Jacopo Brunetti

Antonio Culla

Walter D'ambrogio
email: walter.dambrogio@univaq.it

Annalisa Fregolent

Selection of Interface DoFs in Hub-Blade(s) Coupling of Ampair Wind Turbine Test Bed

Keywords: Experimental Dynamic Substructuring, Interface DoFs, Wind Turbine

Substructure coupling is an important tool in several applications of modal analysis. It is particularly relevant in virtual prototyping of complex systems and responds to actual industrial needs, especially in an experimental context. Furthermore, the reverse problem, the decoupling of a substructure from an assembled system, arises when a substructure cannot be tested separately but only when coupled to neighboring substructures, a situation often encountered in practice. In this paper, the dynamic behavior of the Ampair test bed wind turbine rotor, made by three blades -each one bolted to the hub at three points -is analyzed. The aim is both to identify the dynamic behavior of the rotor starting from the frequency response functions (FRFs) of blades and hub, and to select a reduced set of relevant DoFs to represent the interface between blades and hub. FRFs to be used in the coupling procedure are obtained starting from FE model of each substructure, by using a super-element based computational approach. The decoupling problem, with the aim of identifying the dynamic behavior of each blade from the FRFs of the assembled rotor and of the hub, is also considered.

INTRODUCTION

In experimental dynamic substructuring two main problems can be defined: addition of substructures (coupling) and subtraction of substructures (decoupling). Coupling is important to find out the dynamic behaviour of complex structures from a dynamic description of their components. It is particularly relevant in virtual prototyping of complex systems and responds to actual industrial needs, especially in an experimental context, i.e. when the model of at least one subsystem derives from experimental tests. Decoupling can be important in built-up structures where some components (critical subsystems or joints) cannot be removed or accessed easily. Decoupling involves the identification of the dynamic behaviour of a structural subsystem, starting from information about the remaining part of the structural system (residual subsystem) and from the known dynamic behaviour of the complete system. Addition of substructures (coupling) can be seen as a structural modification problem [START_REF] D'ambrogio | A Unified Approach to Substructuring and Structural Modification Problems[END_REF]. Similarly, the decoupling problem can be seen as a structural modification problem with negative modification. Due to modal truncation problems, in experimental dynamic substructuring, the use of FRFs (Frequency Based Substructuring) is preferred with respect to the use of modal parameters. The main algorithm for frequency based substructuring is the improved impedance coupling [START_REF] Jetmundsen | Generalised Frequency Domain Substructure Synthesis[END_REF] that involves just one matrix inversion with respect to the classical impedance coupling technique that requires three inversions. A general framework for dynamic substructuring is provided in [START_REF] De Klerk | General Framework for Dynamic Substructuring: History, Review, and Classification of Techniques[END_REF][START_REF] De Klerk | Dynamic Response Characterization of Complex Systems through Operational Identification and Dynamic Substructuring[END_REF]: in this context, the so called dual domain decomposition is very useful for experimental application, since it allows to retain the full set of global DoFs by ensuring equilibrium at the interface between substructures. A similar formulation for the decoupling problem is developed and discussed in [START_REF] D'ambrogio | Decoupling procedures in the general framework of Frequency Based Substructuring[END_REF][START_REF] D'ambrogio | The role of interface DoFs in decoupling of substructures based on the dual domain decomposition[END_REF][START_REF] Vormeeren | A dual approach to substructure decoupling techniques[END_REF][START_REF] Voormeeren | A Family of Substructure Decoupling Techniques Based on a Dual Assembly Approach[END_REF].

