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More than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\\n\\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\\n\\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\\n\\nAdditionally, each book published by IntechOpen contains original content and research findings.
\\n\\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Simba Information has released its Open Access Book Publishing 2020 - 2024 report and has again identified IntechOpen as the world’s largest Open Access book publisher by title count.
\n\nSimba Information is a leading provider for market intelligence and forecasts in the media and publishing industry. The report, published every year, provides an overview and financial outlook for the global professional e-book publishing market.
\n\nIntechOpen, De Gruyter, and Frontiers are the largest OA book publishers by title count, with IntechOpen coming in at first place with 5,101 OA books published, a good 1,782 titles ahead of the nearest competitor.
\n\nSince the first Open Access Book Publishing report published in 2016, IntechOpen has held the top stop each year.
\n\n\n\nMore than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\n\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\n\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\n\nAdditionally, each book published by IntechOpen contains original content and research findings.
\n\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\n\n\n\n
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Precision positioning and alignment are critical to an emerging class of small-scale manufacturing and numerous motion control applications. The drive for better performance steers design and control effort into achieving high tolerances and stringent specifications in terms of parameters, such as resolution, range, load-capacity, and bandwidth. Examples of applications needing precision positioning and alignment include (i) high-bandwidth steering of mirrors in telecommunication applications [1], (ii) tool-sample alignment in stamping applications such as imprint lithography [2, 3] and micro-contact printing [4], and (iii) alignment of optically flat surfaces brought in close proximity to characterize fields and forces on small-scales, such as the Casimir force [5, 6].
\n\t\t\tA widely used set of designs for precision applications described above involve compliant mechanisms based on slender beam modules, also referred to as flexures [7, 8]. The advantages flexures offer are mainly smooth elastic motion without non-linearities such as friction or backlash [9]. Flexure-based mechanisms such as the diaphragm flexure involve the payload suspended on a radial or tangential arrangement of flexural beams. Various forms of such flexures have appeared over the past few decades for applications such as MEMS mirrors, and in angle alignment and guidance applications [4, 10, 11]. Analysis of the statics and dynamics of flexure-based mechanisms have been extensively studied [9, 12].
\n\t\t\tWhile flexure-based engineering designs have been around for many decades [13], designing them for dynamic performance has sought little attention. Few publications [14, 15] have appeared in this context. The design for dynamical performance of flexures in the context of mechanical advantage is detailed in [14]. A finite-element approach based on Euler-Bernoulli beam bending theory is formulated for analyzing dynamics in [15] and optimizing the design space for precision flexure-based applications in [16].
\n\t\t\tWe build on the work presented in the literature and integrate models that can enhance the accuracy in predicting the dynamics of a given flexure-based design by including the effects of distributed mass and compliance of the flexures covering shear and rotational effects. These effects are shown to dominate at small flexure lengths. Further, we use the models to characterize design space parameters such as range, load-capacity, and bandwidth. While most of the current literature in flexure-based designs focuses on static values of performance variables such as angular position, or acceleration, we present a state-space approach for characterizing the bounds on these variables in the frequency domain. This is critical for ensuring that performance requirements are met within the usually large bandwidths of operation, an example application being fast steering of mirrors in telecommunications [1].
\n\t\t\tA diaphragm flexure as a parallel kinematic mechanism with a central rigid mass connected by n = 3 flexural beam units to the ground. The dimension D\n\t\t\t\t\t\t0 = 2l+ 2R is referred in this chapter as the footprint of the mechanism. The Z axis is shown pointing out of the page.
This chapter is organized as follows. In Section 2 we assemble lumped parameter models for a diaphragm flexure design. Section 3 covers the closed-form characterization of the design space from the dynamic models. Non-dimensional design plots and the efficacy of the models in capturing shear effects at small flexure lengths are addressed in this section. A state-space approach is used for characterizing key performance variables in Section 4. This section taps into multi-input multi-output (MIMO) analysis tools to develop a framework for mapping design requirements over to the state-space. The effects of manufacturing errors are studied in the context of decoupling and asymmetry in this section. Finally, we conclude with a summary of the contributions of the work.
\n\t\tOur goal is to capture the out-of-plane behavior, i.e. the vertical translation, pitch, and roll degrees of freedom of diaphragm flexures used in precision angle alignment mechanisms. In this section, we assemble dynamic models for a class of diaphragm flexures – namely, those applying radial constraints on a central rigid mass via flexural beam units. We derive lumped parameter models representing the mass and stiffness of the diaphragm flexure. The applications of compliant mechanisms using the simple flexural beam units studied here span multiple scales from MEMS to meso-scale systems. In all these applications, it is desirable to develop accurate models since the mechanisms constitute the plant in the overall closed-loop control system.
\n\t\t\tTo model bending of the flexural beam unit, shown in Fig. 2, we use a Timoshenko beam [17] model since a simple Euler-Bernoulli beam model cannot capture the effects of shear and rotational inertia. As will be shown later, these effects become significant for short beams, which are widely used in compliant mechanisms spanning multiple length scales. To model torsion of the flexural element, we use St. Venant\'s torsion formulation assuming that (i) the effects of restrained warping are negligible and (ii) bending and torsion are decoupled. Further, we assume that the deflections of the flexural beam element are small (an order of magnitude smaller than the beam thickness) and hence, neglect the effects of axial stretching and the resultant stress stiffening along the length of the beam element.
\n\t\t\t\tSchematic diagram showing a flexural beam element with deflection w(x, t), slope Ө(x, t), and twist about the X-axis by an angle ϕ(x, t).
Under the above-mentioned assumptions, the distributed parameter model for the beam is well-documented in the literature [17] as being depicted by a set of partial differential equations in the deflection w(x, t), slope Ө(x, t), and angle of twist ϕ(x, t) listed in Section A. 1 of the Appendix.
