Antonsson Erik K. - Springer Handbook of Mechanical Engineering

Ramin S. Esfandiari 1

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Abstract

This chapter is concerned with fundamental mathematical concepts and methods pertaining to mechanical engineering. The topics covered include complex analysis, differential equations, Laplace transformation, Fourier analysis, and linear algebra. These basic concepts essentially act as tools that facilitate the understanding of various ideas, and implementation of many techniques, involved in different branches of mechanical engineering. Complex analysis, which refers to the study of complex numbers, variables and functions, plays an important role in a wide range of areas from frequency response to potential theory. The significance of ordinary differential equations (ODEs) is observed in situations involving the rate of change of a quantity with respect to another. A particular area that requires a thorough knowledge of ODEs is the modeling, analysis, and control of dynamic systems. Partial differential equations (PDEs) arise when dealing with quantities that are functions of two or more variables; for instance, equations of motions of beams and plates. Higher-order differential equations are generally difficult to solve. To that end, the Laplace transformation is used to transform the data from the time domain to the so-called s -domain, where equations are algebraic and hence easy to treat. The solution of the differential equation is ultimately obtained when information is transformed back to time domain. Fourier analysis is comprised of Fourier series and Fourier transformation. Fourier series are a specific trigonometric series representation of a periodic signal, and frequently arise in areas such as system response analysis. Fourier transformation maps information from the time to the frequency domain, and its extension leads to the Laplace transformation. Linear algebra refers to the study of vectors and matrices, and plays a central role in the analysis of systems with large numbers of degrees of freedom.

Abbreviations

Complex numbers, variables and functions are the main focus of this section. We will begin with complex numbers, their representations, as well as properties. The idea is then extended to complex variables and their functions.