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Abstract

A novel six-direction sensor for absolute vibration motion is proposed in this study for the measurement of absolute motions by employing multi-direction quasi-zero-stiffness property. Scissor-like structures with pre-deformed springs are applied symmetrically in the proposed sensor. Based on the mathematical modeling of the proposed six-direction sensor, it is shown that the stiffness and damping properties of the sensor are adjustable nonlinear functions which are dependent on the structural parameters of scissor-like structures. And then, by utilizing bifurcation theory and perturbation method, the structural parameters are optimized for the improvement of measurement accuracy. For the realization of the approximate fixed point for a moving vibration object, the vibration from measurement object to the sensor is isolated, and thus the sensor could obtain the absolute motions. Utilizing the signals obtained by the proposed sensor directly, better vibration/suppression effectiveness is achieved. The results provide a novel and significant six-direction absolute motion measurement method by utilizing multi-direction quasi-zero-stiffness property with noticeable isolation performance only with passive elements.

Keywords

Motion measurement nonlinear vibration multi-direction sensor active control

Introduction

In the feedback control,^1–4 parameter identification fields,⁵ and so on for dynamic systems, the output signal is the most important feature which reflects the response characteristics in work processing. Recent development in the study of control strategies requires veracious output-feedback signal, which depends on the performance of signal pickup assembly and an appropriate measuring method to achieve beneficial performances such as isolation for moving vehicle^6,7,8 and vibration suppression for precise instrument on micro air vehicle (MAV). The usual signal measurement method contains sensor (i.e. relative displacement sensor and acceleration sensor),⁹ radar (radio detection and ranging),¹⁰ and laser measurement technology.¹¹ The problem is that how to reduce the error and time delay between the signal feedback on the system by active control and the signal assembled.

Since it is found that utilizing absolute displacement or velocity response as the variable of feedback-control strategy can effectively improve performance^1–4,12,13 of vibration isolation system, the output-feedback control is generally designed as a function of displacement and velocity response. But the distortion induced by integral operation and filtration of the feedback-control signals to the real output response cannot reach anticipant control effect using the acceleration sensor which obtains the absolute acceleration of the vibration of a moving object. However, the time delay cannot be ignored in feedback control, which can induce bifurcation and instability of a dynamic system.¹⁴ To obtain the absolute displacement motion or velocity motion of a moving vibration object, an absolute motion sensor based on quasi-zero-stiffness (QZS) isolation property is proposed.^15,16 The measuring signal obtained by the sensor system can be used directly for feedback control and parameter identification.

But the previous research is focused on one-direction measurement, since it could realize isolation in only vertical direction. As the generalization of the previous research, the idea of vibration sensor in this article comes from multi-direction (MD) QZS vibration isolator which can isolate the vibration from the under structure. Because of the remarkable isolation effect of MD QZS property, the MD relative motions between the QZS system and the under structure are close to the absolute motions of the under structure. The most common method of isolating vibrations is to use an isolator whose stiffness is close to zero.^17–21 In previous studies,^17–21 the QZS-VI system is demonstrated to be an ultra-low frequency vibration isolator by adjusting the stiffness characteristics and geometric size of the two horizontal springs. Therefore, in order to decrease the error of the relative motion and absolute motion, it is important to improve the sensor system to obtain much better MD isolation performance. This article utilizes the isolation characteristic of the QZS structure and scissor-like structures (SLSs)^22,23 to approximately obtain the absolute motions of a vibration system in six directions.

The article is organized as follows. First, an MD QZS sensor for absolute motion measurement is proposed in section “Structure and modeling of the proposed sensor.” Second, the measurement accuracy, design, and optimization of the sensor are analyzed by Harmonic balance methods (HBMs) in section “Measurement mechanisms and accuracy,” and thus the optimal structural parameters are obtained. Then, the signal obtained by the QZS vibration sensor is used for vibration control/suppression with adjustable time delay as shown in section “Application in time-delayed control.” Finally, a conclusion is thereafter given in section “Conclusion.”

Structure and modeling of the proposed sensor

Structural diagram of the proposed sensor

A MD sensor for MD absolute motions is proposed, whose structural diagram is shown in Figure 1.

Figure 1.

(a) The structural diagram of the proposed six-direction (6D) sensor for absolute displacement measurement and (b) assembling the proposed sensor on measurement object M₁.

Figure 1 shows the proposed sensor for measurement of absolute motions of a moving and vibrating system M₁ (i.e. a moving vehicle body). The structural diagram of the sensor is shown in Figure 1(a) which utilizes several symmetric SLSs with pre-deformed springs. Assembling the proposed sensor on the measurement object M₁, the absolute motions of M₁ could be measured by three distance sensors fixed on the M₂. The structural parameters of the proposed sensor are listed in Notation. For the structure characteristic of the proposed sensor, there are several adjustable structural parameters (i.e. the pre-deformation of springs in vertical SLS λ_s₁, in horizontal SLS λ_s₂, and the assembly angle θ₁ and θ₂ of the rods in SLSs) which could improve the precision of measurement signals. For pre-extension springs in SLSs, the values of λ_s₁ and λ_s₂ are positive since the springs are stretched in the assembly processing while the values are negative for pre-compression case. Also, the assembly angles θ₁ and θ₂ could adjust the strength of nonlinearity in the two directions. The effect of adjustable structural parameters on vibration properties would be studied in detail.

Modeling of the proposed sensor

The kinetic energy T of the system equals to the sum of kinetic energy of M₁ and M₂, which can be written as

$\begin{array}{l} T = \frac{1}{2} M_{1} [{(\frac{d x_{1}}{d t})}^{2} + {(\frac{d y_{1}}{d t})}^{2}] + \frac{1}{2} J_{1} {(\frac{d φ_{1}}{d t})}^{2} \\ + \frac{1}{2} M_{2} [{(\frac{d x_{2}}{d t})}^{2} + {(\frac{d y_{2}}{d t})}^{2}] + \frac{1}{2} J_{2} {(\frac{d φ_{2}}{d t})}^{2} \end{array}$ (1)

In order to obtain the potential energy V of the elastic components in the system, the deformation of the SLSs in the sensor should be analyzed. Figure 2 shows the vibration motions of mass M₂ and the midlines of the four SLSs assembled in the proposed sensor.

