Structure Parameters and Displacement Equations of Goldberg Mechanisms With the Rank More Than sixStructure Parameters and Displacement Equations of Goldberg Mechanisms With the Rank More Than Six

An infinite number of N-bar Goldberg mechanisms with the rank more than six are proposed. A single closed-loop eight-bar mechanism is first developed by removing the common side links of adjacent loops from the five-loop Bennett mechanism in which all loop frames are collinear. An analysis of trajectory circles of the links shows that the mechanism developed maintains the original constraint conditions and kinematic characteristics, and that its degree of freedom (DOF) is also one. This novel single-loop mechanism is named the eight-bar Goldberg mechanism because of its similar constraint characteristics with the five-bar and six-bar Goldberg mechanisms. The eight displacement parameters of this mechanism are analyzed by using the interior angles of adjacent links, and the five cosine equations for the tangent product of the stagger angles between adjacent links are established in together with one angular displacement equation of odd links and one angular displacement equation of even links. This novel mechanism has one DOF, eight displacement parameters and seven independent displacement equations. The loop rank of the eight-bar Goldberg mechanism is seven, which revises the existing mechanism theory of single closed-loop mechanisms with the rank no more than six. Similarly, a nine-bar Goldberg mechanism is constructed, and its nine displacement parameters are analyzed. This mechanism has eight independent displacement equations and the loop rank is eight, proving that the mechanisms with the rank more than six are not rare. Then, according to the characteristics of the independent displacement equations of the eight-bar and nine-bar Goldberg mechanisms, the N (N>7) displacement parameters of N-bar Goldberg mechanisms are analyzed, and the (N–1) independent displacement equations are deduced with odd and even Ns. It is proved that the loop rank of N-bar Goldberg mechanisms is (N–1), so there exist an infinite number of mechanisms with the rank more than six

Ignaczak equation of elastodynamics

The development of linear elastodynamics in pure stress-based formulation began over half-a-century ago as an alternative to the classical displacement-based treatment that came into existence two centuries ago in the school of mathematical physics in France. While the latter approach – fundamentally based on the Navier displacement equation of motion – remains the conventional setting for analysis of wave propagation in elastic bodies, the stress-based formulation and the advantages it offers in elastodynamics and its various extensions remain much less known. Since the key mathematical results of that formulation, as well as a series of applications, originated with J. Ignaczak in 1959 and 1963, the key relation is named the Ignaczak equation of elastodynamics. This review article presents the main ideas and results in the stress-based formulation from a common perspective, including (i) a history of early attempts to find a pure stress language of elastodynamics, (ii) a proposal to use such a language in solving the natural traction initial-boundary value problems of the theory, and (iii) various applications of the stress language to elastic wave propagation problems. Finally, various extensions of the Ignaczak equation of elastodynamics focused on dynamics of solids with interacting fields of different nature (classical or micropolar thermoelastic, fluid-saturated porous, piezoelectro-elastic) as well as nonlinear problems are reviewed.

Finite Element Based Response Surface Methodology to Optimize Segmental Tunnel Lining

The main objective of this paper is to optimize the geometrical and engineering characteristics of concrete segments of tunnel lining using Finite Element (FE) based Response Surface Methodology (RSM). Input data for RSM statistical analysis were obtained using FEM. In RSM analysis, thickness (t) and elasticity modulus of concrete segments (E), tunnel height (H), horizontal to vertical stress ratio (K) and position of key segment in tunnel lining ring (θ) were considered as input independent variables. Maximum values of Mises and Tresca stresses and tunnel ring displacement (UMAX) were set as responses. Analysis of variance (ANOVA) was carried out to investigate the influence of each input variable on the responses. Second-order polynomial equations in terms of influencing input variables were obtained for each response. It was found that elasticity modulus and key segment position variables were not included in yield stresses and ring displacement equations, and only tunnel height and stress ratio variables were included in ring displacement equation. Finally optimization analysis of tunnel lining ring was performed. Due to absence of elasticity modulus and key segment position variables in equations, their values were kept to average level and other variables were floated in related ranges. Response parameters were set to minimum. It was concluded that to obtain optimum values for responses, ring thickness and tunnel height must be near to their maximum and minimum values, respectively and ground state must be similar to hydrostatic conditions.

