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  1. See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/320766695See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/320766695 Basic Response Functions of Simple Inertoelastic and Inertoviscous Models Articlein Journal of Engineering Mechanics · August 2017 DOI: 10.1061/(ASCE)EM.1943-7889.0001348 CITATION 1 READS 115 1 author: Nicos Makris University of Central Florida 233PUBLICATIONS4,942CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Seismic Protection of Equipment and Building ContentView project Structural ControlView project All content following this page was uploaded by Nicos Makris on 20 November 2017. The user has requested enhancement of the downloaded file.

  2. Basic Response Functions of Simple Inertoelastic and Inertoviscous Models Nicos Makris, M.ASCE1 Abstract: Motivated by the growing interest in suppressing vibrations with supplemental rotational inertia, this paper examines and constructs the basic frequency-response functions and subsequently derives the corresponding causal time-response functions of elementary mechanical networks that involve the inerter, a two-node element in which the force-output is proportional to the relative acceleration of its end-nodes. This is achieved by extending the relationship between the causality of a time-response function and the analyticity of its corresponding frequency-response function to the case of generalized functions. This paper shows that all frequency-response functions that exhibit singularities along the real frequency axis need to be enhanced with the addition of a Dirac delta function or with its derivative, depending on the strength of the singularity. It is shown that because of the inerter, some basic time-response functions exhibit causal os- cillatory response, in contrast to the decaying exponentials that originate from dashpots. Most importantly, the inerter emerges as an attractive response-modification element because in some cases it absorbs the singular response of the solitary spring or dashpot. DOI: 10.1061/ (ASCE)EM.1943-7889.0001348. © 2017 American Society of Civil Engineers. Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. Introduction very similar to the rotational inertia damper initially proposed by Hwang et al. (2007). The main difference is that the Ikago et al. (2012) configuration contains an additional flywheel to accentuate the rotational inertia effect of the proposed vibration control device. About the same time, Takewaki et al. (2012) examined the re- sponse of a SDOFand multi-degree-of-freedom (MDOF)structures equipped with supplemental rotational inertia that is offered from a ballscrew type device that sets in motion a rotating flywheel. These rotational inertia dampers, where the supplemental inertia added to the system is proportional only to the relative acceleration between the two end-nodes of the device, are merely implementa- tions of the inerter element. The term inerter was apparently coined by Smith (2002) to describe a linear mechanical element where the output force is proportional only to the relative acceleration of its end-nodes (terminals), in an effort to establish a completely analo- gous correspondence between mechanical and electrical networks. Accordingly, in a force-current/velocity-voltage analogy, the inerter is the mechanical analog of the capacitor and its constant of pro- portionality is called the inertance with units of mass (M). Smith and his coworkers developed and tested both a rack-and-pinion inerter and a ballscrew inerter (Papageorgiou and Smith 2005; Papageorgiou et al. 2009). Upon its conceptual development and experimental validation, the inerter was implemented to control the suspension vibrations of racing cars, under the name J-damper (Chen et al. 2009; Kuznetsov et al. 2011). Subsequent studies on the response of MDOF structures equipped with supplemental rota- tional inertia have been presented by Marian and Giaralis (2014), Lazar et al. (2014), and Giaralis and Taflanidis (2015) in the con- text of enhancing the performance of tuned mass dampers. More recently, Makris and Kampas (2016) showed that the seismic protection of structures with supplemental rotational inertia has some unique advantages, particularly in suppressing the spectral displacement of long period structures—a function that is not effi- ciently achieved with large values of supplemental damping. How- ever, this happens at the expenses of transferring appreciable forces at the support of the flywheels. The increasing acceptance of the inerter as a response modifi- cation element in both mechanical and civil applications has come about mainly because it complements the traditional supplemental Traditionally, the vibration suppression of civil structures has been achieved primarily with supplemental damping. Accordingly, dur- ing the last 40 years several hysteretic, viscous, nonlinear viscous, or viscoelastic devices have been proposed and implemented to limit the earthquake-induced or wind-induced displacements of structures (Skinner et al. 1973; Aiken and Kelly 1990; Inaudi and Makris 1996; Soong and Dargush 1997; Constantinou et al. 1998; Symans et al. 2008; and references reported therein). Several of these devices, in addition to supplemental damping, add stiffness to the structural system, either because of the preyielding stiffness of the yielding (hysteretic) device (Tena-Colunga 1997; Black et al. 2002, 2003; Chang and Makris 2000) or because of the viscoelastic nature of the energy dissipation material that typically introduces an increasing stiffness as the rate of loading increases (Zhang and Soong 1989; Makris et al. 1993a, b; Fu and Kasai 1998; and others). In addition to the widely accepted viscous, nonlinear viscous, and viscoelastic dampers, during the last decade a growing number of publications have proposed the use of rotational inertia dampers. For instance, Hwang et al. (2007) proposed a rotational inertia damper in association with a toggle bracing for vibration control of building structures. The proposed rotational inertia damper con- sists of a cylindrical mass that is driven by a ball screw and rotates within a chamber that contains some viscous fluid. In this way the vibration reduction originates partly from the difficulty of mobiliz- ing the rotational inertia of the rotating mass and partly from the additional damping due to the shearing of the viscous fluid. Ikago et al. (2012) examined the dynamic response of a single-degree-of- freedom (SDOF) structure equipped with a rotational damper that is 1Professor, Dept. of Civil, Environmental and Construction Engi- neering, Univ. of Central Florida, Orlando, FL 32816. E-mail: Nicos .Makris@ucf.edu Note. This manuscript was submitted on November 12, 2016; approved on May 8, 2017; published online on August 16, 2017. Discussion period open until