Design strategies for controlling damping in micromechanical and nanomechanical resonators
 Surabhi Joshi^{2},
 Sherman Hung^{2} and
 Srikar Vengallatore^{2}Email author
DOI: 10.1186/epjti5
© Joshi et al.; licensee Springer on behalf of EPJ. 2014
Received: 19 December 2013
Accepted: 29 April 2014
Published: 23 May 2014
Abstract
Damping is a critical design parameter for miniaturized mechanical resonators used in microelectromechanical systems (MEMS), nanoelectromechanical systems (NEMS), optomechanical systems, and atomic force microscopy for a large and diverse set of applications ranging from sensing, timing, and signal processing to precision measurements for fundamental studies of materials science and quantum mechanics. This paper presents an overview of recent advances in damping from the viewpoint of device design. The primary goal is to collect and organize methods, tools, and techniques for the rational and effective control of linear damping in miniaturized mechanical resonators. After reviewing some fundamental links between dynamics and dissipation for systems with small linear damping, we explore the space of design and operating parameters for micromechanical and nanomechanical resonators; classify the mechanisms of dissipation into fluid–structure interactions (viscous damping, squeezedfilm damping, and acoustic radiation), boundary damping (stresswave radiation, microsliding, and viscoelasticity), and material damping (thermoelastic damping, dissipation mediated by phonons and electrons, and internal friction due to crystallographic defects); discuss strategies for minimizing each source using a combination of models for dissipation and measurements of material properties; and formulate design principles for lowloss micromechanical and nanomechanical resonators.
Keywords
Dissipation Damping Nanomechanical sensing MEMS NEMS Optomechanical systems Quality factor Atomic force microscopy Structural design OptimizationIntroduction
Although damping has been studied for well over a hundred years, the rational design and control of structural damping has seemed a distant goal to many generations of engineers. By way of illustration, we quote from two excellent articles published in the 1990s: “All structures exhibit vibration damping, but despite a large literature on the subject damping remains one of the least wellunderstood aspects of general vibration analysis” [1], and damping in microcantilevers “is not readily susceptible to engineering analysis” [2].
During the past 15 years, however, there has been a remarkable resurgence of interest in structural damping, especially at small length scales (1 nm to 100 μm), motivated by a host of emerging technologies that include microelectromechanical systems (MEMS), nanoelectromechanical systems (NEMS), nanomechanical sensors, and optomechanical systems. This paper presents an overview of major advances in this field from the viewpoint of device design. Our goal is to collect methods and techniques that can provide designers with guidelines and tools for analyzing, controlling, and minimizing damping in miniaturized resonators.
To this end, the next section reviews the fundamental relationships between dissipation and structural dynamics for systems with small linear damping. Subsequently, we explore the space of design and operating parameters for miniaturized mechanical resonators; review the major mechanisms of dissipation by classifying them into three categories: fluid–structure interactions, boundary damping, and material damping; and discuss a set of strategies for controlling each source of damping by combining models for dissipation with measurements of material properties. The concluding section integrates the various strategies in the form of design principles and presents a casestudy to illustrate an intriguing tradeoff between functionality and performance for bilayered resonators.
Foundations of damping
Let us consider a single mode of oscillations of a linear mechanical resonator. Damping refers to dissipation (that is, the conversion of useful mechanical energy into disordered thermal energy) in the oscillating structure, and to the effects of dissipation on structural dynamics (see, for instance, [3–9]). Dissipation is quantified using two dimensionless measures: (i) the specific damping capacity, Ψ = (ΔW/W); and (ii) the loss factor, η = (ΔW/2π W). Here, ΔW is the energy dissipated during one cycle of vibration, and W is the maximum elastic strain energy during the vibration cycle. It is difficult to measure energy dissipation or entropy generation directly; hence, damping is estimated by monitoring structural dynamics using such techniques as harmonic excitation, free decay, thermomechanical noise, and wave propagation [5]. Harmonic excitation is associated with two dimensionless measures of damping: the loss angle, ϕ, by which the stress (σ) leads the strain (ε); and the quality factor, Q ≡ (ω _{ n }/Δω), where Δω is the halfpower bandwidth of the resonance peak and ω _{ n } is the angular natural frequency. Alternately, the quality factor can be estimated by fitting the resonance peak in the thermomechanical noise spectrum [10–12]. The free decay technique quantifies damping in terms of the logarithmic decrement, δ, and wave propagation techniques measure the attenuation, , of the amplitude of elastic waves with wavelength λ[9].
