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Work and Energy
Published in Osamu Morita, Classical Mechanics in Geophysical Fluid Dynamics, 2019
Let’s take the y-axis vertically downward and the projecting position as the reference point of the potential energy. As gravity is the conservative force, the mechanical energy is conserved at the initial and final positions. Therefore, 12mv02+0=12mv2−mgh,v=v02+2gh.
Energy, Work, and Power
Published in Daniel H. Nichols, Physics for Technology, 2019
Recall that the work done on a mass is the force applied on the mass in the direction of the motion of the mass. A conservative force is a force where the work done on a particle does not depend on the path the mass is pushed or pulled through by the force. An example of a conservative force is gravity, or electric forces, or the force associated with a spring. In the case of gravity or a spring, the change in potential energy and the work done on a particle depends only on the initial and final positions of the system, and not on the path taken. Conservative forces therefore have associated with them a potential energy. Work done by a frictional force is an example of a non-conservative force, because the work depends on the path and not just the end points.
Mechanics and Electromagnetics
Published in Sergey Edward Lyshevski, Mechatronics and Control of Electromechanical Systems, 2017
The energy of a magnetic moment m→ in an externally produced B→ is characterized by the potential energy Π=−m→·B→. Using the magnetization, Π=-12∫M→⋅B→dv. The work done by a conservative force is W = −ΔΠ, where ΔΠ is the change in the potential energy associated with the force. The negative sign indicates that work performed against a force field increases potential energy, while work done by the force field decreases potential energy.
Simulation of vehicle-turnout coupled dynamics considering the flexibility of wheelsets and turnouts
Published in Vehicle System Dynamics, 2023
Jiayin Chen, Ping Wang, Jingmang Xu, Rong Chen
Furthermore, the first order variation of total potential energy δU and the virtual work δW caused by the conservative force in turnout systems are: where, Nf is the number of fasteners; Kri-ver and Ksi-ver represent element stiffness matrices of rails and sleepers in the vertical panel, respectively; Kri-hor denotes the element stiffness matrix of rails in the horizontal plane. zri, yri and βri are the vertical, lateral and pitching dofs of rail nodes; zsi and ysi are the vertical and lateral dofs of sleepers; their first derivative forms represent the corresponding velocity. Kfzi, Kfyi and Kfβi denote the vertical, lateral and pitching stiffness of the ith fastener; Kbzi indicates the vertical stiffness of ballast corresponding to the ith sleeper node, and Kbyirepresents the lateral stiffness of the ith sleeper; Cfzi, Cfyi, Cfβi, Cbzi and Cbyi are the corresponding damping properties.
Enhanced model including moment-rotation dependency for stability of thin-walled structures
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2020
Atef S. Gendy, Samir S. Marzouk
The generalized forces acting on a mechanical system may be classified into two categories: conservative and non-conservative forces. The conservative force can be defined as any generalized force associated with stationary single-value potential function dependent only on the generalized displacement. Its work over compatible actual displacement depends on the initial and final configuration of the system (i.e., path independent); and its virtual work over any admissible virtual displacement can be written as a first variation of the displacement-dependent potential function. Dead loads (displacement-independent) and centrifugal forces (displacement-dependent) are examples of conservative forces. On the other hand, the non-conservative force can be defined as any force processing a velocity- or time-dependent potential. It may be depending explicitly on time which is denoted as instationary force. Furthermore, a non-conservative time independent force (i.e., stationary non-conservative force) may be classified either as velocity-dependent, or as velocity-independent; i.e., purely displacement-dependent (more precisely rotation-dependent). The force in this latter case is denoted as circulatory force which is very common in the engineering practical applications. This force does not possess a potential; therefore, its virtual work for any admissible virtual displacement of the body cannot be written as variational of displacement dependent functional. The present work is limited to the large displacement analysis of systems subjected to conservative or non-conservative force of the circulatory type.
Polymorphic transitions mediated by surfactants in liquid crystal nanodroplet
Published in Liquid Crystals, 2019
Hiroaki Tsujinoue, Takuya Inokuchi, Noriyoshi Arai
In Equation (1), is the conservative force, is the dissipative power, is the random force, is the bonding force, is the bending force and is the external force to realize a spherical confinement. The conservative force () is given by