ArticlePDF Available

The Loss Factor as a Measure of Mechanical Damping

Authors:

Abstract

The problem of damping representation and measurement is investigated. Among the many parameters found in literature, the most comprehensive is loss factor 'l· Several definitions of '7 are feasible, but in linear problems they all should reduce to the ratio of the in-phase and quadrature parts of the associated complex modulus. This work surveys measuring methods for materials and structures with respect to the way they express lJ, either by approximated or correct expressions. Since some commonly used techniques such as the Oberst method do not follow the definition of lJ, special care is required when dealing with damping values obtained by different methods and in different environments. On the whole, nonhomogeneous values must be expected, owing to the physical differences among the phenomena that enable damping measurement.
The$paper$
$
M.$Carfagni,$E.$Lenzi,$M.$Pierini,$“The$Loss$Factor$as$a$Measure$of$Mechanical$Damping,”!Atti!del!
16th!International!Modal!Analysis!Conference,$ Santa$ Barbara,$ CA$(USA),$2-5$febbraio$1998,$Vol.$1,$
pp.$580-584.$
$
Can$be$freely$downloaded$at:$
$
https://sem.org/wp-content/uploads/2016/01/sem.org-IMAC-XVI-16th-Int-161802-The-Loss-
Factor-as-Measure-Mechanical-Damping.pdf$
$
$
$$
$
... The parameter ρ essentially represents the "distance" between the two eigenfrequencies of the KDamper, which is a marker of the absorption frequency band of the oscillator. Since the loss factor (n) can represent more accurately the dynamic response of nonlinear systems compared with the damping ratio, which is defined on the grounds of the linear single degree of freedom (SDOF) viscous model, 35,36 hysteretic damping is introduced indirectly considering complex stiffness elements as ...
... A conservative loss factor of η = 0.1 has been used; nonetheless, nowadays, someone can easily find industrially produced rubber materials with loss factors up to 0.4. 36 The optimized parameters of the KD-IAM mounts are summarized in Table V. Figure 6 presents the enhanced acoustic performance of the KD-IAM mounted panel, with the black dotted line corresponding to the lumped parameter model as derived from the first mode approximation of the simply supported panel. Comparing the STL curves of the KD-IAM mounted panel with those of the simply supported panel, the mounted panel shows a 110 Hz wide absorption band between 25 and 135 Hz, which corresponds to the resonance region of the simply supported case. ...
Article
Full-text available
The sound transmission loss of conventional means of passive acoustic treatment in the low-frequency range is governed by two physical mechanisms: the inertia, as stated by the mass density law, and the local resonances of the structure. Since usual partitions are flexible and lightweight, their acoustic performance is poor, especially below 300 Hz. Although conventional acoustic meta-materials can offer excellent acoustic properties, they also perform poorly in this range. Therefore, novel meta-structures are required to overcome these limitations. This proposed novel absorber optimally combines the concepts of KDamper (KD) and inertial amplification mechanisms (IAMs). The novelty of the KD-IAM absorber lies in the generation of equally deep but significantly wider attenuation bands surpassing the mass density law while requiring only a small fraction of additional mass. The absorber is implemented and demonstrated as an elastic mount for retrofitting existing panels, essentially manipulating the resonant response of the structure by controlling the panel’s boundary conditions. It is also shown that increasing the panel’s rigidity and, consequently, its fundamental eigenfrequency utilizing stiffeners results in further improvements in the bandwidth and depth of noise attenuation. A wide and deep attenuation band is demonstrated in the resonance region below 120 Hz, up to 13 dB above the reference level. An indicative design and implementation for a case study are presented. It is further demonstrated that the same concept can be utilized for the formation of meta-structures by periodic repetition of KD-IAM unit cells, leading to significant additional attenuation of the lowest vibration modes.
... In order to simulate the structural damping, that is crucial for this type of numerical analyses [8]- [12], it has been considered a global structural damping of 0.01 for the yacht structures and 0.1 for the dampers. ...
