Novel Methodology for Inflection Circle based synthesis of Straight Line Crank Rocker Mechanism

Emerging fields like Compact Compliant Mechanisms have created newer/novel situations for application of straight line mechanisms. Many of these situations in Automation and Robotics are multidisciplinary in nature. Application Engineers from these domains are many times uninitiated in involved procedures of synthesis of mechanisms and related concepts of Path Curvature Theory. This paper proposes a predominantly graphical approach using properties of Inflection Circle to synthesize a crank rocker mechanism for tracing a coupler curve which includes the targeted straight line path. The generated approximate straight line path has acceptable deviation in length, orientation and extent of approximate nature well within the permissible ranges. Generation of multiple choices for the link geometry is unique to this method. To ease the selection, a trained Artificial Neural Network (ANN) is developed to indicate relative length of various options generated. Using studied unique properties of Inflection Circles a methodology for anticipating the orientation of the straight path vis-à-vis the targeted path is also included. Two straight line paths are targeted for two different crank rockers. Compared to the existing practice of selecting the mechanism with some compromise due to inherent granularity of the data in Atlases, proposed methodology helps in indicating the possibility of completing the dimensional synthesis. The case in which the solution is possible, the developed solution is well within the design specifications and is without a compromise.

Offset Trajectory Planning of Robot End Effector and its Jerk with Curvature Theory

Some situations that change the parameters of the kinematic structure may cause the robot end effector to deviate [Merlet [2005] Parallel Robots, Vol. 128 (Springer Science & Business Media, Germany)] from the desired trajectory. This effect is called the robustness of the robot by Merlet. One of the ways to correct the robustness is by updating the robot trajectory. The jerk vector of the robot end effector is the third-order positional variation of the TCP and defined as thus the time derivative of the acceleration vector. If there is a high curvature on the transition curve trajectory of robot, then there is a tangential jerk along the trajectory. In this study, the geometrically offset trajectory of the robot end effector from the current trajectory was obtained by using the curvature theory. The angular velocity and angular acceleration of the offset trajectory were calculated. An example of the main trajectory of robot end effector and its offset is given. Also, the jerk of the robot end effector of the offset trajectory was calculated according to the curvature of the trajectory surface in case of a jerk problem caused by a high curvature in the transition curve along the offset trajectory curve.

Two-dimensional black holes in the limiting curvature theory of gravity

In this paper we discuss modified gravity models in which growth of the curvature is dynamically restricted. To illustrate interesting features of such models we consider a modification of two-dimensional dilaton gravity theory which satisfies the limiting curvature condition. We show that such a model describes two-dimensional black holes which contain the de Sitter-like core instead of the singularity of the original non-modified theory. In the second part of the paper we study Vaidya type solutions of the model of the limiting curvature theory of gravity and used them to analyse properties of black holes which are created by the collapse of null fluid. We also apply these solutions to study interesting features of a black hole evaporation.

Anisotropic compact stars in higher-order curvature theory

In this paper we shall consider spherically symmetric spacetime solutions describing the interior of stellar compact objects, in the context of higher-order curvature theory of the $${{\mathrm {f(R)}}}$$ f ( R ) type. We shall derive the non-vacuum field equations of the higher-order curvature theory, without assuming any specific form of the $${{\mathrm {f(R)}}}$$ f ( R ) theory, specifying the analysis for a spherically symmetric spacetime with two unknown functions. We obtain a system of highly non-linear differential equations, which consists of four differential equations with six unknown functions. To solve such a system, we assume a specific form of metric potentials, using the Krori–Barua ansatz. We successfully solve the system of differential equations, and we derive all the components of the energy–momentum tensor. Moreover, we derive the non-trivial general form of $${{\mathrm {f(R)}}}$$ f ( R ) that may generate such solutions and calculate the dynamic Ricci scalar of the anisotropic star. Accordingly, we calculate the asymptotic form of the function $${\mathrm {f(R)}}$$ f ( R ) , which is a polynomial function. We match the derived interior solution with the exterior one, which was derived in [1], with the latter also resulting to a non-trivial form of the Ricci scalar. Notably but rather expected, the exterior solution differs from the Schwarzschild one in the context of general relativity. The matching procedure will eventually relate two constants with the mass and radius of the compact stellar object. We list the necessary conditions that any compact anisotropic star must satisfy and explain in detail that our model bypasses all of these conditions for a special compact star $$\textit{Her X--1}$$ Her X - - 1 , which has an estimated mass and radius $$(mass = 0.85 \pm 0.15M_{\circledcirc }\ and\ radius = 8.1 \pm 0.41~\text {km}$$ ( m a s s = 0.85 ± 0.15 M ⊚ a n d r a d i u s = 8.1 ± 0.41 km ). Moreover, we study the stability of this model by using the Tolman–Oppenheimer–Volkoff equation and adiabatic index, and we show that the considered model is different and more stable compared to the corresponding models in the context of general relativity.

Perturbative S-matrix unitarity (S†S = 1) in Rμν2 gravity

We show that in the quadratic curvature theory of gravity, or simply [Formula: see text] gravity, the tree-level unitarity bound (tree unitarity) is violated in the UV region but an analog for [Formula: see text]-matrix unitarity [Formula: see text] is satisfied. This theory is renormalizable, and hence the failure of tree unitarity is a counter example of Llewellyn Smith’s conjecture on the relation between them. We have recently proposed a new conjecture that [Formula: see text]-matrix unitarity gives the same conditions as renormalizability. We verify that [Formula: see text]-matrix unitarity holds in the matter-graviton scattering at the tree level in the [Formula: see text] gravity, demonstrating our new conjecture.

An Analysis for Magnesium Alloy Curvature Products Formed by Staggered Extrusion (SE) Based on the Upper Bound Method

Staggered extrusion (SE) is a new method to solve the bottleneck of traditional curvature products, such as long manufacturing cycle, many forming processes and difficult quality control. How to quantitatively control the curvature of extruded products is the key to implement this method. Herein, the upper bound method is used to calculate and analyze the power consumption of each characteristic zone in the SE process. The theoretical model of extrusion load and curvature is established. The results show that the staggered distance h has an important influence on the curvature κ. When the staggered distance h increases from 8 mm to 24 mm and other conditions remain unchanged, the curvature κ increases from 0.0546 to 0.1607. Any combination of the staggered distance h and the extrusion ratio λ corresponds to an eccentricity ratio ξ. The eccentricity ratio ξ decreases with the increase of the staggered distance h or the extrusion ratio λ. By comparison, it can be seen that the variation trend of the theoretical predicted value and the FE modelling in the steady-state extrusion stage is consistent. The experimental results are in good agreement with the curvature theory prediction model. These results provide a scientific basis for the formulation of the SE process and precisely controlling magnesium alloy curvature products.

An approach for designing a developable and minimal ruled surfaces using the curvature theory

Developable surfaces are defined to be locally isometric to a plane. These surfaces can be formed by bending thin flat sheets of material, which makes them an active research topic in computer graphics, computer aided design, computational origami and manufacturing architecture. We obtain condition for developable and minimal ruled surfaces using rotation frame. Also, the validity of the theorems is illustrated with examples.

Holographic complexity in general quadratic curvature theory of gravity

In the context of CA conjecture for holographic complexity, we study the action growth rate at late time approximation for general quadratic curvature theory of gravity. We show how the Lloyd’s bound saturates for charged and neutral black hole solutions. We observe that a second singular point may modify the action growth rate to a value other than the Lloyd’s bound. Moreover, we find the universal terms that appear in the divergent part of complexity from computing the bulk and joint terms on a regulated WDW patch.