Heron’s cubic root iteration method with a new approach

Heron’s cubic root iteration formula conjectured by Wertheim is proved and extended for any odd order roots. Some possible proofs are suggested for the roots of even order. An alternative proof of Heron’s general cubic root iterative method is explained. Further, Lagrange’s interpolation formula for nth root of a number is studied and found that Al-Samawal’s and Lagrange’s method are equivalent. Again, counterexamples are discussed to justify the effectiveness of the present investigations.

Turning dynamics analysis of a heavy articulated vehicle by the vehicle planar dynamic model with two steering input signals

The vehicle planar single track dynamic model with two input steering angle parameters is derived by using Lagrange's method with the basis of equations for calculating the tire's force components. Dynamic analysis of a heavy articulated vehicle in case of turing is carried out by the vehicle planar dynamic model, in which two input steering angles are taken into account. Simulation with the selected velocity value to make sure that the stability according to the friction conditions at all axles of the vehicle is satisfied. Turning spacing, lateral forces at each axle of the vehicle are determined and analyzed for all three different cases of steering angles, respectively with steering angle of the semi-trailer is in the same direction, in the opposite direction and is locked or not steered in comparision with the steering angle of the tractor. The obtained results show that the derived model could employ to determine the planar kinematic and dynamic parameters, and analyze the dynamic safety features of an articulated vehicle, too. In addition, the derived mathematical model could also employ to develop a computational model that controls the planar articulated vehicle dynamics.

Reliability assessment for multi-state automatic ticket vending machine (ATVM) through software and hardware failures

PurposeThis paper presents the performance analysis of the automatic ticket vending machine (ATVM) through the functioning of its different hardware and software failures.Design/methodology/approachFrequent failures in the working of ATVM have been observed; therefore, the authors of the paper intend to analyze the performance measures of the same. Authors have developed a mathematical model based on different hardware and software failures/repairs, which may occur during the operation, with the help of the Markov process. The developed model has been solved for two kinds of failure/repair rates namely variable failures (very much similar to real-time failure) and constant failures. Lagrange's method and Laplace transformation are used for the solution of the developed model.FindingsReliability and mean time to failure of the ATVM are determined. Sensitivity analysis for ATVM is also carried out in the paper. Critical components of the ATVM, which affect the performance of the same, in terms of reliability and MTTF are also identified.Originality/valueA mathematical model based on different hardware and software failures/repairs of ATVM has been developed to analyze its performance, which has not been done in the past.

Analysis of generalized force and its influence on ride and stability of railway vehicle

Formulation of a rail vehicle model using Lagrange’s method requires the system’s kinetic energy, potential energy, spring potential energy, Rayleigh’s dissipation energy and generalized forces to be determined. This article presents a detailed analysis of generalized forces developed at wheel–rail contact point for 27 degrees of freedom–coupled vertical–lateral model of a rail vehicle formulated using Lagrange’s method and subjected to random track irregularities. The vertical–lateral ride comfort of the vehicle and the ride index of the vehicle are evaluated based on ISO 2631-1 comfort specifications and stability is determined using eigenvalue analysis. The parameters that constitute the generalized forces and critically influence ride and stability have been identified and their influences on the same have been analysed in this work.

Analysis and Design of Twin Gyroscopes for Ocean-Wave Energy Converters and Ship Roll-Stabilizers

Twin-gyroscopic systems are designed for ocean-wave energy converters and ship roll-stabilizers to double desirable gyroscopic effects and eliminate undesirable reaction torques. In deriving analytical equations of motion, the configuration spaces of gyroscopic systems are defined by using body-attached moving frames. The moving frame of each constituent body is defined by its inertial coordinates of the center of mass and a rotation matrix which expresses the attitude of its coordinate axes from the inertial coordinate axes. Therefore, to utilize powerful Lagrange’s method, it is extended to accommodate rotation matrices in configuration spaces and allow angular velocities as generalized velocities. First, in the paper, to identify undesirable reaction torques of gyroscopic systems and find a scheme to eliminate them, we present the basics of a reaction wheel. Second, to identify the desirable gyroscopic effect, we consider a control moment gyroscope and derive the equations of motion using the extended Lagrange’s method. In addition, the equations of motion are also derived by using the Newton-Euler method, where action and reaction torques are explicitly expressed. The comparison of the resulting equations derived by the two methods reveals the simplicity of Lagrange’s method in treating actuating motor torques and how the effects of reaction torques are implicitly included in the variationally derived equations. Finally, the equations of motion for a twin-gyroscopic system are obtained by incorporating the scheme to eliminate the undesirable reaction torques.

Modeling, Validation and Prototype Development of Electric Sail Tether Deployment Systems for CubeSats

The Electric sail concept is based on a distributed tether satellite system with tether lengths on the order of thousands-of meters. The system must deploy from stowed arrangement into a selected flight configuration in which thrust forces are transmitted through the tether to the satellite body. The system must be stable through deployment procedure and maintain stable, desired configuration during flight operations. Understanding the dynamic behavior of the satellite bodies and distributed, conductive tether are critical to long-range design and development of the Electric Sail concept. This paper’s contribution is the presentation, development and validation of a mathematical model for simulating E-Sail deployment of a prototype system for testing on the MSFC Robotic Flat Floor Facility. A massed tether model is developed using the bead and string concept with equations of motion derived from Lagrange’s Method. The model is validated using infrared motion capture data produced by controlled experiments of a representative tether portion outfitted with IR targets. Further, a prototype is presented which will be used to investigate an E-Sail deployment approach and associated control. The design of this system will allow for deployment on specially designed flat floor facilities at MSFC. The prototype will be used to: 1) gather data for validation of system dynamic model, 2) evaluate alternative deployment strategies, 3) evaluate tether reel-out and damping control strategies.

Dynamic analysis of electrohydraulic cable-driven parallel robots

Dynamic analysis is required for achieving higher efficiency of cable-driven parallel robots. This paper presents the dynamic analysis of the cable-driven parallel robots using the Lagrange’s method, taking cable’s mass and elasticity into account. The Lagrange’s equations of motion are derived and evaluated for the generalized coordinates of the system. The dynamic motion of the parallel robot is expressed by the generalized forces and generalized coordinates to completely specify the configuration of the whole mechanical system as well as every component of the system. The cables are modeled to control and design the motion of each part of the rigid body. The elasticity is determined using the optimal cable’s tensions and lengths. Numerical simulations are performed to obtain the dynamic motion of the cable-driven parallel robots and. Experimental analyses and the effect of the mass of the end-effector on the cable’s tension and elasticity are also investigated. These examples illustrate that the general motion of the rigid body is superior described in terms of a set of independent coordinates. The results indicate that a better speed of the end-effector can be achieved by adding the linear and rotational motions of the electrohydraulic cylinder actuators into the traditional cable-driven parallel robots.