Chapter
The future of transportation means is quickly moving towards green solutions in order to reduce the emission of COx and SOx firstly and, secondly, to progressively abandon the fossil fuels. In this perspective, alternative propulsion such as fully electric engine, biofuels, hydrogen, LNG are now largely used in the automotive field and for mass transportation means. The naval field is now moving on the same trend by using hybrid and fully electric engine especially for pleasure vessels, where the relatively small engine power allows the installation of battery stacks onboard without adding unreasonable weight for only few navigation miles. In this paper, the transformation of a traditional pleasure vessel towards a new hybrid version is proposed; after a more comprehensive view of the modifications that are necessary to install hybrid engine and battery onboard, highlighting all the critical aspects of these new design, a FE numerical analysis of the basement of electric variable speed generators is presented.
... The loss factor is defined in terms of energy dissipated with respect to a steady-state oscillation [41,42]. The calculations of the loss factor for a compressive-tensile test !"# and compressive/tensile loading !/# lead to different results, as shown in (1). ...
Article
This work proposes a new dynamic poroelastic model to describe the vibration transmissibility of auxetic polyurethane (PU) foams. The model includes the pneumatic damping effect provided by the porous structure of the foam, and the viscoelasticity of the PU material. The auxetic foam is manufactured using a simplified and relatively low-cost uniaxially thermoforming compression technique, which allows the production of a foam with transverse isotropic properties. Stress relaxation, quasi-static cyclic compressive-tensile and seismic base vibration tests in the small strains and linear elastic regime have been performed in this work. The permeability and the viscoelastic of auxetic foams with different thermoforming compression ratios along the thermoforming and transverse directions are identified by performing a nonlinear regression from the mechanical and vibration-derived data. The auxetic foams possess lower stiffness along the thermoforming direction and higher modulus along the transverse one. The moduli of different auxetic foams obtained from quasi-static tests are 40%-70% lower than the corresponding dynamic values estimated from the seismic base excitation, while the static loss factors are all around 20%-100% lower than the corresponding dynamic ones. The enhancements of the dynamic stiffness and damping are mainly caused by the pneumatic damping effect and the viscoelasticity of the PU material. The permeability of the auxetic foam loaded along the thermoforming direction is ~2 times larger than the one of the pristine foam, but 50% smaller when the foam is dynamically loaded along the transverse direction, due to the transverse isotropic microstructure of this porous material.
... Considering the exponential decay of acoustic waves in both viscoelastic medium and glass up to Ioffe-Regel frequency, a relation can be derived between the microscopic (Γ) and the macroscopic quantity (Q − 1 ) [21,55,[62][63][64][65] : ...
Article
Full-text available
Structured metamaterials are at the core of extensive research, promising for acoustic and thermal engineering. Nevertheless, the computational cost required for correctly simulating large systems imposes to use a continuous model to describe the effective behavior without knowing the atomistic details. Crucially, a correct description needs to describe both the extrinsic interface-induced and the intrinsic atomic scale-originated phonon scattering, especially when the component material is made of glass, a highly dissipative material in which wave attenuation is strongly dependent on frequency as well as on temperature. In amorphous systems, the effective acoustic attenuation triggered by multiple mechanisms is now well characterized and exhibits a nontrivial frequency dependence with a double crossover of power laws. In this work, we propose a continuum viscoelastic model based on the hierarchical strategy multi-scale approach, able to reproduce well the phonon attenuation in a large frequency range, spanning three orders of magnitude from GHz to THz with a ω2−ω4−ω2 dependence, including the influence of temperature.
... The loss factor is calculated using equation (1), where ∆ is the dissipated energy within one hysteretic loop and is the corresponding elastic energy stored in the material [37,50]. The definition of loss factor in equation (1) involved a coefficient 2 in the denominator in case of tensile-compressive cyclic loading [51]. However, in case of tests that involve only compressive or tensile loading, the coefficient 2 should be eliminated. ...
