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Research Papers

Thermalizing and Damping in Structural Dynamics

[+] Author and Article Information
Arghavan Louhghalam

Department of Civil and
Environmental Engineering,
University of Massachusetts Dartmouth,
285 Old Westport Rd,
Dartmouth, MA 02747
e-mail: arghavan.louhghalam@umassd.edu

Roland J.-M. Pellenq

Massachusetts Institute of Technology,
Department of Civil and
Environmental Engineering,
CNRS-MIT Joint Lab <MSE>2: Multiscale Materials Science for Energy and Environment,
Cambridge, MA 02139
e-mail: pellenq@mit.edu

Franz-Josef Ulm

Professor
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: ulm@mit.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 17, 2018; final manuscript received April 17, 2018; published online May 10, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(8), 081001 (May 10, 2018) (9 pages) Paper No: JAM-18-1223; doi: 10.1115/1.4040080 History: Received April 17, 2018; Revised April 17, 2018

Structural damping, that is the presence of a velocity dependent dissipative term in the equation of motion, is rationalized as a thermalization process between a structure (here a beam) and an outside bath (understood in a broad sense as a system property). This is achieved via the introduction of the kinetic temperature of structures and formalized by means of an extended Lagrangian formulation of a structure in contact with an outside bath at a given temperature. Using the Nosé–Hoover thermostat, the heat exchange rate between structure and bath is identified as a mass damping coefficient, which evolves in time in function of the kinetic energy/temperature history exhibited by the structure. By way of application to a simple beam structure subjected to eigen-vibrations and dynamic buckling, commonality and differences of the Nosé–Hoover beam theory with constant mass damping models are shown, which permit a handshake between classical damping models and statistical mechanics–based thermalization models. The solid foundation of these thermalization models in statistical physics provides new insights into stability and instability for engineering structures. Specifically, since two systems are considered in (thermodynamic) equilibrium when they have the same temperature, we show in the case of dynamic buckling that a persistent steady-state difference in kinetic temperature between structure and bath is but indicative of the instability of the system. This shows that the kinetic temperature can serve as a structural order parameter to identify and comprehend failure of structures, possibly well beyond the elastic stability considered here.

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References

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Figures

Grahic Jump Location
Fig. 1

Eigen-vibration dissipation of a simply supported beam obtained with a Nosé–Hoover thermostat. (a)–(d) Kinetic temperature evolution, T(t¯)/T0, versus dimensionless time, t¯=tω0, for different beam–to–water mass ratios, Γ = 0.1; 1; 10; 100. [ω0=EI/(ρAL4), with EI = beam bending stiffness, ρA = linear beam mass, L = beam length].

Grahic Jump Location
Fig. 2

Master-curve of the evolution in time of the damping coefficient predicted by the Nosé–Hoover beam theory for dissipating eigen-vibrations

Grahic Jump Location
Fig. 3

Eigen-vibration dissipation of a simply supported beam obtained with the classical constant mass damping model. (a)–(d) Kinetic Temperature evolution, T(t¯)/T0, versus dimensionless time, t¯=tω0, for different damping coefficients ζ0=Γ/(2π2) corresponding to beam–to–water mass ratios of Γ = 0.1; 1; 10; 100. [ω0=EI/(ρAL4), with EI = beam bending stiffness, ρA = linear beam mass, L = beam length].

Grahic Jump Location
Fig. 4

Master-curves of eigen-vibration dissipation of a simply supported beam predicted by (a) the Nosé–Hoover thermalization model, and (b) the classical constant mass damping model

Grahic Jump Location
Fig. 5

Dynamic buckling: (a) kinetic temperature evolution T(t¯)/T0 and internal energy evolution T(t¯)/T0+U(t¯)/Ek(0) and (b) damping coefficient (results for P¯=2; Γ = 100)

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