Figure 2.

(a) Deformations and rotational angles of midlines of the SLSs in the sensor and (b) deformations of springs between M₁ and the base.

Figure 2(a) shows the deformations and rotational angles of midlines of the four SLSs used in the sensor for vibration and the relative motions between $M_{2}$ and $M_{1}$ are as ${\hat{y}}_{2} = y_{2} - y_{1}$ , ${\hat{x}}_{2} = x_{2} - x_{1}$ , and ${\hat{φ}}_{2} = φ_{2} - φ_{1}$ ; Figure 2(b) shows the deformations of springs between the vibration object $M_{1}$ and the base, and the relative motions between $M_{1}$ and base are ${\hat{y}}_{1} = y_{1} - z_{y}$ , ${\hat{x}}_{1} = x_{1} - z_{x}$ , and ${\hat{φ}}_{1} = φ_{1} - z_{φ}$ . When the proposed sensor is fixed on the vibration object $M_{1}$ to measure the different-direction absolute displacements of $M_{1}$ , the relative motions between mass $M_{2}$ in the sensor and $M_{1}$ are dependent on the deformations of the SLSs used as elastic components in the sensor. From Figure 2, the deformations of midlines of the four SLSs could be set as $L_{21}, L_{22}, L_{23}, and L_{24}$ , respectively. According to the motions of $M_{2}$ shown in Figure 2, the values of $L_{21}, L_{22}, L_{23}, and L_{24}$ and rotational angles of midlines of SLSs ϕ₁, ϕ₂, ϕ₃, and ϕ₄ could be expressed as

$\begin{matrix} L_{21} = \sqrt{{(2 n_{1} l_{1} \sin θ_{1} - {\hat{y}}_{2})}^{2} + {\hat{x}}_{2}^{2}}, \\ L_{22} = \sqrt{{(2 n_{1} l_{1} \sin θ_{1} + {\hat{y}}_{2})}^{2} + {\hat{x}}_{2}^{2}} \\ L_{23} = \sqrt{{(2 n_{2} l_{2} \sin θ_{2} + {\hat{x}}_{2})}^{2} + {({\hat{y}}_{2} - L_{20} {\hat{φ}}_{2} / 2)}^{2}}, \\ L_{24} = \sqrt{{(2 n_{2} l_{2} \sin θ_{2} - {\hat{x}}_{2})}^{2} + {({\hat{y}}_{2} + L_{20} {\hat{φ}}_{2} / 2)}^{2}} \end{matrix}$ (2)

and

$\begin{matrix} ϕ_{1} = \arctan (\frac{{\hat{x}}_{2}}{2 n_{1} l_{1} \sin θ_{1} - {\hat{y}}_{2}}), \\ ϕ_{2} = \arctan (\frac{{\hat{x}}_{2}}{2 n_{1} l_{1} \sin θ_{1} + {\hat{y}}_{2}}) \\ ϕ_{3} = \arctan (\frac{{\hat{y}}_{2} - L {\hat{φ}}_{2} / 2}{2 n_{2} l_{2} \sin θ_{2} - {\hat{x}}_{2}}), \\ ϕ_{4} = \arctan (\frac{{\hat{y}}_{2} + L {\hat{φ}}_{2} / 2}{2 n_{2} l_{2} \sin θ_{2} - {\hat{x}}_{2}}) \end{matrix}$ (3)

Therefore, the deformations of springs in the four SLSs could be obtained according to the lengths of midlines of the SLSs from equation (2). As an example, the lengths of spring and midline in the right-side SLS after deformation are shown in Figure 3.

Figure 3.

(a) The deformation of the right-side SLS and (b) the deformation of one layer.

From Figure 3, the length of spring in the right-side SLS $l_{r}$ could be written as $l_{r} = 2 \sqrt{l_{20}^{2} - {(L_{24} / 2 n_{2})}^{2}}$ . Similar to the case shown in Figure 3, the lengths of springs in the left-side SLS $l_{l}$ , the up-side SLS $l_{u}$ , and the down-side SLS $l_{d}$ could be written as $l_{l} = 2 \sqrt{l_{20}^{2} - {(L_{23} / 2 n_{2})}^{2}}$ , $l_{u} = 2 \sqrt{l_{10}^{2} - {(L_{21} / 2 n_{1})}^{2}}$ , and $l_{d} = 2 \sqrt{l_{10}^{2} - {(L_{22} / 2 n_{1})}^{2}}$ . Therefore, the potential energy V of the system can be written as

$\begin{matrix} V = & \frac{1}{2} k_{v 1} {(L_{11} - l_{v 0})}^{2} + \frac{1}{2} k_{v 1} {(L_{12} - l_{v 0})}^{2} \\ + \frac{1}{2} k_{h 1} {(L_{13} - l_{h 0})}^{2} + \frac{1}{2} k_{h 1} {(L_{14} - l_{h 0})}^{2} \\ + \frac{1}{2} k_{s 1} {(l_{u} - l_{10})}^{2} + \frac{1}{2} k_{s 1} {(l_{d} - l_{10})}^{2} \\ + \frac{1}{2} k_{s 2} {(l_{l} - l_{20})}^{2} + \frac{1}{2} k_{s 2} {(l_{r} - l_{20})}^{2} \end{matrix}$ (4)

Because the value of pre-deformation of springs used in the vertical SLSs is λ_s₁ and in horizontal SLS is λ_s₂, there are conditions that $2 l_{1} \cos_{1} = l_{10} + λ_{s 1}$ and $2 l_{2} \cos_{2} = l_{20} + λ_{s 2}$ . Lagrange’s principle is used to obtain the mathematical model of the proposed sensor, which can be written as