Article
This work describes large strain quasi-static and impact tests of a new class of low-cost, uniaxially thermoformed, transverse isotropic auxetic foams with compression ratio ranging from 20% to 80%. A custom drop tower rig with high-speed cameras suitable for these soft porous foam materials is designed and used to perform impact tests with energy ranging from 0.38J to 1.90J, corresponding to strain rates from 64/s to 143/s. The thermoformed foams only show auxeticity when loaded along the transverse direction to the uniaxial thermoforming compression, with Poisson's ratio !" reaching as low as-1.5 at 20% strain and then close to 0 at large strains. The samples deform with large shear band deformations, also due to the auxeticity. The negative Poisson's ratio foams along that transverse direction also show enhanced impact energy absorption performance, with normalized peak force reduction as high as ~40% against the pristine foam; the reduction is however 20% along the thermoforming direction. A constitutive model based on Nagy's approach is applied to describe the enhancement of dynamic stress during the impact tests compared with the quasi-static one, due to strain rate effects.
... In order to do that, the main assumption is that the bending stiffness of the deformable plate is in a way in series with the stiffness of the KD-IAM. Moreover, since the loss factor ( ) can represent more accurately the dynamic response of nonlinear systems compared with the damping ratio which is defined on the grounds of the linear single degree of freedom (SDOF) viscous model [18], hysteretic damping is introduced indirectly considering complex stiffness elements as ...
Conference Paper
Full-text available
Sound transmission in passive acoustic treatment at low frequencies, is mainly controlled by two physical mechanisms: The inertial, as stated by the mass density law, and the local resonances of the structure. Consequently, conventional means for noise insulation present severe limitations in this range as the improvement of their acoustic performance is associated with the addition of extra mass. Additionally, such flexible structures have many lightly damped vibration modes at low frequencies, further hindering their effectiveness. Locally resonant acoustic meta-materials may also require the use of significant added masses, while demonstrating narrow bandgaps. This work proposes the utilization of a hybrid concept based on the optimal combination of the KDamper (KD) and Inertial Amplification mechanisms (IAM). The combined action of the negative stiffness element and the inertial amplification effect may provide extreme attenuation properties even at very low frequencies. The theoretical framework for a meta-structure based on the periodic repetition of KD-IAM unit cells is established. It is shown to generate deep and wide attenuation bands in the region of the fundamental resonance of the structure, surpassing the mass density law while utilizing a fraction of the additional mass of comparable concepts. An indicative implementation of the meta-structure is also presented and validated via Finite Element Analysis.
... The energy dissipation ability of RSRMS can be quantified by loss factor η proposed by Carfagni et al. [63], which can be expressed as ...
Article
Utilizing elastic metamaterials for energy dissipation is a promising research hotspot since it is reusable compared with the plastic material. An innovative reusable and self-recoverable metastructure (RSRMS) with tetrahedral motif unit cells (TMUCs) for tri-directional energy dissipation is presented in this paper. TMUC, fabricated via 3D printing, mainly composes of a skeleton, six double curved beams, four near-rigid columns, and four caps. The force-displacement relationship of the midpoint of double curved beams is deduced and its criterion for self-recoverability and reusability is investigated. The mechanical model of the proposed RSRMS is established and corresponding snap-through behavior is investigated. The effects of geometric parameters of RSRMS on energy dissipation are systematically studied through numerical simulations, finite element analysis and experimental investigations. The results demonstrate that RSRMS is able to dissipate energy effectively via snap-through induced hysteretic force-displacement behavior and elastic deformation. The efficiency of energy dissipation relies on the number of TMUCs connected in series and the apex height-to-thickness ratio of the double curved beams. Mechanical properties of RSRMS and TMUC are sensitive to the rigidity of supporting skeleton. The proposed RSRMS has potential applications requiring repetitive energy dissipation.
... With structural damping, the measure of damping in structures is included using a relative measure called the damping loss factor η. The loss factor is proportional to the damping capacity which is the ratio of the dissipated energy to the total energy [24,30]. Experimental methods to identify the damping loss factor from measured vibration data can be referred to [23,31]. The damping model mathematically introduces imaginary terms to the stiffness matrix [32], expressed as K = (1 + iη)K ∈ C n×n . ...