Eq. (1) is a cornerstone of the subject of damping. The various relationships are widely used to: (i) quantify damping in miniaturized mechanical structures; (ii) make connections between dissipation and structural dynamics; (iii) compare measurements from different techniques; (iv) compare theoretical predictions with experimental measurements; and (v) design micromechanical and nanomechanical resonators. Indeed, the quality factor is frequently used to estimate damping even when the identities and frequency dependence of the dominant mechanisms of dissipation are not known. In this context, it is worth highlighting the fact that Eq. (1) is neither exact nor universal. Each relationship is an approximation derived from three simple models for linear damping. The first is the classical system consisting of a mass, spring, and viscous dashpot. Viscous damping has its origins in Rayleigh’s dissipation function and assumes that the damping force is proportional to the instantaneous velocity of the mass [3]. The second model uses the concept of a complex spring, k ^{*} = k _{1} + i k _{2} with tan ϕ = (k _{2}/k _{1}). The real and imaginary parts of k ^{*} are called the storage modulus and loss modulus, respectively [9]. The third model is the standard anelastic solid with a constitutive relationship of the form , where E is the elastic modulus, and τ _{ σ } and τ _{ ϵ } are material parameters [4, 5]. Thus, anelasticity represents an extension of elasticity (as embodied in Hooke’s law, σ = E ε) to incorporate rate effects and relaxation [5]. For these elementary models of viscous, viscoelastic, and anelastic behavior, the approximations in Eq. (1) are within 1% of the exact value for small linear damping with ϕ < 0.01 [5].
Devices, applications, and design parameters
The first microelectromechanical oscillator was demonstrated in 1967 in the form of a device called the resonant gate transistor (RGT) [13]. The RGT consisted of a gold microcantilever (oscillating at 5 kHz with a quality factor of 500) fabricated on a silicon integrated circuit for signal processing applications. Even at this early stage, the benefits of miniaturization for attaining high frequencies and quality factors were well recognized. Soon thereafter, Newell [14] highlighted the value of integrating mechanical structures with microelectronic circuits using microfabrication techniques, and discussed the effects of scale on certain material and structural properties (including fatigue, thermomechanical noise, and viscous air damping). These ideas lay dormant for a decade before exploding into a creative burst of activity that continues unabated to the present day. The invention of the atomic force microscope in 1986 [15], and the rapid evolution of MEMS technologies during the 1990s, provided the motivation and processing capabilities for exploiting miniaturized mechanical resonators for sensing, signal processing, timing, vibration energy harvesting, and precision measurements for fundamental studies in diverse areas of science and engineering [16, 17].
Factors that influence damping in miniaturized mechanical resonators
Operation  Structure  Materials  Processing 

Mode  Shape  Chemistry  Deposition 
Frequency  Size  Alloying  Patterning 
Temperature  Architecture  Residual stress  Etching 
Pressure  Supporting frame  Microstructure  Bonding 
Transduction  Defects  Annealing  
Surfaces  Interconnection  
Interfaces  Packaging 
Review of damping
Classification of linear damping mechanisms
Boundary damping  Fluid–structure interactions  Material damping  

Elastic wave radiation^{■}  Viscous damping^{■}  Thermoelastic damping^{■}  
Microsliding^{▲}  Squeezefilm damping^{■}  Phononphonon interactions^{§}  
Viscoelasticity^{▲}  [9]  Acoustic radiation^{■}  [68]  Phonon–electron interactions^{§}  
Internal flow^{■}  Internal Friction^{▲} 
To gain a qualitative understanding of the various mechanisms, let us consider a generic description of micromechanical and nanomechanical resonators. Typically, the resonators are attached to relatively large supporting frames to facilitate handling and interconnection, and then packaged by mounting the supporting frame inside test stations, instruments, or customized packages. Illustrative examples include commercial silicon microcantilevers (1 to 10 μm thick and 0.1 to 0.5 mm long) used for atomic force microscopy (AFM), and silicon nitride nanomembranes (30 to 200 nm thick with lateral dimensions ranging from 10 μm to 1 mm) used in optomechanical systems [85, 86]. In both cases, the devices are attached to silicon frames with thickness of a few hundred microns and lateral dimensions of a few millimeters. The attachment can be monolithic (exemplified by bulkmicromachined silicon resonators) or engendered by the adhesion of thin films to the supporting frame in the case of surfacemicromachined devices [16].