$\frac{d}{dt} [\frac{d (T - V)}{d {q'}_{i}}] - \frac{d (T - V)}{d q_{i}} = Q = \sum_{N} Q_{i}$ (5)

where the generalized coordinates $q_{i} (i = 1, 2, \dots, 6)$ could be written as ${q_{1}, q_{2}, q_{3}, q_{4}, q_{5}, q_{6}} = {x_{1}, y_{1}, φ_{1}, x_{2}, y_{2}, φ_{2}}$ and generalized forces $Q_{i} (i = 1, 2, \dots, N)$ reflects the energy dissipation induced by viscous friction in the joints for different-direction vibration. Because the rotational angles of the midlines of the SLSs cannot be ignored, the generalized force $Q_{i}$ represents the damping of the system and the viscous friction in the joints where the mass $M_{2}$ , SLSs and the frame connect. Assuming the viscous friction coefficient of the joints connecting $M_{2}$ , SLSs, and the frame equals to c (as shown in Figure 1) and the air damping coefficients in different directions as $c_{x}, c_{y}, and c_{φ}$ , the generalized force Q can be written as

$\begin{matrix} Q = - c_{1 x} \frac{d {\hat{x}}_{1}}{dt} - c_{1 y} \frac{d {\hat{y}}_{1}}{dt} - c_{1 φ} \frac{d {\hat{φ}}_{1}}{dt} - c_{2 x} \frac{d {\hat{x}}_{2}}{dt} - c_{2 y} \frac{d {\hat{y}}_{2}}{dt} - c_{2 φ} \frac{d {\hat{φ}}_{2}}{dt} \\ - 2 c (\frac{d ϕ_{1}}{dt} \frac{\partial ϕ_{1}}{\partial q_{i}} + \frac{d ϕ_{2}}{dt} \frac{\partial ϕ_{2}}{\partial q_{i}} + \frac{d ϕ_{3}}{dt} \frac{\partial ϕ_{3}}{\partial q_{i}} + \frac{d ϕ_{4}}{dt} \frac{\partial ϕ_{4}}{\partial q_{i}}) \end{matrix}$ (6)

With the conditions of relative motions and absolute motions of the sensor as $x_{2} = {\hat{x}}_{2} + x_{1}$ , $y_{2} = {\hat{y}}_{2} + y_{1}$ , and $φ_{2} = {\hat{φ}}_{2} - φ_{1}$ and the measurement object $M_{1}$ as $x_{1} = {\hat{x}}_{1} + z_{x}$ , $y_{1} = {\hat{y}}_{1} + z_{y}$ , and $φ_{1} = {\hat{φ}}_{1} - z_{φ}$ , substituting the generalized force from equation (7), kinetic energy T from equation (1), and potential energy V from equation (5) into equation (6) and utilizing Taylor’s series to simplify the stiffness and damping properties, the dynamic equations of the proposed sensor are

${\begin{matrix} M_{2} \frac{d^{2} {\hat{x}}_{2}}{d t^{2}} + K_{x} {\hat{x}}_{2} + F_{x} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2}) + [C_{x} + D_{x} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2})] \frac{d {\hat{x}}_{2}}{dt} = - M_{2} \frac{d^{2} x_{1}}{d t^{2}} \\ M_{2} \frac{d^{2} {\hat{y}}_{2}}{d t^{2}} + K_{y} {\hat{y}}_{2} + F_{y} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2}) + [C_{y} + D_{y} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2})] \frac{d {\hat{y}}_{2}}{dt} = - M_{2} \frac{d^{2} y_{1}}{d t^{2}} \\ J_{2} \frac{d^{2} {\hat{φ}}_{2}}{d t^{2}} + K_{φ} {\hat{x}}_{2} + F_{φ} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2}) + [C_{φ} + D_{φ} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2})] \frac{d {\hat{φ}}_{2}}{dt} = - J_{2} \frac{d^{2} φ_{1}}{d t^{2}} \\ M_{1} \frac{d^{2} x_{1}}{d t^{2}} + 2 k_{h 1} x_{1} + c_{x 1} \frac{d x_{1}}{dt} = 2 k_{h 1} z_{x} + c_{x 1} \frac{d z_{x}}{dt} \\ M_{1} \frac{d^{2} y_{1}}{d t^{2}} + 2 k_{v 1} y_{1} + c_{y 1} \frac{d y_{1}}{dt} = 2 k_{v 1} z_{y} + c_{y 1} \frac{d z_{y}}{dt} \\ J_{1} \frac{d^{2} φ_{1}}{d t^{2}} + 2 k_{v 1} L_{10}^{2} φ_{1} + c_{φ 1} \frac{d φ_{1}}{dt} = 2 k_{v 1} L_{10}^{2} z_{φ} + c_{φ 1} \frac{d z_{φ}}{dt} \end{matrix}$ (7)

where $K_{x}$ , $K_{y}$ , and $K_{φ}$ are linear stiffness coefficients and $C_{x}$ , $C_{y}$ , and $C_{φ}$ are linear damping coefficients in the three directions; $F_{x}$ , $F_{y}$ , and $F_{φ}$ are nonlinear stiffness functions and $D_{x}$ , $D_{y}$ , and $D_{φ}$ are nonlinear damping coefficients in the three directions, respectively. The stiffness functions $F_{x}$ , $F_{y}$ , and $F_{φ}$ , linear and nonlinear damping coefficients $C_{x}$ , $C_{y}$ , $C_{φ}$ , $D_{x}$ , $D_{y}$ , and $D_{φ}$ are as

