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Coordinate Systems and Vector Algebra
Published in Ahmad Shahid Khan, Saurabh Kumar Mukerji, Electromagnetic Fields, 2020
Ahmad Shahid Khan, Saurabh Kumar Mukerji
The three-dimensional coordinate systems can further be classified into (i) parallel coordinate systems wherein a point is visualized in n-dimensional space as a poly-line connecting points on n vertical lines; (ii) curvilinear coordinate systems that are generalized coordinate systems based on intersection of curves; (iii) circular (polar) coordinate systems that represent a point in a plane by an angle and a distance from the origin; (iv) Plücker coordinate systems, in which lines are represented in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates; (v) generalized coordinate systems that are used in a Lagrangian treatment of mechanics; (vi) canonical coordinate systems that are used in Hamiltonian treatment of mechanics; and (vii) orthogonal coordinate systems wherein the coordinate surfaces meet at right angles. This last category is commonly used in the study of field theory.
Mechanics of Fatigue Crack Growth
Published in Vladimir V. Bolotin, Mechanics of Fatigue, 2020
The main variables in analytical mechanics are generalized coordinates and generalized forces. In addition to common (Lagrangian) generalized coordinates, (later on, for brevity, L-coordinates), additional variables are introduced that describe shape, size and position of cracks. In the paper by Bolotin [14] it was proposed to name these variables, in honor of Griffith, Griffithian generalized coordinates (later on, G-coordinates). Actually, Griffith (1920) was the first to introduce this kind of coordinate (such as crack half-lengths) into analytical consideration. In simple cases, the corresponding generalized forces coincide with the well-known and widely used generalized forces of fracture mechanics. Another fundamental concept is the concept of Griffith’s variations (G-variations), named also in honor of Griffith. They are variations of G-coordinates calculated under certain conditions that will be stated later.
Consistent Mass Method for Frames and Finite Elements
Published in Franklin Y. Cheng, Matrix Analysis of Structural Dynamics, 2017
Finite element formulation and calculation can be based on generalized coordinates or natural coordinates. The generalized coordinate is fundamental and provides the basis of understanding the finite element formulation. The natural coordinate finite element, commonly called the isoparametric element, is more effective in practical analysis. As introductory material, this discussion includes only the triangular element for generalized coordinates and the quadrilateral for natural coordinates. Standard texts on finite elements are available for further study.
How AD can help solve differential-algebraic equations
Published in Optimization Methods and Software, 2018
John D. Pryce, Nedialko S. Nedialkov, Guangning Tan, Xiao Li
The second is the task of solving a, possibly constrained, mechanical system directly from a Lagrangian formulation. Conceptually it has several phases. The motion is defined by a Lagrangian function where is a vector of generalized coordinates , plus possibly a vector of constraints . To set up (phase 1), the equations of motion from L and C one applies partial differentiation and , as well as straight , to L and C. When the result is an index 3 DAE, which must (phase 2) be readied for numerical solution and (phase 3) solved.
Coupled numerical model for the nonlinear dynamics of deepwater single barge installation system
Published in Ships and Offshore Structures, 2023
Naeem Ullah, Mac Darlington Uche Onuoha, Menglan Duan, Muhammad Sajid
It is important to consider the surge, heave and pitch motions when modelling the dynamics of a floating multibody system (MBDS). Thus our contextual case of a single barge with a suspended payload featured the use of time-dependent reference frames to establish coordinate points for the coupled single barge and the suspended payload, where E-frame is fixed at the water’s surface and A-frame and B-frame locate the single barge and suspended payload, respectively. With that, the position vector of the planar multibody (single barge and suspended payload) in a global frame of reference is given below as where signify the surge, heave and pitch motions of the floating barge and suspended payload, respectively. Furthermore, P-frame, C-frame and D-frame are used as the time-dependent frame of reference for the translation and rotation of the coupled single barge and payload system. In multibody system dynamics, the number of degrees of freedom is dependent on the generalised coordinates. Thus the planar kinematic joint at point C as shown in Figure 3 is used to establish the connection between the two bodies. Therefore, to formulate the generalised coordinates of this planar multibody system, the translational payload coordinates are substituted with the planar coordinates. With that, the vector of the generalised coordinates, , of Equation (1) is expressed as follows: where represent the surge, heave and pitch motions of the multibody dynamic system.
Geometrical motion planning for cable-climbing robots applied to distribution power lines inspection
Published in International Journal of Systems Science, 2021
Carlos Henrique Farias dos Santos, Mohamed Hassan Abdali, Daniel Martins, Campos Bonilla Aníbal Alexandre
A very effective scheme for motion planning is obtained by representing the robot as a mobile point in an appropriate space, where the images of the workspace obstacles are also reported. To this end, it is natural to refer to the generalised coordinates of the mechanical system, whose value identifies the configuration of the robot. This associates to each posture of the latter a point in the configuration space, i.e. the set of all configuration that the robot can assume (see Figure 7(A–H)).