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Research Papers

# Modeling of Systems With Position-Dependent Mass Revisited: A Port-Hamiltonian Approach

[+] Author and Article Information
Dimitri Jeltsema

Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlandsd.jeltsema@tudelft.nl

Arnau Dòria-Cerezo

Department Electrical Engineering and Institute of Industrial and Control Engineering,  Universitat Politecnica de Catalunya, 08800 Vilanova i la Geltrú, Spainarnau.doria@upc.edu

From a mathematical point of view, it is often possible to find an alternative Lagrangian that incorporates energy or particle losses and yields the correct dynamics. However, the physical interpretation of such Lagrangians is not trivial [8].

It is important to realize that Eq. 26 differs from Eq. 1 in that the masses of the particles are now depending explicitly on position, i.e., $mk=mk(rk)$, whereas the dependence of Eq. 2 on the generalized position coordinates is only due to a change of coordinates (removal of the holonomic constraints).

In general, a full-rank left annihilator of $b∈Rn×m$, $b⊥$, implies: $b⊥b=0$ and $rank(b⊥)=n-m$.

J. Appl. Mech 78(6), 061009 (Aug 24, 2011) (6 pages) doi:10.1115/1.4003910 History: Received June 22, 2010; Revised March 25, 2011; Published August 24, 2011; Online August 24, 2011

## Abstract

It is known that straightforward application of the classical Lagrangian and Hamiltonian formalism to systems with mass varying explicitly with position may lead to discrepancies in the formulation of the equations of motion. Systems with mass varying explicitly with position often arise from situations where the partitioning of a closed system of constant mass leads to open subsystems that exchange mass among themselves. One possible solution is to introduce additional nonconservative generalized forces that account for these effects. However, it remains unclear how to systematically interconnect the Lagrangian or Hamiltonian subsystems. In this note, systems with mass varying explicitly with position and their properties are studied in the port-Hamiltonian modeling framework. The port-Hamiltonian formalism combines the classical Lagrangian and Hamiltonian approach with network modeling and is applicable to various engineering domains. One of the strong aspects of the port-Hamiltonian formalism is that power-preserving interconnections between port-Hamiltonian subsystems results in another port-Hamiltonian system with composite energy and interconnection structure. The motion of a heavy cable being deployed from a reel by the action of gravity is used as an example.

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## Figures

Figure 1

A cable-reel system

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