A comparative study of vibration isolation performance using negative stiffness and inerter dampers

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Journal of the Franklin Institute 356 (2019) 7922–7946 www.elsevier.com/locate/jfranklin

A comparative study of vibration isolation performance using negative stiffness and inerter dampers Xiang SHI a, Songye ZHU b,∗ a College

of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266580, Shandong, China b Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China Received 10 August 2018; received in revised form 15 January 2019; accepted 22 February 2019 Available online 26 July 2019

Abstract Two types of passive devices, namely, negative stiffness damper (NSD) and inerter damper (ID), have been receiving growing interest in vibration isolation and suppression, because both can produce negative-slope force-displacement relationships that are similar to those associated with active control forces. Despite such a similarity, these two passive dampers possess obvious differences in their mechanical behaviors. This study aims to illustrate the similarity and difference between these two dampers in vibration isolation applications with respect to the H2 and H∞ performance. The comparative study indicates that both dampers can reduce the H∞ norm effectively; the negative stiffness devices can reduce the H2 norm as well, whereas the H2 norm cannot converge under the influence of inerter. This finding explains why a tuned-inerter damper, i.e., an inerter connected in series with a spring with proper frequency tuning, is more commonly adopted in vibration isolation. The pros and cons of both devices were further discussed. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author. E-mail address: [email protected] (S. ZHU).

https://doi.org/10.1016/j.jfranklin.2019.02.040 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction Vibration isolation problems have been extensively studied in the past decades. Vibration isolation techniques are usually classified into three categories: passive, semi-active, and active. Active isolation techniques typically achieve better control performance than passive counterparts do. However, the dependence on external power supply, sensing and feedback systems also makes active isolation techniques more complex, costly and difficult. Meanwhile, the passive isolation solutions are favorable in many practical solutions because of their simplicity, reliability, robustness and cheap cost. A passive vibration isolation technique that can offer the performance comparable to active control is highly desirable in practical applications. Researchers have noted that several representative active control theories (such as ideal skyhook damper and linear quadratic regulator algorithm) may produce a control force– displacement relationship with an evident negative slope, which is believed to benefit vibration isolation performance [1]. This finding inspired the development and study of a variety of negative stiffness dampers (NSDs) that aim to achieve a control performance close to those of active controllers by producing a similar negative stiffness force-displacement relationship. Several researchers attempted to implement such a negative stiffness mechanism in a semiactive manner [2–6]. Semi-active damper, although consuming less power than active control in general, still needs a feedback system involving sensors, controllers, and power supply. Several types of fully passive negative stiffness mechanisms, including devices based on the snap-through behavior of a pre-buckled beam [7], pre-compressed springs [8], a sliding isolator with a convex friction interface [9], and two permanent magnet-based negative stiffness designs [10,11], are considered more practically valuable solutions. Inerter, as another novel type of passive devices, also possesses negative-slope feature in its control force. Smith [12] initially proposed the concept of inerter to complete the forcecurrent analogy between mechanical and electrical networks. Consequently, a new concept of inertance was introduced into the structural dynamics: the control force of an inerter is proportional to the relative acceleration between its two terminals, and this proportionality constant is inertance [12]. The representative designs of inerters include the rack–pinion type [13], ball-screw type [14], hydraulic motor type [15], and fluid type [16,17]. Recently, semiactive inerters have been developed either through hydraulic approach [18] or controllable inertia flywheel approach [19]. Negative stiffness and inerter are both regarded as efficient vibration suppression tools. The benefits of negative stiffness have been illustrated in many vibration control applications, including buildings [3,8,9,20,21], stay cables [5,22–26], cable-stayed bridges [2,27,28], high-speed trains [29–31], vehicle seats [7,32], and isolation tables [33,34]. Similarly, the effectiveness of inerter have been explored in various applications, such as vehicle suspensions [17,18,35–40], train suspensions [41–44], building isolation systems [45–49], and stay cables [50,51]. Despite similar force-displacement relationships, negative stiffness and inerter still exhibit notable differences (Fig. 1): (1) Like any stiffness element, a negative stiffness coefficient does not vary with vibration frequencies (Fig. 1(a)), whereas inerter introduces the forcedeformation relationship whose negative slope depends on vibration frequencies (Fig. 1(b)). (2) To maintain stability, negative stiffness is effective in parallel connection with the original stiffness of a structure, whereas inerter is usually embedded inside a network with a stiffness in a series connection. Specific examples include tuned viscous mass dampers [52,53],

