Development of stiffness-adjustable tuned mass dampers for frequency retuning

Abstract

Tuned mass damper is an attractive strategy to mitigate the vibration of civil engineering structures. However, the performance of a tuned mass damper may show a significant loss due to the frequency detuning effect. Hence, an inerter-induced negative stiffness (apparent mass effect) and magnetic-force-induced positive/negative stiffness are proposed to integrate a stiffness-adjustable vertical tuned mass damper and pendulum tuned mass damper for frequency retuning, respectively. Based on the established differential equations of motion for a vertical tuned mass damper coupled with an inerter and a pendulum tuned mass damper integrated with a magnetic-force-induced positive-/negative-stiffness device, the frequency retuning principles of a vertical tuned mass damper and a pendulum tuned mass damper are, respectively, demonstrated. The frequency retuning strategies for both the vertical tuned mass damper and the pendulum tuned mass damper are confirmed and clarified by model tests. Furthermore, the performance of a retuned vertical tuned mass damper for mitigating vibration of a linear undamped single-degree-of-freedom primary structure is discussed, and the effects of the amplitudes of the pendulum tuned mass damper on magnetic-force-induced stiffness as well as the frequency of the pendulum tuned mass damper are also investigated. Both theoretical analysis and experimental investigations show that the proposed frequency tuning methodologies of tuned mass dampers are efficient and cost-effective with relatively simple configurations.

Keywords

frequency retuning inerter magnetic-force-induced stiffness structural vibration control tuned mass damper

Introduction

Tuned mass dampers (TMDs) are one of the most popular vibration control devices for flexible civil engineering structures (Shen et al., 2018; Wen et al., 2016). They consist of an auxiliary mass attached to the controlled main structure via a spring element providing stiffness and a damper dissipating mechanical energy. When the frequency of the TMD is optimally tuned to the dominant mode of the main structure, the vibration energy of the structure will be transmitted into the TMD and eventually dissipated by the damper or stored by an energy-harvesting circuit (Shen et al., 2012; Zhu et al., 2013). According to the spring element configuration, TMDs can be divided into spring-supported type and pendulum type. The vertical TMD (VTMD) belongs to the typical spring-supported TMD, which is mainly used to suppress vibrations of long-span bridges (Wang et al., 2014), building floors (Nguyen et al., 2012) and so on. The pendulum TMD (PTMD) utilizes a suspended mass instead of a sprung mass, which is effective in reducing horizontal vibration of wind-sensitive structures, especially for flexible towers (Roffel et al., 2013) and high-rise buildings (Liu and Lu, 2014).

Due to the relatively simple configurations and well-established design procedures (Den Hartog, 1956; Rana and Soong, 1998), the TMD has been widely used to mitigate the vibrations of civil engineering structures. However, the TMD is very sensitive to tuning frequency ratio (Spencer and Nagarajaiah, 2003) and often suffers from the problem of frequency detuning (Werkle et al., 2013), which may significantly affect the vibration suppression performance of the TMD, causing occupant discomfort and even structural safety issues (Roffel and Narasimhan, 2015). Frequency detuning occurs primarily due to the changes in operating environment of main structures and the deterioration of main structures or TMDs in the long-term operation (Roffel et al., 2011; Xia et al., 2011) which results in the frequency of the TMD being no longer equal to the desired value. Other factors for detuning include the design and manufacture errors of TMDs and inaccurate estimations of the dominant frequencies of the main structures. Consequently, frequency detuning has become a troublesome issue in the long-term operation of the TMD.

The most straightforward ways for frequency retuning of the TMD are to adjust the stiffness by adding an auxiliary minor spring and to directly change the vibrating mass of the TMD. However, these approaches are tedious to install and disassemble in the operation stage of the TMD. In addition, adding an auxiliary spring can only increase the frequency of the TMD, while for the PTMD, changing the vibrating mass does not work. As for the PTMD, the most popular method employs adjusting the length of the connected cable. For example, the PTMD installed in Chifley Tower (Kwok and Samali, 1995) has utilized a passive tuning frame below the upper support in order to manually move the location of the pendulum pivot, thereby tuning the pendulum length and frequency to desirable values. Similar methods have been also employed in Taipei 101 in Taiwan (Roffel et al., 2011) and Shanghai Tower (Lu et al., 2016). However, in order to ensure the continued control performance of the TMD using these passive methods, periodic field tests are necessary, which are expensive and time-consuming.

