The optimization of the motion of an elastic rod by the method of integro-differential relations

The possibility of constructing a solution for given cost functionals and the optimization of the motion of elastic systems with distributed parameters are investigated. The regular integro-differential approach to a broad class of boundary value problems is developed, and a cost functional for the solution obtained is proposed. For the two-dimensional motions of a uniform straight elastic rod, the case of polynomial control is considered. An algorithm for forming the optimal control, which steers the system to the state of minimum total energy at a final time instant, is developed. The parameters of the problem are adjusted so that the period of the lower order mode is comparable with the interval in which the motions are investigated. The results obtained by using the method of separation of variables and the method of integro-differential relations are analyzed and compared.     

The construction of optimal control for the motion of elastic bodies by using the method of integro-differential relations

The opportunities of modeling and optimization of motion of elastic systems with distributed parameters are investigated. A regular integro-differential approach, which reduces a wide class of linear initialboundary value problems to a conditional minimization of non-negative quadratic functionals is developed, and a cost function of approximate solutions obtained is proposed. For longitudinal motions of a uniform straight elastic rod, the case of polynomial control of the motion of its end is considered. An algorithm of constructing an optimal control that steers the system to the state of a minimal mechanical energy at the final time instant. The parameters of the problem are adjusted so that the time of transition processes would be comparable with the interval, on which the motions are investigated. The analysis and comparison of the results obtained by using the method of integro-differential relations for a one-dimensional model of a thin elastic rod and a proposed approximate three-dimensional model of a prismatic beam.     

CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel

In this paper, a computational method for numerical solution of a class of integro-differential equations with a weakly singular kernel of fractional order which is based on Cos and Sin (CAS) wavelets and block pulse functions is introduced. Approximation of the arbitrary order weakly singular integral is also obtained. The fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations by using the operational matrix of fractional integration of CAS wavelets. The error analysis of CAS wavelets is given. Finally, the results of some numerical examples support the validity and applicability of the approach.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

粘弹性板的非线性动力稳定性分析方法

将微分-积分型参数振动方程组转化成微分型,且基于增量谐波平衡法的一般应用途径,分析了受面内周期激励的粘弹性板的非线性动力稳定特性,揭示了主要动力不稳定区域的整体下移以及缩小和标准线性固体材料的粘性参数、板的振动频率之间的关系．同时给出了增量谐波平衡法直接应用于非线性微分-积分型参数振动方程的简化途径,并通过两种应用途径所得结果的对比,检验了这种简化途径的有效性．     

Note on the ill-conditioned eigenvalue problem in elastic vibrations

Two cases which could cause ill-conditioning of the lower modes and frequencies of an undamped elastic structure are discussed. The first case is when rigid body motions are connected to a relatively small moment of inertia. The second case is when the stiffness matrix is ill-conditioned with respect to the solution of linear equations. It is shown that the first kind of ill-conditioning is not serious provided that an appropriate method of computation is used. The second kind, which is more common and more serious, is dealt with in more detail. It is shown how the results may be improved in various ways by applying the Natural Factor approach of Argyris and Brönlund and computing singular values and vectors.     

Upon sector elements for two-dimensional problems

Different sector elements for two-dimensional problems for linear elastic analysis are developed and their utility for different types of problems is discussed. One of these elements (Type 1) fulfils the rigid body criterion for the two translatory motions as well as interelement compatibility conditions. This element is used together with a triangular element for the solution of Kirsch's problem. A considerable accuracy of the results is obtained with a relatively small number of elements.     

A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.     

Source Estimation by Full Wave Form Inversion

We consider the inverse problem of estimating the parameters describing the source in a seismic event, using time-dependent ground motion recordings at a number of receiver stations. The inverse problem is defined in terms of a full waveform misfit functional, where the objective function is the integral over time of the weighted $L_2$ distance between observed and synthetic ground motions, summed over all receiver stations. The misfit functional is minimized under the constraint that the synthetic ground motion is governed by the elastic wave equation in a heterogeneous isotropic material. The seismic source is modeled as a point moment tensor forcing in the elastic wave equation. The source is described by 11 parameters: the six unique components of the symmetric moment tensor, the three components of the source location, the origin time, and a frequency parameter modeling the duration of the seismic event. The synthetic ground motions are obtained as the solution of a fourth order accurate finite difference approximation of the elastic wave equation in a heterogeneous isotropic material. The discretization satisfies a summation-by-parts (SBP) property that ensures stability of the explicit time-stepping scheme. We use the SBP property to derive the discrete adjoint of the finite difference method, which is used to efficiently compute the gradient of the misfit. A new moment tensor source discretization is derived that is twice continuously differentiable with respect to the source location. The differentiability makes the Hessian of the misfit a continuous function of all source parameters. We compare four different gradient-based approaches for solving the constrained minimization problem; two non-linear conjugate gradient methods (Fletcher–Reeves and Polak–Ribière), and two quasi-Newton methods (BFGS and L-BFGS). Because the Hessian of the misfit has a very large condition number, the parameters must be scaled before the minimization problem can be solved. Comparing several scaling approaches, we find that the diagonal of the Hessian provides the most reliable scaling alternative. Numerical experiments are presented for estimating the source parameters from synthetic ground motions in two different three-dimensional models; one in a simple layer over half-space, and one using a fully heterogeneous material. Good convergence properties are demonstrated in both cases.     

