Robustness of a tuned viscous mass damper (TVMD) controlled system

Abstract

A tuned viscous mass damper (TVMD) is a novel type of vibration absorber which exhibits outstanding vibration control performance. In this study, an experiment was conducted to investigate the robustness of a TVMD for the vibration control of a typical single degree of freedom (SDOF) structural system. To that end, a new eddy-current TVMD (EC-TVMD) was developed as a representative TVMD device. The TVMD-controlled SDOF system was investigated using a series of shaking table tests. The influence of variations in stiffness of the primary structure and TVMD damping (including damping amplitude and damping nonlinearity) was assessed. The variation in TVMD damping exerted a smaller influence on control performance than the variation in primary structure stiffness. The robustness of TVMD control was enhanced by increasing the inertance-to-mass ratio. Parametric analyses using a numerical model further confirmed the experimental observations, and indicated that a TVMD exhibits improved robustness compared with a conventional tuned mass damper (TMD) which is sensitive to the detuning effect. The vibration control mechanism and robustness characteristics of the TVMD were further revealed by a Kelvin-Voigt model. Finally, the influence of damping nonlinearity was verified by the nonlinear time history analysis of a finite element model of the test structure. The results indicate that damping nonlinearity has limited influence on the control of a TVMD with nonlinear damping as long as this TVMD has the same peak displacement amplification ratio as the optimal linear design.

Keywords

tuned viscous mass damper eddy current damping robustness of vibration control vibration absorber damping nonlinearity

Introduction

Tuned mass dampers (TMD) are often used to suppress the structural response under wind- or earthquake-induced excitations (Kaynia et al., 1981; McNamara, 1977). A conventional TMD consists of a mass element, a spring element, and a damping element. A tuned viscous mass damper (TVMD) is another type of vibration absorber, which incorporates an inerter element instead of a physical mass element. The concept of an inerter was first proposed by Smith (2002) based on an analogy between electrical and mechanical networks. Unlike physical mass, the inerter is a two-terminal element which can generate an inertial force proportional to the relative acceleration between its terminals instead of the absolute acceleration. Therefore, the TVMD is a relative motion-driven device, which is different from the TMD that is driven by absolute acceleration. The inerter used in a TVMD is realized by a ball screw system (Saito et al., 2008) that can convert the relative translational motion between the terminals into the rotation of the flywheel. In other studies, inerters can be realized using rack and pinion systems (Saitoh, 2012), fluid motion (Wang et al., 2011), or electronic circuits with capacitors (Gonzalez-Buelga et al., 2015). Owing to the motion transformation and amplification mechanism of the ball screw system, the inerter in a TVMD can generate a large inertial force (or inertial mass) with a small physical mass. A TVMD that achieves an inertial mass of 5400 tons using a physical mass of 1.1 tons has been manufactured and tested by Watanabe et al. (2012). Therefore, the inertance-to-mass ratio of a TVMD (i.e., the ratio of the TVMD’s inertance to the mass of the primary structure) can be much larger than the mass ratio of a TMD which is generally smaller than 0.1. This advantage is one of the reasons for the boost in research on TVMDs and other inerter-based vibration control devices, including tuned inerter dampers (TID) (Lazar et al., 2014; Shen et al., 2019), tuned mass damper inerters (TMDI) (Marian and Giaralis, 2014), and tuned inertial mass electromagnetic damper (TIMED) (Shen et al., 2022).

Several methods have been proposed for the optimal design of a TVMD. Saito et al. (2008) and Ikago et al. (2012) proposed closed-form design formulas for the tuning design of a single degree of freedom (SDOF) system based on fixed point theory which was proposed by Den Hartog for the design of a TMD (Hartog, 1985). The formulas were proposed to minimize the H_∞ norm of the transfer function from ground displacement (Saito et al., 2008) or acceleration excitation (Ikago et al., 2012) to structural displacement response. Subsequently, Ikago et al. (2010) proposed a numerical optimal design method for the tuning design of a multi degree of freedom (MDOF) system based on sequential quadratic programming (SQP). Recently, the authors proposed new design formulas for TVMDs based on H₂ optimization of the structural response (Cheng et al., 2022). Previous studies have also verified the effectiveness of TVMDs in controlling the seismic vibrations of multi-story buildings (Ikago et al., 2013; Ji et al., 2021) and base isolation systems (Nakaminami et al., 2012). Full-scale TVMDs have been utilized in real-world building structures (Ikago et al., 2013).

It is recognized that a TMD loses its control effect when the natural frequency of the primary structure shifts from its designed value. Such a detuning effect is one of the most challenging problems for the robustness of TMD control (Rana and Soong, 1998). Because the TVMD appears to share a design concept similar to a TMD, the robustness of TVMD control can also be called into question. An experimental test by Ikago et al. revealed that a TVMD is robust when its spring stiffness is higher and its damping coefficient is lower than the design values (Ikago et al., 2017). This test considered the variation in spring stiffness and damping coefficient of the TVMD itself. In fact, during the application of TVMDs in real buildings, the stiffness of a TVMD is designed according to the measured natural vibration frequency of the building structure, and the spring in a TVMD is easy to manufacture with relatively accurate stiffness. However, the stiffness of a building structure may vary from its initial value. For example, the installation of infilled non-structural walls leads to an increase in building stiffness, while seismic damage results in a significant decrease in structural stiffness. As such, the variation in primary structure stiffness should be of greater concern than the spring stiffness of TVMD which remains almost unchanged throughout its service period. In addition, the damping coefficient of a TVMD may be different from the design value due to the damping nonlinearity and friction forces in the rotational viscous damper. Therefore, the objective of this study was to investigate the robustness of TVMD control against variations in primary structure stiffness and TVMD damping. To this end, an eddy-current TVMD (EC-TVMD) model was designed and a series of shaking table tests were conducted on EC-TVMD-controlled SDOF (TVMD-SDOF) systems. Two test variables were considered: the variation in stiffness of the primary structure and the variation in TVMD damping, including the damping coefficient value and damping nonlinearity. The primary reason for using eddy current damping (ECD), rather than viscous fluid damping, is that the damping coefficient can easily be adjusted by adding or removing magnet pieces.