$\begin{matrix} F_{x} = \frac{1}{8} [\frac{k_{s 2} (2 l_{1} \cos θ_{1} - λ_{s 1}) \sec^{3} θ_{1}}{l_{1}^{3} n_{1}^{4}} + \frac{k_{s 1} (2 l_{2} \cos θ_{2} - λ_{s 2}) (3 - 2 \cos 2 θ_{2}) \sec^{7} θ_{2}}{l_{2}^{3} n_{2}^{4}}] {\hat{x}}_{2}^{3} \\ + \frac{1}{8} [\frac{k_{s 2} (2 - \cos 2 θ_{1}) (2 l_{2} - λ_{s 1}) \sec^{4} θ_{1}}{l_{1}^{3} n_{1}^{4}} + \frac{k_{s 1} (2 l_{2} \cos θ_{2} - λ_{s 2}) (2 - 2 \cos 2 θ_{2}) \sec^{5} θ_{2}}{l_{2}^{3} n_{2}^{4}}] {\hat{x}}_{2} {\hat{y}}_{2}^{2} \\ + \frac{k_{s 1} L_{10}^{2} (2 l_{2} \cos θ_{2} - λ_{s 2}) (2 - 2 \cos 2 θ_{2}) \sec^{5} θ_{2}}{32 l_{2}^{3} n_{2}^{4}} {\hat{x}}_{2} {\hat{φ}}_{2}^{2} \\ F_{y} = \frac{1}{8} [\frac{k_{s 2} (2 l_{1} \cos θ_{1} - λ_{s 1}) (3 - 2 \cos 2 θ_{1}) \sec^{7} θ_{1}}{l_{1}^{3} n_{1}^{4}} + \frac{k_{s 1} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{l_{2}^{3} n_{2}^{4}}] {\hat{y}}^{3} \\ + \frac{1}{8} [\frac{k_{s 2} (2 - \cos 2 θ_{1}) (2 l_{2} - λ_{s 1}) \sec^{4} θ_{1}}{l_{1}^{3} n_{1}^{4}} + \frac{k_{s 1} (2 l_{2} \cos θ_{2} - λ_{s 2}) (2 - 2 \cos 2 θ_{2}) \sec^{5} θ_{2}}{l_{2}^{3} n_{2}^{4}}] {\hat{x}}_{2}^{2} {\hat{y}}_{2} \\ + \frac{k_{s 1} L_{10}^{3} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{64 l_{2}^{3} n_{2}^{4}} {\hat{φ}}_{2}^{3} + \frac{k_{s 1} L_{10} (2 l_{2} \cos θ_{2} - λ_{s 2}) (2 - \cos 2 θ_{2}) \sec^{5} θ_{2}}{16 l_{2}^{3} n_{2}^{4}} {\hat{x}}_{2}^{2} {\hat{y}}_{2} \\ + \frac{3 k_{s 1} L_{10}^{2} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{16 l_{2}^{3} n_{2}^{4}} {\hat{y}}_{2}^{2} {\hat{φ}}_{2} + \frac{3 k_{s 1} L_{10}^{2} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{32 l_{2}^{3} n_{2}^{4}} {\hat{y}}_{2} {\hat{φ}}_{2}^{2} \\ F_{φ} = \frac{k_{s 1} L_{10} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{16 l_{2}^{3} n_{2}^{4}} {\hat{y}}_{2}^{3} + \frac{k_{s 1} L_{10}^{4} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{128 l_{2}^{3} n_{2}^{4}} {\hat{φ}}_{2}^{3} \\ + \frac{k_{s 1} L_{10}^{2} (2 l_{2} \cos θ_{2} - λ_{s 2}) (2 - 2 \cos 2 θ_{2}) \sec^{5} θ_{2}}{32 l_{2}^{3} n_{2}^{4}} {\hat{x}}_{2} {\hat{φ}}_{2}^{2} \\ F_{y} = \frac{1}{8} [\frac{k_{s 2} (2 l_{1} \cos θ_{1} - λ_{s 1}) (3 - 2 \cos 2 θ_{1}) \sec^{7} θ_{1}}{l_{1}^{3} n_{1}^{4}} + \frac{k_{s 1} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{l_{2}^{3} n_{2}^{4}}] {\hat{y}}^{3} \\ + \frac{k_{s 1} L_{10} (2 l_{2} \cos θ_{2} - λ_{s 2}) (2 - \cos 2 θ_{2}) \sec^{5} θ_{2}}{16 l_{2}^{3} n_{2}^{4}} {\hat{x}}_{2}^{2} {\hat{y}}_{2} + \frac{k_{s 1} L_{10}^{3} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{64 l_{2}^{3} n_{2}^{4}} {\hat{y}}_{2} {\hat{φ}}_{2}^{2} \\ + \frac{k_{s 1} L_{10}^{2} (2 l_{2} \cos θ_{2} - λ_{s 2}) (2 - \cos 2 θ_{2}) \sec^{5} θ_{2}}{32 l_{2}^{3} n_{2}^{4}} {\hat{x}}_{2}^{2} {\hat{φ}}_{2} + \frac{3 k_{s 1} L_{10}^{2} (2 l_{2} \cos θ_{2} - λ_{s 2}) \sec^{3} θ_{2}}{16 l_{2}^{3} n_{2}^{4}} {\hat{y}}_{2}^{2} {\hat{φ}}_{2} \end{matrix}$

$\begin{matrix} C_{x} = c_{2 x} + \frac{2 c^{2} \csc^{2} θ_{1}}{l_{1}^{2} n_{1}^{2}}, C_{y} = c_{2 y} + \frac{2 c^{2} \csc^{2} θ_{2}}{l_{2}^{2} n_{2}^{2}}, \\ C_{φ} = c_{2 φ} + \frac{c^{2} L_{10}^{2} \csc^{2} θ_{2}}{2 l_{2}^{2} n_{2}^{2}} \\ D_{x} = \frac{3 c^{2}}{2} (\frac{\csc^{4} θ_{1}}{l_{1}^{4} n_{1}^{4}} + \frac{\csc^{4} θ_{2}}{3 l_{2}^{4} n_{2}^{4}}) {\hat{y}}_{2}^{2} + \frac{c^{2} L_{10}^{2} \csc^{4} θ_{2}}{8 l_{2}^{4} n_{2}^{4}} {\hat{φ}}_{2}^{2} \\ D_{y} = \frac{3 c^{2}}{2} (\frac{\csc^{4} θ_{1}}{3 l_{1}^{4} n_{1}^{4}} + \frac{\csc^{4} θ_{2}}{l_{2}^{4} n_{2}^{4}}) {\hat{x}}_{2}^{2} - \frac{c^{2} L_{10}^{2} \csc^{4} θ_{2}}{4 l_{2}^{4} n_{2}^{4}} {\hat{φ}}_{2}^{2} \\ D_{φ} = \frac{3 c^{2} L_{10}^{2} \csc^{4} θ_{2}}{8 l_{2}^{4} n_{2}^{4}} {\hat{x}}_{2}^{2} - \frac{c^{2} L_{10}^{2} \csc^{4} θ_{2}}{4 l_{2}^{4} n_{2}^{4}} {\hat{y}}_{2}^{2} \end{matrix}$

Therefore, from equation (7), different-direction motions are coupled in nonlinear terms. It can be seen that the isolation effectiveness in different directions is dependent on the stiffness matrix, damping matrix, and nonlinearity, thus the measurement accuracy of the measurement signals is determined by the isolation effectiveness in different directions.