January 16, 2018; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Mechanics, © ASCE, ISSN 0733-9399. © ASCE 04017123-1 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  3. where Io¼ ð1=2ÞmR2is the moment of inertia of the flywheel about point O. Substitution of Eq. (1) into Eq. (2) offers the con- stitutive equation of the linear mechanical element shown in Fig. 1, called the inerter (Smith 2002) FðtÞ ¼Io 2mR2 ρ½¨ u1ðtÞ − ¨ u2ðtÞ? ¼1 ρ2½¨ u1ðtÞ − ¨ u2ðtÞ? ð3Þ Eq. (3) indicates that the force acting on the inerter is propor- tional to the relative acceleration of the end-nodes (terminals) of the element. The proportionality constant of MR¼ ð1=2ÞmðR2=ρ2Þ is the inertance of the proposed element and it has units of mass (M). The inertance can be amplified by installing two (or more) fly- wheels in series, where the first flywheel is a gearwheel (Smith 2002; Makris and Kampas 2016). Accordingly, the constitutive equation of the inerter is Fig. 1. Physical realization of the inerter Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. damping and stiffness strategies. As a result, the physical realiza- tion of the inerter, as documented in the referenced literature and schematically shown in Fig. 1, extends the well-established visco- elastic behavior (a combination of dashpots and springs) (Ferry 1980; Bird et al. 1987; Tschoegl 1989; Giesekus 1995; Makris and Kampas 2009) to the inertoelastic behavior (a combination of inerters and springs) or even to the inertoviscoelastic behavior (a combination of inerters, dashpots, and springs). The aim of this paper is to examine the basic frequency-response functions of simple inertoelastic and inertoviscous elements and subsequently to derive the associated causal time-response functions that are needed when a time-domain formulation is adopted. The contribution of this paper is that it shows that when the frequency-response function has as a singularity along the real fre- quency axis the reciprocal function or the inverse-square function, then the complex frequency-response function needs to be en- hanced with the addition of a Dirac delta function or with its deriva- tive, depending on the strength of the singularity. In this way, the real and imaginary parts of the corrected frequency-response func- tion are Hilbert pairs, thereby yielding a causal time-response function in the time domain. Upon deriving the causal time-response functions of the solitary inerter, the paper examines the basic response functions of a spring and an inerter connected either in parallel or in series, together with the basic response functions of a dashpot and an inerter connected either in parallel or in series. The study of these simple mechanical networks brings forward most of the subtle mathematical opera- tions associated with the construction of analytic frequency- response functions that result in the associatedcausal time-response functions of mechanical networks that involve inerters. FðtÞ ¼ MR½¨ u1ðtÞ − ¨ u2ðtÞ? ð4Þ When a combination of springs, dashpots and inerters form a mechanical network with a fixed end-node (one terminal fixed), the constitutive equation of the mechanical network has the form X where FðtÞ and uðtÞ = force and displacement, respectively, acting on the other end-node, which is free to move. In Eq. (5), the coefficients amand bnare restricted to real numbers, and the order of differentiation m and n is restricted to integers. The linearity of Eq. (1) permits its transformation in the frequency domain by applying the Fourier transform ! X ! M N dm dtm dn dtn am FðtÞ ¼ bn uðtÞ ð5Þ m¼0 n¼0 FðωÞ ¼ ½K1ðωÞ þ iK2ðωÞ?uðωÞ −∞FðtÞe−iωtdt and uðωÞ ¼ ∫∞ ð6Þ where FðωÞ ¼ ∫∞ Fourier transforms of the force and displacement histories respec- tively; and KðωÞ ¼ K1ðωÞ þ iK2ðωÞ = dynamic stiffness of the model KðωÞ ¼ K1ðωÞ þ iK2ðωÞ ¼½PN The dynamic stiffness, KðωÞ, expressed by Eq. (7) is a transfer function that relates a displacement input to a force output and is the ratio of two polynomials where the numerator is of degree n and the denominator of degree m. Accordingly, KðωÞ has m poles and n zeros. A transfer function that has more poles than zeros (m > n) is called strictly proper and results in a strictly causal time-response function. In a strictly proper transfer function, the output of the model is finite and follows the input modulations. The force output FðtÞ in Eq. (5) can be computed in the time domain with the convolution integral Z where qðtÞ = memory function of the model defined as the resulting force at present time t due to an impulsive displacement input at time τðτ < tÞ, and is the inverse Fourier transform of the dynamic stiffness Z The notation qðtÞ used for the memory function is the same as the notation used by Veletsos and Verbic (1974). The inverse −∞uðtÞe−iωtdt = n¼0bnðiωÞn? ½PM ð7Þ m¼0amðiωÞm? Background Consider the linear element shown in Fig. 1, where a flywheel with radius R and mass m can rotate about point O of the rigid arm O-2. Concentric to the flywheel, there is an attached pinion with radius ρ, engaged to a linear rack that is driven by a force FðtÞ. If there is no slipping between the rack and the pinion, the rotation θðtÞ of the flywheel is t −∞qðt − τÞuðτÞdτ FðtÞ ¼ ð8Þ θðtÞ ¼u1ðtÞ − u2ðtÞ ð1Þ ρ where u1ðtÞ and u2ðtÞ = displacements of the end-nodes (terminals) of the element. Dynamic moment equilibrium of the flywheel about point O gives ∞ qðtÞ ¼1 −∞KðωÞeiωtdω ð9Þ 2π Io¨θðtÞ ¼ FðtÞρ ð2Þ © ASCE 04017123-2 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  4. Fourier transform given by Eq. (9) converges only when ∫∞ only when KðωÞ is a strictly proper transfer function (m > n). When the number of poles is equal to the number of zeros (m ¼ n), the transfer function of the model is simply proper and results in a time-response function, qðtÞ, that has a singularity at the time origin because of the finite limiting value of the associated transfer function at high frequencies. This means that the model instantaneously produces an output at a given input. The inverse of the dynamic stiffness is the dynamic flexibility stiffness, kðtÞ, of the model is finite, whereas the memory function, qðtÞ, has a singularity at the time origin. The inverse of the imped- ance is called in structural mechanics mobility, whereas in the elec- trical engineering literature the term admittance is used (Bode 1959; Smith 2002) −∞jKðωÞjdω < ∞; therefore, qðtÞ exists in the classical sense 1 YðωÞ ¼ Y1ðωÞ þ iY2ðωÞ ¼ ð17Þ Z1ðωÞ þ iZ2ðωÞ The mobility (admittance) is a transfer function that relates a force input to avelocityoutput. When the mobility is a proper trans- fer function, the velocity history of the node that is free to move can be computed in the time domain via the convolution Z where yðtÞ = impulse velocity–response function, defined as the resulting velocity at time t for an impulsive force input at time τ (τ < t), and is the inverse