On the coupled dynamics of small spacecraft and elastic deployable appendages

A thorough investigation of the dynamics of finite-mass satellites with a deployable elastic arm is presented. This work is focused on the interaction between spacecraft rigid body motion and its flexible arm dynamics during the deployment process. The classical Newton–Euler formulation and the Lagrangian approach are applied to the study of the dynamics of spacecraft and its deploying arm. Utilizing a non-Newtonian floating frame to define the arm elastic deformation field, the interactions between the spacecraft and its moving arm have been simulated. Complete equations of motion show that the spacecraft motion induces dynamical stiffness on the arm; in addition, axial and lateral motions of the deploying elastic arm change the spacecraft mass-characteristics and thus influence the spacecraft’s rigid body motions. The overall dynamic behavior is highly dependent on spacecraft mass characteristics in addition to the “arm deployment time (ADT)”. The results of case studies clearly indicate that some assumptions previously applied in appendage dynamic analysis are not conservative and produce erroneous results. This study realistically investigates the dynamics of elastic deploying appendages by considering finite-mass characteristics for small and massy spacecraft. The results reveal that for massive spacecraft the arm’s flexible dynamics is mainly excited through deployment, while for small spacecraft the energy transfers to the arm base and the spacecraft rigid body motion is considerably stimulated. Moreover, this work has further highlighted the effects of ADT in the overall system response. The findings of this work show that the energy distribution between arm’s elastic dynamics and spacecraft rigid body motions is an important factor in the design of any control system to limit unwanted arm-tip motions.     

Seismic analysis of asymmetric building-foundation systems

A method of analysis for the earthquake response of three-dimensional multi-storey asymmetric buildings founded on a flexible foundation is presented. The building-foundation system considered in this study is a linear N-storey asymmetric building on a rigid footing resting on the surface of a linear elastic half-space. The whole system has 3N + 5 displacement degrees of freedom. The governing equations are developed considering the motions of each floor and the motions of the whole system. The governing equations of the floors are first uncoupled in terms of footing displacements using the mode superposition method. Substitution of structural deformations, in combination with the dynamic soil-structure interaction force-displacement relationships proposed by Veletsos and Verbic [ASCE J. Engng Mech. Div. 100, 189–201 (1974)]and Veletsos and Nair [ASCE J. Geotechn. Engng Div. 100, 225–246 (1974)]into the governing equations of the whole system results in five integro-differential equations for footing displacements, which are then solved by numerical step-by-step time-history analysis. A 10-storey asymmetric building on soft soil was subjected to an artificially generated earthquake excitation in order to obtain the soil-structure interaction and eccentricity effects. The results show that soft soil conditions increase the lateral deflections, but reduce the twists, storey shears, and torques. Increasing eccentricity increases the twists and torques, but does not modify the lateral deflections at the centre of mass, and the total storey shears.     

Effective Self-calibration for Camera Parameters and Hand-eye Geometry Based on Two Feature Points Motions

A novel and effective self-calibration approach for robot vision is presented, which can effectively estimate both the camera intrinsic parameters and the hand-eye transformation at the same time. The proposed calibration procedure is based on two arbitrary feature points of the environment, and three pure translational motions and two rotational motions of robot endeffector are needed. New linear solution equations are deduced, and the calibration parameters are finally solved accurately and effectively. The proposed algorithm has been verified by simulated data with different noise and disturbance. Because of the need of fewer feature points and robot motions, the proposed method greatly improves the efficiency and practicality of the calibration procedure.      

Coupling characteristics of rigid body motion and elastic deformation of a 3-PRR parallel manipulator with flexible links

Modeling of multibody dynamics with flexible links is a challenging task, which not only involves the effect of rigid body motion on elastic deformations, but also includes the influence of elastic deformations on rigid body motion. This paper presents coupling characteristics of rigid body motions and elastic motions of a 3-PRR parallel manipulator with three flexible intermediate links. The intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions based on the assumed mode method (AMM). Using Lagrange multipliers, the fully coupled equations of motions of the flexible parallel manipulator are developed by incorporating the rigid body motions with elastic motions. The mutual dependence of elastic deformations and rigid body motions are investigated from the analysis of the derived equations of motion. Open-loop simulation without joint motion controls and closed-loop simulation with joint motion controls are performed to illustrate the effect of elastic motion on rigid body motions and the coupling effect amongst flexible links. These analyses and results provide valuable insight to the design and control of the parallel manipulator with flexible intermediate links.     