Whatever be the used approach, the dynamic behaviour (FRFs) at all the coupling DoFs must be determined. Therefore, if coupling involves transmission of moments, either rotational measurements or techniques involving a suitable selection of translational DoFs are required, such as the equivalent multi point connection [START_REF] De Klerk | Dynamic Response Characterization of Complex Systems through Operational Identification and Dynamic Substructuring[END_REF][START_REF] D'ambrogio | Prediction of substructure properties using decoupling procedures, Structural Dynamics -Eurodyn[END_REF] or the transmission simulator method [START_REF] Mayes | Design Studies for the Transmission Simulator Method of Experimental Dynamic Substructuring[END_REF][START_REF] Mayes | Advancements in hybrid dynamic models combining experimental and finite element substructures[END_REF]. With regard to the decoupling problem, additional FRFs at some internal DoFs of the residual subsystem can be used [START_REF] D'ambrogio | The role of interface DoFs in decoupling of substructures based on the dual domain decomposition[END_REF][START_REF] D'ambrogio | Promises and pitfalls of decoupling procedures[END_REF][START_REF] Sj Õvall | Substructure system identification from coupled system test data[END_REF] to avoid ill-conditioning at some particular frequencies.

In this paper, dynamic substructuring is applied to the rotor of the Ampair 600 wind turbine. This system has been proposed as a test bed by the Society of Experimental Mechanics focus group on experimental dynamic susbstructuring, to enable advancements in experimental dynamic substructuring technology and theory. Several specimens of the turbine have been bought. A description of the turbine with modifications made to the system to make it more linear is presented in [START_REF] Mayes | An Introduction to the SEM Substructures Focus Group Test Bed -The Ampair 600 Wind Turbine[END_REF] together with results from a rudimentary modal test on the whole turbine. The results of initial modal tests on six blades are reported in [START_REF] Harvie | Comparison of Some Wind Turbine Blade Tests in Various Configurations, Topics in Experimental Dynamics Substructuring and Wind Turbine Dynamics[END_REF]. Geometry data about the blades and the hub have also been made available to members of the focus group.

The aim of this paper is to select a reduced set of relevant DoFs to describe the interface between each blade and the hub in order to identify the dynamic behavior of the rotor starting from the FRFs of the blades and the hub. FRFs to be used in the coupling procedure are obtained starting from the FE model of each substructure, by using a super-element based computational approach. The choice of using computational FRFs is dictated by the need of being aware of the effect of each simplifying assumption, by comparing the results obtained at each step with the reference ones. The decoupling problem, with the aim of identifying the dynamic behavior of each blade from the FRFs of the assembled rotor and of the hub, is also considered

THEORETICAL BACKGROUND

The coupled structural system is assumed to be made by two (or more) subsystems joined through a number of couplings (see Fig. 1). The degrees of freedom (DoFs) of the coupled system can be partitioned into internal DoFs (not belonging to the couplings) and coupling DoFs (c).

Addition of subsystems

If addition of subsystems (coupling problem) is considered, all subsystems are assumed to be known whilst the FRF of the coupled system is unknown. In the frequency domain, the equation of motion of a linear time-invariant subsystem r may be written as:

Z (r) (ω)u (r) (ω) = f (r) (ω) + g (r) (ω) (1)
where:

Z (r)
is the dynamic stiffness matrix of subsystem r ; u (r) is the vector of degrees of freedom of subsystem r ; f (r) is the external force vector; g (r) is the vector of connecting forces with other subsystems (constraint forces arising from compatibility conditions). For the sake of simplicity, the explicit frequency dependence will be omitted. Furthermore, the procedure will be developed with reference to two subsystems, bearing in mind that it can be easily extended to more subsystems.

The equation of motion of the subsystems to be coupled can be written in a block diagonal format as:

Zu = f + g i.e.
Z (1) 0 0 Z (2) u (1) u (2) = f (1) f (2) + g (1) g (2) (2)

The compatibility condition at the interface DoFs implies that any pair of matching DoFs u (1) l and u (2) m , i.e. DoF l on subsystem 1 and DoF m on subsystem 2 must have the same displacement, that is u (1) l -u (2) m = 0.

This condition can be generally expressed as:

Bu = 0 i.e. B (1
) B (2) u (1) u (2) = 0

where each row of B corresponds to a pair of matching DoFs. Note that B is, in most cases, a signed Boolean matrix and it can be written by distinguishing the contribution of the different subsystems.