Fourier transform of the mobility (admittance) Z At negative times (t < 0), all four time-response functions given by Eqs. (9), (12), (16), and (19) need to be zero in order for the phenomenological model (mechanical network) to be causal. The requirement for a time-response function to be causal in the time domain implies that its corresponding frequency-response function, ZðωÞ ¼ Z1ðωÞ þ iZ2ðωÞ, is analytic on the bottom-half complex plane (Bendat and Piersol 1986; Papoulis 1987; Makris 1997a, b). The analyticity condition on a complex function, ZðωÞ ¼ Z1ðωÞþ iZ2ðωÞ, relates the real part Z1ðωÞ and the imaginary part Z2ðωÞ with the Hilbert transform (Morse and Feshbach 1953; Papoulis 1987) Z 1 HðωÞ ¼ H1ðωÞ þ iH2ðωÞ ¼ ð10Þ K1ðωÞ þ iK2ðωÞ Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. which is the transfer function that relates a force input to a dis- placement output. From Eqs. (7) and (10), it is clear that when a mechanical network has a strictly proper dynamic flexibility it has an improper dynamic stiffness, and vice versa. Consequently, when the causality of a proposed mechanical model (network) is a con- cern, it is most important to specify what is the input and what is the output. When the dynamic flexibility HðωÞ is a proper transfer function, the displacement, uðtÞ, in Eq. (5) can be computed in the time domain via the convolution integral Z where hðtÞ = impulse-response function, defined as the result- ing displacement at time t for an impulsive force input at time τ (τ < t) and is the inverse Fourier transform of the dynamic flexibility Z The notation hðtÞ for the impulse-response function used by Veletsos and Verbic (1974) is adopted in this article. An equally useful transfer function of a phenomenological model is the impedance, ZðωÞ ¼ Z1ðωÞ þ iZ2ðωÞ, which relates a velocity input to a force output t −∞yðt − τÞFðτÞdτ vðtÞ ¼ ð18Þ ∞ yðtÞ ¼1 −∞YðωÞeiωtdω ð19Þ 2π t −∞hðt − τÞFðτÞdτ uðtÞ ¼ ð11Þ ∞ hðtÞ ¼1 −∞HðωÞeiωtdω ð12Þ 2π Z Z2ðxÞ x − ωdx; Z1ðxÞ x − ωdx ð20Þ ∞ ∞ Z1ðωÞ ¼ −1 Z2ðωÞ ¼1 π π −∞ −∞ FðωÞ ¼ ½Z1ðωÞ þ iZ2ðωÞ?vðωÞ ð13Þ Basic Response Functions of the Linear Spring where vðωÞ ¼ iωuðωÞ = Fourier transform of the velocity time history. For the linear inertoviscoelastic model given by Eq. (5), the impedance of the model is For the linear elastic spring with spring-constant k, Eq. (5) reduces to ½PN n¼0bnðiωÞn? m¼0amðiωÞmþ1? FðtÞ ¼ kuðtÞ ð21Þ ZðωÞ ¼ Z1ðωÞ þ iZ2ðωÞ ¼ ½PM ð14Þ Therefore, its dynamic stiffness kðωÞ ¼ k and its dynamic flexibil- ity HðωÞ ¼ 1=k are real-valued constants. Accordingly, both these transfer functions are simply proper (m ¼ n ¼ 0), and their corre- sponding time-response functions exhibit a singularity at the time origin The force output FðtÞ appearing in Eq. (5) can be computed in the time domain with an alternative convolution integral Z where kðt − τÞ = relaxation stiffness of the model, defined as the resulting force at the present time, t, due to a unit step displacement input at time τ (τ < t), and is the inverse Fourier transform of the impedance Z Eq. (16) indicates that if the dynamic stiffness of a phenomeno- logical model is a simple proper transfer function, then the imped- ance is a strictly proper transfer function; therefore, the relaxation t −∞kðt − τÞ˙ uðτÞdτ FðtÞ ¼ ð15Þ qðtÞ ¼ kδðt − 0Þ ð22Þ and hðtÞ ¼1 kδðt − 0Þ ð23Þ ∞ kðtÞ ¼1 −∞ZðωÞeiωtdω ð16Þ where δðt − 0Þ = Dirac delta function at the time origin (Lighthill 1958; Papoulis 1987). The Fourier transform of Eq. (21) is FðωÞ ¼ kuðωÞ, and by using that vðωÞ ¼ iωuðωÞ, the impedance of the linear spring as defined by Eq. (13) assumes the form 2π © ASCE 04017123-3 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  5. dδðt − 0Þ dt ZðωÞ ¼k iω¼ −ki1 yðtÞ ¼1 ð24Þ ð31Þ ω k The inverse Fourier transform of −i=ω is ð1=2ÞsgnðtÞ (Morse and Feshbach 1953), where sgnðtÞ, is the signum function. Accord- ingly, by using the expression of the impedance of the linear spring as given by Eq. (24), the resulting relaxation stiffness as defined by Eq. (16) is kðtÞ ¼ ðk=2ÞsgnðtÞ, which is clearly a noncausal function. In fact the signum function, sgnðtÞ, indicates that there is as much response before the induced step displacement as the response upon the excitation is induced. Two decades ago, Makris (1997a) resolved this impasse by extending the relation between the analyticity of a transfer function and the causality of the cor- responding time-response function to the case where generalized functions are involved. Given that the Hilbert pair of the reciprocal function 1=ω is −πδðω − 0Þ, Makris (1997a) explained that a Dirac delta function needs to be appended as a real part in Eq. (24) and that the correct expression of the impedance of the linear spring is ? By manually appending the real part, πδðω − 0Þ, in Eq. (24), the Fourier transform of the correct impedance of the linear spring given by Eq. (25) gives Z Eq. (31) shows that the impulse velocity–response function of the linear spring exhibits a strong singularity at the time origin, given that its mobility increases linearly with frequency. Fig. 2 summarizes the basic frequency-response and time- response functions of the linear spring computed in this section. In the interest of completeness, next to these functions Fig. 2 sum- marizes the corresponding functions of the linear dashpot that have been computed by following the same reasoning (Makris 1997a). Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. Basic Response Functions of the Inerter With reference to Fig. 1, when node 2 is fixed (u2¼ 0), Eq. (5) reduces to d2uðtÞ dt2 ? FðtÞ ¼ MR ð32Þ πδðω − 0Þ − i1 ZðωÞ ¼ k ð25Þ ω The Fourier transform of Eq. (32) is FðωÞ ¼ −MRω2uðωÞ ð33Þ Therefore, the dynamic stiffness of the inerter, kðωÞ ¼ −MRω2, is improper to the second degree. Accordingly, the memory function qðtÞ, as defined by Eq. (9), does not exist in the classical sense. Recall that qðtÞ is the resulting force due to an induced impulsive displacement. An impulsive displacement is not physically realiz- able on the inerter, as it is not possible to rotate instantaneously a flywheel with finite rotational inertia; the only thing that can be achieved is to buckle the supports of the flywheel or to fail the gears along the rack-pinion interface. Nevertheless, a mathematical ex- pression for qðtÞ can be reached by using the properties of the higher-order derivatives of the Dirac delta function (Lighthill 1958) Z By employing Eq. (34), the Fourier transform of d2δðt − 0Þ= dt2is Z Consequently, based on Eq. (35), the inverse Fourier transform of the dynamic stiffness of the inerter, kðωÞ ¼ −MRω2, is ? ? Z −∞ZðωÞeiωtdω ¼k ∞ ∞ kðtÞ ¼1 πδðω − 0Þ − i1 eiωtdω 2π 2π ω −∞ ð26Þ By recalling that the Fourier transform of −i=ω is ð1=2ÞsgnðtÞ, Eq. (25) gives ?1 ¼ k ? Z ∞ −∞δðω − 0Þeiωtdω þ1 2þ1 kðtÞ ¼ k 2sgnðtÞ 2 ?1 ? ¼ kUðt − 0Þ 2sgnðtÞ ð27Þ dnδðt − 0Þ dtn fðtÞdt ¼ ð−1Þndnfð0Þ ∞ ð34Þ dtn −∞ where Uðt − 0Þ = Heaviside unit-step function at the time origin; therefore, the relaxation stiffness, kðtÞ ¼ kUðt − 0Þ, is casual. The notation Uðt − 0Þ for the Heaviside unit-step function used by Papoulis (1987) is adopted in this article. The mobility (admittance) of the linear spring is the inverse of the uncorrected impedance given by Eq. (24) YðωÞ ¼ iω d2δðt − 0Þ dt2 ∞ e−iωtdt ¼ ð−1Þ2ð−iωÞ2¼ −ω2 ð35Þ −∞ ð28Þ k and is an improper transfer function. Accordingly, its inverse Fourier transform, the impulse velocity–response function yðtÞ as defined by Eq. (19), does not exist in the classical sense. Never- theless, it can be constructed mathematically with the calculus of generalized functions and, more specifically, with the property of the derivative of the Dirac delta function (Lighthill 1958) Z d2δðt − 0Þ dt2 qðtÞ ¼ MR ð36Þ Eq. (36) shows that the memory function of the inerter exhibits a strong singularity at the time origin, as its dynamic stiffness increases quadratically with frequency. While the dynamic stiffness of the inerter, kðωÞ ¼ −MRω2, is an improper transfer function, its inverse, the dynamic flexibility as defined by Eq. (10), is a proper transfer function Z dδðt − 0Þ dt −∞δðt − 0ÞdfðtÞ dt ¼ −dfð0Þ ∞ ∞ fðtÞdt ¼ − dt dt −∞ ð29Þ 1 1 HðωÞ ¼ − ð37Þ MR ω2 By employing Eq. (29), the Fourier transform of dδðt − 0Þ=dt is ∞ −∞ Consequently, based on Eq. (30), the inverse Fourier transform of the mobility given by Eq. (28) is Z Z dδðt − 0Þ dt Nevertheless, while the dynamic flexibility of the inerter as ex- pressed by Eq. (37) is a proper transfer function, the inverse Fourier transform of −1=ω2is ðt=2ÞsgnðtÞ, where sgnðtÞ is the signum function. Accordingly, the appearance of the signum function alone renders the corresponding time-response function, hðtÞ, noncausal. ∞ e−iωtdt ¼ − −∞δðt − 0Þð−iωÞe−iωtdt ¼ iω ð30Þ © ASCE 04017123-4 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  6. Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 2. Basic frequency-response functions and their corresponding causal time-response functions of elementary inertoelastic and inertoviscous models © ASCE 04017123-5 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  7. Given that the dynamic flexibility of the inerter, as expressed by Eq. (37), is a purely real quantity, this study was in search of the imaginary Hilbert pair of −1=ω2. The Hilbert pair of −1=ω2is constructed by employing the first equation of Eq. (20), with the help of Eq. (29). By letting H2ðωÞ ¼ πdδðω − 0Þ=dω, its Hilbert transform gives H1ðωÞ ¼ −1 π By following the operations described by Eq. (30), the inverse Fourier transform of the impedance of the inerter is dδðt − 0Þ dt kðtÞ ¼ MR ð46Þ Eq. (46) indicates that the relaxation stiffness of the inerter exhibits a strong singularity [not as strong as the memory function, qðtÞ] at the time origin, as it is not physically realizable to impose a step displacement on the inerter. The mobility (admittance) of the inerter is the inverse of its impedance given by Eq. (45) Z −∞πdδðx − 0Þ ∞ 1 x − ωdx ð38Þ dx and with the change of variables ξ ¼ x − ω, dξ ¼ dx, Eq. (38) becomes Z ¼ dδ½ξ − ð−ωÞ? dξ ∞ 1 ξdξ 1 1 i1 ω H1ðωÞ ¼ − Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. MRiω¼ − YðωÞ ¼ ð47Þ MR −∞ ∞ −∞δ½ξ − ð−ωÞ? ? ? Z −1 dξ ¼ −1 The mobility of the inerter given by Eq. (47) is of the same form as the impedance of the linear spring given by Eq. (24). By following the same reasoning described to construct the correct impedance of the linear spring given by Eq. (25), a Dirac delta function is appended as a real part in Eq. (47), and the correct expression of mobility of the inerter is ? By manually appending the real part, πδðω − 0Þ, in Eq. (47), the inverse Fourier transform of the correct mobility of the inertergiven by Eq. (48) is ?1 which is a causal function, because Uðt − 0Þ is the Heaviside unit- step function at the time origin. The eight basic response functions of the inerter computed in this section are summarized in Fig. 2, next to the basic response functions of the linear spring and the viscous dashpot. ð39Þ ξ2 ω2 The result of Eq. (39) indicates that the right-hand side of Eq. (37) cannot stand alone and has to be accompanied by its imaginary Hilbert pair, πdδðω − 0Þ=dω. Consequently, the correct expression of the dynamic flexibility of the inerter is ? Comparing Eqs. (25) and (40) shows that the stronger the sin- gularity of the frequency term (1=ω or 1=ω2), the stronger is the singularity of its Hilbert companion [πδðω − 0Þ or πdδðω − 0Þ= dω]. By manually appending the imaginary part, πdδðω − 0Þ=dω, in Eq. (37), the inverse Fourier transform of the correct dynamic flexibility of the inerter given by Eq. (40) gives Z ¼ MR 2π −∞ By recalling that the Fourier transform of −1=ω2is ðt=2ÞsgnðtÞ, Eq. (41) gives ?t and after employing Eq. (29), the second term in the right-hand side of Eq. (42) gives Z Substitution of the result of Eq. (43) into Eq. (42) gives the causal expression for the impulse-response function of the inerter ?t where Uðt − 0Þ is again the Heaviside unit-step function at the time origin. Eq. (44) indicates that an impulse force on the inerter creates a causal response that grows linearly with time and is inverse pro- portional to the inertance, MR. The impedance of the inerter derives directly from Eq. (32) by using that vðωÞ ¼ iωuðωÞ ZðωÞ ¼ iωMR which is an improper transfer function and is of the same form as the mobility (admittance) of the linear spring given by Eq. (28). ? ? 1 πδðω − 0Þ − i1 ω2þ iπdδðω − 0Þ 1 −1 YðωÞ ¼ ð48Þ HðωÞ ¼ ð40Þ MR ω MR dω ? 1 2þ1 1 Uðt − 0Þ yðtÞ ¼ 2sgnðtÞ ¼ ð49Þ MR MR ∞ hðtÞ ¼1 −∞HðωÞeiωtdω 1 2π ? ? Z ω2þ iπdδðω − 0Þ ∞ 1 −1 eiωtdω ð41Þ dω Basic Response Functions of the Two-Parameter Inertoelastic Solid ? Z dδðω − 0Þ dω 2sgnðtÞ þi ∞ 1 eiωtdω hðtÞ ¼ ð42Þ MR 2 −∞ Upon the derivation and establishment of the correct basic frequency-response functions of the linear spring and the inerter that result in the corresponding causal time-response functions, this study then proceed with the derivation of the basic time-response functions of the two-parameter inertoelastic solid element, that is, a spring and an inerter connected in parallel, as shown in Fig. 3(a). The term solid is used to express that this network sustains a finite displacement under a static load. The constitutive equation of the mechanical network shown in Fig. 3(a) is Z dδðω − 0Þ dω i 2 eiωtdω ¼ −i −∞δðω − 0Þiteiωtdω ¼t ∞ ∞ ð43Þ 2 2 −∞ ? 2sgnðtÞ þt d2uðtÞ dt2 1 1 Uðt − 0Þt hðtÞ ¼ FðtÞ ¼ kuðtÞ þ MR ¼ ð44Þ ð50Þ MR MR 2 and its Fourier transform gives FðωÞ ¼ ½k − MRω2?uðωÞ ð51Þ Eq. (51) indicates that the dynamic stiffness KðωÞ ¼ k − MRω2 of the inertoelastic solid element is improper to the second degree because of the inerter that is connected in parallel with the elastic spring. Given this parallel arrangement, the memory function, qðtÞ, of the inertoelastic solid element is the summation of the memory functions of the elastic spring given by Eq. (22) and that of the inerter given by Eq. (36) ð45Þ © ASCE 04017123-6 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  8. Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 3. (a) Two-parameter inertoelastic solid; (b) two-parameter inertoelastic fluid ? d2δðt − 0Þ dt2 1 1 1 1 − qðtÞ ¼ kδðt − 0Þ þ MR HðωÞ ¼ ð52Þ ω − ωR 2MR ωR ω þ ωR ? Eq. (52) shows that the memory function (inverse Fourier trans- form of the dynamic stiffness) of the inertoelastic solid element exhibits a strong singularity (second order) at the time origin because of the inerter [d2δðt − 0Þ=dt2] and a weak singularity [δðt − 0Þ] because of the linear spring. While the dynamic stiffness of the two-parameter inertoelastic solid, kðωÞ ¼ k − MRω2¼ MRðω2 function, its inverse, the dynamic flexibility, is a proper transfer function þ iπ½δðω þ ωRÞ − δðω − ωRÞ? ð57Þ By manually appending the imaginary part π½δðω − ωRÞ − δðω þ ωRÞ? within the parentheses of Eq. (54), the inverse Fourier transform of the dynamic flexibility of the network shown in Fig. 3(a) is evaluated by rearranging Eq. (57) ? − R− ω2Þ is an improper transfer i i 1 πδðω þ ωRÞ − HðωÞ ¼ 2MR ωR ω þ ωR ?? ? 1 i HðωÞ ¼ − ð53Þ πδðω − ωRÞ − ð58Þ MRðω2− ω2 RÞ ω − ωR in which ω2 in Fig. 3. The poles of the dynamic flexibility given by Eq. (53) are ω ¼ ?ωR, and partial fraction expansion gives 1 2MR ωR R¼ k=MRis the natural frequency of the systems shown The inverse Fourier transform of the first two terms in the bracket of Eq. (58) is Z and with the change of variables, Ω ¼ ω þ ωR, and dΩ ¼ dω, Eq. (59) gives Z ? ? i ∞ ? ? F1ðtÞ ¼1 πδðω þ ωRÞ − eiωtdω ð59Þ 1 1 1 HðωÞ ¼ − − 2π ω þ ωR ð54Þ −∞ ω − ωR ω þ ωR The transfer function given by Eq. (54) is a purely real quantity with singularities at ω ¼ ωRand ω ¼ −ωR; therefore, it faces sim- ilar challenges as those faced by the expression of the impedance of the linear spring given by Eq. (24), which has a singularity at ω ¼ 0. Given that the dynamic flexibility of the two-parameter inertoelastic solid as expressed by Eq. (54) is a purely real quan- tity, this study was in search of the imaginary Hilbert pairs of 1=ðω − ωRÞ and 1=ðω þ ωRÞ. By first setting Ω ¼ ω − ωRand letting H2ðΩÞ ¼ πδðΩ − 0Þ, its Hilbert transform as expressed by the first equation of Eq. (20) gives Z and with the change of variables, ξ ¼ x − Ω and dξ ¼ dx, Eq. (55) becomes Z Eq. (56) indicates that the imaginary Hilbert pair of 1=ðω − ωRÞ is πδðω − ωRÞ. Following the same reasoning shows that the imag- inary Hilbert pair of 1=ðω þ ωRÞ is πδðω þ ωRÞ, and the correct expression of the dynamic flexibility of the two-parameter inertoe- lastic solid is ? ? πδðΩ − 0Þ −i ∞ F1ðtÞ ¼ e−iωRt1 eiΩtdΩ ¼ Uðt − 0Þe−iωRt Ω 2π −∞ ð60Þ where again Uðt − 0Þ is the Heaviside unit-step function at the time origin. The Fourier integral in Eq. (60) is computed by follow- ing the steps outlined in Eqs. (26) and (27). Similarly, the inverse Fourier transform of the last two terms in the brackets of Eq. (58) is Z ? ? i ∞ F2ðtÞ ¼1 πδðω − ωRÞ − eiωtdω ¼ Uðt − 0ÞeiωRt πδðx − 0Þ x − Ω ∞ H1ðΩÞ ¼ −1 ω − ωR 2π dx ð55Þ −∞ π −∞ ð61Þ Substitution of the results of Eqs. (60) and (61) into Eq. (58) gives δ½ξ − ð−ΩÞ? ξ ∞ 1 1 H1ðωÞ ¼ − dξ ¼ − i −Ω¼ ð56Þ 1 Uðt − 0Þðe−iωRt− eiωRtÞ ω − ωR hðtÞ ¼ −∞ 2MR ωR 1 Uðt − 0ÞsinðωRtÞ ¼ ð62Þ MRωR Eq. (62) indicates that the impulse-response function, hðtÞ, of the two-parameter inertoelastic solid shown in Fig. 3(a) is a causal © ASCE 04017123-7 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  9. Z −∞πδðx − 0Þ sinusoidal function (oscillatory response that initiates upon the application of the input impulse force) with the frequency ωR¼ ffiffiffiffiffiffiffiffiffiffiffiffi indicates that when an impulsive force is exerted on a spring that is connected in parallel with an inerter, the impulse-response function is a finite oscillatory response; therefore, the presence of the inerter absorbs the weak singularity, δðt − 0Þ, that exists in the impulse- response function of the solitary spring given by Eq. (23). This ob- servation indicates the potential benefits of the inerter as a response modification device. In the limiting case of a very soft spring (ωR→ 0), the limiting expression for the impulse-response function given by Eq. (62) is ∞ Y2ðωÞ ¼1 1 dx ¼ − ð69Þ x − Ω ω − ωR π p k=MR uously their strain and kinetic energies. Most importantly, Eq. (62) , where the elastic spring and the inerter exchange contin- Eq. (69) indicates that the real Hilbert pair of −1=ðω − ωRÞ is πδðΩ − ωRÞ. Following the same reasoning shows that the real Hilbert pair of −1=ðω − ωRÞ is πδðΩ − ωRÞ and the correct expres- sion for the mobility of the two-parameter inertoelastic solid given by Eq. (68) is ? ? 1 1 1 πδðω − ωRÞ − i þ πδðω þ ωRÞ − i YðωÞ ¼ ω − ωR 2MR ω þ ωR ð70Þ Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. The inverse Fourier transform of Eq. (70) is computed in a sim- ilar way, by computing the inverse Fourier transform of Eq. (58) 1 1 Uðt − 0ÞωRt ¼ Uðt − 0Þt ωR→0hðtÞ ¼ lim ð63Þ MRωR MR 1 and the impulse-response function of the solitary inerter given by Eq. (44) is recovered. The impedance of the two-parameter inertoelastic solid derives directly from Eq. (50) by using that vðωÞ ¼ iωuðωÞ ZðωÞ ¼ −ki Uðt − 0ÞcosðωRtÞ yðtÞ ¼ ð71Þ MR where Uðt − 0Þ is again the Heaviside unit-step function at the time origin. Eq. (71) indicates that the impulse velocity–response function, yðtÞ, of the two-parameter inertoelastic solid shown in Fig. 3(a) is a causal cosine function with frequency ωR¼ Similar to Eq. (62), Eq. (71) indicates that when an impulsive force is exerted on a spring that is connected in parallel with an inerter, the impulse velocity–response function is a finite oscillatory response; therefore, the presence of the inerter absorbs the singu- larity, dδðt − 0Þ=dt, that exists in the impulse velocity–response function of the solitary spring given by Eq. (31). In the limiting case of a very soft spring (ωR→ 0), the limiting expression for the impulse velocity–response function given by Eq. (71) is p ffiffiffiffiffiffiffiffiffiffiffiffi ωþ iωMR ð64Þ k=MR . The first term in Eq. (64) is the impedance of the linear spring given by Eq. (24), and its second improper term is the impedance of the inerter given by Eq. (45). The imaginary reciprocal function, i=ω, is enhanced with its Hilbert pair according to Eq. (25), and the correct expression for the impedance of the two-parameter inertoelastic solid is ? By following the mathematical operations described by Eqs. (26), (27), and (30), the inverse Fourier transform of the impedance of the two-parameter inertoelastic solid is ? πδðω − 0Þ − i1 ZðωÞ ¼ k þ iωMR ð65Þ ω 1 Uðt − 0Þ ωR→0yðtÞ ¼ lim ð72Þ MR and the impulse velocity–response function of the solitary inerter given by Eq. (49) is recovered. The eight basic response functions of the two-parameter viscoelastic solid computed in this section are also summarized in Fig. 2, next to the basic response functions of the linear spring, the viscous dashpot, and the inerter. dδðt − 0Þ dt kðtÞ ¼ kUðt − 0Þ þ MR ð66Þ The mobility (admittance) of the two-parameter inertoelastic solid shown in Fig. 3(a) is the inverse of its impedance expressed by Eq. (64) Basic Response Functions of the Two-Parameter Inertoelastic Fluid iω 1 YðωÞ ¼ − ð67Þ ω2− ω2 MR R This study was next interested in deriving the basic time-response functions of the two-parameter inertoelastic fluid element, that is, a spring and an inerter connected in series, as shown in Fig. 3(b). The term fluid is used to express that this network undergoes an infinite displacement under a static loading. Given that the force, FðtÞ, is common at the spring and the inerter FðtÞ ¼ k½uðtÞ − u3ðtÞ? and at the same time in which ω2 of the mobility given by Eq. (67) are ω ¼ ?