Measurement mechanisms and accuracy

Vibration mechanisms and vibration responses

For an arbitrary excitation from the base, the vibration energy of the vibration system with the proposed sensor is shown in Figure 4.

Figure 4.

The vibration energy transfer processing for the MD vibration system with proposed sensor.

The input and output functions shown in Figure 4 represent the vibration signals for different objects. The input of the vibration energy is from the base excitation; the vibration response of $M_{1}$ is output displacement signal $Z_{1} (ω) = {x_{1} (ω), y_{1} (ω), φ_{1} (ω)}^{T}$ which is the signal required to be measured; and the output displacement signal $Z_{2} (ω) = {x_{2} (ω), y_{2} (ω), φ_{2} (ω)}^{T}$ reflects the vibration of the internal mass $M_{2}$ in the sensor. In order to obtain the influence on vibration property and the measurement accuracy of the proposed sensor, the amplitude-frequency curves of the proposed sensor should be solved. For the nonlinear dynamic equation of the proposed sensor (equation (7)), HBM is utilized. For harmonic excitation, $z_{x} = x_{e} \cos t$ , $z_{y} = y_{e} \cos t$ , and $z_{φ} = φ_{e} \cos t$ . For small-amplitude excitation and response, the absolute motions of the measurement object $x_{1}$ , $y_{1}$ , and $φ_{1}$ and the relative motions ${\hat{x}}_{2}$ , ${\hat{y}}_{2}$ , and ${\hat{φ}}_{2}$ could be set as periodic solutions as

${\begin{matrix} x_{1} = a_{1} \cos (ω t + ϕ_{1 x}) \\ y_{1} = b_{1} \cos (ω t + ϕ_{1 y}) \\ φ_{1} = c_{1} \cos (ω t + ϕ_{1 φ}) \end{matrix}, {\begin{matrix} {\hat{x}}_{2} = a_{2} \cos (ω t + ϕ_{2 x}) \\ {\hat{y}}_{2} = b_{2} \cos (ω t + ϕ_{2 y}) \\ {\hat{φ}}_{2} = c_{2} \cos (ω t + ϕ_{2 φ}) \end{matrix}$ (8)

where $a_{1}$ , $b_{1}$ , $c_{1}$ , $a_{2}$ , $b_{2}$ , and $c_{2}$ are amplitudes of the vibrations of measurement objects $M_{1}$ and $M_{2}$ in sensor for different directions, ω is the frequency from vibration excitation. Substituting the solution of equation (8) into the nonlinear dynamic equation from equation (7), the amplitude-curves for different-direction motions could be theoretically expressed.

According to the definition of the displacement transmissibility function $H_{2} (ω)$ , it could be written as

$\begin{matrix} H_{2} (ω) = {\begin{matrix} H_{2 x} (ω) \\ H_{2 y} (ω) \\ H_{2 φ} (ω) \end{matrix}} = {\begin{matrix} ‖ x_{2} (ω) ‖ / ‖ x_{1} (ω) ‖ \\ ‖ y_{2} (ω) ‖ / ‖ y_{1} (ω) ‖ \\ ‖ φ_{2} (ω) ‖ / ‖ φ_{1} (ω) ‖ \end{matrix}} \\ = {\begin{matrix} \frac{‖ a_{2} \cos (ω t + ϕ_{2 x}) + a_{1} \cos (ω t + ϕ_{1 x}) ‖}{‖ a_{1} \cos (ω t + ϕ_{1 x}) ‖} \\ \frac{‖ b_{2} \cos (ω t + ϕ_{2 y}) + b_{1} \cos (ω t + ϕ_{1 y}) ‖}{‖ b_{1} \cos (ω t + ϕ_{1 y}) ‖} \\ \frac{‖ c_{2} \cos (ω t + ϕ_{2 φ}) + c_{1} \cos (ω t + ϕ_{1 φ}) ‖}{‖ c_{1} \cos (ω t + ϕ_{1 φ}) ‖} \end{matrix}} \\ = {\begin{matrix} \sqrt{1 + 2 \frac{a_{2}}{a_{1}} \cos (ϕ_{1 x} - ϕ_{2 x}) + {(\frac{a_{2}}{a_{1}})}^{2}} \\ \sqrt{1 + 2 \frac{b_{2}}{b_{1}} \cos (ϕ_{1 y} - ϕ_{2 y}) + {(\frac{b_{2}}{b_{1}})}^{2}} \\ \sqrt{1 + 2 \frac{c_{2}}{c_{1}} \cos (ϕ_{1 φ} - ϕ_{2 φ}) + {(\frac{c_{2}}{c_{1}})}^{2}} \end{matrix}} \end{matrix}$ (9)

If the value of each item in the displacement transmissibility function $H_{2} (ω)$ approaches to zero in all frequency bands, the vibration from $M_{1}$ to $M_{2}$ is mostly isolated as $x_{2} \approx 0$ , $y_{2} \approx 0$ , and $φ_{2} \approx 0$ . The approximate fixed point in the space is achieved and the relative motions between $M_{2}$ and $M_{1}$ could describe the absolute motions of $M_{1}$ .