\n\t\t\t\tThe infinite-dimensional behavior governed by the set of partial differential equations can be approximated to that arising for a one-element model using the method of assumed modes [18]. By this method, the infinite-dimensional behavior of the mechanism is approximated to a finite-series made up of spatially varying mode shape functions (or trial functions) with temporally varying mode amplitudes [19]. Since a one-element model is used for the beam, the distributed properties of the beam are lumped to the node at the guided end of the beam; the fixed node of the beam has no lumped mass or stiffness. Hence, from the three displacements assumed for the guided end of the beam, a three-DOF lumped parameter model can be derived.[1] -\n\t\t\t\t
\n\t\t\t\tThe detailed application of the assumed modes method to the set of partial differential equations governing the motion of the Timoshenko beam can be found in parts in [20] and [21]. The key results used in this work are highlighted here. Under the geometric boundary conditions of (i) one end x = 0 of the flexural beam being grounded and (ii) the other end x = lsubject to generalized displacements V(t) = [w(l, t) Ө(l, t) ϕ(l, t)]T, (where w(l, t) is the vertical deflection, Ө(l, t) the slope, and (ϕ,\n\t\t\t\t\n\t\t\t\t\tt) is the angle of twist), the corresponding 3×3 matrices – mass Mf and stiffness Kf are as given below:
\n\t\t\t\twhere the matrix values depend on material properties and geometry, and are tabulated in Section A. 1 of the Appendix. Zeroes in either matrix result from the decoupling assumed between bending and torsion. The lumped mass and stiffness matrices are used as building blocks for assembling dynamic models of mechanisms involving flexural beam units. Note that we need to restrict these matrix building blocks to parallel kinematic configurations since the geometric boundary conditions corresponding to x = 0 have been assumed to be all zero. Formulations for serial kinematic configurations can be developed by altering this set of geometric boundary conditions [20].
\n\t\t\tHere, we formulate the dynamics of parallel kinematic mechanisms that contain a rigid body connected to the ground through a multitude of flexural beam units. We integrate the lumped parameter model for the flexural beam in Section 2 with rigid body dynamical models using appropriate transformations to obtain the global model [15]. These transformations are chosen to ensure the continuity of nodal displacements at the interface between the rigid body and the flexures.
\n\t\t\t\tConsider a parallel kinematic mechanism with a central rigid circular disk centered at the origin and parallel to the horizontal XY plane of the cartesian XYZ space, as shown in Fig. 1. In the rest position, the principal axes of the disk X\', Y\', and Z\' coincide with the cartesians axes X, Y, and Z, respectively. Let the disk be of radius R, thickness T, mass MR, and moments of inertia JRxx and JRyy about the X and Y axes respectively. A number n of slender beam flexures, each of width W, thickness H, and length l, are in the XY plane connecting every peripheral point Pi to the ground. The coordinates of Pi in the X\'Y\' plane are (RcosαI,Rsinαi) with angles αi ∈[0, 2π) for i = 1, 2, 3… n.
\n\t\t\t\tSince the beams provide high axial (and hence in-plane XY) stiffness and low out-of-plane stiffness, we expect that the dominant modes correspond to the out-of-plane motion, namely vertical deflection, pitch, and roll. We hence assume that the out-of-plane motion of the disk is decoupled from the in-plane motion, i.e. the center of the disk always moves only vertically. For small vertical deflection z(t) of the center of mass, and small angular rotations Өx(t) and Өy(t) about the X and Y axes respectively, the principal plane X\'Y\' of the disk moves out of the XY plane to the one depicted by
\n\t\t\t\tFor continuity of displacement at each of the nodes Pi, Eq. (2) can be used to show that the end-displacements Vi(t) of every ith flexure are related to the global generalized (rigid body) displacements VR(t) as follows:
\n\t\t\t\twhere the transformation matrix R=
\n\t\t\t\tBased on the mass and stiffness properties of the individual flexural beam units connected to the central rigid body, we need to derive the mass M and stiffness K properties of the assembly.
\n\t\t\t\tBy formulating the Lagrangian of the assembly in terms of the rigid body displacements VR(t), we develop the lumped mass and stiffness matrices of the overall parallel kinematic mechanism as follows:
\n\t\t\t\twhereMfi and Kfi are the lumped mass and stiffness matrices, respectively, of the individual flexure building blocks given in Section 2.1, and Tables 4 and 5; MR is the mass matrix of the rigid body and is given by:
\n\t\t\t\tThe equations of motion of the lumped parameter representation, for the free response case, is in the form given below:
\n\t\t\t\tNote that we have not presented the modeling of damping matrix B in this chapter. Models such as proportional damping, given by B = bmM+bkK, are widely used in the literature [22], where bm and bk are constants that depend on material properties and are experimentally determined from sine-sweep frequency response measurements. For the design of active or passive damping in flexure mechanisms, a survey and foam-based methods are detailed in [23].
\n\t\t\tIn this section, we use the dynamic models developed from Section 2.3 to examine (i) the influence of geometric arrangement of flexures on coupling between the global generalized displacements or modes, (ii) the best bandwidth possible for a given foot-print of a symmetric diaphragm flexure mechanism, and (iii) the performance trade-offs between parameters such as range, bandwidth, and load-capacity for the same.
\n\t\t\tFrom the equations of motion of the diaphragm flexure derived in Section 2.3, the geometrical layout of flexural constraints that allow for static and dynamic decoupling of the three DOFs can be determined. For static decoupling, the off-diagonal terms in the overall stiffness matrix K should be zero. For dynamic decoupling, the off-diagonal terms in both the mass M and stiffness K matrices should be zero.
\n\t\t\t\tStatic and dynamic decoupling is desirable, for instance, when the diaphragm flexure mechanism is controlled to vertically position the central rigid mass while ensuring low error motions in the other DOFs, namely pitch and roll. Stable decoupled systems tend to be more amenable to low error motions even under open-loop control. It should, however, be noted that perfect decoupling cannot be achieved in practice owing to non-uniformities arising from manufacturing or material properties. Nonetheless, designing a compliant mechanism to be as close to a decoupled dynamic system as possible is desirable [24]. Here, we examine conditions under which such decoupling is possible for a diaphragm flexure mechanism.
\n\t\t\t\tSubstituting the values of R from Eq. (4) into Eqs. (5) and (5), and given Eqs. (1) and (6), the conditions necessary for the off-diagonal terms in the global mass M and stiffness K matrices to be zero are as follows:
\n\t\t\t\tHence, the geometric arrangement of a number n ≥ 3 of flexures around the central rigid mass allows for the overall mechanism to be close to being statically and dynamically decoupled if the above conditions are satisfied. Note here that each individual flexure of the mechanism has its vertical deflection and slope coupled (both statically and dynamically), it is only the parallel combination of three or more of them that allow for the decoupling between the global modes to occur. Some possible design layouts that satisfy Eq. (8) are discussed in Section A. 2 of the appendix. The effect of deviations from perfect symmetry on design requirements such as maximum deflection, velocity, and acceleration are studied in Section 4.