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F

Low frequency High frequency

F

x

(a) Negative stiffness

x

(b) Inerter

Fig. 1. Force vs. displacement relationship.

tuned mass-damper-inerter systems [54–60], tuned inerter dampers (TID) [61], and inerter– based resonant vibration absorbers [62]. Given these differences, the mechanisms of negative stiffness and inerter in vibration isolation warrant a detailed analysis and comparison that has not been reported in the literature. By using the H2 and H∞ norms, the present study investigates the similarities and differences between NSD and inerter damper (ID) in vibration isolation applications. A general single degree–of–freedom (SDOF) system is considered, and frequency- and time-domain analyses are conducted. On the basis of these analyses, the deficiencies of NSD and ID are identified, and proper solutions are discussed. 2. Problem formulation In vibration isolation, the resonant peak control and high-frequency attenuation should be properly balanced and sometimes a tradeoff exists between these two objectives. A skyhook control, which refers to the installation of a damper between a sprung mass and the still reference in the sky, has been regarded as an ideal vibration isolation strategy to isolate the suspended mass from base excitations and eliminate such a tradeoff [63,64]. Despite being an ideal and fictional concept, the skyhook control has motivated many researches in the development of the corresponding semi-active or active control strategies [63,65,66]. In this comparative study, the skyhook control is considered as a reference case of an optimal vibration isolation. The dynamic equations of an SDOF system with various vibration isolation methods are formulated in this section. Fig. 2(a), (b), and (c) illustrate the SDOF systems with NSD, ID, and ideal skyhook control, respectively. 2.1. NSD The governing equation of the SDOF system with an NSD subjected to base excitation is expressed as     mx¨ + cnsd x˙ − x˙g + (k − knsd ) x − xg = 0, (1) where m and k are the mass and stiffness of the SDOF system, respectively; x and xg are the absolute response of the sprung mass and ground motion, respectively; and knsd and cnsd are

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m k

cnsd

knsd

(a) NSD

(b) ID

(c) Skyhook damper

Fig. 2. Passive vibration isolation for the SDOF system.

the absolute values of the stiffness and damping coefficients of the NSD, respectively. The negative sign in front of knsd indicates that the stiffness is negative. The natural frequency of the original SDOF system without any dampers is  ωn = k/m, (2) The transfer function from the base excitation to the absolute displacement of the sprung mass can be obtained from Eq. (1) as Hnsd (s) =

X cnsd s + k − knsd = . 2 Xg ms + cnsd s + k − knsd

(3)

where s is the complex frequency parameter in the Laplace transform. The damper force of the NSD fnsd can be calculated using     fnsd = −knsd x − xg + cnsd x˙ − x˙g , (4) and the corresponding transfer function for the damper force-displacement relationship is given by Fnsd = −knsd + cnsd s, X

(5)

where X = X − Xg is the relative displacement between the sprung mass and the ground base. To facilitate the following discussion, the ratio of the negative stiffness to the structural stiffness is defined as α = knsd /k .

(6)

An extreme case is when α = 1 and cnsd = 0, we have the zero-stiffness isolation in which the structure is completely isolated from the ground motion. However, to maintain the stability of the SDOF system, the absolute value of knsd shall be less than k (α < 1) in reality, i.e., the negative stiffness cannot be stronger than the positive stiffness of the original system.