These problems could be well solved by employing adaptive TMD (Bhowmick and Mishra, 2015; Deng et al., 2006; Deng and Gong, 2008; Shi et al., 2017), including active TMD (Wen and Sun, 2015) and semi-active TMD (Ghorbani-Tanha et al., 2011; Lin et al., 2015; Sun and Nagarajaiah, 2013), whose stiffness properties can be tuned in real-time to match the frequency of the target structure. For example, Weber et al. (2011) presented an adaptive tuned mass damper (ATMD), which can realize the adaptive adjustment of the equivalent stiffness and frequency of the ATMD by controlling the input voltage of the magnetorheological (MR) damper. Roffel et al. (2011) developed an adaptive pendulum length adjustment mechanism, achieving the frequency compensation of the PTMD through a system of stepper motors and a microcontroller. The researches above have confirmed that frequency retuning can be achieved by active/semi-active control solutions. However, these solutions are not as popular as the passive methods due to reliability issues and the economic cost. Other alternative strategies include employing the nonlinear TMD (NTMD; Alexander and Schilder, 2009) or multiple TMDs (Mohebbi et al., 2015) to broaden the frequency bandwidth and enhance TMD efficiency, but these methods cannot fundamentally solve the detuning problem of the TMD.

The moving mass of the TMD can be amplified by an inerter, which is a two-terminal device developing resisting forces proportional to the relative acceleration of its terminals (Smith, 2002). The vibration control devices based on inerter, such as rotational inertia damper (Hwang et al., 2007), electromagnetic inertial mass damper (Nakamura et al., 2014), tuned viscous mass damper (Ikago et al., 2012) and tuned inerter damper (Lazar et al., 2013), have been successfully developed with enhanced control performance. Motivated by the inerter-induced apparent mass effect, an inerter is proposed to retune the frequency of the VTMD in this article. As for the frequency retuning of the PTMD in this work, a stiffness-adjustable PTMD based on magnetic-force-induced positive-/negative-stiffness principle is developed, which has been emphasized in vibration energy harvesting and vibration isolators field (Challa et al., 2008, 2011; Zhu et al., 2012). In civil engineering area, magnetic-induced stiffness has also been proposed to integrate novel dampers with negative-stiffness behaviors for enhancing energy dissipation efficiency (Shi et al., 2017; Shi and Zhu, 2015).

This article is organized as follows. First, the frequency retuning principles of a VTMD and a PTMD are demonstrated. Next, the proposed frequency tuning strategies of the TMDs are then confirmed and clarified by model tests. Practical design considerations for frequency retuning of the TMDs are also theoretically investigated, including the influence of the inclusion of an inerter on control performance of the VTMD and magnetic-force-induced stiffness calculation of the PTMD. Finally, key conclusions are drawn.

Frequency retuning methodology

It is well known that the frequency of a VTMD or a PTMD can both be described by the following equation

f = \frac{1}{2 π} \sqrt{\frac{k}{m}}

(1)

where k and m denote the stiffness and mass of a VTMD or a PTMD, respectively. Equation (1) demonstrates either the stiffness or the mass needs to be adjusted so that the frequency of a VTMD or a PTMD can be tuned. In this article, an inerter-induced apparent mass effect or magnetic-force-induced positive/negative stiffness is proposed to formulate a stiffness-adjustable VTMD or PTMD for frequency retuning, respectively.

Frequency retuning principle for a VTMD

Figure 1 shows the schematic diagram of a VTMD coupled with an inerter. The inerter mainly consists of a changeable flywheel and a ball screw amplification system. The system includes a ball nut, a ball screw, and a thrust bearing. The linear motion of the mass of the VTMD can be converted into high-speed rotational motion of the flywheel through ball screw mechanism. And the rotating flywheel will produce an inertial torque, which will be further amplified to an inertial force through the ball screw system producing the mass amplification effect. Therefore, frequency retuning of the VTMD is achieved as the resulting total mass of the VTMD is changed, which can be described by equations (2) to (7). It should be pointed out that there is no instability problem for a VTMD with an inerter since the inerter does not respond to static load (Shi and Zhu, 2018).

