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Mechanical Structure Including Mechanisms and Load Analysis
Published in Seong-woo Woo, Design of Mechanical Systems Based on Statistics, 2021
Leonard Euler (1707–1783) was one of the earliest experts in mathematics to investigate the mathematics of linkage design (synthesis). Most linkages are planar; their movement is enclosed to a plane. The general study of linkage motions, planar and spatial, is defined as screw theory. Sir Robert Stawell Ball (1840–1913) is deemed the father of screw theory. Near industrial revolution, lots of the weaving of cloth suggested to the need for more complicated machines to convert waterwheels’ rotary motion into complex motions. The design of the steam engine generated a huge necessity for newly designed mechanisms and machines. Lengthy linear motion travel was required to robustly use steam power. Especially, though he did not understand it, James Watt (1736–1819) applied thermodynamics and rotary joints and long links to generate structured straight line motion (Figure 5.5).
Higher order kinematic formulas and its numerical computation employing dual numbers
Published in Mechanics Based Design of Structures and Machines, 2023
R. Peón-Escalante, A. Espinosa-Romero, F. Peñuñuri
Infinitesimal screw theory has been one of the most used methods for higher-order kinematic analysis. Its mathematical framework was developed in the nineteenth century by Robert Stawell Ball (Ball 1900), yet, its extension to higher-order kinematics to include acceleration, jerk, and jounce/snap is relatively new. For instance, an acceleration analysis of kinematic chains was conducted by Rico and Duffy (1996), while the extensions to the analysis of jerk and jounce/snap of spatial chains were presented by Rico, Gallardo, and Duffy (1999); Gallardo-Alvarado (2014); Müller (2014). Some interesting applications of this theory can be seen in Cheng et al. (2016); Müller (2018, 2019); Sun et al. (2020); Yin et al. (2020); Chen et al. (2022); Valderrama-Rodríguez, Rico, and Cervantes-Sánchez (2022). The equations of higher order analysis for kinematic chains until the fourth order are also obtained by computing successive derivatives of the Jacobian matrix in (López-Custodio et al. 2017); these results are in agreement with the formulas obtained by Rico, Gallardo, and Duffy (1999); Gallardo-Alvarado (2014) where screw theory was used. Screw theory is a powerful mathematical tool for higher-order kinematic analysis of kinematic chains. Nevertheless, it is unclear how to directly apply it to compute the kinematic quantities for a given trajectory function. Although equations for the higher order kinematics are presented until the third order by differentiating a trajectory function (the position vector) (Xianwen, Yongguo, and Tingli 1993; Lerbet 1998), the resultant formulas still need to be computed.