The equilibrium condition for constraint forces associated with the compatibility conditions implies that, when the connecting forces are added for a pair of matching DoFs, their sum must be zero, i.e. g (1) l + g (2) m = 0: this holds for any pair of matching DoFs. Furthermore, if DoF k on subsystem 1 (or 2) is not a connecting DoF, it must be g (1) k = 0: this holds for any non-interface DoF.

Overall, the above conditions can be expressed as:

L T g = 0 (4)
where the matrix L is a Boolean localisation matrix. Note that the number of rows of L T is equal to the number of non-interface DoFs plus the number of pairs of interface DoFs.

Eqs. (2-4) can be put together to obtain the so called 3-field formulation [START_REF] De Klerk | General Framework for Dynamic Substructuring: History, Review, and Classification of Techniques[END_REF]:

   Zu = f + g Bu = 0 L T g = 0 (5)
2.1.1 Dual formulation in the frequency domain [START_REF] De Klerk | General Framework for Dynamic Substructuring: History, Review, and Classification of Techniques[END_REF] In the dual formulation, the total set of DoFs is retained, i.e. each interface DoF is present as many times as there are substructures connected through that DoF. The equilibrium condition g (1) l + g (2) m = 0 at a pair of interface DoFs is ensured by choosing, for instance, g (1) l = -λ and g (2) m = λ. Due to the construction of B, the overall interface equilibrium can be ensured by writing the connecting forces in the form:

g = -B T λ (6)
where λ are Lagrange multipliers corresponding to connecting force intensities. Since there is a unique set of connecting force intensities λ, the interface equilibrium condition (4) is satisfied automatically for any λ, i.e.

L T g = -L T B T λ = 0 (7)
Then B T is the nullspace of L T , so Eq. ( 7) is always satisfied and the system of equations (5) becomes:

Zu + B T λ = f Bu = 0 (8)
In matrix notation:

Z B T B 0 u λ = f 0 (9)
that is:

    Z (1) 0 B (1) T 0 Z (2) B (2) T B (1) B (2) 0          u (1) u (2) λ      =      f (1)
f (2) 0

     (10)
Note that B (1) and B (2) extract the coupling DoFs among the full set of DoFs.

By eliminating λ, it is possible to obtain a relation in the form u = H f, which provides the FRF of the coupled system [START_REF] D'ambrogio | The role of interface DoFs in decoupling of substructures based on the dual domain decomposition[END_REF]:

u = Z -1 -Z -1 B T BZ -1 B T -1 BZ -1 f (11)
In expanded notation:

u (1) u (2) =    Z (1) 0 0 Z (2) -1 - Z (1) 0 0 Z (2) -1 B (1) T B (2) T × ×   B (1) B (2) Z (1) 0 0 Z (2) -1 B (1) T B (2) T   -1 B (1) B (2) Z (1) 0 0 Z (2) -1    f (1) f (2) (12)
i.e., by introducing the FRFs H (1) and H (2) at the full set of DoFs instead of [Z (1) ] -1 and [Z (2) ] -1 :

H RU = H (1) 0 0 H (2) -

H (1) 0 0 H (2) B (1) T B (2) T B (1) B (2) H (1) 0 0 H (2) B (1) T B (2) T -1 B (1) B (2) H (1) 0 0 H (2) (13)
With the dual formulation, the rows and columns corresponding to the coupling DoFs appear twice in H RU . Obviously, only independent entries are retained.

Subtraction of subsystems using the dual domain decomposition [6]

If subtraction of subsystems (decoupling problem) is considered, the coupled structural system RU and a residual subsystem R are assumed to be known whilst the FRF of subsystem U is unknown. The unknown subsystem (U) and the residual subsystem (R) joined through a number of couplings (see Fig. 2). The degrees of freedom (DoFs) of the coupled system can be partitioned into internal DoFs (not belonging to the couplings) of subsystem U (u), internal DoFs of subsystem R (r ), and coupling DoFs (c).