This paper is structured as follows. The second section describes the design of the EC-TVMD and the shaking table test setup. The third section presents the experimental results regarding the dynamic properties of the SDOF primary structure and the EC-TVMD. The fourth section presents the robustness characteristics of TVMD control based on the results of the experiment and parametric analyses. The final section reveals the mechanism of TVMD control based on an equivalent Kelvin-Voigt model. This section also discusses the influence of damping nonlinearity based on the results of nonlinear time history analyses.

TVMD device and experimental scheme

Eddy-current TVMD

Eddy current damping has been found to be an effective energy dissipating mechanism (Gay and Ehsani, 2006), and ECD devices have been used in structures to suppress human- or wind-induced vibration (Chen et al., 2017; Lu et al., 2017; Wang et al., 2012). Figure 1(a) presents a photograph of the small-scale EC-TVMD used in the tests. The spring element comprised eight compression springs and five rods (four secondary rods and one main rod). The main rod was used to transfer axial force and torque from the viscous mass damper (hereinafter referred as the viscous mass damper (VMD) element) that includes the inerter and damping element to the spring element. The four secondary rods were used to transfer load and prevent the springs from buckling when compressed. As shown in Figure 1(a), because of the constraint by the four secondary rods, the movement of the plates at location A and C were identical, which is the displacement of TVMD (u). The linear bearing flanges at location B was connected to the main rob, and thus their movement is identical to the displacement of VMD (u_d). During the vibration, when four compression springs on one side were pressed, the other four springs were free. Hence, the total stiffness of the spring element was four times the stiffness of a single spring. The original length of each compression spring was 40 mm.

Figure 1.

Eddy-current TVMD: (a) Photographs of the eddy-current TVMD, (b) Section view of the VMD element.

The VMD element was formed by a ball screw system. The inertial mass of the inerter was increased by attaching additional flywheels to the nut of the ball screw. The diameter of the screw shaft was 8 mm and the lead length was 12 mm. The inertial mass of inerter (m_r) was determined by the moment of inertia of the rotating parts (I) and the lead length (L_d), as formulated in equation (1) (Ikago et al., 2012). Two EC-TVMDs were used in this study: Specimen-A and Specimen-B. The mass, moment of inertia, and inertial mass of rotating parts of the two specimens are listed in Table 1.

m_{r} = {(\frac{2 π}{L_{d}})}^{2} I

(1)

Table 1.

Inertial mass of rotating parts in TVMD specimens.

Items		Mass (g)	Moment of inertia (g·mm²)	Inertial mass (g)
Outer tube (with end panels)		49.0	9635.6	2639.0
Ball bearings		49.1	3472.0	950.9
Nut of ball screw		20.6	3182.2	871.5
Additional flywheel (Specimen-B)		40.5	11,318.0	3099.7
Total	Specimen-A	118.7	16,289.8	4461.4
Total	Specimen-B	159.2	27,607.8	7561.2

Eddy currents are typically generated in conductors in a time-varying magnetic field. They can also be caused by the motion of a conductor in a constant magnetic field generated by permanent magnets (Wang et al., 2012). The repulsive force created by the eddy currents is generally proportional to the velocity of the conductor which makes the force act like a damping force. The ECD element used in EC-TVMD is formed mainly by two tubes and permanent arc magnets. As shown in Figure 1(a), an aluminum alloy outer tube acted as the conductor, and the magnetic field was generated by the permanent arc magnets attached to the iron inner tube. The outer tube was connected with the nut of the ball screw (as shown in Figure 1(b)), and therefore rotated when the EC-TVMD had the relative translational motion between its two terminals. The material chosen for the outer tube was aluminum alloy because its low resistance generated high current and repulsive forces. The iron inner tube acted like a back iron which increased the intensity of the magnetic field outside the inner tube, and further amplified the impulsive forces. In addition, this arrangement also suppressed the intensity of the magnetic field inside the inner tube, thus preventing the screw shaft from being attracted by magnets. The magnets were radial-magnetized, and the magnetic poles of adjacent magnets were in the opposite direction. Two sizes of arc magnets were used with projection areas of the arc segments of 20.5 mm × 12.5 mm and 7.2 mm × 5 mm, respectively. The thickness of the magnets was 3 mm, and the air gap was 2 mm thick. The damping coefficient was easily adjusted by changing the number of attached magnets.

The parameters of TVMDs in these two test specimens were optimally designed based on the fix point theory. The inertance-to-mass ratios of Specimen-A and Specimen-B were 0.13 and 0.22, respectively. The dynamic parameters of the TVMD were designed using Ikago et al.’s closed-form design formulas (equations (2) and (3)) (Ikago et al., 2012). The designed stiffness and damping coefficient of each specimen are listed in Figure 2. The mechanical model of the TVMD-SDOF system is depicted in Figure 2, where the symbols are defined in the Notation in the appendix.

β = \frac{1}{\sqrt{1 - μ}}

(2)

ξ = \frac{1}{2} \sqrt{\frac{3 μ}{2 - μ}}

(3)

Figure 2.

A TVMD-SDOF system.

Test setup

A TVMD-SDOF test platform was established to investigate the dynamic behavior and robustness of the TVMD-controlled structure. A photograph of this test platform is presented in Figure 3. The primary SDOF structural system was mimicked by steel blocks representing the mass, and tension springs representing the stiffness. The steel blocks were supported by linear guides and connected to the shaking table by tension springs from both sides. The lateral stiffness of the SDOF system was adjusted by altering the number of tension springs. Although the linear guides had a low friction coefficient, they provided a non-negligible friction force to the steel blocks.

Figure 3.

TVMD-SDOF shaking table test platform.

The EC-TVMD was used to connect the steel blocks and shaking table. Three laser displacement sensors (LDSs) were used to measure the displacement. LDS-1 measured the displacement of the VMD element (u_d). LDS-2 measured the displacement of the primary structure (i.e., the displacement of the TVMD u) relative to the shaking table. LDS-3 measured the displacement of the shaking table (u₀). The displacement of the spring element was calculated as u_s = u - u_d. The axial force of the EC-TVMD was measured by a load cell.