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2.2. ID When installed with an ID, the governing equation of the SDOF system is       mx¨ + cid x˙ − x˙g + k x − xg + b x¨ − x¨g = 0,

(7)

where b and cid are the inertance and damping coefficients of the ID, respectively. The transfer function from the ground motion to the absolute displacement response of the structure is Hid (s) =

X bs2 + cid s + k = . Xg (m + b)s2 + cid s + k

The damping force of the ID (fid ) is     fid = b x¨ − x¨g + cid x˙ − x˙g ,

(8)

(9)

and the corresponding transfer function for the damper force-displacement relation is Fid = bs2 + cid s. (10) X To facilitate the following discussion, the ratio of the inertance to the structural mass is defined as β = b/m.

(11)

2.3. Skyhook damper The governing equation of the SDOF system with an ideal skyhook damper subjected to base excitation is expressed as   mx¨ + csky x˙ + k x − xg = 0, (12) where csky is the damping coefficient of the skyhook damper. The transfer function from the base excitation to the absolute displacement of the sprung mass can be obtained from Eq. (12) as Hsky (s) =

X k = . 2 Xg ms + csky s + k

(13)

The control force of the skyhook damper fsky can be calculated using fsky = csky x˙.

(14)

From Eqs. (13) and (14), the transfer function of the damper force with respect to the relative displacement X is Fsky −csky k = . X ms + csky

(15)

When csky → ∞, the asymptotic solution of Eq. (15) is −k, implying that the structure is completely isolated with the performance similar to the zero-stiffness case. The damping ratio of SDOF system with the skyhook damper is defined as csky ξsky = √ . (16) 2 mk

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2.4. Comparison of damper forces The control forces of NSD and ID are generated by the relative response between the base and the sprung mass. Fig. 3(a), (b) and (c) compares the damper forces of the skyhook damper, NSD (α = 20%), and ID (β = 30%) when the SDOF system is subjected to a harmonic base excitation Xg = A sin (ωt ) with the excitation frequencies ω = 0.5ωn , ωn , and 2ωn , respectively, wherein ωn is the natural frequency of the uncontrolled structure. The slope of the NSD under three excitation frequencies is particularly shown by three straight lines √ in Fig. 3. The damping coefficients in all three cases are set identical (c = 0.5 mk ). The force-displacement relationships of all three dampers are rotated ellipses with negative slope. However, the negative–slope of the NSD case does not vary with the excitation frequency; the slope of the ID case increases with the excitation frequency; and the slope of the skyhook damper case decreases with the excitation frequency. The magnitude and phase angle of the damper forces can be obtained by substituting s = jω into Eqs. (15), (5) and (10) Skyhook

2 csky k Fsky mk csky ω =− 2 + 2 j X csky + m2 ω2 csky + m2 ω2

tan sky = − NSD

Fnsd = −knsd + cnsd ω j X tan nsd = −

ID

mω csky

cnsd ω knsd

Fid = −bω2 + cid ω j X

(17a) (17b) (18a) (18b) (19a)

cid (19b) bω In Eqs. 17(a), 18(a), and 19(a), the real parts are always negative and the phase angles always show phase-lead. Apparent differences can be observed among three cases: the real part of the NSD case is independent with the excitation frequency; the real part of the ID case increases quadratically with the increasing excitation frequency; the real part of the skyhook damper is inversely proportional to the square of the excitation frequency. The bode graphs of the control forces of the skyhook damper, NSD and ID with respect to the relative displacement (i.e., F/X/k) are shown in Fig. 4. The parameters of the three dampers are tuned so that their force magnitudes and phase angles are the same when ω = ωn , which implies that their control performance at ω = ωn is identical. At a low frequency, the control force of NSD Fnsd is mainly determined by the negative stiffness, and thus the phase angle with respect to the relative displacement X is close to 180°; while at a high frequency, the control force of NSD is mainly contributed by the damping part, and thus the phase angle is nearly 90°. The control force of skyhook damper has the same phase angle as that of NSD and changes from 180° at a low frequency to 90° at a high frequency. However, the phase angle of ID is completely different from those of skyhook damper and NSD. Considering the relationship of X = X − Xg , the transfer function from the ground displacement to the tan id = −

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Fig. 3. The control forces of the skyhook damper, √ NSD (α = 20%), and ID (β = 30%) under harmonic excitation with various frequencies (csky = cnsd = cid = 0.5 mk ) (The straight lines show the constant negative slope of NSD).

