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Research Papers

# Lagrangians for Damped Linear Multi-Degree-of-Freedom Systems

[+] Author and Article Information

Professor
Departments of Aerospace and Mechanical Engineering,
Civil Engineering,
Mathematics,
Systems Architecture Engineering,
and Information and Operations Management,
University of Southern California,
430 K Olin Hall,
Los Angeles, CA 90089

Hancheol Cho

Department of Aerospace and Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089

Since we cannot differentiate the function $L(t,q,q·)$, with respect to a dependent variable, say $q·i$, by $∂L(t,q,q·)/∂q·i$ we mean $∂L(t,s,r)/∂ri|s=qr=q·$, where $t$, $s$, and $r$ are considered independent variables; similarly, by $∂L(t,q,q·)/∂qi$, we mean $∂L(t,s,r)/∂si|s=qr=q·$.

There is a slight abuse of notation here, since in Eqs. (11)–(14) the variables t, $q$, and $q·$ are considered to be independent, while in the equations of motion, $x$ and $x·$ (see Eq. (8)) are considered to be functions of time t.

Manuscript received March 1, 2012; final manuscript received October 27, 2012; accepted manuscript posted November 19, 2012; published online May 23, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 80(4), 041023 (May 23, 2013) (10 pages) Paper No: JAM-12-1087; doi: 10.1115/1.4023019 History: Received March 01, 2012; Revised October 27, 2012; Accepted November 19, 2012

## Abstract

This paper deals with finding Lagrangians for damped, linear multi-degree-of-freedom systems. New results for such systems are obtained using extensions of the results for single and two degree-of-freedom systems. The solution to the inverse problem for an n-degree-of-freedom linear gyroscopic system is obtained as a special case. Multi-degree-of-freedom systems that commonly arise in linear vibration theory with symmetric mass, damping, and stiffness matrices are similarly handled in a simple manner. Conservation laws for these damped multi-degree-of-freedom systems are found using the Lagrangians obtained and several examples are provided.

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## Figures

Fig. 1

Linear 2-DOF mass-spring-damper system with b = 2 m

Fig. 2

General 2-DOF mass-spring-damper system

Fig. 3

General 3-DOF mass-spring-damper system

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