Loading protocols

The experimental tests consisted of two phases. The first phase involved examining the dynamic properties of the EC-TVMD. In this phase, the steel block (i.e., the mass of primary structure) was securely fixed to the actual lab fix ground using a magnetic base, and thus the table movement was the displacement demand applied to the EC-TVMD. A series of sinusoidal motions with a variety of exciting frequencies (from 0.1 Hz to 2.2 Hz or 2.5 Hz, with an increment of 0.1 Hz) were used as the shaking table input. The amplitude of the table displacement was 10 mm for all loading cases. The second phase of the tests involved examining the dynamic properties and responses of the TVMD-SDOF system. Similarly, a series of sinusoidal motions with frequencies ranging from 0.5 Hz to 2.2 or 2.8 Hz (with an increment of 0.1 Hz) were used as the table input. The amplitude of the table displacement was 4 mm for tests of Specimen-B, but was reduced to 3 mm for Specimen-A tests to ensure the EC-TVMD displacement did not exceed its stroke capacity. In addition, free vibration tests were conducted to assess the dynamic properties of the primary SDOF structure.

In each phase of the tests, the stiffness of primary structure and damping of the TVMD were the test variables used to investigate the robustness of the TVMD-controlled system. Table 2 summarizes the parameters of the primary structure and EC-TVMD in the tests. With the combination of test phases and variation of parameters, a total of approximately 700 loading cases were included in the tests.

Table 2.

Parameters of the primary structure and the EC-TVMD.

	Case No.	TVMD state	Structure state	Structure parameters				TVMD parameters					Parameter variation
	Case No.	TVMD state	Structure state	m (kg)	k (N/m)	ζ (%)	f₀ (Hz)	m_r (kg)	k_b (N/m)	c_d (N·s/m)	ξ (%)	f_d (Hz)	Parameter variation
Specimen-A (μ = 0.13)	A1-OPT	Opt.	Init.	34.1	3404	1.5	1.59	4.46	520.2	23.5	24.4	1.72	0%
	A2-K-	Opt.	K-	34.1	2633	1.7	1.40	4.46	520.2	23.5	24.4	1.72	−23%
	A3-K--	Opt.	K--	34.1	1831	2.5	1.17	4.46	520.2	23.5	24.4	1.72	−46%
	A4-K+	Opt.	K+	34.1	4143	1.2	1.75	4.46	520.2	23.5	24.4	1.72	22%
	A5-K++	Opt.	K++	34.1	4979	1.1	1.92	4.46	520.2	23.5	24.4	1.72	46%
	A6-C-	C-	Init.	34.1	3404	1.5	1.59	4.46	520.2	20.9	30.1	1.72	−11%
	A7-C+	C+	Init.	34.1	3404	1.5	1.59	4.46	520.2	29.0	30.1	1.72	23%
	A8-C++	C++	Init.	34.1	3404	1.5	1.59	4.46	520.2	35.0	36.3	1.72	49%
	A9-C-F	C-F	Init.	34.1	3404	1.5	1.59	4.46	520.2	24.5	25.4	1.72	4%

Specimen-B (μ = 0.22)	B1-OPT	Opt.	Init.	34.1	3404	1.5	1.59	7.56	1079	56.0	31.0	1.90	0%
	B2-K-	Opt.	K-	34.1	2633	1.7	1.40	7.56	1079	56.0	31.0	1.90	−23%
	B3-K--	Opt.	K--	34.1	1831	2.5	1.17	7.56	1079	56.0	31.0	1.90	−46%
	B4-K+	Opt.	K+	34.1	4143	1.2	1.75	7.56	1079	56.0	31.0	1.90	22%
	B5-K++	Opt.	K++	34.1	4979	1.1	1.92	7.56	1079	56.0	31.0	1.90	46%
	B6-C-	C-	Init.	34.1	3404	1.5	1.59	7.56	1079	32.1	19.6	1.90	−43%
	B7-C--	C--	Init.	34.1	3404	1.5	1.59	7.56	1079	24.0	14.7	1.90	−57%
	B8-C+	C+	Init.	34.1	3404	1.5	1.59	7.56	1079	82.9	50.8	1.90	48%
	B9-C++	C++	Init.	34.1	3404	1.5	1.59	7.56	1079	95.0	58.1	1.90	70%
	B10-C-F	C-F	Init.	34.1	3404	1.5	1.59	7.56	1079	55.8	30.9	1.90	0%

Note: Opt indicates that the parameters of TVMD were optimally designed; Init (Initial) means the parameters of structure were the design values; The “-” or “+” represents the decrease or increase in the values of a parameter of K (the stiffness of primary structure) and C (the damping coefficient of TVMD). The parameter variation represents the ratio of the variation in the structural stiffness/TVMD damping to its initial value. The TVMD state with C-F corresponds to the configuration where the eddy current damping was replaced by Coulomb damping. A9-C-F and B10-C-F were conducted to study the influence of damping nonlinearity.

Dynamic properties of the primary structure and eddy-current TVMD

Dynamic properties of the SDOF primary structure

The mass of the steel blocks was 34.1 kg. The stiffness and damping ratio of the primary structure were obtained from the free vibration tests. Figure 4 presents the free vibration response of the primary structure. The damped vibration period was determined as the average time interval between peak-to-peak responses. The structural stiffness was calculated by the measured mass and natural vibration frequency. The damping of the primary structure was caused by the viscous resistance of the grease and dry friction in the linear guide. The former is characterized as a linear viscous damping model where the free vibration response decays exponentially, while the latter is characterized as a Coulomb damping model (Chopra, 1995) in which the free vibration response decays linearly. Figure 4 indicates that when the displacement response amplitude was larger than 5 mm, the response decay was dominated by linear viscous damping. When the displacement was less than 5 mm, the Coulomb damping appeared to dominate the damping behavior that exhibited linear response decay. During the tests of harmonic excitation, the displacement responses had a magnitude ranging from 5 to 15 mm. Therefore, using the logarithmic decrement method (Chopra, 1995), the damping ratio was estimated from the envelope curve of the free vibration decay data. The estimated damping ratio ranged from 1.1% to 2.5% for all test cases, as listed in Table 2.