It is required to find the FRF of the unknown substructure U starting from the FRF of the coupled system RU.

The subsystem U can be extracted from the coupled system RU by cancelling the dynamic effect of the residual subsystem R. This can be accomplished by adding to the coupled system RU a fictitious subsystem with a dynamic stiffness opposite to that of the residual subsystem R and satisfying compatibility and equilibrium conditions. According to this point of view, the interface between the coupled system RU and the fictitious subsystem should not only include the coupling DoFs between subsystems U and R, but should as well include the internal DoFs of subsystem R. However, it can be shown that the problem can be solved by considering a number of interface DoFs at least equal to the number of coupling DoFs n c . Therefore, three options for interface DoFs can be considered:

• standard interface, including only the coupling DoFs (c) between subsystems U and R;

• extended interface, including also some internal DoFs (i ⊆ r ) of the residual substructure;

• mixed interface, including some coupling DoFs (d ⊆ c) and some internal DoFs (i ⊆ r ) of the residual substructure.

In the framework of the dual formulation in the frequency domain (see Section 2.1.1), the union between the coupled system RU and the fictitious subsystem can be written (see Eq. 10) as:

    Z RU 0 B RU T 0 -Z R B R T B RU B R 0          u RU u R λ      =      f RU f R 0      (14)
Following the same procedure used in Section 2.1.1, it is possible to obtain the FRF of the unknown subsystem U.

H U = H RU 0 0 -H R - H RU 0 0 -H R B RU T B R T B RU B R H RU 0 0 -H R B RU T B R T -1 B RU B R H RU 0 0 -H R (15)
Note that H RU and H R are the FRFs at the full set of DoFs of the coupled system and the residual subsystem.

CASE STUDY

In this paper the dynamic behavior of the rotor of the Ampair 600 wind turbine is analyzed. The rotor is made by three blades clamped to the hub by three bolts and two plates that sandwich the blade. The blades are made of glass reinforced polyester and they are coated with a white epoxy. The hub to which the blades are attached has an overspeed mechanism to vary the pitch of the blades: the mechanism is activated by centrifugal masses that are fixed through bolts in the holes at the greatest radial distance from the rotation axis of the hub.

One of the most interesting features in the study of coupling and decoupling problem is the possibility to use both experimental and numerical data. In this case a Finite Element Model of the whole rotor is developed to extract the Frequency Response Functions needed to apply the Frequency Based Substructuring methods. In this model all the alteration to the commercial turbine introduced to reduce the nonlinear behavior are considered. The overspeed mechanism was potted to reduce the rattling and the free play between the different parts of the mechanism and it is not considered in this modeling. Moreover, the centrifugal masses that activate the mechanism are not taken into account and lastly the internal shaft is considered to be directly bonded to the hub.

Four substructures are defined inside the whole rotor: the central hub and the three blades which are identical to each other, so only two different substructures have to be modeled to find the dynamic behavior of the whole system. The hub is made of isotropic material whit a negligible damping. For the sake of simplicity it is modeled entirely in Aluminum Alloy, keeping the chance to eventually consider different materials for each component. The Finite Element Model of the hub (Fig. 3) is made of around 130000 tetrahedral elements with linear shape function. The blades are made of Fiber Reinforced Polyester but the mechanical properties are unknown. Here a linear material is considered and the Finite Element Model of each blade (Fig. 4) is made of around 50000 tetrahedral elements. The whole rotor, composed by the three blades fixed on the central hub, is also modeled (Fig. 5) with the aim to compare its behavior to the coupling results and to approach the decoupling of the system. Although the connection between the substructures is ensured by contacting surfaces, efforts to simplify as much as possible the interface are undertaken, with the aim to minimize the DoFs involved in the coupling without losing important information about the dynamic behavior of the interface. After several analysis the connection between the substructures is modeled by enforcing compatibility only at the translational DoFs of the master nodes of the Rigid Body Elements (RBEs) which are realized on both edges of each hole (Fig. 6). This 6 points connection (18 DoF) introduces a relative error lower than 3% in the determination of the first modal frequencies in the range 0-250 Hz compared to the modal frequencies when the compatibility is enforced on each node on the contact surface. Moreover this highlights the low influence of the rotational DoFs in the connecting points considered.