The materials, methods, and designs used for packaging vary widely depending on the device and application. For example, silicon microcantilevers are loaded into AFMs using clamps, springloaded clips, or adhesive gels, but other devices require customdesigned packages for electrical connections, thermal interfaces, microfluidic manifolds, or vacuum packaging to maintain the resonator at low pressure [16]. The dissipation due to all components associated with the supporting frame and package is collectively called boundary damping. There are three main mechanisms of boundary damping: (i) support losses or anchor losses due to the radiation of elastic waves (or stress waves) from the resonator into the supporting frame [20–35]; (ii) microsliding at the interfaces between the resonator and supporting frame, and between the supporting frame and package [61]; and (iii) viscoelasticity in the gels and adhesives used to bond the supporting frame to the package [9]. (The term clamping loss is frequently encountered in the literature. Depending upon the context, it can refer either to stresswave radiation or to microsliding).
The application dictates the fluidic environment of the resonator. When a miniaturized mechanical structure oscillates in a chamber containing a gas or liquid, energy is dissipated during every cycle of vibration due to the conversion of ordered structural energy into the thermal energy of the molecules of the fluid. Fluid–structure interactions (FSI) have been studied extensively using experiment, theory, and computation because they are a major source of damping. Indeed, the immense literature on FSI in microscale and nanoscale resonators deserves a dedicated fullfledged review in its own right. A detailed understanding of viscous damping, squeezedfilm damping, and acoustic radiation has been obtained for numerous devices including plates, membranes, beams, torsional resonators, and hollow resonators containing internal microfluidic channels. The dependence of damping on material properties, fluidic properties, geometry, size, confinement, mode, frequency, pressure, and temperature has been captured well by models (see, for example, [14, 16, 40–42, 64, 68–70]).
Material damping refers to all dissipative mechanisms that operate within the volume (bulk), at the free surfaces, and at the internal interfaces of the resonator. Composite architectures are ubiquitous in MEMS and NEMS technologies; hence, several types of interfaces (layer boundaries, grain boundaries, and phase boundaries) can be encountered in miniaturized resonators. The creation of interfaces is often unintentional and, in some cases, undesirable. For example, the free surfaces of silicon, titanium, and aluminum are invariably covered with ultrathin coatings of native oxide under typical processing and operating conditions. Similarly, internal oxide layers and interfaces can be created due to unintentional oxidation during deposition [87].
Material damping can be divided into two categories: fundamental damping and internal friction[6]. The former is a set of mechanisms (thermoelastic damping, phononphonon interactions, and phonon–electron interactions) that set the ultimate lower limit on damping. These mechanisms have their origin in the electronic, atomic, and molecular structure of materials, and operate even in the idealized limit of perfectly engineered devices (for example, highquality singlecrystal materials with negligible defect density). Internal friction refers to damping caused by the irreversible motion of crystallographic defects (for example, vacancies, divacancies, interstitial atoms, substitutional atoms, surface adatoms, edge dislocations, screw dislocations, grain boundaries, phase boundaries, layer boundaries, and precipitates) [4], and each type of defect can give rise to several mechanisms of dissipation [5].
