Vector Mechanics For Engineers Dynamics Solutions

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The following list of lectures is only indicative and should be considered an example of delivery of the course. Introduction and Math RecapL1. Introduction to the course. L2. Mathematical methods for fluid mechanics: revision of vector total and partial derivatives, application to fluid mechanics, introduction to Einstein notation and application to differential operations, revision of vector calculus (gradient, divergence, Stokes and Green s theorem), complex variable calculus and Fourier and Laplace transforms.Governing Equations of Fluids L3. Derivation of the continuity equation.L4. Definition of the stresses and of the strain rate tensor; derivation of the momentum Cauchy equation.L5. Constitutive equation for Newtonian fluids, derivation of the Navier-Stokes equation.L6. Exact and integral solutions of the Navier-Stokes equation. L7. Derivation of the nondimensional form of the Navier-Stokes equation.Potential flowL8. The basics of potential flow: introduction of vorticity and the velocity potential and derivation of the conservation laws governing incompressible irrotational flow, including Bernoulli's law.L9. The building blocks of potential flow: introduction to the elementary solutions to the Laplace equation, the principle of linear superposition and application to explain applied fluid dynamics problems.L10. Forces on objects in potential flow: flow past a rotating circle, the Magnus effect and the d'Alembert's paradox, Kelvin s circulation theorem and Kutta-Joukowsky s theorem.L11. How to reconcile potential flow with rotational flow: the link between circulation and vorticity, bound circulation and free vortices. L12. Introduction to thin airfoil theory: key assumptions and basic results. Turbulent FlowL13. Phenomenology of turbulent flow, Reynolds-averaged Navier-Stokes equation.L14. Reynolds stress tensor, wall scales, Boussinesq hypothesis, turbulent viscosity.L15. Derivation of the universal law of the wall and taxonomy of wall bounded flow.L16. Moody diagram, k-type and d-type roughness.Boundary LayerL17. Phenomenology and taxonomy of boundary layer flow, von Karman integral of the boundary layer and definition of the displacement and momentum thickness.L18. Derivation of the boundary layer equations, summary of results of the Blasius solution of the laminar boundary layer equations, and summary of results of the solutions of the power law for turbulent flow.Turbulent StatisticsL19. The statistical approach: ensemble, moments, stationarity and homogeneity.L20. Correlations, integral scale, spectra, Kolmogorov s scales.Tutorial classesT1. Mathematics revisionT2. Navier-Stokes equationT3. Navier-Stokes equationT4. Potential flowT5. Potential flowT6. Mock examT7. Turbulent flowT8. Turbulent flowT9. Boundary layerT10. Turbulent statisticsAHEP outcomes: SM1m, SM2m, SM3m, SM5m, SM6m, EA1m, EA2m, P1, G1, G2.

Today, the scope of the mechanical engineering discipline is ever-expanding. Mechanical engineers work in industries that include, but are not limited to, the aerospace, bio-pharmaceutical, civil, computer and cyber, biomedical, industrial, materials and manufacturing industries. They provide innovative solutions for contemporary problems and address problems yet identified. For example, 3-D printed components are readily being used in manufactured components, as part of medical implants and devices and even in structural applications. The mechanical performance of the components will likely vary dramatically from the ideal laboratory environment in which they were produced. Mechanical engineers are needed to characterize these aggressive environments in which they may be used, design test matrices to study their performance, and determine environmentally-based mechanical properties needed for design.

The probability density of the response state vector of a nonlinear system under the excitation of Gaussian white noises is governed by a parabolic partial differential equation, called the Fokker-Planck equation. Exact solutions to such equations are difficult especially when both parametric (multiplicative) and external (additive) random excitations are present. In this paper, methods of solution for response vectors at the stationary state are discussed under two schemes based on the concept of detailed balance and the concept of generalized stationary potential, respectively. It is shown that the second scheme is more general and includes the first scheme as a special case. Examples are given to illustrate their applications. 153554b96e

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