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Fig. 4. Bode graph of the control forces (F/X/k) of three dampers.

Fig. 5. The normalized transfer function from the ground motion to the control force (F/Xg ) of three dampers.

control force (i.e., F/Xg ) can be obtained and its magnitude is shown in Fig. 5. The skyhook damper has the peak control force corresponding to the frequency ωn and produces smaller control force at a higher frequency, which explains why the skyhook damper only reduces resonant responses and does not affect high-frequency response adversely. The magnitude of the control forces of NSD and ID generally increases with vibration frequency, except the frequencies near ωn . In the case of NSD, the negative stiffness force is independent with the frequency, and thus the increase is mainly due to the damping force that is proportional to the excitation frequency (Eq. (5)). In the case of ID, the inertance force is proportional to the square of the excitation frequency (Eq. (10)), and thus the magnitude of the ID force increases much faster than the NSD force. When installed in a SDOF system, the control force of ID is mainly contributed by the inerter part at a high frequency. These differences significantly influence the performance of the NSD, ID, and skyhook damper, which will be discussed in the following sections with respect to the H2 and H∞ norms.

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√ Fig. 6. Bode graph of the system response with respect to ground excitation X/Xg with an NSD (cnsd = 0.1 mk ).

3. H2 and H∞ control performances System norms are important quantitative measurements for the assessment of vibration control performance. The H2 performance of the system can be interpreted as an average response under dynamic loadings over all frequencies, whereas the H∞ performance of the system represents the peak response of the system under dynamic loadings in different frequencies with the same energy level. 3.1. Frequency response functions (FRFs) By substituting s = jω into Eqs. (3), (8) and (13), the FRFs of the SDOF system with different types of dampers can be obtained as follows Hnsd ( j ω) = Hid ( jω ) =

X k − knsd + j cnsd ω = . Xg k − knsd − mω2 + j cnsd ω

X k − bω2 + j cid ω = . Xg k − (m + b)ω2 + j cid ω

Hsky ( jω ) =

X k = . 2 Xg k − mω + j csky ω

(20) (21) (22)

Figs. 6–8 show the bode graphs of the SDOF responses with three different types of dampers, namely, NSD, ID, and skyhook damper, respectively.

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√ Fig. 7. Bode graph of the system response with respect to ground excitation X/Xg with an ID (cid = 0.1 mk ).

Fig. 8. Bode graph of the system response with respect to ground excitation X/Xg with a skyhook damper.

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3.2. H2 performance The SDOF structure excited by ground motions is a stable single-input, single-output (SISO) system, wherein the input and output are the base excitation and the absolute displacement response of the sprung mass, respectively. The corresponding H2 norm is defined as  1/2  +∞ 1 H 2 = |H ( jω)|2 dω . (23) 2π −∞ If the transfer function, H, can be expressed in a special form as H ( j ω) =

B0 + j ω B1 , A0 + j ω A1 − ω 2 A2

then the integration in Eq. (23) can be computed as [67]:   2

B  +∞ π A00 A2 + B12 |H ( jω)|2 dω = , A1 A2 −∞

(24)

(25)

Assume the base excitation Xg is a white-noise random excitation, then the power spectral density (PSD) of Xg is uniform. Sxg (ω) = S0

(26)

Therefore, the PSD of the displacement response of the SDOF system when subjected to white-noise base excitation is Sx (ω) = |H ( jω)|2 Sxg (ω) = |H ( jω)|2 S0 . The corresponding RMS displacement is  1/2  1/2  +∞  +∞ √ 1 1 |H ( jω)|2 S0 dω xrms = Sx (ω )dω = = H 2 S0 . 2π −∞ 2π −∞

(27)

(28)

Combine Eqs. (20)–(22) with Eq. (23), we have the H2 norm of the SDOF structure with different types of dampers as   1/2 1 k − knsd cnsd Hnsd 2 = + . (29) 2 cnsd m Hid 2 = ∞. Hsky = 2



k 2csky

(30) 1/2

.

