Vibration analysis of a planar multilink manipulator using a reduced-order matrix formulation

The main scope of this paper is both to provide an efficient formulation for deriving natural frequencies and mode shapes of a multilink flexible manipulator in an arbitrary posture and discuss the importance of introducing axial deformations along with transversal ones in the same postures. The free vibration problem is solved through a global transfer matrix formulation, which is obtained by opportunely assembling the transfer matrices of links and joints associated with the desired configuration of the robot. The formulation has been chosen in order to keep the same matrix order of the problem, regardless of the number of links of the manipulator. Manipulator links are treated as Timoshenko beams by accounting for both axial loading capability and mass/inertial effects due to the joint actuators. Numerical comparisons between the solutions derived herein and those obtained through existing formulations highlight the importance of the proposed model within the frame of multilink flexible manipulators. A forced vibration analysis is finally presented for a two-link manipulator.     

平面问题极坐标下几何方程的一种解析法

经典弹性力学教材中都是通过单元的几何变形来推导平面问题极坐标下的几何方程. 这里, 用解析法从直角坐标系下的几何方程推出极坐标下的几何方程.     

A velocity transformation method for the nonlinear dynamic simulation of railroad vehicle systems

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.     

A BEM formulation for transient dynamic elastoplastic analysis via particular integrals

This paper presents the particular integral formulation for two (2D) and three (3D) dimensional transient dynamic elastoplastic analyses. The elastostatic equation is used for the complementary solution. The particular integrals for displacement, traction and stress rates are obtained by introducing the concept of a global shape function to approximate acceleration and initial stress rate terms of the inhomogeneous equation. The Houbolt time integration scheme is used for the time-marching process. The Newton–Raphson algorithm for plastic multiplier is used to solve the system equation. The developed program is integrated with the pre- and post-processor. Numerical results for four example problems are given to demonstrate the validity and accuracy of the present formulation.     

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

We proposed a mesh-free method, the called node-based smoothed point interpolation method (NS-PIM), for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection. This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations, but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.     

A FE formulation for elasto-plastic materials with planar anisotropic yield functions and plastic spin

In this article a stress integration algorithm for shell problems with planar anisotropic yield functions is derived. The evolution of the anisotropy directions is determined on the basis of the plastic and material spin. It is assumed that the strains inducing the anisotropy of the pre-existing preferred orientation are much larger than subsequent strains due to further deformations. The change of the locally preferred orientations to each other during further deformations is considered to be neglectable. Sheet forming processes are typical applications for such material assumptions. Thus the shape of the yield function remains unchanged. The size of the yield locus and its orientation is described with isotropic hardening and plastic and material spin.The numerical treatment is derived from the multiplicative decomposition of the deformation gradient and thermodynamic considerations in the intermediate configuration. A common formulation of the plastic spin completes the governing equations in the intermediate configuration. These equations are then pushed forward into the current configuration and the elastic deformation is restricted to small strains to obtain a simple set of constitutive equations. Based on these equations the algorithmic treatment is derived for planar anisotropic shell formulations incorporating large rotations and finite strains. The numerical approach is completed by generalizing the Return Mapping algorithm to problems with plastic spin applying Hill’s anisotropic yield function. Results of numerical simulations are presented to assess the proposed approach and the significance of the plastic spin in the deformation process.     

Using nodal coordinates as variables for the dimensional synthesis of mechanisms

The method of the lower deformation energy has been successfully used for the synthesis of mechanisms for quite a while. It has shown to be a versatile, yet powerful method for assisting in the design of mechanisms. Until now, most of the implementations of this method used the dimensions of the mechanism as the synthesis variables, which has some advantages and some drawbacks. For example, the assembly configuration is not taken into account in the optimization process, and this means that the same initial configuration is used when computing the deformed positions in each synthesis point. This translates into a reduction of the total search space. A possible solution to this problem is the use of a set of initial coordinates as variables for the synthesis, which has been successfully applied to other methods. This also has some additional advantages, such as the fact that any generated mechanism can be assembled. Another advantage is that the fixed joint locations are also included in the optimization at no additional cost. But the change from dimensions to initial coordinates means a reformulation of the optimization problem when using derivatives if one wants them to be analytically derived. This paper tackles this reformulation, along with a proper comparison of the use of both alternatives using sequential quadratic programming methods. In order to do so, some examples are developed and studied.     