Figure 4.

Free vibration results for the SDOF system.

Properties and responses of the eddy-current TVMD

The mechanical model of the EC-TVMD has been depicted in Figure 2. When subjected to harmonic excitation u = Usin(ωt), the steady-state response of VMD displacement was denoted as u_d = U_dsin(ωt+ϕ). The frequency response function from u to u_d is given by equation (4) (Ji et al., 2021). This equation can be reorganized as equation (5) with the non-dimensional parameters defined in the Notation.

\frac{U_{d}}{U} = \frac{k_{b}}{\sqrt{{(k_{b} - m_{r} ω^{2})}^{2} + {(c_{d} ω)}^{2}}}

(4)

\frac{U_{d}}{U} = \frac{1}{\sqrt{{(1 - γ^{2})}^{2} + λ^{2} γ^{2}}}

(5)

For EC-TVMDs in the tests, the inertial mass was calculated using equation (1). The spring stiffness was obtained by linear fitting of force-displacement responses of the spring element measured in the tests. Figure 5 presents the frequency response function of U_d/U (this ratio is referred to as the displacement amplification ratio hereinafter) for the EC-TVMD with optimally designed parameters. The circle plots represent the steady-state response amplifications obtained by the harmonic vibration test data. The solid lines are the fitted curves from the test data (i.e., the circle plots) using equation (4). The damping coefficient obtained by the curve fitting is listed in Table 2. The dash lines in the figure represent the theoretical solution of the transfer function of U_d/U with optimally designed parameters. The test data correlated well with the optimally designed results, indicating that the damping in EC-TVMD was close to its designed value.

Figure 5.

Transfer function of EC-TVMDs: (a) Specimen-A, (b) Specimen-B.

The hysteretic responses of the EC-TVMD under sinusoid excitation are plotted in Figure 6. Each cycle of the plots represents one cycle of a steady-state hysteretic response of harmonic vibration under a given exciting frequency (Specimen-A: 0.1 Hz, 0.2 Hz, …, 2.2 Hz; Specimen-B: 0.1 Hz, 0.2 Hz, …, 2.5 Hz). The thick dash curves represent the steady-state response at an exciting frequency of 1.6 Hz which is close to the tuning frequency (the natural vibration frequency of the SDOF primary system). Figure 6(a) and (d) depict the linear response of the spring element, despite the slight energy dissipation caused by the friction of linear bearings. Figure 6(b) and (e) display the hysteretic response of the VMD element, indicating the obvious energy dissipation contributed by the ECD. The pseudo-negative stiffness of VMD was induced by the inerter with a value of -ω²m_r, and its magnitude increased in line with exciting frequency. Figure 6(c) and (f) illustrate the force versus displacement relationship of the EC-TVMD. The equivalent stiffness and energy dissipation of the EC-TVMD varied with exciting frequency. At an exciting frequency of 1.6 Hz (i.e., tuning frequency), the EC-TVMD displayed significantly large energy dissipation while the equivalent stiffness was close to zero.

Figure 6.

Hysteretic responses of the EC-TVMDs: (a) Responses of spring (A1-OPT), (b) Responses of VMD (A1-OPT), (c) Responses of TVMD (A1-OPT), (d) Responses of spring (B1-OPT), (e) Responses of VMD (B1-OPT), (f) Responses of TVMD (B1-OPT).

Figure 7(a) plots the relationship of steady-state force versus velocity response of VMDs at u_d = 0. Each point in Figure 7(a) represents the result obtained from a steady-state hysteresis of a certain frequency. The lines for $c_{d} {\dot{u}}_{d}$ (c_d was obtained from the transfer function fitting) are also plotted in this figure. In the sinusoid response, the acceleration and corresponding inertial force was zero at u_d = 0; thus, the VMD force was only contributed by the damping force. This indicates that the damping of VMD was dominated by linear viscous damping at the testing velocity region, although dry friction (corresponding to the Coulomb damping) led to an increased equivalent damping coefficient at small displacement responses.

Figure 7.

Behavior of damping and inerter: (a) Damping behavior, (b) Inerter behavior.

Figure 7(b) plots the relationship of steady-state force versus the acceleration response of VMDs at ${\dot{u}}_{d} = 0$ where the acceleration was calculated as $- ω^{2} u_{d}$ . The lines for $m_{r} {\ddot{u}}_{d}$ (m_r was obtained from equation (1)) are also plotted in this figure. The plots depict the high linearity of the force-acceleration relation, and confirm the accuracy of the calculated inertial mass.

Robustness characteristics of TVMD control

Influence of structural stiffness on TVMD-SDOF responses

During the application of TVMDs in real buildings, their parameters (e.g. the stiffness and inertial mass) are typically designed using the measured natural frequency of the building structure. The changes in stiffness of the primary structure induce changes in the natural vibration frequency which might further influence the control performance of TVMDs. To study the influence of the variation in structural stiffness, the stiffnesses of primary structures in systems A2-K- to A5-K++ and B2-K- to B5- were designed to shift from their original values. The range in stiffness variation was ±46%, and was determined by the previous test results. On the one hand, a full-scale shaking table test indicated that the non-structural walls, including drywall partitions and autoclaved lightweight concrete panels, contribute approximately 10% of the inter-story stiffness of steel frame buildings (Ryan et al., 2016). Other experimental tests revealed that the infilled walls can increase the stiffness of the steel frame by more than 20% (Liu and Li, 2005). On the other hand, when the structure sustains cracking and damage during an earthquake event, the structural stiffness decreases. A full-scale substructure test for a high-rise steel structure in E-Defense revealed a stiffness degradation of 22% when subjected to a maximum considered earthquake (MCE) (Ji et al., 2011). Another full-scale test for a four-story RC shear wall building structure indicated a stiffness degradation of 55% when subjected to severe seismic motion shaking (Nakashima et al., 2010). The system identification of a nine-story RC building revealed that the cracks of structural components in a moderate earthquake event reduced the building stiffness by approximately 53% (Chopra, 1995). Such tests and observations suggest that a stiffness decrease of 50% is quite possible for an RC structural building in an earthquake event.

