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Vehicle Structure Crash Dynamics
Published in Donald E. Struble, John D. Struble, Automotive Accident Reconstruction, 2020
Donald E. Struble, John D. Struble
Plowing ahead with a linear structure, the model then looks a lot like the usual spring-mass system, or harmonic oscillator, as seen in Figure 17.6, except that the spring is a compression-only device. The single mass represents the vehicle, its occupants and its cargo. The load–deflection “curve” represents all of the crushable structure, and is just a straight line having a slope—or stiffness—of k, which is in consistent units (lb/ft, for example). The mass moves in the positive or negative X direction without friction (no tire forces). At time zero (initial contact), its velocity is Vi in the positive X direction and its displacement is zero. Since the barrier does not move, the crush of the structure is equal to the displacement X (again in consistent units—ft, for example) of the mass. The time derivative (i.e., the rate of change with respect to time) of X is the velocity, denoted by X˙, measured in consistent units (ft/sec, for example). The time derivative of the velocity is the acceleration, denoted by X¨, and measured in consistent units (for example, ft/sec2). The acceleration of gravity is 32.2 ft/sec2 in the example units.
Analytical Solutions of Rectangular Laminated Plates Using CLPT
Published in J. N. Reddy, Mechanics of Laminated Composite Plates and Shells, 2003
The set of three equations in (6.7.2), for any fixed m and n, can be solved exactly using either the Laplace transforin method or the modal analysis methods. Both methods are algebraieallv complicated and require the determination of eigenvalues and eigenfunctions, as in the state-space method. Therefore we will not attempt them here. Alternatively, wc scck numerical solutions to Eq. (6.7.2) using the well-known family of Newmark's integration schemes for second-order differential equations (see Reddy (27]). In this numerical integration method. the time derivatives are approximated using difference approximations (or truncated Taylor's series), and therefore solution is obtained only for discrete times and not as a continuous function of time.
Mathematical Preliminaries
Published in William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain, of Physical Systems, 2020
William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain
Integration over various domains of derivatives of a function changes the scale of these derivatives. Many of the integration theorems relate integrals of partial time derivatives to time derivatives of integrals over volumes, surfaces, and curves. These partial derivatives are taken fixed to a position in space, on a surface, or along a curve. When the time derivative is moved outside the integral, the differentiation is such that it applies to the integration region, a region that may be moving through space. This feature must be accounted for in the development of relations among various time derivatives.
Neutronics Calculation Advances at Los Alamos: Manhattan Project to Monte Carlo
Published in Nuclear Technology, 2021
Avneet Sood, R. Arthur Forster, B. J. Archer, R. C. Little
In deterministic methods, the neutron transport equation (or an approximation of it, such as diffusion theory) starts with an integral-differential equation, discretized, and is solved as a system of algebraic equations where the spatial, angular, energy, and time variables are discretized. Spatial variables are typically discretized by meshing the geometry into many small regions. Angular variables can be discretized by discrete ordinates and weighting quadrature sets (used in the Sn methods) or by functional expansion methods such as spherical harmonics (used in Pn methods). Energy variables are typically discretized by the multigroup method, where each energy group uses flux-weighted constant cross sections with the scattering kinematics calculated as energy group-to-group transfer probabilities. The time variable is divided into discrete time steps, where the time derivatives are replaced with difference equations. The numerical solutions are iterative since the discretized variables are solved in sequence until the convergence criteria are satisfied.