A velocity formulation for flow past a symmetric profile with a wake

A model having velocity components as basic unknowns is presented for calculation of two-dimensional flow past a symmetric profile with a wake in a channel. A modified least squares functional is used for the finite element solution of velocities. The determination of the position of the free streamline is treated as an optimum design problem. The concepts of cost function, geometry parameter and sensitivity derivative are employed. Numerical results are compared with published results obtained with streamfunction formulations.     

A new formulation for fictitious mass of viscous dynamic relaxation method

Abstract

In this article, a new relationship is proposed for the fictitious mass of viscous dynamic relaxation (DR) method. First, incremental equations are derived for DR steps. Using transformed Gershgörin theory, a new relationship is achieved for fictitious mass of viscous DR by formulating modified time step ratio. This procedure presents a new algorithm for the viscous DR method. To evaluate the numerical efficiency of the proposed method, some 2D and 3D truss and frame structures are analyzed with elastic linear and geometrically nonlinear behaviors. Results show that by using the proposed algorithm for fictitious mass, the convergence rate of the viscous DR method is improved so that the proposed algorithm presents the structural response with lower iterations in comparison with other common DR techniques.

Communicated by Joerg Fehr.     

Accuracy limits for planar measurements of flow field velocity, temperature and pressure using Filtered Rayleigh Scattering

 We present experimental results using Filtered Rayleigh Scattering to make planar measurements of velocity, temperature and pressure in ambient air and in a Mach 2 free jet. The ambient air measurements are used to identify and calibrate experimental uncertainties. The Mach 2 measurements demonstrate the usefulness of the technique for making accurate planar measurements in a high speed flow. The measured values for velocity, temperature and pressure in the Mach 2 jet ranged, through a shock system, from 205 to 235 m/s, 150 to 170 K and 700 to 1000 torr, with estimated uncertainties of ±5.4 m/s, ±3.2 K and ±38 torr (±2 to 3%, ±2% and ±4–5%, respectively). Received: 10 December 1996/Accepted: 14 July 1997     

Dynamic stability analysis of composite plates including delaminations using a higher order theory and transformation matrix approach

A refined higher order shear deformation theory is used to investigate the dynamic instability associated with composite plates with delamination that are subject to dynamic compressive loads. Both transverse shear and rotary inertia effects are taken into account. The theory is capable of modeling the independent displacement field above and below the delamination. All stress free boundary conditions at free surfaces as well as delamination interfaces are satisfied by this theory. The procedure is implemented using the finite element method. Delamination is modeled through the multi-point constraint approach using the transformation matrix technique. For validation purposes, the natural frequencies and the critical buckling loads are computed and compared with three-dimensional NASTRAN results and available experimental data. The effect of delamination on the critical buckling load and the first two instability regions is investigated for various loading conditions and plate thickness. As expected, the natural frequencies and the critical buckling load of the plates with delaminations decrease with increase in delamination length. Increase in delamination length also causes instability regions to be shifted to lower parametric resonance frequencies. The effect of edge delamination on the static and dynamic stability as well as of delamination growth is investigated.     

