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Damping of Materials and Structures

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Handbook of Experimental Structural Dynamics

Abstract

Damping is a phenomenon that can be observed in connection with all kind of materials: solid, liquid, or gaseous. Any kind of time-dependent change in stresses or strains of the material results in a loss of mechanical energy, which in most cases is transformed into thermal energy. However, all the other mechanisms such as the conversion into electrical energy or any kind of radiation over the system’s boundaries play a role. Typical observations that can be made in connection with damping are the occurrence of creep and relaxation processes or hysteresis curves in the case of cyclic loadings. The overall damping is influenced by a variety of mechanisms, especially for structures assembled from different components.

No matter whether the presence of damping is sought or should be avoided in technical applications, for any kind of tuning or optimization of a system under consideration, a basic understanding of the underlying physics is needed. This is especially true if calculations or simulations have to be run in order to predict the dynamical behavior of a system.

This chapter intends to introduce the reader into the subject and provide an extensive overview on the different aspects of damping regarding the fundamentals, mathematical, and numerical models as well as experimental techniques for the detection of damping properties. It shall give an overview of the state of knowledge and experience gathered in various fields of application and research. For further information, the reader is referred to various publications and textbooks whenever needed. This chapter is organized as follows: Sect. 1 provides an extensive overview on the topic, the classification of damping phenomena, and some remarks on computer-based programs. Section 2 refers to the damping of solids, while Sect. 3 extends the view on structures assembled from different components. Section 4 deals with different mathematical models toward the description of damping and relevant numerical approaches. Experimental techniques for the detection of the damping parameters needed for calculations are described in Sect. 5. This includes possible instrumentation as well as analytical methods. Finally, in Sect. 6, an application of the whole subject covering the detection of damping properties, its mathematical representation, and parameter identification along with a numerical simulation is presented as an example. Conclusions from this chapter are drawn in Sect. 7.

Lothar Gaul: deceased.

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Abbreviations

α :

Order of derivative

β :

Order of derivative

Φ :

Matrix of eigenvectors

B :

Spatial derivatives of matrix of shape functions

H :

Matrix of shape functions

K :

Stiffness matrix

M :

Mass matrix

u :

Relative displacement vector

χ :

Material loss factor

\(\dot {\gamma }\) :

Shear speed

γ :

Pure shear distortion

κ :

Curvature

λ :

Wavelength

ν :

Poisson’s ratio, order of derivative

ω 0 :

Natural frequency

ω d :

Natural damped frequency

Ψ(t):

shift function

ρ :

Mass density

σ :

Normal stress

τ :

shear stress

\( \underline {\boldsymbol {\sigma }}\) :

Complex stress

\( \underline {\boldsymbol {\varepsilon }}\) :

Complex strain

\( \underline {\boldsymbol {E}}\) :

Complex elastic modulus

\( \underline {\boldsymbol {G}}\) :

Complex shear modulus

\( \underline {\boldsymbol {K}}\) :

Complex bulk modulus

ε :

Strain

Ω:

Angular frequency

𝜗 :

damping ratio

ζ :

Angular phase difference

E :

Young’s modulus, spring constant

e :

Deviatoric strain

E(t):

Relaxation function

E ′ :

Storage modulus

E ′′ :

Loss modulus

F :

Force

H k :

Coulomb element

i :

Unit imaginary number

J(t):

Creep compliance function

K(t):

Decay function

P :

Active power

p :

Spring-pot coefficient

R k :

damping constant

So :

Sommerfeld number

T :

Temperature, time interval

t :

Time

U :

Potential energy

V :

Volume

W :

Mechanical work

W D :

dissipated mechanical energy

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Authors and Affiliations

  1. Institut für Nichtlineare Mechanik, Universität Stuttgart, Stuttgart, Germany

    Lothar Gaul

  2. Institut für Nichtlineare Mechanik, Universität Stuttgart, Stuttgart, Germany

    André Schmidt

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  1. Lothar Gaul
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  2. André Schmidt
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Correspondence to André Schmidt .

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Editors and Affiliations

  1. Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA

    Randall Allemang

  2. Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, Lowell, MA, USA

    Peter Avitabile

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Gaul, L., Schmidt, A. (2022). Damping of Materials and Structures. In: Allemang, R., Avitabile, P. (eds) Handbook of Experimental Structural Dynamics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4547-0_19

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