Research on Displacement Transfer Characteristics of a New Vibration-Isolating Platform Based on Parallel Mechanism

Based on the parallel mechanism theory, a new vibration-isolating platform is designed and its kinetic equation is deduced. Taylor expansion is used to approximately replace the elastic restoring force expression of vibration-isolating platform, and the error analysis is carried out. The dynamic-displacement equation of the vibration-isolating platform is studied by using the Duffing equation with only the nonlinear term. The dynamic characteristics of the vibration-isolating platform are studied, including amplitude-frequency response, jumping-up and jumping-down frequency, and displacement transfer rate under base excitation. The results show that the lower the excitation amplitude, the lower the initial vibration isolation frequency of the system. The influence of the platform damping ratio ζ on displacement transfer rate is directly related to the jumping-down frequency Ωd and the external excitation frequency. The vibration-isolating platform is ideally suited for high-frequency and small-amplitude vibrations.

Stability to the dissipative Reissner–Mindlin–Timoshenko acting on displacement equation

In this paper, we show that there exists a critical number that stabilises the Reissner–Mindlin–Timoshenko system with frictional dissipation acting only on the equation for the transverse displacement. We identify that the Reissner–Mindlin–Timoshenko system has two speed characteristics v12 := K/ρ1 and v22 := D/ρ2 and we show that the system is exponentially stable if only if \begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*}In the general case, we prove that the system is polynomially stable with optimal decay rate. Numerical experiments using finite differences are given to confirm our analytical results. Our numerical results are qualitatively in agreement with the corresponding results from dynamical in infinite dimensional.

Workspace Modeling and Analysis for Dexterous Hands

The workspace is a fundamental feature for a dexterous hand to grasping plan, motion control, and mechanical design. Although the graphic, numerical, or analytical methods are valid to generate the workspace of dexterous hands, but there exist several defects for these methods to describe the workspace characteristics, such as forms, boundaries, volumes, and intersection workspaces. We propose a combined modeling approach to visualize and analyze the workspace of YWZ dexterous hand, which has five fingers, 20 degrees of freedom (Dofs). The proposed approach is also fit for similar dexterous hands to generate their precise workspaces. After a brief mechanism introduction of YWZ dexterous hand, a displacement equation of its index finger is deduced to calculate the fingertip positions in three-dimension (3D) Euclidean space. Then, a two-dimension (2D) boundary figure enclosing the flexion–extension motion field of the finger is drawn by the displacement equation. Taking this boundary figure as a sketch in a CAD modeling environment, we further model a vivid 3D finger workspace, which is relative with the abduction–adduction motion of the finger. Besides, we generate the whole workspace of YWZ dexterous hand and analyze its volumes and intersection workspaces. The kinematic simulations and physical prototype experimentations are carried out to validate this approach can construct perfect workspace of dexterous hands. Compared with the Monte Carlo method, the form and boundary of the 3D finger workspace generated by our proposed approach have more accurate, integrated, vivid, and intuitional.

Solution of Statically Indeterminate Beam with Straight Axis by Section-Conversion Method

For computation of reaction, internal force and displacement of a beam, the displacement equation from Conversion Method is used in establishing compatibility condition of deformation. In the process of Section-Conversion Method, the joints without sway is set as the coordinate criterion. Cutting the part between two joints as a base element with the method of section, it becomes a simply supported beam in form. The internal forces at the section cut are those which equivalent forces from other side of the beam. According to the axiom of action and reaction, the equivalent forces react to the other part of the beam. The displacement equations are used for all parts one by one. The precision resolutions of reaction, internal force and displacement of a beam are achieved.

Element for Beam Dynamic Analysis Based on Analytical Deflection Trial Function

For beam dynamic finite element analysis, according to differential equation of motion of beam with distributed mass, general analytical solution of displacement equation for the beam vibration is obtained. By applying displacement element construction principle, the general solution of displacement equation is conversed to the mode expressed by beam end displacements. And taking the mode as displacement trial function, element stiffness matrix and element mass matrix for beam flexural vibration and axial vibration are established, respectively, by applying principle of minimum potential energy. After accurate integral, explicit form of element matrix is obtained. The comparison results show that the series of relative error between the solution of analytical trial function element and theoretical solution is about1×10-9and the accuracy and efficiency are superior to that of interpolation trial function element. The reason is that interpolation trial function cannot accurately simulate the displacement mode of vibrating beam. The accuracy of dynamic stiffness matrix method is almost identical with that of analytical trial function. But the application of dynamic stiffness matrix method in engineering is limited. The beam dynamic element obtained in this paper is analytical and accurate and can be applied in practice.