的影响,得到圆板材料拉压弹性模量相差较大时,其挠度计算不宜采用相同弹性模量经典薄板理论,而应该采用双模量薄板理论的结论.     

双圆弧圆柱齿轮传动系统的振动分析

建立了双圆弧圆柱齿轮传动系统的周向—径向—轴向—扭摆耦合的动力学模型,通过实例分析计算了其振动响应,并和实测结果进行了对比。结果表明：二者的变化规律是基本吻合的。为开展双圆弧圆柱齿轮传动的动态设计提供了依据。     

An investigation on effects of traveling mass with variable velocity on nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions

In this paper, the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a traveling mass with variable velocity is investigated. The nonlinear coupled partial differential equations of motion for the bending rotation of cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. These nonlinear coupled PDEs are solved by applying Galerkin’s method to obtain dynamic response of the beam under the act of a moving mass. The appropriate parametric studies by taking into account the effects of the magnitude of the traveling mass, the velocity of the traveling mass with a constant acceleration/deceleration and effect of different beam’s boundary conditions are carried out. The beams’ large deflection has been captured by including the stretching effect of its mid-surface. It was seen that the existence of quadratic-cubic nonlinear terms in the governing coupled PDEs of motion renders hardening/stiffening behavior on the dynamic responses of the beam when traversed by a moving mass. In addition, the obtained nonlinear results are compared with those from the linear analysis.     

Aeroelastic behavior of a slender body considering free fittings

This paper presents dynamic and vibration analysis of a flight vehicle with consideration of the free fitting between its two sections. Using the Lagrangian approach, a general analytical model is developed for a non-spinning elastic vehicle in planar motion. The model contains rigid body motions and bending deformations of two sections of the flight vehicle and a nonlinear rotational spring that models the freeplay between the two sections. To express bending deformation, the mode summation method is applied. It is shown that freeplay in the joints significantly affects the trajectory of the flight vehicle. Numerical examples reveal the effect of a joint’s nonlinearity on the trajectory and stability of the flight vehicle.     

A new formulation for the torsional vibration analysis of rotating cantilever rods

This paper presents a new modeling method for the analysis of torsional vibration of rotating cantilever rods. The natural frequency and mode shape variations due to the rotational motion could be accurately estimated with the modeling method, and the coefficient for the well-known Southwell equation could be obtained, as well. This method has couple of advantages compared to previous conventional modeling methods. Different from the previous modeling methods, the equations of motion of the rotating cantilever rod were derived with consistent linearization in a rigorous way. An eigenvalue problem to obtain the coefficients of the Southwell equation of rotating rods were derived from the original eigenvalue problem.     

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

In this research, mechanical buckling of circular plates composed of functionally graded materials (FGMs) is considered. Equilibrium and stability equations of a FGM circular plate under uniform radial compression are derived, based on the higher order shear deformation plate theory (HSDT). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations are established. A buckling analysis of a functionally graded circular plate (FGCP) under uniform radial compression is carried out and the results are given in closed-form solutions. The results are compared with the buckling loads of plates obtained for FGCP based on the first order shear deformation plate theory (FSDT) and classical plate theory (CPT) given in the literature. The study concludes that HSDT accurately predicts the behavior of FGCP, whereas the FSDT and CPT overestimates buckling loads.     

Stability and response analysis of symmetrical single-disk flexible rotor-bearing system

The motion equations have been established for symmetrical single-disk flexible rotor-bearing system, and non-linear oil-film forces of finite journal bearings are calculated. The rotor's stiffness and damping are considered. The motions of journal and disk have been simulated with fourth-rank Runge-kutta method. The threshold speed of the system based on linear oil-film forces has been derived. Non-linear transient simulation and unbalanced responses are investigated. Moreover, linear oil-film forces are used for computing unbalanced responses, and the feasibility is discussed.     