ωR, and partial fraction expansion gives ? The transfer function given by Eq. (68) is a purely imaginary quantity with singularities at ω ¼ ωRand ω ¼ −ωR; therefore, it faces similar challenges as those faced by the expression of the impedance of the linear spring given by Eq. (24), which has a sin- gularity at ω ¼ 0. Given that the mobility (admittance) of the two- parameter inertoelastic solid, as expressed by Eq. (68), is a purely imaginary quantity, this study was in search of the real Hilbert pairs of 1=ðω − ωRÞ and 1=ðω þ ωRÞ. By first setting Ω ¼ ω − ωRand letting Y1ðΩÞ ¼ πδðΩ − 0Þ, its Hilbert transform as expressed by the second equation of Eq. (20) gives [Eqs. (55) and (56)] R¼ k=MR= natural frequency of the system. The poles ? i 1 1 YðωÞ ¼ − þ ð68Þ ω − ωR 2MR ω þ ωR ð73Þ d2u3ðtÞ dt2 FðtÞ ¼ MR ð74Þ where u3ðtÞ = nodal displacement of the internal node, 3, where the spring and the inerter connect. Upon differentiating Eq. (73) two times and substituting d2u3ðtÞ=dt2from Eq. (74), the constitutive equation of the mechanical network shown in Fig. 3(b) is expressed © ASCE 04017123-8 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  10. ? ? ω2þ iπdδðω − 0Þ d2FðtÞ dt2 d2uðtÞ dt2 HðωÞ ¼1 1 −1 FðtÞ þ1 ¼ MR kþ ð81Þ ð75Þ MR dω ω2 R The inverse Fourier transform of the correct dynamic flexibility of the two-parameter inertoelastic fluid given by Eq. (81) is com- puted by following the steps outlined in Eqs. (41)–(44) and results in a causal impulse-response function The Fourier transform of Eq. (75) gives ω2 FðωÞ ¼ ω2 RMR uðωÞ ð76Þ ω2− ω2 R hðtÞ ¼1 1 kδðt − 0Þ þ Uðt − 0Þt ð82Þ and by recalling that k ¼ ω2 mechanical network that results from Eq. (76) is KðωÞ ¼ kω2=ðω2− ω2 ing the constant value, k, at the high-frequency limit. By separating the high-frequency limiting value, k, the dynamic stiffness of the two-parameter inertoelastic fluid that results from Eq. (76) is expressed RMR, the dynamic stiffness of the MR where Uðt − 0Þ is again the Heaviside unit-step function at the time origin. Eq. (82) indicates that when an impulsive force is exerted on the two-parameter inertoelastic fluid, the impulse is initially taken entirely by the elastic spring, δðt − 0Þ=k; subsequently, the inerter enters a causal response that grows linearly with time, during which the idle spring is carried along. The impedance of the two-parameter viscoelastic fluid shown in Fig. 3(b) derives directly from Eq. (76) by using that vðωÞ ¼ iωuðωÞ RÞ, which is a simple proper transfer function, reach- Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. ? ? ω2 R KðωÞ ¼ k 1 þ ð77Þ ω2− ω2 R The second term within the parentheses in Eq. (77), R=ðω2− ω2 two-parameter inertoelastic solid given by Eq. (53). Accordingly, the purely real term, ω2 ω ? ωR, needs to be enhanced with its corresponding imaginary delta functions. By following a procedure similar to the one out- lined by Eqs. (55)–(58), the correct expression of the dynamic stiff- ness of the two-parameter viscoelastic fluid described by Eq. (75) is iω ZðωÞ ¼ −k ð83Þ ω2 RÞ, is of the same form as the dynamic flexibility of the ω2− ω2 R R=ðω2− ω2 RÞ, in Eq. (77), which has poles at Eq. (83) is of the same form as the mobility (admittance) of the two-parameter inertoelastic solid given by Eq. (67); therefore, by following the steps outlined by Eqs. (68)–(70), the correct expres- sion of the impedance of the two-parameter inertoelastic fluid is ? ? ð84Þ ZðωÞ ¼k i i πδðω − ωRÞ − þ πδðω þ ωRÞ − ? KðωÞ ¼ k þkωR ω − ωR ω þ ωR 1 1 2 − ω − ωR ω þ ωR 2 ? þ iπ½δðω − ωRÞ − δðω þ ωRÞ? ð78Þ The inverse Fourier transform of Eq. (84) is computed in a sim- ilar way by computing the inverse Fourier transform of Eq. (58) By manually appending the imaginary part π½δðω − ωRÞ − δðω þ ωRÞ? in Eq. (77), the inverse Fourier transform of the dy- namic stiffness of the two-parameter viscoelastic fluid shown in Fig. 3(b) is evaluated by following a procedure similar to the one outlined in Eqs. (59)–(61); the resulting memory function is kðtÞ ¼ kUðt − 0ÞcosðωRtÞ ð85Þ where Uðt − 0Þ is again the Heaviside unit-step function at the time origin. Eq. (85) indicates that the imposed unit-step displacement is initially accommodated entirely by the elastic spring, which sub- sequently sets the inerter in motion with the elastic and kinetic en- ergies being continuously interchanged between the elastic spring and the inerter. The mobility (admittance) of the two-parameter inertoelastic fluid shown in Fig. 3(b) is the inverse of its impedance expressed by Eq. (83) qðtÞ ¼ k½δðt − 0Þ − ωRUðt − 0ÞsinωRt? ð79Þ Under an impulsive displacement input, the two-parameter vis- coelastic fluid shown in Fig. 3(b) initially behaves like the linear spring, given that the inerter provides infinite resistance at the ini- tiation of motion. Accordingly, at the time origin, the memory func- tion given by Eq. (79) is singular, as is the memory function of the solitary elastic spring given by Eq. (22); subsequently, the inerter engages in motion and follows a causal sinusoidal response with frequency ωR¼ Eq. (76) also indicates that the dynamic flexibility HðωÞ ¼ uðωÞ=FðωÞ of the two-parameter inertoelastic fluid is a simply proper transfer function kiω −ω2 i ω¼1 i ω YðωÞ ¼1 1 R kiω − ð86Þ k MR p ffiffiffiffiffiffiffiffiffiffiffiffi Eq. (86) is of the same form as the impedance of the two- parameter inertoelastic solid given by Eq. (64). It includes an improper term, iω=k, which results from the mobility of the solitary linear spring given by Eq. (28) together with the reciprocal func- tion, 1=ω, which results from the mobility of the solitary inerter given by Eq. (47). Consequently, the imaginary reciprocal function, i=ω, is enhanced with its Hilbert pair according to Eq. (25) and the correct expression for the mobility of the two-parameter inertoelas- tic fluid is k=MR . k−ω2 HðωÞ ¼1 ω2¼1 1 1 1 R k− ð80Þ k MR ω2 The first term, 1=k, in Eq. (80) is the dynamic flexibility of the elastic spring, and the second term is the dynamic flexibility