Abstract
The law of mechanics shows that Euler's equation must be equivalent to Lagrange's equation concerning the rotational motion of a rotor. However, the research papers heretofore in issue are divided, as to equations of motion of a rotating shaft when the rotational speed varies, being accompanied by the terms due to the change of rotational speed. Since these terms consist of the second-order quantities of the shaft deformation, it cannot be demonstrated until the second-order quantities are perfectly expressed that the equations of motion are identical in essence independent of their method of derivation. The equations of motion in this paper hold an accuracy to the second-order, and explain the appearance of the second-order terms. The Lagrangian requires the deformation energy of the shaft to be expressed with an accuracy of the third order, which is derived by applying the theory of elasticity to bending and torsion.
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