Mechanism  Relaxation strength  Relaxation time 

Thermoelastic damping (TED) 


Akhiezer damping 


Nominal properties of common metals and ceramics at 300 K[98]
Material  E (GPa)  ρ (kg/m^{3})  k (W/m/K)  α (K^{−1})  C (J/m^{3}/K)  v (m/s)  γ 

Aluminum  70  2.7 × 10^{3}  220  24.0 × 10^{−6}  2.4 × 10^{6}  5.1 × 10^{3}  1.7 
Copper  120  8.9 × 10^{3}  400  20.0 × 10^{−6}  3.8 × 10^{6}  3.6 × 10^{3}  1.9 
Gold  82  1.9 × 10^{4}  320  15.0 × 10^{−6}  2.5 × 10^{6}  2.1 × 10^{3}  2.9 
Nickel  210  8.9 × 10^{3}  92  13.0 × 10^{−6}  3.9 × 10^{6}  4.8 × 10^{3}  1.8 
Silicon  160  2.3 × 10^{3}  150  2.6 × 10^{−6}  1.6 × 10^{6}  8.3 × 10^{3}  0.4 
Silicon carbide  400  3.2 × 10^{3}  70  3.0 × 10^{−6}  3.0 × 10^{6}  1.1 × 10^{4}  1.1 
Silicon oxide  70  2.2 × 10^{3}  1  0.5 × 10^{−6}  1.5 × 10^{6}  5.6 × 10^{3}  1.9 
Silicon nitride  250  3.2 × 10^{3}  8  3.0 × 10^{−6}  3.0 × 10^{6}  8.8 × 10^{3}  1.2 
Silver  76  1.1 × 10^{4}  430  18.0 × 10^{−6}  3.0 × 10^{6}  2.7 × 10^{3}  2.4 
Techniques for measuring internal friction in thin films were developed over 30 years ago [72], but many classes of materials, structures, and defects still remain largely unexplored. The sparse literature on this topic can be classified into two broad categories. The first set focuses on the effects of temperature on dissipation in an effort to measure relaxation peaks and study the underlying defect interactions (see, for example, [72–77]). The second set explores the effects of processing conditions (including postdeposition annealing), residual stress, and frequency on internal friction (see, for example, [78–83]). In general, the dissipation spectra rarely exhibit the Debye peak predicted by Eq. (2); instead, internal friction is often a weak monotonic function of frequency [9]. Explanations for this behavior range from a distribution of activation energies and relaxation times [5, 9], to a hierarchically constrained sequence of serial relaxation processes in which the fast degrees of freedom (involving the motion of single atoms) must relax before the slower degrees of freedom involving the coordinated motion of groups of atoms [84].
Strategies for controlling damping
Fluid–structure interactions (FSI)
where M is the molar weight of the gas, R is the universal gas constant, and P is the gas pressure. When the pressure is reduced further, fluidic damping becomes negligible at a critical value which is a function of the size, shape, and mode of the resonator. The critical pressure has been measured to range from 0.1 Pa to 10^{3} Pa for miniaturized mechanical resonators [14, 29, 40–42].
Elastic wave radiation
where h _{ s } is the thickness of the support and λ is the wavelength of the elastic wave propagating in the support [22].
Support losses can be reduced by using analytical and numerical models for stresswave radiation to guide the selection of materials, structures, dimensions, modes, and frequencies. Alternately, the generation and propagation of elastic waves can be disrupted by contacting the resonator at its nodal points using anchors or tethers [26–31] and incorporating phononic bandgap structures [32–35], acoustic reflectors [36, 37], and vibration isolators [38, 39]. In each case, wellestablished models from vibrations and elasticity are available to guide design.
Microsliding and viscoelasticity
The other sources of boundary damping face challenges that are common to many aspects of thinfilm adhesion and packaging of MEMS [16]. The variety of designs, processes, techniques, and materials makes it difficult to develop general guidelines for a large class of devices and applications. Models for adhesion and friction can provide useful qualitative insights into the underlying mechanisms but improving thinfilm adhesion and reducing microsliding remains an art. Nevertheless, several general strategies can be formulated: (i) minimize deformations and strains at the interface between the supporting frame and package: for example, by employing the antisymmetric modes of dualcantilever beams [60] or doublepaddle oscillators [62, 63], and acoustically isolating the resonator from the supporting frame and package; (ii) use precisionmachined clamps and avoid springloaded clips, gels, polymerbased adhesives, and die bonds for packaging; and (iii) ensure good adhesion between the resonator and supporting frame by activating the substrate before depositing structural thin films, employing adhesion promoters in the form of ultrathin films of Ti or Cr, and using ionbeam assisted deposition techniques [88].
Thermoelastic damping
Thermoelastic damping (TED) is a rare example of a dissipative mechanism that can be computed accurately from first principles [4]. Consider a beam or a plate that is subjected to timeharmonic bending forces or moments. The elastic stresses in the structure give rise to elastic strains (which are in phase with the stress) and thermal strains due to thermoelastic coupling. In general, the thermal strains are not in phase with the driving elastic stresses; hence, energy is dissipated during every cycle of vibration. Alternately, TED can be viewed as the result of oscillating temperature gradients generated by oscillating stress gradients in thermoelastic solids. Heat conduction across finite temperature gradients leads to entropy generation and energy dissipation [43–48].