\n\t\t\tThe best –3 dB bandwidth possible for a closed-loop system depends on many factors, including the natural frequencies or poles of the open-loop plant. Fig. 3 shows the plots of undamped natural frequencies of the first three modes of the symmetric diaphragm flexure of Fig. 1. The plots show the variation corresponding to diaphragm flexure configurations with flexural beam length, l, varied in the range of about 0.5 in to 3 in, while keeping the footprint 2l+2R at a constant value of 7 in. This constant footprint is chosen as a scaling factor for the length dimension and will be used in Section 3.3 to normalize all lengths in the design to formulate a non-dimensional study.
\n\t\t\t\tThe plots of Fig. 1 show values of the undamped natural frequencies obtained from models based on St. Venant\'s torsion theory and one of two distinct beam bending theories – either (i) Timoshenko beam bending theory, or (ii) Euler Bernoulli beam bending theory. As explained earlier, the former beam bending theory accounts for shear and rotational effects versus, while the latter does not. In the plots of the figure, the frequency values obtained from a commercial FEA package are superimposed for comparison of the chosen models.
\n\t\t\t\tPlot of undamped natural frequencies of first three modes of the symmetric diaphragm flexure of Fig. 1 for flexure beam length l varying from about 0.2 in to 3 in, keeping foot-print D\n\t\t\t\t\t\t\t0 = 2l+2R at a constant value of 7in. Other parameter values used in the simulation are: beam width W = 0.75 in, beam thickness H = 0.1 in, central rigid disk thickness T = 2.5 in, poisson\'s ratio v = 0.33, elastic modulus E = 69 GPa, density ρ = 2700\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t.
The trends observed for the variation of natural frequencies for small flexural beam lengths is as expected, since small beam lengths result in large stiffness. Since the footprint is maintained constant, smaller beam lengths also imply large radius of the central disk and hence larger moving mass. However, the cubic dependence of stiffness on beam length dominates over the square dependence of mass on radius of the disk; hence the large natural frequencies at short beam lengths. For large beam lengths, the radius of the central disk is small, and hence the moving mass.[1] - This effect is marginally larger than the loss in stiffness and hence the slight increase in natural frequency at large beam lengths.
\n\t\t\t\tFor flexural beam lengths smaller than the shear approximation length factor c ≈ 0.6 in, the Timoshenko model matches the trend from the FEA data better than the Euler-Bernoulli model. This confrms the prediction that shear effects dominate at small beam lengths and agrees with similar observations supporting the Timoshenko beam bending models for depicting the natural frequencies of short AFM cantilevers in [25].
\n\t\t\t\tClosed-form expressions for the natural frequency of the first three-dominant modes in the decoupled case, for large flexure lengths, are presented in Table 1. These expressions can be used as part of formulating an optimization problem, or to gain useful insights from parametric dependencies in designing a precision angular alignment setup based on diaphragm flexures.
\n\t\t\t\tClosed-form expressions for natural frequencies of first three modes of diaphragm flexure.
The design space for utilizing flexure-based precision angular alignment mechanisms can be characterizing in terms of key parameters such as the range, payload capacity, and bandwidth. Fig. 4 shows the variation of the key non-dimensionalized performance parameters as a function of the non-dimensional flexural beam length l for all diaphragm flexures with a constant footprint of D = 2l + 2R. The performance parameters plotted in the figure are (i) the natural frequencies of the first three modes, namely deflection z and the two rotations Өx and Өy, (ii) the maximum load-capacity, Fmax, defined as the load that causes the resultant axial stress in the flexural beams to reach the yield strength, σY, of the material within a safety factor ɳ, and (iii) the maximum vertical deflection δmax, i.e. range under a given load. The normalization factors used for non-dimensionalizing the parameters are tabulated in Table 3, where ρ and E are the density and elastic modulus, respectively, of the material constituting the diaphragm flexure σ.
\n\t\t\t\tFlexural Building blocks Comparison
Normalization factors used for Design Parameters in Fig. 4.\n\t\t\t\t\t\t
The trade-off between load-capacity and range at different flexural beam lengths is evident from the figure. Small beam lengths allow for large load capacity and low range, whereas large beam lengths allow for low load capacity and large range. Natural frequencies are relatively low for intermediate beam lengths. The trade-off between natural frequencies and range is evident for small beam lengths, since stiffer beams have smaller deflections. The compromise between natural frequency and range is relatively mild at large beam lengths. An extension of the dynamic performance tradeoff characterization to the case of n = 6 flexural beams is presented in Fig. 5. From kinematic exact-contraint theory, it is known that only three constraints are needed to curb the three in-plane degrees of freedom of the rigid body. Thus for the case n = 6, three of the remaining constraints are redundant.
\n\t\t\t\tNon-dimensional plots capturing the key dynamic performance parameters for diaphragm flexures of Fig. 3 with n = 3 flexural beams of the same footprint D, but different flexure lengths ranging in the approximate range 0-0.45 D. The parameters of interest are (i) the natural frequency of first three modes \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tӨ\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t \n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tӨ\n\t\t\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t, all normalized by f\n\t\t\t\t\t\t\t0, (ii) load capacity Fmax normalized by F\n\t\t\t\t\t\t\t0 and (iii) static vertical range δmax normalized by δ\n\t\t\t\t\t\t\t0.
Design parameter variations (shown as solid lines) captured for the case of double the number of flexures, i.e. n = 6, as compared to the case of n = 3 (shown as dashed lines) captured in Fig. 4.
The redundancy comes with added features which are desirable and facilitate improving the performance as follows. Since the load is distributed between larger number of flexures, the load-capacity envelope in this case is pushed higher to double that of the n = 3 case. The stiffnesses sum up since the flexural beams are arranged in a parallel combination. The resultant range is hence unchanged from that of the case of n = 3. The added slight benefit is that there is no reduction in the natural frequencies. Instead, there is an almost \n\t\t\t\t\t\t
Characterization of parameters such as maximum displacement (or range), maximum velocity and acceleration is critical for precision motion control applications. A procedure for identifying the static (ω = 0) values of these parameters is developed for precision control of ball-screw drives in [26]. In this section, we present a state-space approach for determining the design performance parameters – maximum values of deflection (or range), velocity, and acceleration that are possible not only for static (ω = 0) but for a large range of operating frequencies.
\n\t\t\tWe follow the approach for the case of a diaphragm flexure used for precision angle alignment. To account for the case when symmetry cannot be ensured for the diaphragm flexure, we assume a coupled multi-input multi-output (MIMO) model, as against a collection of independent single-input single-output (SISO) models. We focus our analyses to parameters such as maximum vertical and angular displacement (range), velocity, and acceleration. The presented approach can be extended to map other design parameters to the state-space. Further, while the ideas presented here are general and applicable to the case when state or output feedback control is used as well, we focus our analysis on just the open-loop system.