Figure 1.

Schematic diagram of a vertical tuned mass damper (VTMD) with an inerter.

The equation of motion of a VTMD coupled with an inerter can be expressed as

(m + b) \frac{d^{2} u}{d t^{2}} + c \frac{d u}{d t} + ku = 0

(2)

where m, c, and k denote the mass, damping coefficient, and stiffness coefficient of the VTMD, respectively; u denotes the displacement of the VTMD; and b denotes the apparent mass of the flywheel with the expression as

b = {(\frac{2 π}{L_{d}})}^{2} I_{f}

(3)

where L_d denotes the lead of the ball screw and I_f denotes the moment of inertia of the flywheel.

The frequency f of a VTMD with an inerter can be obtained by the equation of

f = \frac{1}{2 π} \sqrt{\frac{k}{m + b} (1 - ζ^{2})}

(4)

where ζ denotes the damping ratio of the VTMD with the expression as

ζ = \frac{c}{2 \sqrt{(m + b) k}}

(5)

when the actual frequency f_Vactual of the VTMD is higher than the target frequency f_Vtarget, the required apparent mass b can be obtained as

b = ({(f_{Vactual} / f_{Vtarget})}^{2} - 1) m

(6)

Finally, the moment of inertia of the flywheel I_f can be determined by

I_{f} = \frac{L_{d}^{2}}{4 π^{2}} b

(7)

However, equation (5) implies that inerter-induced apparent mass will decrease the damping ratio of the VTMD, and it may lead to mitigation performance deterioration of the VTMD, which needs to be further investigated.

Frequency retuning principle for a PTMD

Figure 2 shows the schematic diagram of a stiffness-adjustable PTMD, which consists of a mass, two cables, and a stiffness adjusting device (SAD). The proposed SAD is combined with moving and static cylindrical permanent magnets. The core idea of the SAD is that the moving and static permanent magnets are, respectively, fixed on the mass of PTMD and the exterior frame connected to the main structure, and the induced attractive or repulsive magnetic force can alter the restoring force of the PTMD. For example, to lower the frequency of the PTMD, 2n moving permanent magnets are symmetrically mounted on both sides of the mass moving together with the mass, and other 2n static permanent magnets with opposite polarity are symmetrically fixed on the corresponding exterior frame, as shown in Figure 2(a). The generated attractive magnetic force will be applied on both sides of the mass, while resultant magnetic force F_mag applied on the mass is in the opposite direction of the restoring force. As can be seen from Figure 2(b), the slope of corresponding magnetic force–displacement curve is always negative, demonstrating a negative-stiffness behavior of the SAD, which will inevitably decrease the effective stiffness and frequency of the PTMD. Similarly, when the frequency of the PTMD needs to be retuned higher, repulsive force between permanent magnets should be provided by the SAD, as shown in Figure 2(c). Hence, force–displacement relationship of the SAD with positive-stiffness behavior can be depicted in Figure 2(d), which will increase both the effective stiffness and frequency of the PTMD.

Figure 2.

Frequency tuning principle of a PTMD with an SAD: (a) an SAD with attractive magnetic force, (b) negative stiffness induced by magnetic force, (c) an SAD with repulsive magnetic force, and (d) positive stiffness induced by magnetic force.

Figure 3 shows the coordinates and force diagram of a PTMD with an SAD. Based on D’Alembert principle, the differential equation of motion of the combined PTMD-SAD system can be derived as

m l^{2} \frac{d^{2} θ}{d t^{2}} + mgl \sin θ = - (F_{p} - F_{n}) l \cos θ

(8)

where m, l, and θ represent the moving mass, pendulum length, and pendulum angle of the PTMD, respectively; F_p and F_n represent positive force and negative force with respect to the restoring force of the PTMD, respectively.

Figure 3.

Coordinates and force diagram of a PTMD with an SAD.