Chapter
The paper presents a practical approach to deploy Krylov-based model order reduction techniques for industrial vibroacoustic problems. The numerical analysis of frequency-domain transfer functions provides valuable insights into the model’s behavior even at an early stage of product development. Model order reduction is a promising approach to yield faster computations while analyzing large-scale vibroacoustic models, where an expensive evaluation is performed for a significantly large number of frequency points. However, reducing a full-order model including damping mechanisms requires special attention so as to efficiently gain from the reduction process with respect to accuracy and performance. Therefore, one main focus of this contribution is to identify methods and strategies to efficiently reduce vibroacoustic models incorporating different damping mechanisms. The identified and optimized model reduction approach is finally applied to an industrial problem, where a real automotive structure is reduced and evaluated for the coupled system response using a combined substructuring and model reduction scheme. Moreover, through the example, the efficiency of the Krylov-based techniques is demonstrated with respect to localized damping.
Article
Dissipating kinetic energy from shock and vibration is an urgent requirement for various applications in aerospace to mechanical engineering. This paper proposes a series of innovative metamaterials with planar and cylindrical patterns for elastic energy dissipation and shock isolation. The planar unit cell with two axes of reflectional symmetry mainly consists of two V-shaped regions, four identical right triangle regions, and narrow regions. The stereometric and cylindrical unit cells are obtained through rotating and convolving methods. The mechanics of energy dissipation and shock isolation are systematically investigated with finite element analysis (FEA), and experiments. Results show that various mechanical responses of unit cells are obtained through tailoring combined geometric parameters and additional boundary conditions, and the planar unit cell has stronger bistability than the cylindrical one. Then, the designed multilayers metamaterials exhibit considerable energy dissipation via the snap-through induced hysteric force-displacement behaviors, whose performances are influenced by geometric parameters and the number of layers. Lastly, the designed metamaterials could effectively suppress the acceleration responses via the snap-through behaviors induced by elastic instability. The shock response process and corresponding deformation mechanics are investigated experimentally and numerically. The designed metamaterials have potential in shock and vibration engineering.
Article
This paper gives an intuitive numerical multi-scale method to estimate damping in anisotropic viscoelastic hybrid composite structures using finite element analysis. Different CFRP-R (CFRP with Rubber) architectures, both microstructural and macrostructural, are studied and compared in order to maximize damping but also to minimize rigidity loss. Homogenization by virtual DMA in frequency domain is performed on representative volume elements (RVE) to obtain the effective viscoelastic behaviour of every hybrid microstructure. The effective behaviours are used to define mechanical behaviour of laminates on which vibratory simulations are performed. Interesting and advanced simulations are discussed regarding materials parameters and geometrical aspects and are compared to experimental results.
Article
Several methods for measuring damping are summarized in this article with respect to their advantages and disadvantages. The use of Digital Filters (DF) and the Fast Fourier Transform (FFT) are compared. In general, FFT analysis is best suited for heavily damped structures, while it is advantageous to use DF analysis when dealing with lightly damped structures.
Article
The operator form of the constitutive equation containing fractional derivatives leads to an expression for the complex modulus which is a ratio of polynomials of fractional order in reduced frequency. A ratio of factored polynomials is developed by use of Bode diagrams; another related form arises from the generalized fractional Maxwell model. Bode diagrams are used to determine parameter values. Interconversion to other mechanical properties is outlined. The results potentially form the basis of a new theory.
Article
The definition of loss factor in terms of energy quantities is reexamined, particularly as it applies to composite viscoelastic systems. A restatement of this definition in terms of a corresponding viscoelastic spring is used to show that this definition is extremely useful for massless (ideal viscoelastic spring) systems, but may be applied unambiguously to spring systems with a single attached mass only at resonance. Simple relations are derived which express the loss factors of series‐parallel arrays of massless viscoelastic springs in terms of properties of the individual components; application of these relations to damping analyses of composite structures should result in considerable economy of effort.