The analysis of TED is conceptually straightforward for heat conduction in the diffusive regime (that is, the mean free path of thermal phonons is much less than the characteristic length scale of the resonator [46]). The formula in Table 3 is due to a celebrated analysis by Zener in 1937. More recently, Zener’s analysis of homogenous, isotropic beams has been extended in multiple directions and there are now over 350 significant publications on this topic. A selection of the literature includes models to account for the effects of structural boundaries [47, 48]; polycrystalline grain structure [49, 50]; layered composite architecture [51–53]; electrostatic actuation [54]; and geometry (plates [54, 55], slotted, channeled, and hollow beams [56–58], doublepaddle oscillators [63], and bulkmode, ringmode, and discmode resonators [59]). Using these models, the design space can be explored to formulate detailed guidelines for selecting geometries, structures, modes, materials, and frequencies to minimize thermoelastic damping.
Internal friction
The magnitude of internal friction is governed by the type, distribution, density, and mobility of defects, and by the interactions between different classes of defects. Quantifying these details, especially in micrometer and nanometer scale structures, requires extensive experimental studies using a suite of microscopic and spectroscopic techniques (see, for example, [82, 87, 89]). Even when such information is available, it is difficult to model the dynamics of defects over multiple scales of length, time, and energy. Therefore, in common with many other material properties, design guidelines are derived by measuring internal friction, characterizing the microstructure, and formulating processstructure–property relationships[5, 9, 71].
There are three main strategies for controlling dissipation due to internal friction. The first is to control the type, distribution, and density of defects. Highquality singlecrystals can be used for resonators but fabrication challenges currently limit this option to a small set of materials (for example, commercial silicon wafers and epitaxial thin films grown using molecular beam epitaxy). More practically, defects can be controlled by an appropriate selection of the deposition technique, processing conditions, and postdeposition heat treatment. For example, when sputterdeposited aluminum films were annealed at 450°C for 1 hour in an inert atmosphere, the average grain size increased from 100 nm to 390 nm, and the roomtemperature values of internal friction reduced from 0.05 to 0.02 [82].
The second strategy is to reduce the mobility of defects by materials selection and materials design. Defect mobility is a function of several variables including atomic bonding, crystal structure, microstructure, melting temperature, operating temperature, and frequency. The lattice selfdiffusivity at the melting point is ~10^{16} m^{2}/s for covalently bonded diamondcubic materials, ~5×10^{14} m^{2}/s for oxides, and ~5×10^{13} m^{2}/s for facecentered cubic metals [90]. All else being equal, defects are less mobile in brittle ceramics (Si, SiO_{2}, quartz, SiC, TiO_{2}, Al_{2}O_{3}, and Ta_{2}O_{5}) than in common metals and alloys [6]. Thus, for precision measurements requiring ultrahigh quality factors, multilayer stacks of dielectric films consisting of alternate layers of amorphous silica (SiO_{2}) and amorphous tantala (Ta_{2}O_{5}) are preferred over metallic thin films for optical coatings [17]. Furthermore, small quantities of alloying additions can have a disproportionately large influence on defect mobility. For instance, damping in aluminum alloy Al 5056 is an order of magnitude less than that in pure aluminum over a broad range of temperatures and frequencies [6, 74].
Finally, internal friction can be altered by changing the operating temperature but the implementation of this strategy must take into account the nonmonotonic behavior of dissipation. For example, internal friction in bulk quartz crystals increased by over two orders of magnitude (from 10^{−7} to 4 × 10^{−5}) upon cooling from 300 K to 50 K, and then reduced precipitously to a remarkably low value of 2 × 10^{−10} when cooled further to 2 K [91].
Stressengineered resonators
Micromechanical and nanomechanical resonators are commonly subjected to a prestress (or residual stress) originating from intrinsic stresses generated during the growth of thin film materials [92] and differential expansion caused by thermal excursions during processing [16]. The prestress can affect several properties (stiffness, natural frequencies, mode shapes, linearity, and damping) and reliability (fracture under tension, buckling under compression, inelastic deformation, and stress relaxation by creep [93]). Therefore, the control and mitigation of residual stresses is a major consideration in the design of surfacemicromachining processes [16].