A vorticity formulation for computational fluid dynamic and aeroelastic analyses of viscous flows

The paper deals with a novel computational formulation for the analysis of viscous flows past a solid body. The formulation is based upon a convenient decomposition of the flow field into potential and rotational velocity contributions, which has the distinguishing feature that the rotational velocity vanishes in much of, if not all, the region in which the vorticity is negligible. Contrary to related formulations implemented by the authors in the past, in the proposed approach, discontinuities of the potential and rotational velocity fields across a prescribed surface emanating from the trailing edge (such as the wake mid-surface) are eliminated, thereby facilitating numerical implementations. However, the main novelty is related to the application of the boundary condition: first, the expression for the velocity used for the condition on the body boundary is consistent with that for the velocity in the field; also—contrary to related formulations used by the authors in the past—in the proposed approach, the condition on the body boundary does not require the evaluation of the total vorticity (inside and outside the computational domain). The proposed algorithm, valid for three-dimensional compressible flows, is validated—as a first step—for the case of two-dimensional incompressible flows. Specifically, numerical results are presented for the aerodynamic analysis of two-dimensional incompressible viscous flows past a circular cylinder and past a Joukowski airfoil. In order to verify the desirable absence of artificial damping, we present also results pertaining to the flutter (i.e., dynamic aeroelastic) analysis of a spring-mounted circular cylinder in a viscous flow, free to move in a direction orthogonal to the unperturbed flow. In both cases (aerodynamics and aeroelasticity), the results are in good agreement with existing literature data.     

On the formulation of the planar ANCF triangular finite elements

In this paper, new planar isoparametric triangular finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are developed. The proposed ANCF elements have six coordinates per node: two position coordinates that define the absolute position vector of the node and four gradient coordinates that define vectors tangent to coordinate lines (parameters) at the same node. To shed light on the importance of the element geometry and to facilitate the development of some of the new elements presented in this paper, two different parametric definitions of the gradient vectors are used. The first parametrization, called area parameterization, is based on coordinate lines along the sides of the element in the reference configuration, while the second parameterization, called Cartesian parameterization, employs coordinate lines defined along the axes of the structure (body) coordinate system. The fundamental differences between the ANCF parameterizations used in this investigation and the parametrizations used for conventional finite elements are highlighted. The Cartesian parameterization serves as a unique standard for the triangular FE assembly. To this end, a transformation matrix that defines the relationship between the area and the Cartesian parameterizations is introduced for each element in order to allow for the use of standard FE assembly procedure and define the structure (body) inertia and elastic forces. Using Bezier geometry and a linear mapping, cubic displacement fields of the new ANCF triangular elements are systematically developed. Specifically, two new ANCF triangular finite elements are developed in this investigation, namely four-node mixed-coordinate and three-node ANCF triangles. The performance of the proposed new ANCF elements is evaluated by comparison with the conventional linear and quadratic triangular elements as well as previously developed ANCF rectangular and triangular elements. The results obtained in this investigation show that in the case of small and large deformations as well as finite rotations, all the elements considered can produce correct results, which are in a good agreement if appropriate mesh sizes are used.     

A multi level approach to synthesis of planar mechanisms

A multi level approach to synthesis of planar mechanisms is presented. The approach covers both structural and dimensional synthesis of planar rigid body mechanisms containing revolute and translational joints. The synthesis is based on four different criteria. Firstly the type of mechanism is chosen with a view to get the simplest mechanism that satisfactorily fulfills the remaining three criteria. Two of these criteria are formulated as constraints on the kinematic behavior and the total area occupied by the mechanism, respectively. The fourth criteria is simply the desired minimization of the reactive forces/moments that appear in the mechanism. The desired kinematic behavior is based on a finite number, typically 1, ..., 6, of points in time (positions of the mechanism) where the position and orientation of up to two output bodies may be prescribed. The constraints on occupied areas are labelled territory constraints and formulated as a number of restricted areas (boxes). A synthesis is automatically performed at five levels. At the first level the structure of the mechanism is decided. At the second level initial dimensions for the given type of mechanism are found by random checking. At the third level the constraints on the kinematic behavior is fulfilled. At the fourth level the territory constraints are taken into account and, finally, at the fifth level the minimization of reactions is carried out. The entire approach has been implemented in a software package SYNMEC that runs on PCs and constitutes a way of performing the synthesis of a mechanism that is general and flexible with respect to both the type of mechanism that may be synthesized as well as the desired behavior upon which the synthesis is based.