Metin Gürgöze

Classical variational formulations are typically presented for systems of constant total mass, which can obscure how to treat problems where a convenient subsystem gains or loses mass. We consider a guided ``falling chain'' variant in which a uniform chain of length L interacts with a horizontal disc rotating at constant angular speed ω. A single generalized coordinate r(t) describes the chain length on the disc while the remaining length L-r(t) hangs vertically. We derive the same equation of motion using fixed-mass Lagrangian, D’Alembert, variable-mass and Lagrange-multiplier formulations, thus comparing energy-based, force-based, subsystem-based and constraint-based viewpoints within a single problem. A compact nondimensional form shows that the dynamics depends on a single parameter β = ω2L/(2g), yields a closed-form equilibrium relation for the on-disc fraction, and allows a simple local analysis showing that this equilibrium is linearly unstable in the present idealized model. The example is intended as a short worked comparison problem for advanced analytical mechanics teaching, illustrating how subsystem choice, generalized forces, constraint reactions, and interface momentum flux must be handled in order for fixed-mass and variable-mass descriptions to remain consistent.

The first author has been teaching the postgraduate course, “The Dynamics of Mechanical Systems” in The ITU Faculty of Mechanical Engineering for nearly 20 years. He has observed that students frequently have problems in obtaining the equations of motion of the vibrating systems which were placed on moving bases. Starting from this observation, he has found that the homework stated below, which was given to the students occasionally, was very helpful in learning the subject. The main idea of the homework is the derivation of the equations of motion, with the help of formulating the Lagrange’s equations with respect to a moving set of axis for a vibration system with two degrees of freedom which consists of a horizontal table rotating with a constant angular velocity around a vertical axis. The students were also asked to solve the same problem with a different method of their choice and to determine the reaction forces as well. We want to share this problem with the reader, which we have assessed as very instructive and appropriate from the viewpoint of applicability of different methods.

In a recent study, the eigenvalue sensitivities of a combined system consisting of a linear structure with several lumped attachments were analyzed. An efficient approach was developed for the eigenvalue sensitivities with respect to the parameters of the attachments and their locations by using the implicit function theorem. In the study mentioned above, the theory was applied to a Bernoulli–Euler beam with several attachments. Motivated by this study, the methodology established there will be applied, in the present paper, to an axially vibrating rod with several lumped attachments. Hence, the utility of the methodology is justified through a quite different type of vibrational system. The method is shown as an efficient tool that can be taught in related graduate-level courses for the eigenvalue estimation of complex linear systems with several lumped attachments.

The current study deals with in-class observations made during the teaching of a graduate-level mechanical engineering course entitled ‘Dynamics of Mechanical Systems’. The specific discussion is about the students' various approaches to the application of Lagrange equations in the context of a homework assignment. The analytical solution requested from the class deals with the determination of the vertical component of the reaction force between a spherical pendulum and its hanging point.