In some cases, however, large tensile residual stresses can be used to reduce damping, as exemplified by highQ devices fabricated using amorphous silicon nitride films grown by lowpressure chemical vapor deposition (see, for example, [83, 86, 94–96]). Indeed, quality factors of 50 million have been obtained at room temperature with highlystressed (~1 GPa) nanomembrane resonators [94]. In general, the effects of the tensile prestress are governed by the relative magnitudes of two factors: (i) increase in the elastic stored energy (both extensional and flexural), and (ii) stressinduced changes in dissipation. The former can be obtained by analyzing the elastic deformation of the resonator (see, for example, [95–97] for expressions of the quality factors for stretchedstring resonators), but the latter has been analyzed explicitly only for a few mechanisms. Notable examples include thermoelastic damping in plates [54] and membranes [94]; in both cases, the magnitude of TED can be reduced by applying an inplane tensile prestress.
Design principles
Miniaturized mechanical resonators are enabling a remarkably large and diverse set of applications ranging from sensing, timing, and signal processing to scanning probe microscopy and precision measurements for fundamental studies in several fields of engineering and science. New concepts continue to emerge; devices are growing in sophistication, performance, and functionality; and there is a growing emphasis on transitioning from proofsofconcept to fullfledged commercialization. All these trends have created an urgent need for developing effective and rational methods for design, analysis, and optimization.
When microresonator technologies began to emerge in the 1980s, damping was controlled using iterative cycles of fabrication, testing, and analysis at the device level [2]. Unfortunately, the design space is much too vast and complex to be probed efficiently in this manner, especially considering the time and resources required for integrating new materials and structures into micromachining and nanofabrication process flows [16]. It is therefore useful and timely to develop methods that account for damping at every stage of the design cycle. In this section, we present an approach that is based on identifying the mechanisms of damping, and then developing strategies for controlling each mechanism with a judicious combination of models for dissipation and measurements of material properties.
Using the standard conceptembodimentdetail description of the design process [98], consider a typical problem that begins by translating market needs and application requirements into a set of device specifications, which are then used to generate a set of preliminary designs. At this stage, a careful estimation of fundamental material losses can establish the ultimate limits of dissipation and serve as a criterion for ranking the various designs. The device can approach the ultimate limits only to the extent that all other sources of dissipation (FSI, boundary damping, and internal friction) are eliminated. The optimal solutions are to: (i) eliminate FSI by operating at a sufficiently low pressure (ranging from 0.1 Pa to 10^{3} Pa, depending on the size, shape, and mode of the resonator); (ii) eliminate boundary damping by decoupling the resonator from the supports and package by using nodal supports, phononic bandgaps, and acoustic isolators; and (iii) eliminate internal friction by using highquality singlecrystals or materials with low defect mobility. In the literature, we can now find a small, but steadily growing, set of devices that can approach the ultimate limits of damping (see, for example, [6, 17, 28, 63, 80]).
In many cases, however, the optimal solutions cannot be implemented because the design is constrained by the application, functionality, and actuation, and by the limitations of processing and packaging technologies. As an example, consider the class of bilayered metalcoated ceramic resonators that are widely used in MEMS and NEMS. The ceramic structure can be designed to oscillate with low damping approaching the fundamental limits. The coating adds valuable functionality by enhancing the optical reflectivity and electrical conductivity, but severely degrades the quality factor and performance of the device due to the relatively large internal friction in polycrystalline metallic films. Thus, the design problem is to identify an optimal tradeoff between performance and functionality.
where (W _{film}/W _{bilayer}) is obtained by computing the elastic deformation of the film and bilayer. Eq. (5) suggests two distinct strategies for increasing the quality factor. The first is to control internal friction in the film using processstructure–property relationships. For sputtered aluminum films (which are widely used as coatings for numerous applications including commercial probes for atomic force microscopy), internal friction can be reduced by postdeposition annealing to increase grain size [82], reducing film thickness [80], and alloying aluminum with small amounts (5%) of magnesium [6, 74]. Alternately, aluminum can be replaced with gold [80] or multilayer dielectric stacks (for example, alternate layers of silica and tantala [17]). The second strategy is to minimize the ratio of elastic strain energies by confining the metallic coating to regions of low deformation and strain [99, 100]. For microcantilevers oscillating in the fundamental bending mode, internal friction due to metallization can be made negligible by coating only the tip of the beam [99].
Declarations
Acknowledgements
This paper is based on experience gained over several years of work on damping. During that period, we benefited greatly from valuable discussions with a number of generous colleagues around the world. We owe a debt of gratitude to all of them. Financial support from the Canada Research Chairs program and the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
Authors’ Affiliations
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