\n\t\t\tWe begin with a state vector x containing the generalized coordinates depicting the equations of motion of the system. One choice of state variables could be the generalized displacements and their first-order derivatives.
\n\t\t\t\tThe goal here is to find the maximum values of displacements, velocities, and acceleration for any set of inputs (which can be oriented in any direction in the input space). That means we need to compute the upper bounds on the amplification of a scalar component xi, which is derived as:
\n\t\t\t\twhere the ith element of the row vector Ei is 1 and the rest of the m – 1 elements are zero. The component xi can be any design variable, such as angular velocity, or vertical deflection of the diaphragm flexure.
\n\t\t\t\tFor a chosen control law, in the Laplace domain, the following relations hold between the state vector X(s)m\n\t\t\t\t\t×1, its ith component Xi(s), and the input vector U(s):
\n\t\t\t\tThe maximum amplification [27] of the component Xi(s) for a given input U(s) can be expressed as the 2-induced (Euclidean) norm of the gain matrix Ei\n\t\t\t\t\t1×\n\t\t\t\t\tmG(s)m\n\t\t\t\t\t×\n\t\t\t\t\tr. For the choice of Ei, the gain matrix reduces to the ith row of G(s). Hence, its 2-induced norm reduces to a vector norm, and is given by its lone singular value. This singular value of Ei\n\t\t\t\t\t1×\n\t\t\t\t\tmG(s)m\n\t\t\t\t\t×\n\t\t\t\t\tr is always smaller than or equal to the singular values of the matrix G(s) and hence provides a tighter bound on the amplification of Xi(s).
\n\t\t\tWe now apply the above formulation to the case of the diaphragm flexure of Fig. 3 to derive tight upper bounds[1] - on the amplification of state vector components, such as vertical deflection, or say, maximum angular velocity of the diaphragm flexure in a given control situation. We do not consider the feedback control problem here; however, the proposed method can be extended to that case as well.
\n\t\t\t\tThe state vector x(t) and the input vector u(t) for a configuration with three linear actuators pushing down on the central rigid body at three locations Qi(RcosβI,Rsinβi), for i = 1, 2, 3, are given as:
\n\t\t\t\tThe matrix A is assembled from components of M-1K from Eq. (5), and accounting for the derivative relationship pairs between the components of the state vector. By determining the force and moment components, the matrix B is given as
\n\t\t\t\tWith the choice of Ei as described earlier, the maximum bound on each of the components of the state vector are found as shown in Fig. 6 for the diaphragm flexure of Fig. 3 containing three flexural beam units arranged symmetrically around the central rigid mass, and with linear actuators located at angles, β\n\t\t\t\t\t1 = 0, β\n\t\t\t\t\t2 =\n\t\t\t\t\t\t
The maximum amplification of states for an unit input vector (along any direction in the position space) is plotted for the case of three flexural beam units arranged symmetrically around the central rigid mass.
The same analysis is repeated for the case when a 1º misalignment of one of the flexural beam units is caused by a manufacturing error. The system is now coupled, with deflection and angular position being dependent on each other, as seen from the two resonance peaks appearing in the variation of the singular values. This coupling implies that the angular position can assume exceedingly large values at a resonance frequency lower than that expected when perfect symmetry is ensured. The input directions that correspond to the maximum bound on a component Xi(jω) at a chosen frequency ω lie along the right eigen vectors of the matrix \n\t\t\t\t\t\t
In summary, the benefits of using this approach for specifying the design performance variables are two fold – (i) it is applicable in case of deviations from perfect symmetry, allowing to analyze the effects of the deviations, and (ii) it gives the bounds not only for the static case (ω= 0) but also for the desired frequency range of interest. This approach can be incorporated into the design decision-making process, along with other important considerations, such as constraints imposed by physical limits, for example, saturation of the actuators, or limit stops in the path of a motion stage.
\n\t\t\tWe have examined the need for diaphragm flexures in precision angular positioning applications. To accurately characterize the dynamics, we assembled lumped parameter models from mass and stiffness matrices for individual flexural building blocks and from the connected rigid body, as is done in typical finite element methods. Unlike previous
\n\t\t\tThe maximum amplification of states for an unit input vector (along any direction in the position space) is plotted for the case when a 1º misalignment of one of the flexural beam units is caused by a manufacturing error. Note that the system is now coupled as seen from the variation of the singular values.
works on flexure modeling, we found that Timoshenko beam bending models capture shear effects that dominate for short flexural beam lengths. We identified the key performance trade-offs in range, load-capacity, and natural frequencies of the first three modes of the diaphragm flexure. Redundancy in constraints was exploited to improve on load-capacity while ensuring the range or natural frequency requirements are met. Further, a control analysis based on singular value decomposition was formulated to capture the maximum values of performance variables such as linear or angular position and their derivatives. While perfect symmetry ensures decoupling between the modes, it was found that the amplification signature changes significantly in the presence of slight asymmetry caused by manufacturing errors. The dynamic modeling and state-space performance analysis detailed in this chapter are intended to serve as design tools for implementing high-precision motion control applications, and in particular, angular alignment based on diaphragm flexures.
APPENDIXThe set of partial differential equations governing Timoshenko beam bending and St. Venant\'s torsion under the assumptions stated in Section 2.1 can be written in terms of the deflection w(x, t), slope Ө(x, t), and angle of twist ϕ(x, t) as follows [21, 20]:
\n\t\t\t\twhere ρ, E, GJxx are the density, elastic modulus, and torsional rigidity, respectively; A, Iyy, and к are the cross-sectional area, area moment of inertia about the neutral axis Y, and a geometry-dependent shear-factor, respectively. For a rectangular cross-section к assumes a value of 0.833 [25].
\n\t\t\t\tThe component values of the mass M and stiffness K matrices are listed in Tables 4 and 5. The parameters used in the tables are \n\t\t\t\t\t\t
Mass matrix component values
Stiffness matrix component values
To identify the designs of diaphragm flexures that are close to being statically and dynamically decoupled, numerical methods can be used to solve the conditions given in Eq. (8). A geometric interpretation of the first two conditions of Eq. (8) is presented as:
\n\t\t\t\twhere j is the imaginary number \n\t\t\t\t\t\t
For odd values of n, n ≥ 3, a possible solution subset is \n\t\t\t\t\t\t\t\t
For even values of n, n ≥ 4, there is an \n\t\t\t\t\t\t\t\t
A solution subset to the case when n = 6, corresponding to three-fold symmetry with angle between any two constraints being 2α = 60º. The dashed lines denote the axes of symmetry. If the flexure beams on either side of the horizontal axis of symmetry are brought symmetrically closer by an angle ∆α, they still satisfy the decoupling conditions of Eq. (8).