Modified membrane finite element formulation for sheet metal forming analysis of planar anisotropic materials

A finite element formulation is derived for sheet metal forming analysis of planar anisotropic materials. The formulation incorporates membrane elements whereas it takes the bending effect into account explicitly. The strain energy term in the formulation is decomposed into the membrane energy term for mean stretching and the bending energy term for pure bending. This procedure needs careful evaluation for the orientation of the anisotropic axes. The formulation is then combined with an effective algorithm to calculate distribution of the blank holding force in each step according to the thickness in the flange region. The calculation employs a special relation between the thickness and the blank holding force. The simulation examples demonstrate the validity and versatility of the developed computer code by showing that the thickness variation in the flange region redistributes the blank holding force during the deformation. The present algorithm can predict accurate deformed shapes and thickness strain distribution with the anisotropy of materials and the variable blank holding force.     

翼型叶片精确参数化建模的研究及应用

从理论上分析了抛物线、单圆弧和双圆弧型翼型中心线方程,推导并更正了双圆弧中心线翼型的前、后缘方向角的取值规律。阐述了翼型叶片截面的参数化建模方法,利用微分的思想,通过中心线分割点的弧长值求得了对应横坐标,从而获得了更加精确的翼型叶片上、下轮廓控制点坐标参数化计算方法。在此基础上,开发了翼型叶片截面参数化绘图程序,实现了叶型截面的精确自动绘制。     

State-space approach for statics and dynamics of angle-ply laminated cylindrical panels in cylindrical bending

Elasticity method is developed to study the bending and free vibration of simply-supported angle-ply laminated cylindrical panels in cylindrical bending. The present analysis is completely based on the state-space formulations, which is very effective for laminated structural analysis. In particular, exact solution is derived for the static bending problem by virtue of a variable substitution technique. Approximate analytical solution is derived for dynamic problem by employing the layerwise method. Numerical results are finally presented. The current study should serve as a necessary supplement to related works in the literature.     

Recursive newton-euler formulation for flexible dynamic manufacturing analysis of open-loop robotic systems

The main objective of this paper is to develop a recursive formulation for the flexible dynamic manufacturing analysis of open-loop robotic systems. The nonlinear generalized Newton-Euler equations are used for flexible bodies that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars, vectors and matrices that depend on the spatial coordinates as well as the assumed displacement fields. These time invariant quantities represent the dynamic manufacturing couplings between the rigid body motion and elastic deformation. This formulation applies recursive procedures with the generalized Newton-Euler equations for flexible bodies to obtain a large, loosely coupled system equation describing motion in flexible manufacturing systems. The techniques used to solve the system equations can be implemented in any computer system. The algorithms presented in this investigation are illustrated using cylindrical joints for open-loop robotic systems, which can be easily extended to revolute, slider and rigid joints. The recursive Newton-Euler formulation developed in this paper is demonstrated with a robotic system using cylindrical mechanical joints.     

Nonlinear modeling and dynamic analysis of flexible structures undergoing overall motions employing mode approximation method

This paper presents a nonlinear modeling method for dynamic analysis of flexible structures undergoing overall motions that employs the mode approximation method. This method, different from the naive nonlinear method that approximates only Cartesian deformation variables, approximates not only deformation variables but also strain variables. Geometric constraint relations between the strain variables and the deformation variables are introduced and incorporated into the formulation. Two numerical examples are solved and the reliability and the accuracy of the proposed formulation are examined through the numerical study.     

Recursive Newton–Euler formulation for flexible dynamic manufacturing analysis of open-loop robotic systems

The main objective of this paper is to develop a recursive formulation for the flexible dynamic manufacturing analysis of open-loop robotic systems. The nonlinear generalized Newton–Euler equations are used for flexible bodies that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars, vectors and matrices that depend on the spatial coordinates as well as the assumed displacement fields. These time invariant quantities represent the dynamic manufacturing couplings between the rigid body motion and elastic deformation. This formulation applies recursive procedures with the generalized Newton–Euler equations for flexible bodies to obtain a large, loosely coupled system equation describing motion in flexible manufacturing systems. The techniques used to solve the system equations can be implemented in any computer system. The algorithms presented in this investigation are illustrated using cylindrical joints for open-loop robotic systems, which can be easily extended to revolute, slider and rigid joints. The recursive Newton–Euler formulation developed in this paper is demonstrated with a robotic system using cylindrical mechanical joints.