of the solitary inerter given by Eq. (37). By following the same pro- cedure as the one presented by Eqs. (38)–(40), the correct expres- sion for the dynamic flexibility of the two-parameter inertoelastic fluid is ? ? YðωÞ ¼1 1 πδðω − 0Þ − i1 kiω þ ð87Þ MR ω By following the mathematical operations described by Eqs. (26), (27), and (30), the inverse Fourier transform of the © ASCE 04017123-9 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  11. mobility of the two-parameter inertoelastic fluid given by Eq. (87) is Although the dynamic stiffness of the rotational inertia damper, kðωÞ ¼ Ciω − MRω2, is an improper transfer function, its inverse, the dynamic flexibility, is a proper function dδðt − 0Þ dt yðtÞ ¼1 1 Uðt − 0Þ þ ð88Þ 1 ¼1 1 k MR HðωÞ ¼ ð92Þ ωð−ωMR iωC − ω2MR C Cþ iÞ The eight basic response functions of the two-parameter visco- elastic fluid computed in this section are summarized in Fig. 2, next to the basic response functions of the two-parameter viscoelastic solid. where the ratio MR=C = retardation time of the network and has units of time (T). The poles of the dynamic flexibility given by Eq. (92) are ω ¼ 0 and ω ¼ iC=MR; therefore, partial fraction expansion of the right- hand side of Eq. (92) gives ? ? Basic Response Functions of the Rotational Inertia Damper (Dashpot and Inerter in Parallel) ω−MR Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. HðωÞ ¼1 −i1 1 ð93Þ 1 þ iωMR C C C About a decade ago, Hwang et al. (2007) proposed a rotational inertia damper that consists of a cylindrical mass that is driven by a ball screw and rotates within a chamber that contains some vis- cous fluid. In this way, the resistance to the driving force originates partly from the difficulty of mobilizing the rotational inertia of the rotating mass and partly from the viscous stresses that develop on the circumference of the rotating mass as it shears the viscous fluid. Consequently, the mechanical analog of the rotational inertia damper proposed by Hwang et al. (2007) and later by Ikago et al. (2012) is essentially a dashpot and an inerter connected in parallel, as shown in Fig. 4(a). The constitutive equation of the mechanical network shown in Fig. 4(a) is The first term within the parentheses in Eq. (93) is the imaginary reciprocal function, i=ω, which faces the same challenges as the impedance of the linear spring given by Eq. (24); therefore, it is enhanced with its Hilbert pair according to Eq. (25). The second term within the brackets in Eq. (93) is a strictly proper transfer function; therefore, its inverse Fourier transform converges in the classical sense. Consequently, the correct expression for the dy- namic flexibility of the rotational inertia damper is (? ) ? −MR HðωÞ ¼1 πδðω − 0Þ − i1 1 ð94Þ 1 þ iωMR C ω C C The inverse Fourier transform of Eq. (94) is evaluated by fol- lowing the mathematical operations described by Eqs. (26) and (27) for the term πδðω − 0Þ − i=ω, whereas the last term is integrated by employing the method of residues (Morse and Feshbach 1953). Accordingly FðtÞ ¼ CduðtÞ d2uðtÞ dt2 þ MR ð89Þ dt and its Fourier transform gives FðωÞ ¼ ðCiω − MRω2ÞuðωÞ ð90Þ hðtÞ ¼1 C½Uðt − 0Þ − e−ðC=MRÞt? ð95Þ With the parallel arrangement, the memory function, qðtÞ, of the rotational inertia damper shown in Fig. 4(a) is the summation of the memory function of the dashpot [see Makris 1997a, Fig. 2, and the Fourier transform of Eq. (28) given by Eq. (31)] and that of the inerter given by Eq. (36) where again Uðt − 0Þ = Heaviside unit-step function at the time origin. Eq. (95) indicates that the impulse-response function of the rotational inertia damper assumes the value of 1=C at the time origin and subsequently decays exponentially to zero. The impedance of the rotational inertia damper shown in Fig. 4(a) derives directly from Eq. (90) by using that vðωÞ ¼ iωuðωÞ ZðωÞ ¼Ciω − MRω2 iω qðtÞ ¼ Cdδðt − 0Þ d2δðt − 0Þ dt2 þ MR ð91Þ dt The improper dynamic stiffness appearing in Eq. (90) indicates that it is not physically realizable to impose an impulse displace- ment on a rotational inertia damper. Nevertheless, a mathematical expression of its memory functions can be reached via the use of the derivatives of the Dirac delta function, as shown by Eq. (91). ¼ C þ iωMR ð96Þ where the constant C is the impedance of the viscous dashpot and iωMRis the impedance of the inerter given by Eq. (45). With this Fig. 4. (a) Rotational inertia damper, which is an inertoviscous model in parallel; (b) inertoviscous model in series © ASCE 04017123-10 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  12. parallel arrangement, the inverse Fourier transform of the imped- ance of the rotational inertia damper is Accordingly, the memory function of a dashpot and an inerter connected in series is ?dδðt − 0Þ The dynamic flexibility of a dashpot and an inerter connected in series is the inverse of the dynamic stiffness given by Eq. (102) ? dδðt − 0Þ dt −C MR δðt − 0Þ þC2 e−ðC=MRÞt kðtÞ ¼ Cδðt − 0Þ þ MR qðtÞ ¼ C ð104Þ ð97Þ dt M2 R The mobility of the rotational inertia damper is the inverse of its impedance given by Eq. (96) 1 þ iωMR ω2 1 ¼1 1 1 1 ω2−1 1 Ci1 YðωÞ ¼ ð98Þ C HðωÞ ¼ − ¼ − ð105Þ 1 þ iωMR C þ iωMR C MR MR ω C The first term in the right-hand side of Eq. (105) is the dynamic flexibility of the inerter as expressed by Eq. (37); the second term is the dynamic flexibility of the linear dashpot (Makris 1997a). Con- sequently, both terms are enhanced with their Hilbert pairs as shown by Eqs. (25) and (40), and the correct expression of the dy- namic flexibility given by Eq. (105) is ? The transfer function given by Eq. (98) is a strictly proper trans- fer function, and its inverse Fourier transform is evaluated with the method of residues Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. 1 e−ðC=MRÞt yðtÞ ¼ ð99Þ MR ? ? ? ω2þ iπdδðω − 0Þ The eight basic response functions of the rotational inertia damper (dashpot and inerter connected in parallel) are summarized in Fig. 2 next to the inertoelastic elements examined in this article. 1 −1 þ1 πδðω − 0Þ − i1 HðωÞ ¼ MR dω C ω ð106Þ By manually appending the Hilbert pair singularities πdδðω − 0Þ=dω for the inerter and πδðω − 0Þ for the dashpot, the inverse Fourier transform of the dynamic flexibility given by Eq. (106) gives the causal impulse-response function ? where Uðt − 0Þ = Heaviside unit-step function at the time origin. Eq. (107) indicates that the causal impulse-response function of a dashpot and an inerter in series assumes a constant value, 1=C, at the time origin and subsequently grows linearly with time. The impedance of a dashpot and an inerter connected in series derives directly from Eq. (102) by using that vðωÞ ¼ iωuðωÞ ZðωÞ ¼ MR Basic Response Functions of a Dashpot and Inerter Connected in Series This study was next interested in deriving the basic time-response function of a dashpot and an inerter in series, as shown in Fig. 4(b). Given that the force, FðtÞ, is common at the dashpot and the inerter ?duðtÞ and at the same time, FðtÞ satisfies Eq. (74). In Eq. (100), u3ðtÞ is the nodal displacement of the internal node, 3, where the dashpot and the inerter connect. Upon differentiating Eq. (100) and substi- tuting d2u3ðtÞ=dt2from Eq. (74), the constitutive equation of the mechanical network shown in Fig. 4(b) is expressed ? C hðtÞ ¼1 Uðt − 0Þ t 1 þ ð107Þ C MR ? −du3ðtÞ FðtÞ ¼ C ð100Þ dt dt iω ð108Þ 1 þ iωMR C FðtÞ þMR dFðtÞ dt d2uðtÞ dt2 ¼ MR The transfer function given by Eq. (108) is a simple proper transfer function, reaching the constant value, C, at the high- frequency limit. By separating the high-frequency limit, C, the impedance of the mechanical network shown in Fig. 4(b) as results from Eq. (108) is expressed ð101Þ C The Fourier transform of Eq. (101) gives ω2 FðωÞ ¼ −MR uðωÞ ð102Þ ! 