Measurement accuracy and the design of structural parameters

Since the displacement transmissibility function $H_{1} (ω)$ reflects the vibration amplitudes ratio between base and the measurement object $M_{1}$ and $H_{2} (ω)$ reflects the vibration amplitudes between $M_{1}$ and $M_{2}$ , the measurement accuracy of the proposed sensor is dependent on the vibration of $M_{2}$ . The closer the $H_{2} (ω)$ approaches zero, the smaller vibration energy is transferred to the sensor. Thus, the relative motions between $M_{1}$ and $M_{2}$ have better accuracy with the absolute motions of $M_{1}$ . Therefore, the structural parameters and assembly processing of SLSs in the sensor need appropriate design and optimization for significant isolation of vibration in a wide frequency bands. Therefore, the MD QZS property could be utilized to achieve the requirement of isolation effectiveness. The expression of $K_{x}$ , $K_{y}$ , and $K_{φ}$ are as

${\begin{matrix} K_{x} = \frac{2 k_{1} \tan^{2} θ_{2}}{n_{2}^{2}} - \frac{k_{2} λ_{s 1} \sec θ_{1}}{l_{1} n_{1}^{2}} - \frac{k_{1} λ_{s 2} \sec^{3} θ_{2}}{l_{2} n_{2}^{2}} \\ K_{y} = \frac{2 k_{2} \tan^{2} θ_{1}}{n_{1}^{2}} - \frac{k_{2} λ_{s 1} \sec^{3} θ_{1}}{l_{1} n_{1}^{2}} - \frac{k_{1} λ_{s 2} \sec θ_{2}}{l_{2} n_{2}^{2}} \\ K_{φ} = - \frac{k_{1} L^{2} λ_{s 2} \sec θ_{2}}{4 l_{2} n_{2}^{2}} \end{matrix}$ (10)

If it needs $K_{φ} = 0$ , the pre-deformation λ_s₂ should be as $λ_{s 2} = 0$ . And to achieve the QZS property in vertical and horizontal directions as $K_{x} = K_{y} = 0$ , the adjustable structural parameters θ₁ and λ_s₁ should be fixed as the values shown in Figure 5.

Figure 5.

Pre-deformation versus assemble angle diagrams for (a) $n_{1} = 2$ and $n_{2} = 2$ , (b) $n_{1} = 2$ and $n_{2} = 3$ , and (c) $n_{1} = 1$ and $n_{2} = 2$ . Red lines stand for $K_{x} = 0$ and black lines stand for $K_{y} = 0$ .

In Figure 5, the red lines stand for $K_{x} = 0$ and black lines stand for $K_{y} = 0$ , thus the intersection points of the two curves stand for $K_{x} = K_{y} = 0$ . Because $K_{y}$ is independent on the parameter $n_{2}$ for $λ_{s 2} = 0$ , the black lines are invariant as Figure 5. The values of structural parameters satisfying $K_{x} = K_{y} = 0$ are on the black lines. The values of adjustable structural parameters θ₁ and λ_s₁ for $K_{x} = K_{y} = 0$ for different number of layers $n_{1}$ and $n_{2}$ of SLSs utilized in different directions are listed in Table 1. For other values of structural parameters, the optimization values of the adjustable parameters θ₁ and λ_s₁ for designing MD QZS property could also be obtained by setting $K_{x} = K_{y} = 0$ as equation (10). For fixing $λ_{s 2} = 0$ because of $K_{φ} = 0$ , the first equation of equation (10) is set as $K_{x} = 0$ , it has the condition as $2 k_{1} ta n^{2} θ_{2} / n_{2}^{2} = {k_{2}}_{s 1} \sec θ_{1} / (l_{1} n_{1}^{2})$ , thus the second equation of equation (10) is as $K_{y} = 2 k_{2} ta n^{2} θ_{1} / n_{1}^{2} - 2 k_{1} ta n^{2} θ_{2} se c^{2} θ_{1} / n_{2}^{2}$ . For $K_{x} = K_{y} = 0$ , it could obtain the condition as $\sin θ_{1} = n_{1} / n_{2} \sqrt{k_{1} / k_{2}} \tan θ_{2}$ and $λ_{s 1} = l_{1} si n^{2} θ_{1} \cos θ_{1}$ .

Table 1.

The values of the adjustable structural parameters λ_s₁ and θ₁ in the sensor system for different layers $n_{1}$ and $n_{2}$ as $k_{s 1} = 500$ , $k_{s 2} = 1000$ , θ₂ = π/4, $l_{1} = 0.1$ , and $l_{2} = 0.2$ .

Number of layers, n₁	Number of layers, n₂	Pre-deformation, λ_s₁	Angle, θ₁
n ₁ = n	n ₂ = n	0.07071	0.7854
n ₁ = 1	n ₂ = 2	0.02338	0.3614
n ₁ = 1	n ₂ = 3	0.0108	0.2379
n ₁ = 2	n ₂ = 3	0.0392	0.4909

The transmissibility function $H_{2} (ω)$ for different adjustable structural parameters is shown in Figure 6. The situation of Black lines in Figure 6 is defined as the first case, which is as $c_{2 x} = c_{2 y} = c_{2 φ} = 2$ , $c = 0.1$ , θ₁ = π/4, and $λ_{s 1} = λ_{s 2} = 0$ ; The situation of red lines in Figure 6 is defined as the second case, which is set as θ₁ = 0.07071, λ_s₁ = 0.7854, and λ_s₂ = 0 for $K_{x} = K_{y} = K_{φ} = 0$ and with the same damping coefficients as the first case; And the situation of blue lines in Figure 6 is defined as the third case, which is for increasing the damping coefficients as $c_{2 x} = c_{2 y} = c_{2 φ} = 10$ , $c = 0.2$ and with the same structural parameters as the first case.

Figure 6.

The vibration transmissibility function $H_{2} (ω)$ between M₂ in sensor and the measurement object M₁ for different structural parameters: (a) x-direction displacement transmissibility, (b) y-direction, and (c) ϕ-direction. Black lines (the first case) are for $c_{2 x} = c_{2 y} = c_{2 φ} = 2$ , $c = 0.1$ , θ₁ = π/4, and λ_s₁ = λ_s₂ = 0; red lines (the second case) are for $c_{2 x} = c_{2 y} = c_{2 φ} = 2$ , $c = 0.1$ , θ₁ = 0.07071, λ_s₁ = 0.7854, and λ_s₂ = 0; blue lines (the third case) are for $c_{2 x} = c_{2 y} = c_{2 φ} = 10$ , $c = 0.2$ , θ₁ = π/4, and λ_s₁ = λ_s₂ = 0.