Figure 8 presents the transfer functions from the base acceleration to the structural displacement for the TVMD-SDOF specimens with varied structural stiffness. The markers represent the transfer functions obtained from the sinusoid-excitation tests. The curves represent the theoretical transfer function which was calculated using equation (6) based on the parameters defined in the Notation.

| H | = \sqrt{\frac{{(- (1 + μ β^{2}) {γ^{*}}^{2} + β^{2} - η β {γ^{*}}^{2} λ)}^{2} + {((1 + μ β^{2}) β λ γ^{*} + (β^{2} - {γ^{*}}^{2}) η γ^{*})}^{2}}{{((1 - {γ^{*}}^{2}) (β^{2} - {γ^{*}}^{2}) - μ β^{2} {γ^{*}}^{2} - η β {γ^{*}}^{2} λ)}^{2} + {((1 + μ β^{2} - {γ^{*}}^{2}) β γ^{*} λ + (β^{2} - {γ^{*}}^{2}) η γ^{*})}^{2}}}

(6)

Figure 8.

Transfer functions of TVMD-SDOF: Influence of variation in primary structure stiffness: (a) Specimen-A, (b) Specimen-B.

Figure 8 reveals that the theoretical results tracked the test results well, except for cases of low exciting frequencies. At a low-frequency excitation, the velocity of TVMD was small, and the equivalent damping coefficient of VMD was larger than the values in Figure 2 (see Figure 7(a)). Consequently, the test results were lower than the theoretical ones. The transfer function of system A1-OPT had two identical peak values as expected for the optimal tuning design results, while that of system B1-OPT did not reflect such perfect tuning characteristics. The reason is that the actual stiffness of the supporting spring of TVMD in Specimen-B was approximately 11% larger than the optimally designed value. However, such errors in the manufacture of TVMD do not impact the following discussion and conclusions regarding the robustness of TVMD control.

When the stiffness of primary structure is changed, the structural response of a TVMD-SDOF system will be affected by the combined effect of the changes in the dynamic characteristic of the SDOF system itself and the detuning effect of the TVMD. To decouple the influence of the detuning effect from the combined effect, a response mitigation ratio h is defined as:

h = (1 - \frac{{| | H_{TVMD - SDOF} | |}_{\infty}}{{| | H_{SDOF} | |}_{\infty}}) \times 100 %

(7)

where H_TVMD-SDOF is the transfer function of the TVMD-controlled SDOF system, and H_SDOF is the transfer function of the corresponding uncontrolled SDOF system. The decrease of h indicates a degradation of vibration control. It is important to note that because the steady-state response of the uncontrolled SDOF system exceeded the stroke limit of the experimental setup, the value of H_SDOF was obtained from the theoretical transfer function rather than from the test.

The test results suggest a possible detuning effect of the TVMD control caused by the variation in primary structure stiffness. As shown in Table 3, the structure response increased when the structural stiffness shifted from its original value. The vibration control of TVMD in Specimen-B (which had a larger inertance-to-mass ratio) was more robust than Specimen-A. For Specimen-A, the response mitigation ratio h decreased by 12.8% when the stiffness decreased by 46% (i.e., from A1-OPT to A3-K--). For Specimen-B, the 46% decrease in primary structure stiffness (i.e., from B1-OPT to B3-K--) only resulted in a 3.9% decrease of h.

Table 3.

Quantitative comparisons of the influence of structural stiffness variation.

Structural stiffness (N/m)	Specimen-A (μ = 0.13)			Specimen-B (μ = 0.22)
Structural stiffness (N/m)	Case No.	Max(\|H\|) (s²)	h (%)	Case No.	Max(\|H\|) (s²)	h (%)
3404 (0%)	A1-OPT	0.0317	90.5	B1-OPT	0.0241	92.8
2633 (↓23%)	A2-K-	0.0479	85.6	B2-K-	0.0292	91.2
1831 (↓46%)	A3-K--	0.0744	77.7	B3-K--	0.0371	88.9
4143 (↑22%)	A4-K+	0.0436	86.9	B4-K+	0.0235	92.9
4979 (↑46%)	A5-K++	0.0537	83.9	B5-K++	0.0253	92.4

Influence of TVMD damping on TVMD-SDOF responses

The damping coefficient of a TVMD might deviate from its design value due to friction action and manufacture errors. In addition, TVMDs often use the viscous fluid to provide viscous damping, while the working temperature changes the damping coefficient (Symans and Constantinou, 1998). In this study, two aspects of variations in TVMD damping were considered, namely damping magnitude and damping nonlinearity.

Specimens A6-C- to A8-C++ and B6-C- to B9-C++ were used to investigate the influence of damping amplitude variation. The damping magnitude was adjusted by changing the number of arc magnets. In this study, the variation in the damping coefficient ranged from −57% to 70%. Because specimen A1-OPT had a small damping coefficient, only one level of damping decrease was considered.

Figure 9 presents the transfer function of TVMD-SDOF systems obtained in harmonic excitation tests. Quantitative comparisons of the influence of damping variation are detailed in Table 4. The results indicate that the variation in the damping coefficient of TVMD had a slight influence on the response mitigation ratio value of a TVMD-SDOF system.

Figure 9.

Transfer functions of TVMD-SDOF: Influence of the variation in the TVMD damping coefficient: (a) Specimen-A, (b) Specimen-B.

Table 4.

Quantitative comparisons of the influence of TVMD damping variation.