(31)

In the case with NSD, Hnsd 2 decreases with the increasing knsd , where knsd is the absolute value of negative stiffness. It can be observed in Fig. 6 as well, that given a stronger negative stiffness value, the total area below the FRF curve becomes smaller. Moreover, once the negative stiffness coefficient knsd is fixed, the optimal cnsd that minimizes Hnsd 2 can be determined from Eq. (29) as  cnsd−opt = m (k − knsd ). (32)

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The optimal cnsd-opt for Hnsd 2 decreases with the increasing value of the negative stiffness knsd . Ideally when knsd = k and cnsd = 0, the sprung mass is completely isolated from the base, and the zero-stiffness isolation leads to Hnsd 2 = 0. In the case with ID, Hid 2 approaches to infinity, implying that the ID leads to nonconvergent response given a white-noise excitation with ideally infinity bandwidth. Even though actual random excitations always have limited bandwidth and input energy, the ID tends to amplify the responses of high-frequency components. The installation of the inerter introduces a quadratic term into the numerator of Eq. (8), and thereby makes the orders of the numerator and denominator in the FRF function equal. Consequently, the base excitation at high frequencies cannot be filtered and the ID will amplify the absolute response of the sprung mass under a high-frequency base excitation. A similar observation was made by Hu et al. [68] and Makris [69] in the analytical study of a parallel-connected inerter. In the case with the skyhook damper, Hsky 2 decreases with the increase in ξ sky . When ξ sky approaches infinity, the sprung mass is fixed to the sky and Hsky 2 approaches zero. At this extreme case, the Fsky /X = −k , which is identical to the ideal NSD case. Fig. 8 shows the bode graph of the SDOF system with a skyhook damper, in which the magnitude at all frequencies decreases with the increase in ξ sky . This observation is consistent with the prediction by Eq. (31). In summary, the NSD and skyhook damper can reduce the H2 norm of an SDOF structure subjected to base excitations, whereas the ID make the H2 norm non-convergent. 3.3. H∞ performance The H2 norm is a characterization of the average gain of a system, whereas the H∞ norm provides a measure of the worst-case system gain. The H∞ norm is the resonant peak of the FRF curve of the SDOF structure H ∞ = sup |H ( jω )|. ω

(33)

The H∞ performances of the NSD, ID, and skyhook damper can be evaluated by the peak magnitude in the bode graph presented in Figs. 6–8, correspondingly. As an ideal isolation method, the skyhook damper reduces the peak response without changing the structural natural frequency (Fig. 8). The NSD and ID can reduce the peak response effectively; however, both reduce the natural frequency of the structure apparently (Figs. 6 and 7). The NSD reduces structural frequency more rapidly by comparison. In summary, the NSD, ID, and skyhook damper are all effective with regard to the H∞ performance in vibration isolations; whereas only the former two tend to lower the structural resonant frequency. 3.4. Dynamic force components When a damper is installed in the SDOF system, the mass is subjected to the damper force, the stiffness force of the original spring, and the inertia force, simultaneously. The bode graphs of every force component that connects the base and sprung mass are examined to investigate the work mechanisms of NSD and ID. Fig. 9 √ shows the bode graph of different force components in the NSD case (α = 25%, cnsd = 0.1 mk ), wherein the negative stiffness value is smaller than structural positive stiffness. The gap between the positive and negative stiffness forces are constant at all frequencies.

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√ Fig. 9. Bode graph of different force components in the SDOF system with an NSD (α = 25%, cnsd = 0.1 mk ).