When attractive force between moving and static cylindrical magnets at each side of the PTMD mass is applied, F_p and F_n can be expressed as (Challa et al., 2008)

\begin{array}{l} F_{n} = n [\frac{B_{r}^{2} A_{m}^{2} {(δ + r)}^{2}}{π μ_{0} δ^{2}}] \\ [\frac{1}{{(d_{0} - l θ)}^{2}} + \frac{1}{{(d_{0} - l θ + 2 δ)}^{2}} - \frac{2}{{(d_{0} - l θ + δ)}^{2}}] \end{array}

(9)

\begin{array}{l} F_{p} = n [\frac{B_{r}^{2} A_{m}^{2} {(δ + r)}^{2}}{π μ_{0} δ^{2}}] \\ [\frac{1}{{(d_{0} + l θ)}^{2}} + \frac{1}{{(d_{0} + l θ + 2 δ)}^{2}} - \frac{2}{{(d_{0} + l θ + δ)}^{2}}] \end{array}

(10)

where B_r is the residual flux density of the magnets, A_m is the common area between the cylindrical magnets, δ is the length of the magnets, r is the radius of the magnets, μ₀ is the permeability of the intervening medium, n is the group number of magnets, and d₀ is the initial distance between the moving and static magnets on the same side of the PTMD mass.

Since θ is very small, the following approximations can be obtained

\sin θ \approx θ, \cos θ \approx 1

(11)

Consequently, equation (8) can be further simplified as

m \frac{d^{2} θ}{d t^{2}} + \frac{1}{l} (mg θ + F_{p} - F_{n}) = 0

(12)

Hence, the resulting overall restoring force of the PTMD can be given as

p (θ) = \frac{1}{l} (mg θ + F_{p} - F_{n})

(13)

However, instability problems of the PTMD may be induced since F_n is always larger than F_p. To ensure the stability of the PTMD, the overall restoring force should satisfy

p (θ) > 0

(14)

In other words, the following condition should be satisfied

F_{n} - F_{p} < mg θ

(15)

On the basis of equation (13), corresponding effective stiffness of the PTMD can be derived as

\begin{array}{l} k_{e} = \frac{δ p (θ)}{δ θ} = \frac{δ [\frac{1}{l} (m g θ + F_{p} - F_{n})]}{δ θ} \\ = \frac{1}{l} m g + \frac{1}{l} \frac{δ (F_{p} - F_{n})}{δ θ} \end{array}

(16)

Equation (16) can be rearranged as

k_{e} = k_{l} + k_{mag}

(17)

where k_l denotes the initial stiffness provided by the pendulum with an expression as

k_{l} = \frac{mg}{l}

(18)

and k_mag denotes the magnetic-force-induced stiffness with an expression as

k_{mag} = k_{p} - k_{n}

(19)

where k_p and k_n can be, respectively, derived as

k_{p} = | n [\frac{B_{r}^{2} A_{m}^{2} {(δ + r)}^{2}}{π μ_{0} δ^{2}}] [- \frac{2}{{(d_{0} + l θ)}^{3}} - \frac{2}{{(d_{0} + l θ + 2 δ)}^{3}} + \frac{4}{{(d_{0} + l θ + δ)}^{3}}] |

(20)

k_{n} = | n [\frac{B_{r}^{2} A_{m}^{2} {(δ + r)}^{2}}{π μ_{0} δ^{2}}] [- \frac{2}{{(d_{0} - l θ)}^{3}} - \frac{2}{{(d_{0} - l θ + 2 δ)}^{3}} + \frac{4}{{(d_{0} - l θ + δ)}^{3}}] |

(21)

Since k_mag = k_p − k_n < 0 and k_e = k_l + k_mag < k_l, the effective stiffness of the PTMD would be always smaller than the initial stiffness of the PTMD. Hence, the frequency of the PTMD will be reduced.

Similarly, when repulsive force between moving and static magnets is applied, k_p and k_n can be, respectively, obtained as

k_{p} = | n [\frac{B_{r}^{2} A_{m}^{2} {(δ + r)}^{2}}{π μ_{0} δ^{2}}] [- \frac{2}{{(d_{0} - l θ)}^{3}} - \frac{2}{{(d_{0} - l θ + 2 δ)}^{3}} + \frac{4}{{(d_{0} - l θ + δ)}^{3}}] |

(22)

k_{n} = | n [\frac{B_{r}^{2} A_{m}^{2} {(δ + r)}^{2}}{π μ_{0} δ^{2}}] [- \frac{2}{{(d_{0} + l θ)}^{3}} - \frac{2}{{(d_{0} + l θ + 2 δ)}^{3}} + \frac{4}{{(d_{0} + l θ + δ)}^{3}}] |

(23)

and the effective stiffness of the PTMD will be larger than the initial stiffness of the PTMD. In this case, the frequency of the PTMD will be increased.