This work was supported by funding grants from the Manufacturing Systems and Technology program under the Singapore MIT Alliance. The first author would like to thank Ajay A. Deshpande and Mythili R. Vutukuru for their discussions on parts of the analysis.
\n\t\tThe Japanese Islands are mainly composed of the Eurasian (EUR) and the North American (NA) plates, and a number of small islands are on the Philippine Sea (PHS) and the Pacific (PAC) plates (Figure 1). The PHS and PAC oceanic plates are subducting beneath the EUR and the NA plates. A number of earthquakes occurred both at the plate interfaces and within the plates.
\nName of plates and location.
After the Kobe earthquake in January 1995, the Japanese government enacted the Special Measure Law on Earthquake Disaster Prevention in July 1995. This was to promote a comprehensive national policy on earthquake disaster prevention. Based on this goal, the National Research Institute for Earth Science and Disaster Resilience (NIED) contracted the deployment of the nationwide high-sensitivity seismograph network (Hi-net) [1] since NIED had already accumulated the experience for the Tokyo metropolitan deep borehole array and operated the Kanto-Tokai seismic network since 1979. NIED operates the Hi-net with approximately 800 stations since 2000 [2] and the full range seismograph network (F-net) [3] with approximately 70 stations composed of broadband seismographs since 1994 [4]. The Japan Meteorological Agency (JMA), the national universities, and other institutes operate other seismic networks with a total of approximately 600 stations for the detection of microseismicity. NIED operates ocean-bottom seismic stations beneath the Sagami Bay, while the JMA operates offshore the Tokai and Boso regions. The Earthquake Research Institute, University of Tokyo, operates the network offshore Sanriku, and the Japan Agency for Marine-Earth Science and Technology (JAMSTEC) operates offshore Kushiro and Muroto networks. JAMSTEC started the construction of the Dense Oceanfloor Network System for Earthquakes and Tsunamis (DONET) [5] off Kii and Muroto Peninsulas near the Nankai Trough in 2010, and they started operation networks offshore Kii (in 2014) and Muroto (in 2016) Peninsulas. NIED deployed the Seafloor Observation Network for Earthquakes and Tsunamis along the Japan Trench (S-net) [6] after the 2011 offshore Tohoku Earthquake (the Tohoku-oki event), which began operating in 2016 [7, 8]. DONET was transferred to NIED from April 2016. NIED started the operation of Monitoring of Waves on Land and Seafloor (MOWLAS) composed of Hi-net, F-net, S-net, DONET, strong-motion seismograph networks (K-NET and KiK-net) [9], and Volcano Observation Network (V-net) [10].
\nNIED S-net and DONET teams manually pick the arrival time data at the oceanic seismic stations after NIED Hi-net team has determined the hypocenters using the land stations. We confirm the difference of shallow hypocenters between the determination by only NIED Hi-net and that by NIED Hi-net and NIED S-net. Stars in Figure 2 show the hypocenters at depths shallower than 20 km beneath the PAC plate determined by NIED Hi-net from September 11, 2017, to the end of 2018. The shallow hypocenters near the main island tend to remain shallow; however, hypocenters more than 200 km off the coast shifted significantly deeper to 40–80 km depth when including the S-net arrival time data (Figure 2). Deep events determined by NIED Hi-net on the east side of a longitude of 144°E are also shifted shallower. This suggests that it is important to include the S-net data for reliable hypocenter locations of offshore events.
\nComparison of hypocenters determined by the NIED (a) Hi-net and (b) Hi-net and S-net. Stars denote hypocenters determined at depths shallower than 20 km by only Hi-net in (a) and redetermined by Hi-net and S-net in (b).
Three-dimensional (3D) seismic velocity structure beneath the whole Japanese Islands has been studied using the vast data of seismic stations within the Japanese Islands maintained by NIED, JMA, national universities, and the other national and local governmental institutes (e.g., [11, 12, 13, 14]). These studies used data obtained mainly at land-based seismic stations with a very few seismic stations on the sea floor such as Sagami Bay, off Kushiro, Sanriku, Boso, and Tokai regions. Reference [14] investigated the structure beneath the PAC plate at depths of 30–50 km using events that occurred under the Pacific Ocean (PO) with focal depths determined by NIED F-net. However, that study was not able to clarify the shallow structure beneath the PO at depths of 0–20 km because of the lack of seismic stations on the seafloor of the PO. The seismic ray takeoff angles proceed downward from the events to the seismic stations on land, and they do not pass through the shallow zone beneath the ocean since the distance from the hypocenter to the seismic stations is usually over 150 km. We investigated the 3D seismic velocity structure of and around Japanese Islands including the Sea of Japan and PO by the seismic tomographic method. We added the arrival time data detected in the S-net, the DONET, and the Hi-net datasets, operated by NIED, as well as other datasets, operated by multiple organizations, after 2016 in addition to the data used in [14]. Then we applied the seismic tomography to these datasets.
\nThe target region, 20–48°N and 120–148°E, covers the whole Japanese Islands from Hokkaido to Okinawa and the seismic stations both Hi-net on land and S-net and DONET beneath the ocean. In addition to the arrival time data used by [14], 1,782,425 P- and 1,528,733 S-wave arrival times from 32,952 earthquakes recorded at approximately 2000 stations including NIED S-net and DONET from April 2016 to June 2018 were selected. A total of 7,853,757 P-wave arrival data and 4,604,780 S-wave arrival data from 112,631 events are available after merging the new datasets (Figure 3).
\nDistribution of hypocenters and seismic stations used for seismic tomography.
We used the seismic tomographic method [15, 16] with spatial velocity correlation and station corrections to the original code by [11]. Grid nodes were placed with half of the spatial resolution. We performed smoothing in order to stabilize the solution for the inverse problem with the LSQR algorithm [17] since arbitrary damping matrix with combination of diagonal and smoothing matrices could be assumed.