Given a 180° difference in the phase angle between them, the negative stiffness cancels part of the positive stiffness. As a result, the total excitation force is reduced. Fig. 10 shows the bode √ graph of different force components in the SDOF system with an ID (β = 30%, cid = 0.1 mk ). More complicated mechanism is shown in the ID case. The inertance force also has a phase angle difference of 180° with the structural stiffness force regardless of the frequencies; however, these two force components vary with the frequency in different trends. At a low frequency, the stiffness force is larger than the inertance force. At a high frequency, however, the inertance force becomes significantly larger than the stiffness force; consequently, the inertance force is nearly in phase with the system displacement response (Fig. 10). Thus, the ID has an adverse effect in the conditions with high frequency base excitations. 4. The pole-zero plot analysis This section further analyzes and compares the performance of the NSD, ID, and skyhook damper in the frequency domain. Fig. 11 plots the system poles and zeros under the influence of various dampers, wherein the system poles and zeros are calculated according to Eqs. (20)–(22). The loci show how system poles and zeros change when the negative stiffness or inertance of the dampers increases and the damping coefficient remains constant. Fig. 12 demonstrates the corresponding changes in the undamped natural frequency (ω0 ) and

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√ Fig. 10. Bode graph of different force components in the SDOF system with an ID (β = 30%, cid = 0.1 mk ).

the damping ratio (δ).The system pole (λ), the damping ratio (δ) and the undamped natural frequency (ω0 ) are correlated as follows.  λ = −δω0 ± j · ω0 1 − δ 2 (34) The skyhook damper only increases the damping ratio δ, without affecting the natural frequency ω0 of the system. As shown by the blue line in Fig. 11(a), the system poles move along a circle with a radius of ω0 toward the real axis when the ξ sky increases. The NSD partially cancel the positive stiffness of the SDOF system; consequently, as the strength of negative stiffness increases, the poles move to the real axis, which results in the undamped natural frequency ω0 decreases and the damping ratio δ increases (Fig. 12). When the poles meet at the real axis, the system achieves the critical damping (i.e., the damping ratio equal to 100%). With the increase in negative stiffness, two poles will move along the real axis in two opposite directions. If knsd > k, one pole will be located on the right side of the plan, implying the system will be unstable. As shown in Fig. 11, the installation of ID drags the system pole toward the origin. Similar to the negative stiffness, increasing the inertance tends to decrease the undamped natural frequency (ω0 ) (Fig. 12(a)); meanwhile, the inertance decreases the system damping ratio (δ) when the damping coefficient is kept constant (Fig. 12(b)), which is different from the negative stiffness. The decreasing rates of the natural frequency ω0 and damping ratio δ become slower with a high inertance value. Consequently, the ID will not lead to the

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√ Fig. 11. Locus of system poles and zeros with an increase in α, β and ξ sky (cnsd = cid = 0.5 mk ).

instability of the system. If the inertance approaches infinity, the natural frequency (ω0 ) will approach zero.

5. Time domain analysis The responses in the time domains of SDOF systems with NSD (α = 20%), ID (β = 30%), and skyhook damper are analyzed in this section to verify the above findings. The natural frequency of the uncontrolled SDOF system is set as 1 Hz, and various base excitations are

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√ Fig. 12. Modal parameters of the SDOF system with vibration isolation (cnsd = cid = 0.5 mk ).