1 þ iωMR C 1 1 − ZðωÞ ¼ C ð109Þ 1 þ iωMR Eq. (102) indicates that the dynamic stiffness of a dashpot and an inerter connected in series is −MRω2=ð1 þ iωMR=CÞ; therefore, it is an improper transfer function. By first separating the improper high-frequency limit, iωC, and subsequently separating the simple- proper high-frequency limit −C2=MR, the dynamic stiffness of the mechanical network shown in Fig. 4(b) as results from Eq. (102) is expressed C The inverse Fourier transform of a constant is the Dirac delta function, whereas the inverse Fourier transform of the strictly proper second term within the parentheses of Eq. (109) is evaluated with the method of residues. Accordingly, the relaxation stiffness of a dashpot and an inerter connected in series is ? The mobility (admittance) of a dashpot and an inerter connected in series is the inverse of the impedance as expressed by Eq. (108) ? δðt − 0Þ −C e−ðC=MRÞt KðωÞ ¼ iωC −C2 þC2 MR kðtÞ ¼ C 1 ð110Þ MR ð103Þ 1 þ iωMR MR C The inverse Fourier transform of the first term, which is the dynamic stiffness of the linear dashpot (Makris 1997a), is eval- uated by employing Eq. (30), whereas the inverse Fourier trans- form of the constant C2=MR results in a weak singularity ðC2=MRÞδðt − 0Þ. The last term in Eq. (103) is a strictly proper transfer function; therefore, its inverse Fourier transform converges in the classical sense and is evaluated with the method of residues. 1 þ iωMR iω 1 ¼1 1 i1 ω C C− YðωÞ ¼ ð111Þ MR MR The first constant term, 1=C, in the right hand side of Eq. (111) is the mobility of the linear dashpot (Makris 1997a), whereas © ASCE 04017123-11 J. Eng. Mech. J. Eng. Mech., 2017, 143(11): 04017123

  13. in the causal time-response function; that is, the memory function (or relaxation stiffness) of the rotational inertia damper is the sum- mation of the memory functions (or relaxation stiffnesses) of the solitary dashpot and that of the solitary inerter. In the case of a dashpot and inerter in series, the superposition of the individual response functions applies to the dynamic flexibility and admit- tance; therefore, it also applies to the corresponding impulse- response function and impulse velocity–response function. Finally, the basic response functions derived in this paper, in association with the mathematical operations outlined in this work, extend the well-established theory of linear viscoelasticity to the inertoelastic and inertoviscoelastic behavior (combination of inerters, dashpots and springs) and introduce the subject of inertoviscoelasticity. the second term is the mobility of the inerter given by Eq. (47). Consequently, the correct expression for the mobility of the mechanical network shown in Fig. 4(b) is ? ByfollowingthemathematicaloperationsdescribedbyEqs.(26) and (27), the inverse Fourier transform of the mobility given by Eq. (112) is ? YðωÞ ¼1 1 πδðω − 0Þ − i1 Cþ ð112Þ MR ω yðtÞ ¼1 1 Cδðt − 0Þ þ Uðt − 0Þ ð113Þ MR Downloaded from ascelibrary.org by University of Central Florida on 08/19/17. Copyright ASCE. For personal use only; all rights reserved. To this end, the eight basic response functions of a dashpot and an inerter connected in series are summarized in Fig. 2, next to the basic response functions of the rotational inertia damper. Acknowledgments The assistance of Dr. H. Alexakis with the management of the electronic document is appreciated. Conclusion This paper examined and constructed the basic frequency-response functions and subsequently derived the corresponding causal time- response functions of elementary mechanical networks that involve the inerter, a two-node element in which the force-output is propor- tional to the relative acceleration of its end-nodes. This was achieved by extending the relationship between the causality of a time-response function and the analyticity of its corresponding frequency-response function to the case of generalized functions. The paper showed that when the frequency-response function has as a singularity the reciprocal function, 1=ðω − ωRÞ (with ωR¼ constant or zero) or the inverse-square function, 1=ω2, along the real frequency axis, then the complex frequency-response function needs to be enhanced with the addition of a Dirac delta function, δðω − ωRÞ, or with its derivative, depending on the strength of the singularity. In this way, the real and imaginary parts of the correct frequency-response function are Hilbert pairs, thereby yielding a causal time-response function in the time domain. Because of the participation of the inerter some basic time- response functions exhibit a causal oscillatory response, and this behavior complements the decaying-exponential response due to the participation of the dashpot. Most importantly, the paper showed that the inerter emerges as an attractive response modifi- cation element, given that in some cases it absorbs the singular response of the solitary spring or dashpot. Fig. 2 presents an important result—mechanical networks that involve inerters exhibit symmetries similar to those of the classical mechanical networks that involve only springs and dashpots. For instance, the dynamic stiffness (or impedance) of the inertoelastic solid (spring and inerter in parallel) is the summation of the dy- namic stiffnesses (or impedances) of the solitary spring and that of the solitary inerter. The outcome of this superposition is reflected in the causal time-response functions, that is, the memory function (or relaxation stiffness) of the inertoelastic solid that is the summation of the memory functions (or relaxation stiffnesses) of the solitary spring and that of the solitary inerter. In the case of the inertoelastic fluid (spring and inerter in series) the superposition of the indi- vidual response functions applies to the dynamic flexibility and admittance; therefore, it also applies to the corresponding impulse- response function and impulse velocity–response function. The same superposition pattern applies to the inertoviscous elements. 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