To have an idea of the dynamic behavior of the system and of the subsystems, the natural frequencies of the blade subsystems, of the hub subsystem and of the coupled rotor system are shown in Table 1. For the coupled system, the values are referred to the simplified connection described above.

Receptance matrices to be used for the coupling with Frequency Based Substructuring techniques are obtained through a superelement based computational approach (Fig. 7). In this way, the receptance matrix is determined only on a subset of physical points that are relevant to solve the coupling problem. First, superelements are defined using a Craig-Bampton reduction of the Finite Element Model; then, the receptance matrix is directly computed using the mass and stiffness matrices of the reduced system. In the Craig-Bampton method, the reduced mass M and stiffness K matrices are defined for a subset of physical DoFs and a set of fictitious DoFs representing the modes of the structure constrained at the connecting points. After condensation, the reduced mass and stiffness matrices are extracted from the FEM and imported in Matlab environment to find the receptance matrix of the subsystem at each frequency of interest Figure 7: SuperElement computational approach for the determination of the receptance matrix.

H(ω) = K -ω 2 M -1 .

Coupling one blade with the rest of the rotor

Coupling between substructures is performed using the dual formulation. The coupled receptance matrix H C is defined as:

H C = H -HB T (BHB T ) -1 BH ( 16
)
where H is the receptance matrix of the whole system assembled using the dual approach, and B is the signed boolean matrix that defines the compatibility condition.

The receptance matrix of the hub substructure (H H ) is determined at 18 coupling nodes (6 for each blade, see Fig. 8a), by considering 16 modes for the Craig Bampton reduction. The receptance matrix of the blade substructure (H B ) is determined at 18 nodes (6 coupling nodes and 12 internal nodes, see Fig. 8b), by considering also in this case 16 modes for the CB reduction. To analyze the single coupling interface, a substructure composed by the hub and 2 blades is also considered, in this case the receptance matrix (H H+2B ) is defined at 42 nodes (6 coupling nodes and 36 internal nodes, see Fig. 8c). In this case, due to the more complex behavior of the substructures, 20 modes are considered for the CB reduction.

The reference system of the blade is different from the reference system of the hub, so a rotation is applied to the receptance matrix of the blade, in order that the two reference systems are coincident and the matrix B is boolean.

H = H H+2B [126×126] 0 [126×54] 0 [54×126] H B [54×54] (17)
The matrix B that defines the compatibility condition is:

B = I [18×18] 0 [18×114] -I [18×18] 0 [18×36] (18)
where matrices I [n×n] and O [n×m] are respectively the square diagonal unit matrix of size n and the null matrix of size [n × m].

The receptance matrix of the assembled system is computed using Eq. (17) with a frequency resolution of 0.1 Hz. The natural frequencies (estimated from peaks of the receptance modulus) are determined very accurately: in fact, the coupling procedure itself does not introduce any error in the natural frequencies, when compared to the natural frequencies obtained from the FEM with the simplified interface shown in Fig. 6.

However, a large number of measurements should be performed to determine the receptance matrices of the substructures at all the translational DoFs of the coupling nodes. Therefore, efforts are undertaken to reduce the number of DoFs involved in the coupling procedure. The aim is to find a subset of relevant DoFs to solve the coupling problem without losing information.

After several analyses, a subset of 9 relevant DoFs is found (see Fig. 9a). In order to have a limited error, the selected DoFs should be able to represent the type of connection that in this case is a rigid joint, enforcing constraints on 6 DoFs. So all the translational and rotational actions should be reproduced through forces acting on the selected DoFs.