Specimen-A (μ = 0.13)				Specimen-B (μ = 0.22)
Case No.	c_d (N·s/m)	Max(\|H\|) (s²)	h	Case No.	c_d (N·s/m)	Max(\|H\|) (s²)	h
A1-OPT	23.5 (0%)	0.0317	90.5%	B1-OPT	56.0 (0%)	0.0241	92.8%
A6-C-	20.9 (↓11%)	0.0314	90.6%	B6-C-	32.1 (↓43%)	0.0316	90.5%
—	—	—	—	B7-C--	24.0 (↓57%)	0.0508	84.7%
A7-C+	29.0 (↑23%)	0.0336	89.9%	B8-C+	82.9 (↑48%)	0.0255	92.3%
A8-C++	35.0 (↑36%)	0.0383	88.5%	B9-C++	95.0 (↑70%)	0.0279	91.6%
A9-C-F	24.5 (↑4%)	0.0324	90.3%	B10-C-F	55.8 (↓0%)	0.0228	93.2%

The damping nonlinearity was mimicked by increase of friction action that is featured by Coulomb damping, while removal of the ECD. Application of the pressure on the ball bearing increased the normal contact force and associated friction force between the ball bearing and inner/outer tube. The test systems A9-C-F and B10-C-F were used to investigate the influence of damping nonlinearity. By adjusting the magnitude of pressure on the ball bearing, the transfer functions U_d/U of the TVMDs with nonlinear damping had peak values identical to those of the ED-TVMDs that were used in A1-OPT and B1-OPT. Figure 10 depicts the transfer function U_d/U of the TVMDs in A9-C-F and B10-C-F systems, compared with that of the EC-TVMDs in A1-OPT and B1-OPT. It is indicated that the damping nonlinearity clearly affected the U_d/U function of Specimen-B (i.e., TVMD in system B10-C-F) but had a limited influence on that of Specimen-A (i.e., TVMD in system A9-C-F).

Figure 10.

Transfer functions of TVMDs with nonlinear damping: (a) Specimen-A, (b) Specimen-B.

The damping behavior of A9-C-F and B10-C-F was caused by the dry friction in the ball bearings and ball screw, and the viscous resistance of the grease. The latter can be approximated as linear viscous damping. Therefore, the damping characteristic of TVMDs in these configurations was captured by a combined model of the Coulomb damping and linear viscous damping, as shown in Figure 11(a). In this model, the Coulomb damping is rendered equivalent to linear damping using equation (8) through (10) (Constantinou et al., 1998) where α is the damping coefficient of the Coulomb damping, u₀ is the displacement amplitude of VMD, c equals the amplitude of friction force, and Γ represents the gamma function. The detailed steps for calculating the equivalent damping coefficient of nonlinear damping are listed in Figure 11(b). The dot lines in Figure 10 represent the frequency response obtained by the modified model which tracked the test results well. The solid green lines in Figure 10 represent the ratio of linear viscous damping coefficient c_l to the total apparent damping coefficient c^*. It is indicated that the linear viscous damping accounted for a larger portion in A9-C-F, while Coulomb damping accounted for a larger portion in B10-C-F.

c_{f} = \frac{2 m_{r} E_{d}}{4 π E_{s 0}} = \frac{m_{r} E_{d}}{π k_{b} u_{0}^{2}}

(8)

E_{d} = π β_{α} c ω_{n}^{α} u_{0}^{1 + α}

(9)

β_{α} = \frac{2^{2 + α} Γ^{2} (1 + α / 2)}{π Γ (2 + α)}

(10)

Figure 11.

Modification of the Coulomb damping TVMD model: (a) Modified VMD model, (b) Steps to calculate transfer functions of the modified model, (c) 0.1 Hz test for Specimen-B.

Figure 12 compares the transfer functions of the SDOF system controlled by TVMDs with ECD (dominated by linear damping) and Coulomb damping (exhibited nonlinearity). The nonlinearity had no significant impact on the vibration control of TVMD. The difference between the maximum values of transfer functions of the A9-C-F and A1-OPT or B10-C-F and B1-OPT was within 5%. The damping nonlinearity had a small influence on the vibration control in this test.

Figure 12.

Influence of damping nonlinearity: (a) Specimen-A, (b) Specimen-B.

Comparison of the robustness between TMD and TVMD control

This subsection compares the robustness of the TVMD control with that of TMD control based on a parametric analysis of their transfer functions. Figure 13 depicts the configurations of the TVMD-SDOF system and TMD-SDOF system. For the TMD-SDOF system, the parameters of TMD can be designed using the equations of fixed point theory (Hartog, 1985). Given that the mass ratio (μ_m, the ratio of TMD mass to structural mass) of a TMD is generally small, μ_m was taken as 0.1. Because the TVMD can achieve a large inertance-to-mass ratio using the amplification mechanism, the inertance-to-mass ratios μ = 0.2 and 0.3 were also considered besides μ = 0.1. In this analysis, the inherent damping ratio of the primary structure was set to be 2%.

Figure 13.

Configuration of the SDOF system controlled by TMD and TVMD.

The surface plots in Figures 14 and 15 illustrate the transfer functions of TMD- or TVMD-SDOF systems with variations in the stiffness of the primary structure and damping coefficient of TMD/TVMD, respectively. In these figures, κ represents the ratio of the actual stiffness of the primary structure to the initial stiffness, and τ represents the ratio of the actual damping coefficient of TVMD/TMD to its optimal designed value. The black solid lines on the surface are the transfer functions for the TVMD-SDOF system with the original stiffness (i.e., κ = 1) or optimally designed damping (i.e., τ = 1). The 2D plots depicts the response mitigation ratios h of the systems with different parameter values compared to that with optimally designed parameter values.

Figure 14.

Parametric analyses of the influence of structural stiffness: (a) TMD-SDOF system (μ_m = 0.1), (b) TVMD-SDOF system (μ = 0.1), (c) TVMD-SDOF system (μ = 0.2), (d) TVMD-SDOF system (μ = 0.3).

Figure 15.

Parametric analyses of the influence of the damping coefficient: (a) TMD-SDOF system (μ_m = 0.1), (b) TVMD-SDOF system (μ = 0.1), (c) TVMD-SDOF system (μ = 0.2), (d) TVMD-SDOF system (μ = 0.3).