Frequency retuning for a VTMD

Frequency retuning test

A test model of a VTMD with an inerter is shown in Figure 4. The VTMD mass with 99.7 kg is supported by a spring with the stiffness coefficient 73.8 kN/m. The initial frequency of the VTMD identified by the test is 4.34 Hz. An inerter is incorporated to the VTMD for frequency retuning, which consists of a thrust bearing, a ball nut, a ball screw as well as a flywheel. The frequency of the VTMD is retuned by changing flywheel with different moment of inertia. Assuming that the frequency of the VTMD is deviated 10% from the optimal value (i.e. 3.91 Hz), the moment of inertia and inertial mass are calculated as 145.00 kg·mm² and 22.41 kg according to equations (6) and (7), respectively. The mass of the VTMD is released manually from the same location for each test, and the acceleration responses are recorded by an acceleration sensor installed at the top of the moving mass.

Figure 4.

Test model of a VTMD coupled with an inerter.

The acceleration response comparisons for the detuned and retuned cases of the VTMD are shown in Figure 5, where the frequency of the VTMD is identified by the fast Fourier transform of the acceleration time history response. The results show that the frequency of the VTMD is retuned to 3.94 Hz, which agrees well with the target value. In conclusion, the frequency retuning of the VTMD is efficient and cost-effective with the inclusion of an inerter.

Figure 5.

Acceleration response comparisons for the detuned and retuned cases: (a) time history response for detuned case, (b) time history response for retuned case, (c) frequency response for detuned case, and (d) frequency response for retuned case.

The influence of an inerter inclusion on control performance of a VTMD

As pointed out previously, the optimum damping ratio of the VTMD has been varied during the frequency retuning, and it is still not sure that whether the inclusion of an inerter will lead to vibration mitigation performance reduction of a VTMD. Consider a linear undamped single-degree-of-freedom (SDOF) dynamical system (primary structure) modeled by a linear spring with stiffness k₁ and a mass m₁ and excited by a harmonic force f with circular frequency ω, as shown in Figure 6. To suppress the vertical vibrations of this primary structure, a classical VTMD is attached, which consists of a vibrating mass m₂, a linear spring with stiffness k₂ and a damper with damping coefficient c₂. An inerter is also inserted between the mass m₂ and the primary structure for frequency retuning of the VTMD.

Figure 6.

An SDOF primary structure coupled with a VTMD and an inerter.

The motion equation of an SDOF primary structure coupled with a VTMD and an inerter is written as

\begin{array}{l} [\begin{matrix} m_{1} + b & - b \\ - b & m_{2} + b \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{x}}_{1} \\ {\overset{\cdot\cdot}{x}}_{2} \end{matrix}} + [\begin{matrix} c_{2} & - c_{2} \\ - c_{2} & c_{2} \end{matrix}] {\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}} \\ + [\begin{matrix} k_{1} + k_{2} & - k_{2} \\ - k_{2} & k_{2} \end{matrix}] {\begin{matrix} x_{1} \\ x_{2} \end{matrix}} = {\begin{matrix} F e^{{j ω}^{t}} \\ 0 \end{matrix}} \end{array}

(24)

where j is the imaginary unit, expressing the response as

{\begin{matrix} x_{1} \\ x_{2} \end{matrix}} = {\begin{matrix} X_{1} {e^{j ω}}^{t} \\ X_{2} {e^{j ω}}^{t} \end{matrix}}

(25)

The solution to X₁ is given by

X_{1} = \frac{F [k_{2} - (m_{2} + b) ω^{2} + j c_{2} ω]}{{[k_{1} - (m_{1} + b) ω^{2}] [k_{2} - (m_{2} + b) ω^{2}] - (m_{2} - b) k_{2} ω^{2} - b^{2} ω^{4}} + j c_{2} ω (k_{1} - m_{1} ω^{2} - m_{2} ω^{2})}