Trying to further reduce the number of interface DoFs, using for instance the set of 7 DoFs shown in Fig. 9b, drastically worsens the results. The Frequency Response Function for the selected subset of 9 DoFs and for the further reduced subset composed of 7 DoFs is shown in Fig. 10 compared to the reference model. The subset composed by 7 DoFs introduces a much larger error compared to the subset of 9 DoFs. Analyzing the FRFs and reading the values of the natural frequencies in Table 2, the whole FRF appears to be shifted to the left: this is an effect of a drop in the stiffness of the joint, due to the removal of some coupling DoFs.

Coupling the three blades with the hub

The study carried out on a single blade is extended to the problem of coupling the three blades with the hub. Now, three coupling interfaces have to be considered. To manage all coupling interfaces in a similar manner, a different reference system is introduced for each coupling interface, having the y axis parallel to the longitudinal direction of the blade, the z axis perpendicular to the coupling surface, and the x axis defined accordingly. Therefore, the receptance matrix considered for any of the three blades is always the same, whereas the receptance matrix of the hub, rotated in these references systems, highlights the axial symmetry of the hub.

H H =    H I H II H III H III H I H II H II H III H I    (19)
The receptance matrices of the hub and the blades have the the following sizes: H H = [54 × 54] and

H B = [54 × 54].
In the formulation of the coupling problem by considering all the coupling DoFs, the matrices H and B are as follows:

H =       H H [54×54] 0 [54×54] 0 [54×54] 0 [54×54] 0 [54×54] H B [54×54] 0 [54×54] 0 [54×54] 0 [54×54] 0 [54×54] H B [54×54] 0 [54×54] 0 [54×54] 0 [54×54] 0 [54×54] H B [54×54]       (20) B =    I [18×18] 0 [18×18] 0 [18×18] -I [18×18] 0 [18×36] 0 [18×18] 0 [18×36] 0 [18×18] 0 [18×36] 0 [18×18] I [18×18] 0 [18×18] 0 [18×18] 0 [18×36] -I [18×18] 0 [18×36] 0 [18×18] 0 [18×36] 0 [18×18] 0 [18×18] I [18×18] 0 [18×18] 0 [18×36] 0 [18×18] 0 [18×36] -I [18×18] 0 [18×36]    (21)
The FRF of the whole system coupled by means of the 9 DoFs selected above is shown in Fig. 11 compared to the reference FRF of the rotor. Moreover, natural frequencies of the coupled system are shown in Table 3, compared to the reference ones.

Decoupling

In this case, the dynamic behavior of the 3 blades is found starting from the dynamic behavior of the whole rotor (Fig. 5) and of the central hub (Fig. 3). The problem is tackled in the simplest way, by using the standard interface: only the DoFs that physically connect the residual substructure (the hub) and the unknown subtructure (the 3 blades) are used as interface DoFs. In this way, all decoupling techniques bring to the same results. To decouple the systems in order to find the dynamic behavior of the blades, the dual decoupling technique is used. It is expressed by the following relation that allows to find the receptance matrix of the unknown system:

H 3B = H R 0 0 -H H - H R 0 0 -H H   B R T B H T     B R B H H R 0 0 -H H   B R T B H T     + B R B H H R 0 0 -H H (22)
where H R is the receptance matrix of the whole rotor and H H is the receptance matrix of the hub. The previous matrices are both defined only for the standard interface DoFs, which connect the residual substructure and the unknown substructure to each other and their size is [54 × 54]. In this analysis, 60 modes for the whole rotor and 16 modes for the hub are taken into account. Furthermore:

B R B H = B R B H =    I [18×18] 0 [18×18] 0 [18×18] -I [18×18] 0 [18×18] 0 [18×18] 0 [18×18] I [18×18] 0 [18×18] 0 [18×18] -I [18×18] 0 [18×18] 0 [18×18] 0 [18×18] I [18×18] 0 [18×18] 0 [18×18] -I [18×18]    (23)
Also in this case, three different reference systems -one for each blade -are introduced to express the FRFs in coupling nodes. The three blades that represent the unknown structure are not connected together, so that the receptance matrix of the unknown system H 3B is expected to be a block diagonal matrix composed by three matrices nominally identical to each other, because of the radial symmetry:

H 3B =    H B1 ≈ 0 ≈ 0 ≈ 0 H B2 ≈ 0 ≈ 0 ≈ 0 H B3    (24)
The decoupling problem is solved using all the translational DoFs of the reduced interface. The identified dynamic behavior (Fig. 12) of the blade reproduces exactly the reference behavior of the blade except for frequencies between 0 and 13 Hz. This is probably due to numerical problems related to the computation of rigid body modes, that are often troublesome for FE software. The dashed line represents the FRF between two DoFs belonging to different blades. As stated previously, it should be theoretically zero. Actually, due to the approximations involved in the use of the reduced interface, it is not zero, but it is several orders of magnitude lower than other FRFs, except at resonances, whose amplitude is not significant being the model undamped.

SUMMARY AND DISCUSSION

In this paper, the rotor of the Ampair 600 wind turbine is considered as a test bed for dynamic substructuring, with the aim of selecting a reduced set of relevant DoFs to describe the interface between each blade and the hub. The connection between these two items is ensured by mechanical contact, and therefore compatibility should be enforced between all facing nodes on the contact surface. A first significant reduction is achieved by enforcing compatibility only at the translational DoFs of 6 points, representing the edges of the holes trough which the blade is bolted to the hub. Such simplified connection (18 DoFs for each blade) introduces a relatively low error in the natural frequencies in the range 0-250 Hz. A second reduction is achieved by enforcing compatibility only at 9 DoFs, selected so as to obtain a behavior similar to that of a rigid joint: the additional error introduced by this operation is quite low. Any further attempt to reduce the number of interface DoFs drastically increases the error.

The reduced 9 DoFs interface provides good results in both the considered coupling cases: (a) hub plus two blades coupled with one blade and (b) hub coupled with three blades. In the decoupling case, good results are obtained using the 18 DoFs interface, meaning that the implementation of the procedure is correct. The next step will consider the use of the reduced 9 DoFs interface.

Further developments are envisaged in several directions. First of all, the model of the blade is not very precise, since it considers a homogeneous and isotropic fiber reinforced polyester. On the contrary, it has been pointed out that the reinforcing glass fabric is placed only in the outer part of the blade, and therefore a non homogeneous orthotropic material should be considered. A second line of development is to consider noise polluted FRFs to check the robustness of the technique. Thirdly, experimentally measured FRFs should be used for sub structuring. In fact, the use of computational FRFs is justified by the need of checking the effect of each simplifying assumption, but in view of the use of experimental FRFs.

Figure 1 :

 1 Figure 1: Scheme of the substructuring problem.

Figure 2 :

 2 Figure 2: Scheme of the decoupling problem.

Figure 3 :

 3 Figure 3: Assembled Hub FE model (a), a cross section of the model showing the different parts of the assembly (b) and the first four modal shapes (c,d,e,f).

Figure 4 :

 4 Figure 4: FEM model of the blade (a) and first four modal shapes (b,c,d,e).

Figure 5 :

 5 Figure 5: FEM model of the rotor (a) and first four modal shapes (b,c,d,e).

Figure 6 :

 6 Figure 6: RBE at the interface.

Figure 8 :

 8 Figure 8: Relevant nodes for coupling in the hub (a), in each blade (b) and in the subsystem composed by the hub and 2 blades (c).