As illustrated in Figure 14, the control of a TVMD was more robust than that of a TMD when the primary stiffness decreased, which is beneficial to seismic response control. The detuning effect caused by stiffness degradation on the TMD-controlled structural system was approximately three times that for the TVMD-controlled system. The detuning effect became less severe as the inertance-to-mass ratio increased. When the inertance-to-mass ratio was larger than 0.2, the response amplification caused by the detuning effect was smaller than 10% with a stiffness ratio κ ranging from 0.4 to 1.5. The influence caused by the variation of TMD or TVMD damping (shown in Figure 15) was less significant. The response mitigation ratio only decreased by up to 10% when the TVMD damping coefficient changed by 60%.

The difference in the influence of stiffness degradation on the control performance of TVMD and TMD was caused by the mobilization mechanism. TMD is mobilized by the absolute acceleration of the installation floor. If its vibration frequency is lower than the TMD frequency, floor acceleration cannot effectively actuate the TMD oscillation. By contrast, the TVMD is actuated by the relative movement at its two ends, and thus is always mobilized and provides additional damping if the structure undergoes lateral vibration. This is the primary reason why the TVMD control exhibited significantly improved robustness compared with the TMD control.

Mechanism of TVMD control

The discussion in the preceding section indicates that the TVMD control was generally robust, particularly for the TVMD with a large inertance-to-mass ratio. This section unravels the mechanism of TVMD control and explains the robustness characteristics observed in this study.

Equivalent Kelvin-Voigt model of TVMD

As shown in Figure 6(c) and (f), a TVMD exhibits viscoelastic hysteretic behavior which can be characterized by frequency-dependent stiffness and damping. This behavior can be represented by the Kelvin-Voigt model presented in Figure 16(a). When subjected to a sinusoid displacement excitation Ue^i(ωt), the resistance force of a TVMD, Fe^iωt, is given by:

F e^{i ω t} = (k_{b} \frac{(λ^{2} - 1) γ^{2} + γ^{4}}{{(1 - γ^{2})}^{2} + λ^{2} γ^{2}} + k_{b} \frac{λ γ}{{(1 - γ^{2})}^{2} + λ^{2} γ^{2}} i) U e^{i ω t}

(11)

Figure 16.

The equivalent Kelvin-Voigt model of a TVMD: (a) Equivalent Kelvin-Voigt model, (b) Test verification (Specimen-B), (c) Behavior of TVMD (Specimen-B).

Equation (11) can be derived by multiplying the transfer function U_k/U by k_b, the derivation of which can be found in a previous study (Ji et al., 2021). Equation (11) can be rearranged in the form of a Kelvin-Voigt model as:

F e^{i ω t} = (k^{*} (γ) + c^{*} (γ) ω i) U e^{i ω t}

(12)

where

k^{*} (γ) = k_{b} \frac{(λ^{2} - 1) γ^{2} + γ^{4}}{{(1 - γ^{2})}^{2} + λ^{2} γ^{2}}

(13)

c^{*} (γ) = c_{d} \frac{1}{{(1 - γ^{2})}^{2} + λ^{2} γ^{2}}

(14)

The steady-state responses calculated using equation (12) can be verified by the test results presented in Figure 16(b). The behavior of a TVMD is displayed in Figure 16(c) where the equivalent stiffness k* varies from 0 to negative values and then back to positive values when the exciting frequency increases from zero to infinity. Therefore, the low-frequency hysteretic response reflects the behavior of a pseudo-negative stiffness-viscoelastic damper while the hysteretic response under excitation with high frequency reflects the behavior of a viscoelastic damper. When the exciting frequency is close to the natural frequency of the SDOF system (γ = γ₀), k* is close to 0, c* approaches its maximum value. This result is consistent with the observations in the 1.6 Hz harmonic tests. Points A through E are key points regarding the behavior of TVMD and can be explicitly obtained using equations (13) and (14). The explanation and expressions of these key points are listed in Figure 16(c).

As illustrated in Figure 16, the vibration control of TVMD is attributed to two mechanisms. One is to provide additional equivalent stiffness (k^*) which would affect the natural vibration frequency of the structural system. Another is to provide additional equivalent damping (c^*) which increases the energy dissipation of the system.

The contribution of each mechanism was further quantified. Firstly, the response of the SDOF system with the additional stiffness of k^* at a given exciting frequency was calculated. Based on the results at a series of exciting frequencies, the response of the SDOF system was obtained. Similarly, the response of the SDOF system with additional damping coefficient c^* was also assessed. In this case study, the inherent damping ratio of the primary structure was 1.5%. Figure 17 presents the response mitigation ratio h value contributed by TVMD, and the contributions by additional k^* and c^*. This indicates that the response mitigation of TVMD was mainly achieved by additional damping c*, and therefore the control of TVMD was primarily contributed by the damping amplification mechanism for the TVMD. It is important to note that because the response mitigation ratio h defined in this paper is related to the peak value of the system transfer function at a wide frequency range, this conclusion is applied for the seismic response control of a structure. For vibration control under forced excitation at a fixed frequency, the additional stiffness k^* may make a significant contribution.

Figure 17.

Contribution of k^* and c^*.

Influence of TVMD damping coefficient variation

As illustrated in the preceding subsection, the TVMD control effect is mainly contributed by the damping amplification effect. The damping amplification effect is associated with the displacement amplification effect of the dashpot (Zhang et al., 2020). According to equation (5), the parameter λ (or damping ratio ξ) affects the displacement amplification U_d/U of TVMD. A series of displacement amplification transfer functions (equation (5)) with a variety of damping ratios are compared in Figure 18. The displacement amplification effect becomes smaller in line with an increase in ξ which is proportional to the damping coefficient. This mechanism compensates for the increase in TVMD damping. When the damping coefficient decreases, this mechanism can also compensate for the decrease in additional damping. Therefore, the influence of TVMD damping is generally small, and the control of TVMD is robust.

Figure 18.

Damping amplification effect.

Influence of inertance-to-mass ratio

Figure 18 indicates that the increase in damping ratio ξ renders the displacement amplification ratio U_d/U of a TVMD less dependent on exciting frequency, which improves the robustness of control. According to the design formula for the optimal damping ratio (equation (3)), the damping coefficient of an optimally designed TVMD is positively related to the inertance-to-mass ratio. When the inertance-to-mass ratio is increased, the designed optimal damping ratio becomes larger and the TVMD control becomes more robust which is consistent with the observations in the test.