(26)

To facilitate further discussions, the following notations are introduced as

\begin{matrix} ω_{2}^{2} = \frac{k_{2}}{m_{2} + b}, ζ = \frac{c_{2}}{2 (m_{2} + b) ω_{2}}, ω_{1}^{2} = \frac{k_{1}}{m_{1}}, μ = \frac{m_{2}}{m_{1}} \\ g = \frac{ω_{2}}{ω_{1}}, λ = \frac{ω}{ω_{1}}, β = \frac{b}{m_{1}}, δ_{st} = \frac{F}{k_{1}} \end{matrix}

(27)

where ω₂ and ζ are the natural circular frequency and the damping ratio of a VTMD with an inerter, respectively; ω₁ is the natural circular frequency of the primary structure; µ denotes mass ratio; g denotes the frequency ratio; λ denotes the ratio of excitation frequency to main structure; β denotes the ratio of apparent mass of the inerter to the mass of primary structure; and δ_st denotes the displacement of primary structure under a static force with magnitude F.

From the definitions above, the magnitude displacement dynamic amplification factor $| X_{1} / δ_{st} |$ of the primary structure can be derived by

| \frac{X_{1}}{δ_{st}} | = \sqrt{\frac{A^{2} + B^{2}}{C^{2} + D^{2}}}

(28)

where

\begin{matrix} A = (μ + β) g^{2} - (μ + β) λ^{2}, B = 2 ζ (μ + β) λ g \\ C = [1 - (1 + β) λ^{2}] [(μ + β) g^{2} - (μ + β) λ^{2})] \\ - (μ + β) (μ - β) g^{2} λ^{2} - β^{2} λ^{4} \\ D = 2 ζ (μ + β) λ g [1 - (1 + μ) λ^{2}] \end{matrix}

(29)

To minimize the response of the primary structure for β = 0, analytical closed-form optimum design parameters of a traditional VTMD for an undamped SDOF system are given as (Den Hartog, 1956)

g = \frac{1}{1 + u}, ζ^{opt} = \sqrt{\frac{3 μ}{8 {(1 + μ)}^{3}}}

(30)

According to equation (28), the displacement dynamic amplification factors of the primary structure with different VTMDs are shown in Figure 7. As expected, the displacement dynamic amplification factors $| X_{1} / δ_{st} |$ for the detuning case are quite larger than those of original optimum design case and retuning case, which means that the frequency detuning of a VTMD will lead to a significant performance deterioration. The displacement dynamic amplification factors $| X_{1} / δ_{st} |$ for retuning case with an inerter shown in Figure 7(a) are slightly larger than those of the original optimum design case as the mass amplification effect of the inerter makes the damping ratio of the VTMD deviate from its optimum value. In other words, the inclusion of an inerter will slightly reduce the vibration mitigation performance of the VTMD. However, if the damping ratio of the VTMD with an inerter can also be retuned and optimized, the displacement dynamic amplification factors $| X_{1} / δ_{st} |$ shown in Figure 7(b) will be smaller than those of the original optimum design case. Therefore, the control performance is superior to the original optimum design of the VTMD when the damping and frequency of the VTMD are both retuned.

Figure 7.

The variations of $| X_{1} / δ_{st} |$ with the λ of the VTMD for three different case (μ = 0.02).

Frequency retuning for a PTMD

Frequency retuning test

A test model of a PTMD with an SAD is shown in Figure 8. The PTMD mass is 10 kg suspending by four cables with 1.83 m length set from a stiff frame. The initial frequency of the PTMD is experimentally identified as 0.3865 Hz. A planar eddy-current damper (ECD; Wang et al., 2012), where two rectangular permanent magnets are mounted to the bottom of the suspended mass with a copper plate and a steel plate below, is adopted and installed at the bottom of the mass of the PTMD. Consisting of two moving and static cylindrical permanent magnets at each side of the PTMD mass, the SAD is arranged along the moving direction of the PTMD, where the combination of a moving and static cylindrical permanent magnet at each side is defined as one group of magnets in this article. The NdFeB permanent magnets with grade 35 are adopted in the ECD and SAD, and the specific properties of the SAD are shown in Table 1. The effect of the magnetic force on the frequency of the PTMD is experimentally investigated by varying the groups and polarity of permanent magnets in the SAD. The mass of the PTMD is horizontally pulled to one side and then released manually from the same location for each test. The free vibration response of the mass was recorded by an 891-4 type velocity sensor set on the top center of the mass.