considered, including random excitation with limited bandwidth, step, impulse, and sine sweep inputs. 5.1. Random excitation The SDOF systems with skyhook damper, NSD, and ID are excited by limited-band whitenoise ground displacement. The absolute displacement of the sprung mass and the corresponding FFT graphs are shown in Fig. 13(a) and (b), respectively. The bandwidth of the random base excitation (i.e., ground displacement) is 0–50 Hz, and the base input is presented by the gray solid lines. The blue solid lines, red dashed lines and green dotted lines present the absolute responses of the sprung mass controlled by skyhook damper, NSD and ID, respectively. As shown in Fig. 13(a), the absolute displacement responses of the mass controlled by the NSD and skyhook damper are very close, and both are much smaller than the response controlled by the ID. The root mean square (RMS) of the absolute displacements are 0.130 mm, 0.133 mm and 0.210 mm for the skyhook damper, NSD and ID cases, respectively. As clearly observed in Fig. 13(b), the ID adversely affect the vibration isolation at high frequencies. Among all the three methods, the skyhook damper provides the best performance at high frequencies isolation. 5.2. Step input The absolute responses of SDOF systems with skyhook damper, NSD, and ID subject to a step input are shown in Fig. 14. The overshoots can be calculated with these system responses in the time domain. The performances of NSD, ID, and skyhook damper are close. The overshoots of the SDOF systems with the skyhook damper, NSD, and ID are equal to 44.4%, 47.3%, and 41.3%, respectively.

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√ Fig. 13. Response of the SDOF system under random excitation of limited bandwidth (cnsd = cid = csky = 0.5 mk ).

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Fig. 14. Response of the SDOF system under a step load.

Fig. 15. Response of the SDOF system under an impulse load.

5.3. Impulse input Fig. 15 shows the free-vibration response when the system is subjected to an impulse load. The first response peak in the ID case is the smallest, because the ID provides additional inertance to the system. NSD can provide the optimal damping performance. The first peak

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Fig. 16. Spectrum envelope of the SDOF system under sine sweep load.

of NSD is the largest among three cases; however, the response peak of NSD drops to the smallest in the following cycles due to a larger damping ratio. 5.4. Sine sweep input The spectrum envelope of SDOF systems with skyhook damper, NSD, and ID subjected to sine sweep input are shown in Fig. 16. The start and end frequency are 0 and 5 Hz, respectively, and the increasing rate is 0.01 Hz/s. At low frequencies, the performances of skyhook damper, NSD and ID are basically identical. For the frequency range above resonant frequencies of the systems, the responses of the systems controlled by skyhook damper and NSD decreases as the excitation frequency increases, and the responses of skyhook damper case decrease faster than the responses of NSD case. However, the responses of ID case do not decay with respect to excitation frequencies. 6. Discussion on deficiencies Some other deficiencies associated with the NSD and ID, together with some potential solutions, are discussed in this section. 6.1. Re-centering of NSD When the NSD partially cancels the structural stiffness, the overall stiffness of the whole system is reduced. Consequently, the displacement responses under static forces acting on the mass may be significantly amplified. A small increase in the payload (e.g., the passenger weight on a seat) may lead to a very large deflection or even instability.

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Fig. 17. Re-center the zero-displacement location of NSD.

Fig. 18. TID for the vibration isolation of the SDOF system.

This deficiency can be overcome by re-centering zero-displacement location of the NSD to the static equilibrium position under the static payload. Fig. 17 shows the re-centering concept of the vibration isolation with the NSD. The static load is completely carried by the original stiffness of the structure; therefore, the static deflection will not be adversely amplified. After the static load is applied, the negative stiffness is introduced or connected through a switch. Consequently, the zero-displacement location of the NSD can be re-centered to static equilibrium position every time. The negative stiffness will only respond to dynamic input, and the static load-carrying capacity of the isolation system will not be affected by negative stiffness. 6.2. Connection of TID In view that the inerter may amplify the system response at high frequencies, using the inerter to directly connect the mass and base was not recommended in the previous studies. Alternatively, the ID is typically connected to a spring in series (Fig. 18), which composes

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√ Fig. 19. Bode graph of the SDOF system with TID (β = 30%,cid = ctid = 0.1 mk ).

a TID with a resonant frequency. The optimal tuning rule of TID was developed by Lazar et al. [61]. Fig. 19 shows the comparison of the SDOF systems with TID and ID, wherein √ both the inertance ratios are β = 30%, and the damping coefficient cid = ctid = 0.1 mk , and the stiffness of TID is optimally tuned. Due to the resonant characteristics, the TID provide the large control force in a narrow bandwidth. Consequently, the peak response and the responses at high frequencies can be effectively lowered by the TID, given the TID is properly tuned. In addition to ID and TID, several other isolation systems with inerter devices were systematically compared in Ref. [60]. 7. Conclusion This study systematically investigates and compares the passive vibration isolation performance of NSD and ID with respect to the H2 and H∞ norm. Although both produces the force-displacement relationships with negative slopes, the NSD and ID exhibit different vibra-