From Figure 6, it can be seen that the optimal values of structural parameters for isolating the vibration transferred from $M_{1}$ to $M_{2}$ . It reveals that there is a resonant peak when the pre-deformation of springs equals to zero as the first case. For increasing the damping coefficients in different directions $c_{2 x}, c_{2 y}, and c_{2 φ}$ and the viscous friction of joints c as the second case shown in Figure 6, although the vibration peak is reduced, the displacement transmissibility is larger than 1 for low frequency band and the vibration is increased in high frequency band. However, as the second case shown in Figure 6, by adjusting the structural parameters λ_s₁ and θ₁ for achieving $K_{x} = K_{y} = K_{φ} = 0$ , the vibration of $M_{2}$ could be mostly isolated in a large frequency band since the MD QZS property is realized.

As the adjustable structural parameters in SLSs are designed for the condition $K_{x} = K_{y} = K_{φ} = 0$ , the MD QZS property could be achieved. Thus, the vibration of $M_{2}$ could be mostly isolated and the transmissibility function $H_{2} (ω)$ approaches to zero in all frequency bands. Therefore, an approximate fixed point is realized and the relative motions between $M_{2}$ and $M_{1}$ measured by distance sensors could describe the absolute motions of $M_{1}$ in different directions. Figure 7 is the measurement effectiveness of the absolute motions of $M_{1}$ for harmonic excitation, multi-frequency excitation, and random excitation, respectively.

Figure 7.

The comparison between the absolute motions $x_{1}$ , $y_{1}$ , and φ₁ (black solid lines) and the relative motions $- {\hat{x}}_{2}$ , $- {\hat{y}}_{2}$ , and $- {\hat{φ}}_{2}$ (red dashed lines) for (a) harmonic base excitation, (b) multi-frequency base excitation, and (c) random base excitation.

The amplitude of motion should be a small vibration for the measurement accuracy, and thus the amplitude of excitation used in Figure 7 is in the range of [0, 0.05] (m). For the measurement of MD vibration of $M_{1}$ , from Figure 7, it has proved that the vibration of $M_{2}$ could be mostly isolated with appropriate values of structural parameters, and thus the relative motions ${\hat{x}}_{2}$ , ${\hat{y}}_{2}$ , and ${\hat{φ}}_{2}$ are approximately equal to $- x_{1}$ , $- y_{1}$ , and −φ₁, respectively. However, for the multi-frequency excitation and random excitation, by numerical simulation, Figure 7 shows the comparison between the absolute motions $x_{1}$ , $y_{1}$ , and φ₁ of the measurement object $M_{1}$ and the relative motions ${\hat{x}}_{2}$ , ${\hat{y}}_{2}$ , and ${\hat{φ}}_{2}$ which could be measured easily.

Application in time-delayed control

Feedback signals

Since the vibrations of $M_{2}$ for different directions are close to zero for the MD QZS property induced by SLSs with pre-deformed springs, the relative motions between $M_{1}$ and $M_{2}$ could approximately reflect the absolute motions of $M_{1}$ . Three distance sensors are fixed on M₂ in the sensor to obtain the relative motions between $M_{2}$ and $M_{1}$ . The signals measured by the distance sensors for different directions are shown in Figure 8.

Figure 8.

The signals measured by the distance sensors fixed on $M_{2}$ for multi-direction vibration.

As shown in Figure 8, the signals measured by the distance sensors are $S_{10} - S_{1}$ , $S_{20} - S_{2}$ , and $S_{30} - S_{3}$ where $S_{10}$ , $S_{20}$ , and $S_{30}$ are the original distance at rest which are $S_{10} = S_{20} = 2 n_{1} l_{1} \sin θ_{1}$ , $S_{30} = 2 n_{2} l_{2} \sin θ_{2}$ , and $S_{1}$ , $S_{2}$ , and $S_{3}$ are the distances for vibration at any time. The values of $S_{10} - S_{1}$ , $S_{20} - S_{2}$ , and $S_{30} - S_{3}$ are dependent on the relative motions between $M_{1}$ and $M_{2}$ . Therefore, the distances measured by the sensors could be written as

${\begin{matrix} S_{10} - S_{1} = {\hat{y}}_{2} - (L_{20} + {\hat{x}}_{2}) {\hat{φ}}_{2} \\ S_{20} - S_{2} = {\hat{y}}_{2} + (L_{20} - {\hat{x}}_{2}) {\hat{φ}}_{2} \\ S_{30} - S_{3} = {\hat{x}}_{2} - 2 n_{1} l_{1} \sin θ_{1} {\hat{φ}}_{2} \end{matrix}$ (11)

The second equation in equation (11) minus the first equation in equation (11), we could obtain $2 L_{20} {\hat{φ}}_{2} = S_{1} - S_{2} + S_{20} - S_{10}$ , and since $S_{10} = S_{20}$ , the relative motion ${\hat{φ}}_{2}$ could be expressed as

${\hat{φ}}_{2} = \frac{(S_{1} - S_{2})}{(2 L_{20})}$ (12)

Then, from the third equation, the relative motion ${\hat{x}}_{2}$ could be expressed as

${\hat{x}}_{2} = 2 n_{2} l_{2} \sin θ_{2} - S_{3} + n_{1} l_{1} \sin θ_{1} (S_{1} - S_{2}) / L_{20}$ (13)

and then substituting ${\hat{x}}_{2}$ and ${\hat{φ}}_{2}$ into the first equation of equation (11), the relative motion ${\hat{y}}_{2}$ could be expressed as

$\begin{array}{l} {\hat{y}}_{2} = 2 n_{1} l_{1} \sin θ_{1} - \frac{S_{1} + S_{2}}{2} + \frac{(2 n_{2} l_{2} \sin θ_{2} - S_{3}) (S_{1} - S_{2})}{2 L_{20}} \\ + \frac{n_{1} l_{1} \sin θ_{1}}{2 L_{20}^{2}} {(S_{1} - S_{2})}^{2} \end{array}$ (14)

The relative motions ${\hat{x}}_{2}$ , ${\hat{y}}_{2}$ , and ${\hat{φ}}_{2}$ , which describe the absolute motions of the vibration system $M_{1}$ with high precision, could be obtained by the simple calculation from equations (12) to (14) via the signals measured by the distance sensors $S_{1}$ , $S_{2}$ , and $S_{3}$ .

