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Motor Vibration and Acoustic Noise
Published in Wei Tong, Mechanical Design and Manufacturing of Electric Motors, 2022
The polar coordinate system uses r as the radial coordinate and ϕ as the angular coordinate. Thus, the Cartesian coordinate system (x, y) can be related to the polar coordinate system (r, ϕ) as x = rcosϕ and y = rsinϕ. Consequently, the equation of the rotating elliptic rotor can be rewritten as res2cos2(ϕ−ωrt)a2+rer2sin2(ϕ−ωrt)b2=1
Microcontroller Software
Published in Syed R. Rizvi, Microcontroller Programming, 2016
René Descartes (1596–1650) was a French philosopher, mathematician, physicist, and writer, one of the key figures in the Scientific Revolution. One of Descartes’ influences in mathematics was the Cartesian coordinate system, which is named after him. Descartes published a philosophical and mathematical treatise titled Discourse on the Method in 1637. (Its full name is Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences.) Descartes notes down in the Discourse on the Method the quote that begins this chapter. In today’s software engineering world, the best practice of decomposing a problem is what Descartes was talking about. Best practices in software engineering employ the divide-and-conquer strategy in order to decompose the problem or the customer requirements into smaller and easy-to-handle problems or product requirements. At another place in the same thesis, Descartes points to the importance of enumeration as “In every case to make enumerations so complete, and reviews so general that I might be assured that nothing was omitted.” In software engineering’s state of the art practices, enumeration holds the key to problem-solving methodologies. It is essential since it not only helps in project planning but also aids in keeping track of the completeness of the proposed solution.
Polar coordinates
Published in W. Bolton, Mathematics for Engineering, 2012
The Cartesian coordinate system enables the position of a point on a plane to be specified by stating its x and y coordinates, i.e. the displacement of the point relative to two perpendicular axes. There is another way of specifying the position of a point on a plane and that is to use polar coordinates. A reference axis is specified as emanating from some fixed point O, often referred to as the pole. A line is then drawn from O to the point being specified. The location of the point is then specified by the angle θ that line makes with the reference axis and the distance along that line to the point, i.e. the radius r of a circle with its centre at O and which would pass through the point concerned (Figure 16.1). The polar coordinates are then (r, θ). The angle θ is a positive angle when measured anticlockwise from the reference axis and negative when measured in a clockwise direction. The angles can be specified in either degrees or radians. This chapter is a consideration of the polar coordinate method of specifying points on a plane and the relationship between polar and Cartesian coordinates.
Variable thermal conductivity and diffusivity impact on forced vibrations of thermodiffusive elastic plate
Published in Journal of Thermal Stresses, 2021
An infinite elastic cartesian layer of finite width 2d, which is homogeneous isotropic and thermally conducting, is taken. Initially, the plate is assumed to be unstrained, isoconcentrated and unstressed at uniform temperature T0. The origin of the cartesian coordinate system is at the point of the middle plane which coincides with the x–y plane, and the z-axis points in the thickness direction. The boundary surface (z = d) of the plate is subjected to mechanical, thermal and mass concentration loadings, and other surface () is rigidly fixed, insulated and impermeable. On restricting our analysis to the x–z plane, the various field quantities depend on only x, z, t and Therefore, displacement components, temperatures and concentration are as follows:
Control volume finite element method for entropy generation minimization in mixed convection of nanofluids
Published in Numerical Heat Transfer, Part B: Fundamentals, 2019
The physical configuration for the natural and mixed convection study is illustrated in Figures 1 and 2. A square enclosure of height and width filled with water-based nanofluids, is shown in Figure 1. A Cartesian coordinate system is referenced so that the x-axis represents the horizontal axis, and the y-axis refers to the vertical axis in the upward direction. It is assumed that the nanoparticles and the working fluid are in thermal equilibrium, and there are no chemical reactions. Radiative heat transfer and heat transfer due to nanoparticle motion relative to the working fluid are neglected. The buoyancy-induced flow is assumed to be two dimensional of a Newtonian fluid. The thermophysical properties of the nanofluid are assumed to be constant. The variation of density in the buoyancy force term is determined based on the Boussinesq approximation [38].