Structural Analysis And Topology Optimization Design For Bandwidth Extension of Magnetoelectric Seismometer

7 Magnetoelectric seismometers can measure earthquake information and play an important role in earthquake monitoring. 8 Aiming at the wider effective frequency bandwidth of magnetoelectric seismometers, a novel seismometer based on 9 topology optimization structural pendulum is reported. The topology optimization of leaf spring structure in 10 magnetoelectric seismometer is designed, the natural frequency and spurious frequency characteristics of the novel 11 seismometer are analyzed. Based on variable density theory, the Solid Isotropic Material with Penalization (SIMP) model 12 of the seismometer is established, and the Method of Moving Asymmetric (MMA) is adopted to obtain the optimal 13 topology structure. The finite element analysis using ANSYS shows that novel seismometer after topology optimization 14 structure is characteristic with lower natural frequency and higher spurious frequency than that of before optimization 15 seismometer. The real vibration experimental results indicate that after topology optimization, the effective frequency 16 bandwidth of seismometer is increased by 55.50%, improving from [1s, 51Hz] to [4s, 78Hz].


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Seismometers, as ground motion measurement instruments, have been widely used in geological hazard predictions, 20 earthquake early warning, engineering exploration and nuclear explosions monitoring [1][2][3][4]. A seismometer can turn 21 ground velocity within a certain bandwidth into an electric output signal without distortion [5]. The response signal of 22 seismometer will decline or become distorted when it falls below natural frequency or near spurious frequency. Therefore, 23 seismometers with a lower natural frequency and a higher spurious frequency play an important role in ground motion 24 measurements with wider bandwidth, ensuring the reliability of quality data acquisition.

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The structural changes in leaf spring of seismometer can search for lower natural frequency and higher spurious 26 frequency with wider effective frequency bandwidth to a certain extent. Faber and Maxwell alter the leaf spring structure 27 of a seismometer to increase spurious frequency which can expand its frequency bandwidth [6]. Woo invents inner and 28 outer annular rings connected by spring arm for a seismometer, which improves the ratio between spurious frequency and 29 natural frequency to broaden bandwidth and reduce signal distortion [7]. Wielandt and Streckeisen propose a 30 seismometer with a rectangular leaf spring instead of a zero-length helical spring, which has a broader bandwidth [8]. 31 Yang et al. adopt multiple pairs of leaf springs in a suspension system for a seismometer, which increases the ratio of 2 spurious frequency to natural frequency in order to expand its frequency range [9]. Xin et al. design a multilayer 33 spiral-corrugated cantilever beam for a piezoelectric seismometer, which has a lower natural frequency [10]. Yao et al.

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propose a novel leaf spring as a substitute for traditional spring to expand effective bandwidth of a seismometer [11]. The 35 existing topology structure of original leaf spring can search for lower natural frequency and higher spurious frequency 36 in a certain extent. However, the topology structure of leaf spring cannot be optimized which has a large space for 37 expanding bandwidth of a seismometer.  The ground motion measurement can directly sense by the mechanical pendulum of seismometer [12]. It suspends an 50 inertial reference mass structure from a rigid fixed shell structure, which is connected by a leaf spring. When local 51 seismic waves occur, the fixed shell structure coupled with the ground vibrates immediately. The inertial reference mass 52 still tends to remain static, so the relative movement is generated between mass and shell. Using this relative motion, the 53 ground motion can be measured. As shown in Fig. 1    The finite element analysis model of mechanical pendulum is built to study the influence of leaf spring size on the 88 natural frequency and spurious frequency. After solid modeling, defining material property and meshing grid, the unit 89 and node number in finite element model is set to 84349 and 148061, respectively. When seismometer measures the 90 ground vibration, the hammer end connected by the leaf spring swing freely and the pressure plate end connected by the 91 leaf spring is fixed. So it has fixed constraints on the end of pressure plate. Loading the stand earth gravity, the finite 92 element analysis of mechanical pendulum is carried out. The first order modal shape and the second order modal shape 93 of mechanical pendulum is shown in Figure 3. It can be seen from Fig. 3(a) that the main vibration of the first order 94 modal of mechanical pendulum is bending motion, while the main vibration of the second order modal of mechanical pendulum is the up and down variation, while the twisting motion is the unexpected motion. Moreover, the excessive 97 twisting motion could destroy the mechanical pendulum, so it is necessary to has a lower natural frequency and higher 98 spurious frequency of the mechanical pendulum.
7 Wherein x is the design variable, i x is the unit density, and m in x is the lower limit of the unit density. N 136 represents the element number, V  is the allowable volume of material,  represents eigenvalue, and f is the 137 natural frequency of mechanical pendulum.

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The sensitivity of natural frequency in mechanical pendulum can be shown as follows,

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The steps of MMA to solve seismometer topology optimization problem are that:

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Step 1: Selecting the initial iteration point of design variables. The size of initial finite element in design 153 domain is 9562.

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Step 2: Calculating values of the natural frequency function and constraint function, and sensitivity of the 155 natural frequency in mechanical pendulum at the current iteration point of design variable.

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Step 3: Solving next iteration point of design variables. Generating MMA sub-problem with adding 8 Step 4: Judging the termination conditions. If the convergence condition is met, acquiring the optimal 160 topology structure; if not, return to Step 2. 199 Table 1 shows the natural frequency and spurious frequency of two kinds of pendulum. The natural frequency of novel 200 pendulum with topology optimization leaf spring is decreased with before optimization pendulum from 6.24 Hz to 201 3.21Hz, which is 48.55% reduction. Meanwhile, the spurious frequency of novel optimization pendulum is increased 202 with that of before optimization pendulum from 105.47Hz to 124.14Hz, which is 17.70% increase. The finite element 203 analysis shows that novel seismometer with topology optimization structure is characteristic with lower natural frequency 204 and higher spurious frequency than that of before optimization seismometer.
205 Table 1 Natural frequency and spurious frequency of two kinds of pendulum.

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The authors declare that they have no known competing financial interests or personal relationships that could have 293 appeared to influence the work reported in this paper.