\nWe placed 3D grid nodes to construct the velocity (slowness) structure with the grid spacing shown in Table 1 and adopted the 1D structure used in the routine determination of hypocenters at the Hi-net and S-net [18] as the initial velocity model (Figure 4). No velocity discontinuities such as Moho discontinuities or the plate boundary between the EUR and PAC or PHS plates were assumed in this study. This is because there were enough data to estimate the steep velocity gradient to represent plate boundaries so that velocity discontinues in the model were not necessary [13, 16, 19]. The total number of unknowns, 4,417,505, for P-wave slowness is the same as those for S-wave slowness. We solved the P- and S-wave slowness at each grid node from more than 10 associated rays.
\nDepth | \nGrid interval | \nResolution/checkerboard pattern | \n||
---|---|---|---|---|
Horizontal | \nVertical (km) | \nHorizontal | \nVertical (km) | \n|
0–10 | \n0.1° | \n2.5 | \n0.2° | \n5 | \n
10–40 | \n5 | \n10 | \n||
40–60 | \n10 | \n20 | \n||
60–180 | \n15 | \n30 | \n||
180–300 | \n20 | \n40 | \n||
300− | \n25 | \n50 | \n
Grid interval and resolution size.
Seismic velocity structures of the initial model and the average of the final 3D model.
First, we inverted the P- and S-wave seismic velocities using the initial hypocenter location. Second, both hypocenters and 3D seismic velocity structure were inverted simultaneously. We included the arrival times from the events beneath the ocean before 2015 in addition to the data used by [14]. Focal depths of offshore events were determined by NIED F-net or [20] since offshore events determined by only NIED Hi-net are not reliable. For these offshore events, only epicenters are inverted by the 3D seismic velocity structure, while hypocenter depths are fixed. We do not fix any condition for the events after 2016 detected by NIED S-net and DONET and the events within 50 km of the onshore seismic networks before 2015 during the inversion.
\nResiduals are improved to within 0.5 s for P-wave and 0.6 s for S-wave in the travel time inversion. In the final iteration, we used 6,356,481 P-wave arrival data and 3,534,482 S-wave arrival data to solve for the P-wave slowness at 1,135,165 grid nodes and the S-wave slowness at 1,103,525 grid nodes. The inversion reduces RMS of the P-wave travel time residual from 0.561 to 0.192 s and that of the S-wave data from 0.812 to 0.239 s after 11 iterations.
\nWe conducted a checkerboard resolution test to evaluate the reliability of our solution [21]. We assumed a ± 5% checkerboard pattern and calculated synthetic travel times with random noise of 0 mean and standard deviations of 0.13 and 0.24 s for P- and S-waves, respectively. The standard deviations for random noise were derived from the average of the estimated uncertainty of the manually picked arrival times. The weight of data is inversely proportional to each width of picking error. The damping factors for the P-wave inversion are twice those for the S-wave inversion, since the average standard deviation of P-wave picking errors is almost half of that of S-wave.
\nFigure 5 shows the results of checkerboard resolution test. We calculate the recovery rate and stability with surrounding grid nodes in order to confirm well-resolved area [15]. The resolutions of Vp and Vs at depths of 5–30 km beneath main four islands are good. At depths of 40–60 km, resolutions are not good along the Sea of Japan coast because there are few deep earthquakes that can be used for inversion.
\nMap views of checkerboard resolution test for Vp and Vs. green line surrounds the well-resolved area.
NIED S-net data increase the resolution at depths of 10–60 km from Honshu to the Japan Trench (Figure 5). Reference [14] used the offshore events such as aftershocks of the Tohoku-oki earthquake. The presence of a seismic station above the events is extremely important for the estimation of velocity structure as well as the determination of hypocenters. The resolutions at depths of 0 and 5 km are still not good in spite of the use of S-net data because the incident angle to the S-net stations are mainly steep and ray paths do not run horizontally because of the lack of shallow earthquakes. Resolutions near the triple junction of Japan Trench and Sagami Trough where three plates, PAC, PHS, and EUR, meet are good at depths of 20–30 km. This is an advantage of using NIED S-net.
\nBeneath the DONET area, the resolution at depths of 10–60 km is good for Vp, and those at depths of 5–40 km are good for Vs. The resolved zone extends to the Nankai Trough since there is sufficient seismicity in this area.
\nWe calculated the average 1D model from the final 3D velocity structure (Figure 4). We also showed the perturbation from these average velocities (Figure 6).
\nMap views of Vp and Vs perturbation and Vp/Vs. Colored area is the resolved area. Broken white lines at depths of 10 and 20 km denote the median tectonic line.
At a depth of 5 km, low-Vp and low-Vs regions are located along the PAC coast beneath southeastern Hokkaido, northeastern Honshu, most of Kanto, Sagami Bay, southern Kinki, and southern Shikoku regions. A low-Vs region extends beneath the entire Shikoku and southern Chugoku regions. A low-Vp/Vs region runs along the Ou backbone range in northeastern Japan and central Japan. Other regions have high Vp/Vs.
\nAt a depth of 10 km, low-Vp regions extend beneath the active volcanoes in the northeastern and central Honshu and Kyushu regions. Low-Vs regions are almost the same as those at a depth of 5 km. High-Vp/Vs regions are distributed at central Hokkaido and coastal area in northeastern Japan. Low-Vp/Vs covers the other regions.
\nAt a depth of 20 km, low-Vp regions lie beneath volcanoes in Hokkaido, central Honshu, and Kyushu. Low-Vs regions extend beneath the volcanoes and back-arc side of Honshu. Both low-Vp and low-Vs regions extend from central Kinki to Kyushu region across central Shikoku. This low-V zone remains the same as at a depth of 5 km. High-Vp/Vs regions cover the Ou backbone range and back-arc side of northeastern Honshu.
\nAt a depth of 30 km, low-Vp extends beneath the northeastern Honshu, central and southwestern Honshu, and northern Kyushu regions. Low-Vs regions extend beneath most of Honshu, Kyushu, and northern Shikoku regions. High-Vp/Vs regions cover almost all Japanese Islands except the central Hokkaido.
\nAt a depth of 40 km, low-Vp regions exist beneath the volcanoes in southeastern Hokkaido and northeastern and central Honshu regions. The low-Vp regions beneath the volcanoes in the northeastern Japan extend to back-arc side. Low-Vs regions are clarified beneath the volcanoes in southeastern Hokkaido and central Honshu regions. Low-Vs regions beneath the northeastern Honshu can be found east of the volcanic front as are low-Vp regions. Low-Vp/Vs regions cover the central mountains across Hokkaido and northeastern and central Honshu.