Figure 8.

Test model of a PTMD with an SAD: (a) prototype, (b) SAD, and ECD.

Table 1.

Variables descriptions and values in an SAD.

Symbol	Description	Value	Units
B _r	Residual flux density	1.17	T
r	Radius of the magnet	15	mm
M _mag	Mass of the magnet	53.7	g
A _m	Common area between magnets	353.25	mm²
δ	Length of the magnet	10	mm
μ ₀	Permeability of intervening medium	1.256 × 10⁻⁶	H m⁻¹
A	Maximum amplitude of the pendulum	0.1	m
d ₀	Initial distance between the moving and static magnets	0.18	m

SAD: stiffness adjusting device.

Figure 9 shows typical free response histories of PTMD’s velocity. The test model has a low damping, which results in an ignorable vibration attenuation between two adjacent peaks. Hence, the time interval between two adjacent peaks could be taken as an approximate vibration period of the PTMD.

Figure 9.

Time history response of the PTMD velocity for different cases: (a) nothing; (b) attractive force, two groups; and (c) repulsive force, two groups.

As can be seen from the relationship between the frequencies and the amplitudes of the PTMD (Figure 10), the frequencies of the PTMD with attractive magnetic force will be lower than those without attractive magnetic force, and the frequency continuously decreases with the increase in the amplitude or the magnet groups. On the contrary, the frequencies of the PTMD with repulsive magnetic force will be larger than those of without repulsive magnetic force, while the frequency continuously increases with the increase in the amplitude or the magnet groups. The effective stiffness and the frequency of the PTMD will be reduced when the attractive force between magnets is applied. On the contrary, when the repulsive force between magnets is applied, the effective stiffness and the frequency of the PTMD will be increased.

Figure 10.

Frequency versus displacement amplitude of the PTMD at each test case.

However, the frequency of the PTMD with an SAD is no longer a constant as contrasted to a classical PTMD, and there appears a small variation with varying amplitude. Actually, such PTMDs with amplitude-dependent stiffness, utilizing wire-rope springs as stiffness element, have been successfully developed and behaved well (Gerges and Vickery, 2003). Since the variation ratio of the frequency with respect to optimal design frequency is less than 5% (Rana and Soong, 1998), it will not significantly reduce the performance efficacy of the PTMD. There are also some benefits for such frequency variation, such as avoiding the fatigue damage caused by small continuous vibrations and achieving automatic stroke limitation of the PTMD.

Practical design considerations for the magnetic-force-induced stiffness calculation

To achieve the frequency retuning of the PTMD, the magnetic-force-induced stiffness k_mag by one group of magnets should be accurately determined. Equations (19) to (23) demonstrate that k_mag depends on the magnetic flux density, the common area between the magnets, the length of the magnets, the initial distance between the moving and static magnets on the same side of the PTMD mass, and the displacement of the PTMD mass. Given the properties of the SAD in Table 1, the relationship between k_mag and the displacement x of the PTMD is shown in Figure 11. As expected, when attractive magnetic force between magnets is applied, the magnetic-force-induced stiffness generated is always negative and decreases with the increase in displacement x, while the stiffness reaches the maximum at the maximum amplitude. Accordingly, the effective stiffness of the PTMD will decrease with the displacement x. Similarly, when repulsive magnetic force between magnets is applied, the magnetic-force-induced stiffness is always positive and increases with the increase in displacement x, while the stiffness also reaches the maximum at the maximum amplitude. In this case, the effective stiffness of the PTMD will increase with the increasing displacement x.

Figure 11.

The magnetic-force-induced stiffness versus displacement x of the PTMD with different SAD.