However, an increase in the inertance-to-mass ratio also mitigates the displacement amplification effect. As depicted in Figure 18, the peak of the transfer function decreases as the damping ratio increases. When the inertance-to-mass ratio is larger than 0.8, the designed damping ratio is larger than $\sqrt 2 / 2$ and the peak of the transfer function vanishes. In this case, the equivalent damping coefficient c^* of the TVMD will always be lower than c_d which demonstrates that the energy dissipation capability of the TVMD is lower than a VD with the same damping coefficient, and that the tuning design is ineffective. This phenomena relates to the tuning efficiency of a TVMD, and was discussed previously by the authors (Cheng et al., 2022).

Influence of damping nonlinearity

The experimental study of A9-C-F and B10-C-F has primarily revealed that the damping nonlinearity of a TVMD has limited impacts on the effect of vibration control. To verify this observation, and to study the behavior of TVMDs with other damping exponents, a group of nonlinear time history analyses were conducted on the finite element analysis (FEA) platform OpenSees. The damping element and spring element were simulated by the Truss element with ViscousDamper material and Elastic material, respectively. The inerter was simulated by the InertiaTruss element which was previously presented in the work of the authors (Ji et al., 2020). The properties of the TVMD model and TVMD-SDOF model were identical to the TVMD in A1-OPT configuration (as listed in Table 2). The damping exponent α varied from 0.2 to 1.0 with an increment of 0.2. For each case, the value of the damping coefficient c of the nonlinear damping was determined such that the peak value of transfer function U_d/U (i.e., displacement amplification ratio) of the TVMD with nonlinear damping was identical to that of the optimal linear TVMD. Furthermore, the transfer function U_d/U for a nonlinear TVMD is related to the excitation amplitude, and thus, the designed values of the nonlinear damping coefficient c were related to the displacement amplitude of TVMDs. The damping parameters and hysteretic responses of the nonlinear damping element in the TVMD models are presented in Figure 19(a) in which the exciting frequency is 1.6 Hz and the displacement amplitude of the damping element is 21 mm.

Figure 19.

FEA of TVMD models: (a) Damping parameters of the nonlinear damping element; (b) Validation of TVMD model; (c) Frequency responses of displacement amplification ratio.

As shown in Figure 19(b), the TVMD finite element model was verified by the result of the TVMD performance test of A1-OPT. Figure 19(c) illustrates the frequency responses of the amplification ratio for the models with different damping exponents. The FEA results were obtained from the steady-state responses of nonlinear time history analyses with the same loading protocols as the test. The maximum amplification ratios were almost identical, following the assumption in the design of nonlinear damping parameters. When the exciting frequency shifted away from the natural frequency of the TVMD, the displacement of the inerter became smaller, increasing the equivalent damping of the TVMD and resulting in a smaller amplification ratio. This effect became more obvious with a smaller α value.

To reveal the influence of damping nonlinearity on the structural response of the TVMD-SDOF system, nonlinear time history analyses were also conducted on the TVMD-SDOF models. The displacement amplitude (U_table) was taken as U_table = 3 mm for FEA which was identical to the input of the shaking table test. The exciting frequency varied from 0.1 Hz to 2.8 Hz with an increment of 0.1 Hz. The steady-state frequency responses of the TVMD-controlled SDOF system are shown in Figure 20. This indicates that, although the damping exponent α varied from 0.2 to 1.0, the damping nonlinearity had almost no impact on the structural response which is similar to the observations in the test. The FEA, together with the test results, demonstrate that the damping nonlinearity of a TVMD has limited influence on the structural dynamic response as long as the nonlinear TVMD under the target displacement amplitude has a peak displacement amplification ratio identical to that of the optimal linear TVMD.

Figure 20.

FEA results of the TVMD-SDOF system.

Conclusions

In this study, a small sized EC-TVMD was manufactured and tested using a shaking table to examine the robustness of TVMD control. The influence of variations in the stiffness of primary structures and TVMD damping were assessed. The vibration control mechanism of the TVMD was discussed based on a Kelvin-Voigt model. FE analyses were conducted to further reveal the influence of damping nonlinearity. Based on the results of the experimental tests and theoretical analyses, the following conclusions can be drawn:

1. The test results indicate that the response mitigation factor of the TVMD reduced by 14% when the structural stiffness decreased by 46%. The variation in the TVMD damping coefficient had an even smaller influence on the response mitigation factor. An increase in the inertance-to-mass ratio further improved the robustness of the TVMD control.

2. The robustness of TVMD can be quantitatively estimated by parametric analysis. The TVMD exhibited better robustness than the conventional TMD, when the stiffness of primary structure was decreased. The detuning effect caused by stiffness degradation on the TVMD-controlled structural system was approximately one third of that for the TMD-controlled system.

3. The behavior of a TVMD can be represented by a Kelvin-Voigt model where a TVMD is regarded as the combination of an additional dynamic stiffness and an additional damping. For a TVMD designed using fixed-point theory, its control effect is primarily due to the addition of damping to the system, which relies on the TVMD damping coefficient and damping amplification effect. Because those two coefficients have an inverse relationship, the control effect remains stable against the variation in the TVMD damping coefficient.

4. Both test results and FEA results reveal that damping nonlinearity has a small influence on the vibration control of TVMD when the TVMD with nonlinear damping has the same peak displacement amplification ratio as the optimal linear TVMD.

Footnotes

Acknowledgements

The authors gratefully acknowledge the sponsorship of the Beijing Natural Science Foundation (Grant No. JQ18029), the Institute for Guoqiang of Tsinghua University (Grant No. 2020GQC0003), and Tsinghua University Initiative Scientific Research Program (Grant No. 20193080019). The writers wish to express their sincere gratitude to the sponsors.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Beijing Natural Science Foundation (Grant No. JQ18029), the Institute for Guoqiang of Tsinghua University (Grant No. 2020GQC0003), and Tsinghua University Initiative Scientific Research Program (Grant No. 20193080019).

ORCID iD

Xiaodong Ji

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