Time-delayed control effectiveness

Because the sensor could obtain the approximate absolute motions ${\hat{x}}_{1}, {\hat{y}}_{1}, and {\hat{φ}}_{1}$ of the measurement object $M_{1}$ which is described by $- {\hat{x}}_{2}, - {\hat{y}}_{2}, and - {\hat{φ}}_{2}$ . The signal could be utilized as a time-delayed feedback signal, and the time delay could be adjusted in a wide range since the integral processing is reduced. Figure 9 shows the time-delayed feedback control for the vibration system $M_{1}$ by utilizing the measurement signals of the relative sensors directly.

Figure 9.

MD vibration system with time-delayed control.

Based on the signals measured with the three sensors for relative motions ${\hat{x}}_{2}, {\hat{y}}_{2}, and {\hat{φ}}_{2}$ , the time-delayed feedback control is introduced. Providing control signal $g_{1} {\hat{x}}_{2} (t - τ)$ to controller 1, signal $g_{2} {\hat{y}}_{2} (t - τ) + g_{3} {\hat{φ}}_{2} (t - τ)$ to controller 2, and signal $g_{2} {\hat{y}}_{2} (t - τ) + g_{3} {\hat{φ}}_{2} (t - τ)$ to controller 3, the controlled dynamic equation of the system is as

${\begin{matrix} M_{2} \frac{d^{2} {\hat{x}}_{2}}{d t^{2}} + K_{x} {\hat{x}}_{2} + F_{x} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2}) + [C_{x} + D_{x} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2})] \frac{d {\hat{x}}_{2}}{dt} = - M_{2} \frac{d^{2} x_{1}}{d t^{2}} \\ M_{2} \frac{d^{2} {\hat{y}}_{2}}{d t^{2}} + K_{y} {\hat{y}}_{2} + F_{y} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2}) + [C_{y} + D_{y} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2})] \frac{d {\hat{y}}_{2}}{dt} = - M_{2} \frac{d^{2} y_{1}}{d t^{2}} \\ J_{2} \frac{d^{2} {\hat{φ}}_{2}}{d t^{2}} + K_{φ} {\hat{x}}_{2} + F_{φ} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2}) + [C_{φ} + D_{φ} ({\hat{x}}_{2}, {\hat{y}}_{2}, {\hat{φ}}_{2})] \frac{d {\hat{φ}}_{2}}{dt} = - J_{2} \frac{d^{2} φ_{1}}{d t^{2}} \\ M_{1} \frac{d^{2} x_{1}}{d t^{2}} + 2 k_{h 1} x_{1} + c_{x 1} \frac{d x_{1}}{dt} = 2 k_{h 1} z_{x} + c_{x 1} \frac{d z_{x}}{dt} + α_{x} {\hat{x}}_{2} (t - τ) \\ M_{1} \frac{d^{2} y_{1}}{d t^{2}} + 2 k_{v 1} y_{1} + c_{y 1} \frac{d y_{1}}{dt} = 2 k_{v 1} z_{y} + c_{y 1} \frac{d z_{y}}{dt} + α_{y} {\hat{y}}_{2} (t - τ) \\ J_{1} \frac{d^{2} φ_{1}}{d t^{2}} + 2 k_{v 1} L_{10}^{2} φ_{1} + c_{φ 1} \frac{d φ_{1}}{dt} = 2 k_{v 1} L_{10}^{2} z_{φ} + c_{φ 1} \frac{d z_{φ}}{dt} + α_{φ} {\hat{φ}}_{2} (t - τ) \end{matrix}$ (15)

where $α_{x} = 2 g_{1}$ , $α_{y} = 2 g_{2}$ , $α_{φ} = 2 g_{3} L_{10}$ , and τ is the adjustable time delay introduced into the feedback control. Figure 10 shows the control effectiveness of the feedback control with/without time delay.

Figure 10.

Vibration transmissibility from base to the controlled object $M_{1}$ for different control parameters: (a) horizontal direction, (b) vertical direction, and (c) rotational direction.

The transmissibility function $H_{1} (ω)$ reflects the vibration energy transferred from the base to $M_{1}$ . Figure 10 shows the vibration of $M_{1}$ with/without active control. For increasing the control strength g, the resonance peak of frequency is reduced and the effective isolation frequency band is increased; also, with the increasing of time delay τ, the resonance peak is reduced since time delay could adjust the equivalent stiffness and damping properties simultaneously. Since the feedback-control signals are the relative motions ${\hat{x}}_{2}, {\hat{y}}_{2}, and {\hat{φ}}_{2}$ which are very approach to $x_{1}$ , $y_{1}$ , and −φ₁ and utilized directly as the control signals in equation (15), the time delay in integration and computation can be much reduced. Therefore, the time delay could be added into control processing as adjustable control parameters for better vibration suppression effectiveness.

Conclusion

A sensor for MD absolute motion measurement based on its significant vibration isolation effectiveness is proposed for vibration system in its processing of moving (i.e. a moving vehicle and isolation platform in spacecraft) in this article. The MD QZS property is utilized for the realization of quasi-fixed point, and thus the sensor can measure the MD absolute motions approximately by the relative motion between the quasi-fixed point in space and the measurement object. Adjusting the structural parameters of the proposed sensor system, the relative motion measured is closed to the absolute motion of the measurement object in that the MD QZS system has remarkable isolation characteristic, which means that the isolation object of the MD QZS system can almost remain in steady state with small magnitude vibration. The better the isolation of the QZS system, the smaller the relative error between the signal measured and the absolute motion. For different styles of vibration, the proposed MD QZS vibration sensor is effective and convenient to measure the absolute motion of a MD vibration system. The analysis provides a useful insight into the design, optimization, and application for time-delayed control of the MD QZS vibration sensor.

Footnotes

Academic Editor: Mohammad Razzaque

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 Shanghai Sailing Program (16YF1408000),the Natural Science Foundation of Shanghai (16ZR1423600),and National Natural Science Foundation of China (11602141).

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