\nAt a depth of 60 km, low-Vp and low-Vs regions extend beneath the volcanoes in Honshu and central Honshu. High-Vp and Vs regions extend beneath the Kinki, Shikoku, and eastern Kyushu regions where the PHS plates subduct. High-Vp/Vs regions are distributed across western Hokkaido, central Honshu, and central Shikoku regions.
\nAt a depth of 90 km, low-Vp and low-Vs regions exist beneath the volcanoes beneath Hokkaido and Honshu. High-Vp and Vs regions extend to the east of northeastern Japan where the PAC plate subducts. High-Vp/Vs regions cover northern and southwestern Hokkaido, central Honshu, and central Kyushu regions.
\nAt a depth of 10 km, a low-Vp and low-Vs zone extends along the coast of the PO in the northeastern Honshu. A high-Vp and high-Vs zone exists between the longitudes of 142 and 143°. East of longitude of 143° (Figure 6A and B), low-Vp, and low-Vs zone shows again. Vp/Vs is generally low except in some small regions.
\nAt a depth of 20 km, a high-Vp and high-Vs zone extends along the coast of the PO in the northeastern Honshu, in contrast to the structure at a depth of 10 km. Low-V zones extend to the east of the high-Vp zone; however, some high-Vp zones exist among the low-Vp zones (Figure 6C). High-Vs zones are mixed with minor low-Vs zone off the east of northeastern Honshu, extending to a longitude of 143.5° (Figure 6D). This pattern can be seen with Vp at a depth of 10 km. Vp/Vs is also broadly low, and this pattern of Vp/Vs can be seen when the depth is 10 km except in some regions.
\nAt a depth of 30 km, low-Vp zone extends off the east of northeastern Honshu between longitudes of 142 and 143.5 and to the region off the southeast of Hokkaido. High-Vp zone can be seen along the Japan Trench. Two patches of low-Vs zones exist in the east of northeastern Honshu at latitude of 37–40° and longitude of 142–143° and at latitude of 35–36° and longitude of 141–142°. High-Vp/Vs region is bounded by the low-Vp/Vs region, a north–south “stripe” pattern.
\nAt a depth of 40 km, low-Vp and low-Vs zones extend between longitude of 142–143° and latitudes of 37–41°. These low-Vp and low-Vs zones extend to the west of the Hidaka Mountains. High-Vp and high-Vs zones can be seen on the east of the low-V zone and reach the east of the Japan Trench. Vp/Vs in this area is moderate except for some low-Vp/Vs regions with north–south trend.
\nAt a depth of 60 km, low-Vp and low-Vs zones extend just off the coast of the PAC in the northeastern Honshu. High-Vp and high-Vs zones extend broadly on the east of the narrow low-V zone. Vp/Vs in this area is high.
\nAt depths of 20 and 30 km, low-Vp zones extend around the hypocenters of the large events with magnitude 6.9 and 7.4 that occurred on September 5, 2004. Low-Vs zones partly exist within the low-Vp zone. We cannot resolve the continuous structure from Honshu at depths of 5–10 km since the number of events for seismic tomography beneath the DONET stations is small. This is because the DONET picked data are basically added after the Hi-net manual picking. The seismic tomography will be recalculated when the microearthquake data triggered at DONET stations become available.
\nThe station corrections for the final model are shown in Figure 7. Red stations denote positive O-C travel times. It means that the modeled velocity is too high due to thick sediment or low-V materials since the calculated travel time is too small. It also depends on the depth of borehole of Hi-net stations. The seismometers of the Hi-net stations are typically deployed at depths of around 100–200 m, and low-Vp sediment materials are estimated beneath the backbone range and back-arc side of Japan. Large station corrections are estimated along the Sea of Japan coast in northeastern Honshu since there are thick sediments, while borehole stations are relatively shallow. For Vs, there are many blue-colored stations meaning that the velocity model is too slow. Large station corrections are also estimated on the Sea of Japan side of northeastern Honshu.
\nStation corrections for (a) Vp and (b) Vs.
For S-net stations, blue stations can be seen near the coast and the Japan Trench. Red stations are shown between them for both Vp and Vs. It suggests that the seismic velocity model is too slow near the coast and the Japan Trench and too fast between them.
\nFor DONET stations, red stations are shown near the coast and blue stations reside off the coast. It means that the modeled seismic velocity is too high near the coast.
\nFigure 8 shows the histogram of the epicentral movement during the iterations. Epicenters determined by NIED F-net and [20] are shifted over 50 km after the inversion. Epicenters determined by NIED Hi-net or by NIED Hi-net, S-net, and DONET are mainly less than 10 km in spite of 11 iterations of inversion.
\nHistogram of the earthquake epicentral movements during the inversion. The initial epicenters are determined by (a) NIED Hi-net; (b) NIED Hi-net, S-net, and DONET; (c) NIED F-net; and (d) Ref. [20]. The Hi-net system also uses the seismic stations operated by the other organizations.
Ref. [14] also clarified the seismic velocity structure beneath the PO at depths of 30–50 km; however, that study could not resolve the shallow structure at depths of 0–20 km since the ray paths, such as head waves, from the oceanic event to the land seismic stations pass through the deep zone. The ray paths from the events to NIED S-net stations run through the shallow part of the PO. In this study, we can clarify the structure at depths of 10–60 km and even east of the Japan Trench at depths of 20–30 km (Figures 5 and 6). This is a major improvement enabled by including NIED S-net data
\nOne important feature is the probable Mesozoic rift structure trending NS from the coast of Tohoku to the west of Hidaka Collision Zone. The recent 2018 Hokkaido Eastern Iburi earthquake (M6.7) (Iburi earthquake) occurred at a depth of around 32 km, which is much deeper than the usual inland crustal earthquake. Unfortunately, the structure beneath the PO between the Honshu and Hokkaido islands at a depth of 20 km is not clear; however, a low-Vp zone at a depth of 30 km in north–south direction between 142 and 143° (Figure 9) is resolved. Low-Vp zones also exist west of the Hidaka Mountains and between the Honshu and Hokkaido at the northern extension of this low-V zone, although the high-Vp zone parallel to the Japan Trench along the coast of Honshu and Hokkaido invades the low-Vp zone. The high-Vp zone is consistent with the large positive Bouguer gravity anomaly [22] and large positive aeromagnetic anomaly zones [23]. It implies that high-V mantle mafic material is located in the shallow zone. The depth of the Moho is also shallow near the coast of northern Honshu [24]. The Iburi earthquake may be related to the reactivation of the rift related to the structure in the upper mantle to the lower crust, where it is marked by high-Vp.