As demonstrated in equations (19) to (23) and Figure 11, the magnetic-force-induced stiffness shows nonlinear behavior. As a result, the supplemental stiffness of the PTMD is displacement dependent. However, the equivalent supplemental stiffness at the design amplitude needs to be determined for initial design of frequency retuning. In view of this, a theoretical calculation schematic of equivalent supplemental stiffness shown in Figure 12 is proposed with the stiffness-displacement area equivalent principle. Hence, the theoretical equivalent supplemental stiffness can be calculated by

k_{mag, e} = \frac{1}{2 A} \int_{- A}^{A} k_{mag} (x) d x = \frac{1}{2 \times 0.1} \int_{- 0.1}^{0.1} k_{mag} (x) d x = 9.76 N / m

(31)

where k_{mag, e} denotes the equivalent supplemental stiffness by one group of magnets.

Figure 12.

Theoretical calculation schematic of the equivalent supplemental stiffness of an SAD.

For the convenience of design, the equivalent supplemental stiffness can be also expressed as

k_{mag, e} = η k_{mag} (A) = η \times 40.959 N / m

(32)

where η is a dimensionless coefficient and k_mag(A) is the supplemental stiffness at the maximum amplitude of the PTMD. Finally, η is determined as 0.24 based on equations (31) and (32).

Table 2 gives the comparisons between the theoretically estimated and experimentally identified frequencies of the PTMD under design amplitude. It shows that the error is around 3.62%–6.76%, and the experimentally identified supplemental stiffness is always smaller than the theoretical counterpart. There are two main sources of error. One is that the direction of magnetic force and the tangent direction of moving mass will cause significant offset when the amplitude of the PTMD gets larger, and the other is that the effects of permanent magnet magnetization and hysteresis have been neglected in the theoretical model. Therefore, it is recommended that the magnetic-force-induced stiffness should be reduced during the design stage of an SAD, and the corresponding reduction coefficient can be determined as

β = \arg min ‖ β k_{theo} - k_{test} ‖_{2}

(33)

where k_theo and k_test represent the theoretical and experimental supplemental stiffness induced by magnetic force at each case in Table 2, respectively.

Table 2.

Comparisons between theoretical and experimental frequencies of the PTMD under design amplitude.

Case	Experimental frequencies (Hz)	k _test (N/m)	Theoretical frequency (Hz)	k _theo (N/m)	Frequency error (%)
1	0.3685	–	0.3684	–	0.03
2	0.3867	6.001	0.4007	9.76	3.62
3	0.4134	14.298	0.4304	19.52	4.11
4	0.3453	−5.950	0.3332	−9.76	3.5
5	0.3150	−13.729	0.2937	−19.52	−6.76

Finally, the reduction coefficient defined in equation (33) is determined as 0.70 in this model test. For practical engineering application, the reduction coefficient can be conservatively taken as 0.50 to broaden the frequency bandwidth of the PTMD. Moreover, the frequency of the PTMD can be well tuned to the exact target by adjusting either the initial distance between magnets or magnet groups at the design or debugging stage.

Conclusion

In this article, an inerter-induced negative stiffness and magnetic-force-induced positive/negative stiffness are proposed to formulate a stiffness-adjustable VTMD and PTMD and solve the frequency detuning issue of the TMDs, respectively. The feasibilities and the effectiveness of two frequency retuning strategies are investigated via both theoretical analysis and model test. Some practical design considerations for frequency retuning of the TMDs have been investigated and highlighted. The main conclusions of this study are summarized as follows.

The incorporation of an inerter in a VTMD can reduce the frequency of the VTMD. As a consequence, the frequency retuning of the VTMD can be achieved by adjusting the moment of inertia of the attached flywheel. However, the mass amplification effect of the inerter makes the damping ratio of the VTMD deviate from its optimum value, which will slightly reduce the vibration mitigation performance of the optimum VTMD.

The frequency of a PTMD could be retuned to be either reduced or increased through applying either attractive or repulsive magnetic force. However, the frequency of the PTMD is amplitude dependent due to the nonlinearity of magnetic force, which has both negative and positive effects on the control performance of the PTMD. On one hand, the control performance of the PTMD will be slightly reduced. On the other hand, the retuned PTMD helps to broaden frequency bandwidth and achieve an automatic stroke control.

Footnotes

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: The authors greatly acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos 51308214 and 51578151) and the National Basic Research Program of China (973 Program; Grant Nos 2015CB060000 and 2015CB057702).

ORCID iD

Zhihao Wang

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