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tion isolation performance. The NSD reduces the system stiffness and minimizes the dynamic excitation input, and thus it demonstrates the effective vibration isolation in a wide frequency range. However, the introduction of negative stiffness tends to amplify static deflection under a variable payload. This deficiency can be solved by the re-centering function that re-centers the zero-displacement location of the NSD to the static equilibrium position under a variable payload every time. The control force of the ID increases rapidly with the excitation frequencies. Consequently, the inertance forces of the ID become the major excitation source at high excitation frequencies, and thus amplify the system responses at high frequencies. To avoid this performance deficiency at high frequencies, the TID comprising an inerter connected in series with a spring is often adopted in vibration isolation. Although the presented comparative study is based on the analyses of an SDOF structure, the above conclusions will certainly shed light on the applications to more complicated structures. The detailed comparison of vibration isolation performance of two dampers in multi-degree-of-freedom structures will be investigated in the future study. Acknowledgment The second author is grateful for the financial supports from the Research Grants Council of Hong Kong through the GRF grant (Project no. PolyU 152222/14E) and from the Innovation and Technology Commission of the HKSAR Government to the Hong Kong Branch of National Rail Transit Electrification and Automation Engineering Technology Research Center (Project no. 1-BBYB). The first author is grateful for the financial support from the China University of Petroleum (East China) (Project no. Y18050010). The findings and opinions expressed in this paper are solely those of the authors and not necessarily the views of sponsors. References [1] H. Iemura, M.H. Pradono, Simple algorithm for semi-active seismic response control of cable-stayed bridges, Earthq. Eng. Struct. Dyn. 34 (4–5) (2005) 409–423. [2] H. Iemura, M.H. Pradono, Passive and semi-active seismic response control of a cable-stayed bridge, J. Struct. Control 9 (3) (2002) 189–204. [3] H. Iemura, A. Igarashi, M.H. Pradono, A. Kalantari, Negative stiffness friction damping for seismically isolated structures, Struct. Control Health 13 (2–3) (2006) 775–791. [4] J. Høgsberg, The role of negative stiffness in semi-active control of magneto-rheological dampers, Struct. Control Health 18 (3) (2011) 289–304. [5] F. Weber, C. Boston, Clipped viscous damping with negative stiffness for semi-active cable damping, Smart Mater. Struct. 20 (4) (2011) 045007. [6] F. Weber, H. Distl, Semi-active damping with negative stiffness for multi-mode cable vibration mitigation: approximate collocated control solution, Smart Mater. Struct. 24 (11) (2015) 115015. [7] C.M. Lee, V.N. Goverdovskiy, A.I. Temnikov, Design of springs with “negative” stiffness to improve vehicle driver vibration isolation, J. Sound Vib. 302 (4) (2007) 865–874. [8] D.T.R. Pasala, A.A. Sarlis, S. Nagarajaiah, A.M. Reinhorn, M.C. Constantinou, D. Taylor, Adaptive negative stiffness: new structural modification approach for seismic protection, J. Struct. Eng. 139 (7) (2013) 1112–1123. [9] H. Iemura, M.H. Pradono, Advances in the development of pseudo-negative-stiffness dampers for seismic response control, Struct. Control Health 16 (7–8) (2009) 784–799. [10] X. Shi, S. Zhu, Magnetic negative stiffness dampers, Smart Mater. Struct. 24 (7) (2015) 072002. [11] X. Shi, S. Zhu, Simulation and optimization of magnetic negative stiffness dampers, Sens. Actuat. A Phys. 259 (2017) 14–33.

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