Using Pseudo-Parabolic and Fractional Equations for Option Pricing in Jump Diffusion Models

In mathematical finance a popular approach for pricing options under some Lévy model would be to consider underlying that follows a Poisson jump diffusion process. As it is well known this results in a partial integro-differential equation (PIDE) that usually does not allow an analytical solution, while a numerical solution also faces some problems. In this paper we develop a new approach on how to transform the PIDE into a class of so-called pseudo-parabolic equations which are well known in mathematical physics but are relatively new for mathematical finance. As an example we will discuss several jump-diffusion models which Lévy measure allows such a transformation.     

A new computational method for solution of non-linear Volterra–Fredholm integro-differential equations

An approximation method is developed for the solution of high-order non-linear Volterra–Fredholm integro-differential (NVFID) equations under the mixed conditions. The approach is based on the orthogonal Chebyshev polynomials. The operational matrices of integration and product together with the derivative operational matrix are presented and are utilized to reduce the computation of Volterra–Fredholm integro-differential equations to a system of non-linear algebraic equations. Numerical examples illustrate the pertinent features of the method.     

基于优化的DTW算法的人体运动数据检索

随着大量三维人体运动数据库的建立,使得在数据库中实现基于内容的三维人体运动检索面临着诸多困难,文中提出一种分阶段的动态时间变形(DTW)优化算法的人体运动数据检索技术,可有效检索出逻辑上相似的运动。该算法首先对齐两个运动序列的坐标位置,基于窗口距离构造距离矩阵。其次采用基于全局和局部约束的DTW优化算法进行相似度匹配,得到两个运动间的对应关系。最后通过归一化相似度和DTW平均距离分阶段判断运动的相似性。实验结果表明,分阶段的DTW优化算法在提高效率的同时对长度不等的运动能取得较好的检索结果。     

Rigid-elastic modeling of meshing gear wheels in multibody systems

In many applications in mechanical engineering, gear wheels are used to transmit power between rotating shafts and, therefore, the ability to incorporate them into multibody systems and to simulate contact between them has become an essential topic in multibody dynamics.However, in some applications gear wheels may not be considered as being perfectly rigid. Due to the effect of contact forces there occur relevant deformations in meshing teeth and it is required for a high quality of the analysis to introduce some elasticities in the model of meshing gear wheels. Therefore, in this work elastic elements between the teeth and the body of each gear wheel are considered. This approach is especially well suited for multibody systems since it is a compromise between a totally rigid model and a fully elastic model allowing the simulation of large motions with many revolutions while still important elasticities are considered. The teeth and the body of each gear wheel are still modelled as being rigid but they are connected to each other by elastic elements. In doing so, an efficient and physically motivated algorithm is described and implemented in order to find the effects of multi-tooth contact as well as backlash and left and right hand side contact of the meshing teeth. Some examples compare the simulation results of rigid, partially elastic and fully elastic models.     

Finite element calculation of the anomalous skin effect in a homogeneous,unmagnetized cylindrical plasma column

An integro-differential equation which describes the anomalous penetration of an oscillating axial magnetic field into a homogeneous, unmagnetized cylindrical plasma column is solved numerically by a finite element expansion. Test cases show the good convergence properties of the method. Available experimental data are compared with numerical results obtained using the measured parameters.     

Homotopy analysis method for higher-order fractional integro-differential equations

In this paper, we present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of higher-order fractional integro-differential equations with boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions. The comparison of the results obtained by the HAM with the exact solutions is made, the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of series solution.     

Optimal design of elastic plastic frames accounting for seismic protection devices

The optimal design of elastic perfectly plastic steel frames with or without suitable protection devices and subjected to static as well as seismic loadings is studied. Two minimum volume problem formulations are proposed, on the grounds of the so-called statical approach, accounting for three different resistance limits: the purely elastic limit, the (elastic) shakedown limit and the instantaneous collapse limit. The adopted load combinations are characterized by the presence of fixed loads, of quasi-static perfect cyclic loads and dynamic (seismic) loads. The linear elastic effects of the dynamic actions are studied by utilizing a modal technique. The proposed treatment is referred to the most recent Italian code related to the structural analysis and design. The solution of the optimization problem is reached by using an appropriate linearization iterative technique specialized to the proposed formulations. Flexural frames and cross-braced frames are studied, and the related minimum volume structures are reached for assigned features of the base isolation device. The Bree diagrams of the obtained optimal designs are also determined in order to characterize their structural behaviour.