\nMap views of (a) Moho depth, (b) aeromagnetism, (c) Bouguer gravity anomaly, Vp perturbation at depths of (d) 20 km and (e) 30 km beneath northern Japan.
We clarified the seismic velocity structure beneath the Sea of Japan at depths of 10–20 km from offshore Hokkaido to Wakasa Bay (Figure 6). The Vp beneath the Okushiri and Sado Islands is low at a depth of 10 km; however, Vp beneath the Sea of Japan is high at depths of 10–35 km. Vp along the coast of Sea of Japan in western Japan gives moderate value. The lithospheric velocity structure in this region is strongly affected by the Mid-Tertiary breakup and formation of the Sea of Japan. Through the reactivation of the younger compressed tectonic terrain, tsunamigenic source faults have been developed. The lithospheric structure provides essential information to infer the structure of faults.
\nRef. [25] imaged the bending-shaped low-Vp oceanic crust of PAC plate subducting from the Japan Trench at latitudes of 38–38.5° offshore Miyagi where the rupture of large interplate earthquakes propagated. In this study, low-Vp material is imaged at depths of 40–50 km bounded by the high-Vp materials with a number of earthquakes surrounded with red ellipse in Figure 10. It indicates the subducting oceanic crust of the PAC plate
\nVertical cross section beneath the Pacific Ocean off Miyagi in WNW-ESE direction. Black circle shows the relocated hypocenters used for seismic tomography in this study.
The isovelocity contour of Vp = 7.0 km/s lies around depths of 25–40 km. Active-source seismic experiments off Sanriku region imaged the same contour lying at depths of 20–35 km [25] on the west side of Japan Trench, at depths of 15–30 km at the Japan Trench [26], and at depths of 15–25 km in NS direction between Honshu and Japan Trench [27]. The seismic velocity model of this study is relatively slower than those models derived from seismic experiments. The difference may depend on the initial velocity model of the oceanic region being set as the same as the land area in this study. The Moho depth becomes shallower with the EUR crust toward the Japan Trench. The oceanic crust of the PAC plate has also thinner crust than the EUR island arc crust.
\nFigure 11 shows the Vp perturbation just above the upper boundary of the PAC plate within the overriding EUR plate. The plane with the upper side at surface has a dip angle of 15°. Reference [28] also showed the Vp perturbation [29] above the upper boundary of the subducting PAC slab and three low-V zone offshore Sanriku, Miyagi, and Ibaraki. In our results, we obtain velocity structure in fine scale; however, we do not estimate the shallow structure along the Japan Trench. We obtain the broad low-Vp and low-Vp/Vs zone within the overriding EUR plate between the Japan Trench and Honshu. A high-Vp and slightly high-Vp/Vs zone exists on the west side of the low-Vp and low-Vp/Vs zone. There are some small high-V zones within the low-V zone near the hypocenter of the Tohoku-oki event.
\nVp perturbation on the plane just above the upper boundary of the PAC plate within the overriding EUR plate. The plane has strike with S17degW from the point with a longitude of 144.5 and a latitude of 41.0 with dip angle of 12 deg. The depth of the upper edge of the plane is 10 km.
Figure 12 also shows the Vp perturbation and Vp/Vs on the coseismic plane of the Tohoku-oki event [30]. We do not obtain the shallow structure along the Japan Trench although the extremely large slip of the Tohoku-oki event is estimated near the Japan Trench. The western edge of the large slip zone is consistent with the high-Vp zone; however, the surrounding region has low-Vp and low-Vp/Vs. Low-Vp/Vs material is difficult to deform so that it can generate large elastic waves if it fails. Low-Vp/Vs on the coseismic slip region may be one of the reasons for the extreme size of the Tohoku-oki event.
\n(a) Vp perturbation and (b) Vp/Vs on the coseismic slip plane [30].
We conducted the seismic tomography for entire Japanese Islands including oceanic area. This is the first tomographic study to use the data from NIED S-net. The hypocenters of oceanic events are greatly improved using the S-net data. We also obtain the detailed seismic velocity structure beneath the PO at depths of 10–60 km. Low-Vp and low-Vs zones are revealed between 142 and 143° at a depth of 30 km and in western Hokkaido where the Eastern Iburi Earthquake in 2018 occurred. The lithospheric velocity structure on the coast of Sea of Japan on Honshu is strongly affected by the Mid-Tertiary breakup and formation of the Sea of Japan. Tsunamigenic source faults have been developed through the reactivation of the younger compression. Subducting low-V oceanic crust is imaged within the mantle of overriding EUR and subducting oceanic PAC plate. The coseismic slip plane of the Tohoku-oki event has low-Vp/Vs; however, the shallow structure along the Japan Trench will be improved in the future with increased data. Previous seismic reflection and refraction studies found the oceanic crust at the uppermost part of the PAC plate with Vp of approximately 6–7 km/s; however, the seismic tomography with NIED S-net clarified the 6–7 km/s Vp zone at depths of 25–40 km. The result may depend on the initial velocity model beneath the PO, which was the same initial model as the land area in this study. Applying the initial velocity model derived from the refraction or reflection seismology would improve the results beneath the ocean in the future.
\nWe used the seismic data provided by the National Research Institute for Earth Science and Disaster Resilience, the Japan Meteorological Agency, Hokkaido University, Hirosaki University, Tohoku University, the University of Tokyo, Nagoya University, Kyoto University, Kochi University, Kyushu University, Kagoshima University, the National Institute of Advanced Industrial Science and Technology, the Geographical Survey Institute, Tokyo Metropolis, Shizuoka Prefecture, Hot Springs Research Institute of Kanagawa Prefecture, Yokohama City, and Japan Agency for Marine-Earth Science and Technology. This study was supported by the project on the Operation of Seismograph Networks for NIED. We thank academic editor Masaki Kanao for checking and commenting on our manuscript. We also thank David Shelly and Tomoko E. Yano for their helpful comments and improvement of our manuscript. Some of the figures were drawn using Generic Mapping Tools software [31] and the software for viewing 3D velocity structures beneath whole Japan Islands [32]. This work was financially supported in part by Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and by the Council for Science, Technology and Innovation (CSTI) through the Cross-ministerial Strategic Innovation Promotion Program (SIP), entitled “Enhancement of societal resiliency against natural disasters” (Funding agency: Japan Science Technology Agency).
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