Figure 9 :Figure 10 :

 910 Figure 9: Interface DoFs selected for coupling.

Figure 11 :

 11 Figure11: FRF of the coupled system: reference (-), after coupling 3 blades (-).

Figure 12 :

 12 Figure 12: FRF of a blade: reference (-), after decoupling (-); FRF between two different blades (---).

TABLE 1 :

 1 Natural frequencies of the systems[Hz]

			System
	Mode	Blade	Hub	Rotor
	1-6	0	0	0
	7	42,64	1977,77	21,43
	8	123,93	1977,77	28,77
	9	238,6	2284,54	28,77
	10	251,83	3091,28	73,34
	11	329,22	3091,29	76,62
	12	430,24	3424,74	76,63
	13	532,22	3643,71	124,47
	14	702,64	3643,79	124,47
	15	781,74	3643,81	165,08
	16	942,22	4863,77	175,97

TABLE 2 :

 2 Natural frequencies of the coupled system (1 blade with the rest of the rotor).

	Mode	Reference		7 DoF		9 DoF
	number	[Hz]	[Hz]	[Err%]	[Hz]	[Err%]
	7	21,4	21,2	-0,93%	21,3	-0,47%
	8-9	28,8	28,4	-1,39%	28,6	-0,69%
	10	73,3	71,4	-2,59%	72,4	-1,23%
	11-12	76,6	73,8	-3,66%	75,2	-1,83%
	13-14	124,5	114,5	-8,03%	120,8	-2,97%
	15	165,1	158,6	-3,94%	162,6	-1,51%
	16	176	170,6	-3,07%	174,9	-0,62%
	17-18	185,6	181,1	-2,42%	182,6	-1,62%

TABLE 3 :

 3 Natural frequencies of the 3 blades coupled system[Hz]

	Mode	Reference	Coupl. 3 Blades
	number	[Hz]	[Hz]	[Err%]
	7	21,4	21,1	-1,40%
	8-9	28,8	28,6	-0,69%
	10	73,3	71,3	-2,73%
	11-12	76,6	74,2	-3,13%
	13-14	124,5	119,2	-4,26%
	15	165,1	158,8	-3,82%
	16	176	172,8	-1,82%
	17-18	185,6	179,4	-3,34%

ACKNOWLEDGEMENTS

This research is supported by grants from University of Rome La Sapienza and University of L'Aquila.


The linguistic archeology of the Ryukyu Islands

Thomas Pellard (cnrs-crlao) 1 What, if anything, is Ryukyuan?

What are the Ryukyuan languages?1 How and whence did these languages arise and diversify? The history of the evolution of the Ryukyuan languages is a fundamental issue that still remains by and large unsolved.

, and though it is often not possible to establish a straightforward equation between a people, a culture, and a language, the history of languages is obviously inseparable from that of their speakers. Since each discipline has its own limitations, the only viable approach to a holistic reconstruction of the past of human populations is an interdisciplinary one, a "new synthesis" (Renfrew 1992) that triangulates the different kinds of evidence offered by archaeology, anthropology, and linguistics.

Traditionally, the term "Ryukyuan" designates the native languages of the chain of islands ranging from Amami Ōshima and Kikai in the north to Hateruma and Yonaguni in the southwest. But are these really a valid subgroup of closely related languages or just a collection of lookalikes that happen to be spoken in the same area? Are they sister languages of Japanese, do they belong to a Western branch of Japanese, or something else? What, if anything, is Ryukyuan? Unfortunately, this fundamental question has not been fully investigated as yet.

Appearances can be deceptive, and both biology and linguistics sometimes revise classifications based on similarities. Whales are not fishes, and birds, although they are usually not considered to be reptiles, are more closely related to crocodiles than lizards are. Similarly, Vietnamese is not related to Chinese or Thai, though all three of them are mainly monosyllabic and have similar tone systems. Species, both living and linguistic, can undergo similar evolutions independently, and languages are also known to be prone to converge along the same evolutionary paths as their neighbors.

1. I take here the practical view that there are five distinct Ryukyuan languages: Amami, Okinawan, Miyako, Yaeyama, and Dunan (Yonaguni). However, several other languages should probably be recognized, especially within the Yaeyama group, as some varieties are reported to be not mutually intelligible (Reiko Aso p.c. 2011, personal field notes).

1.1 Trees and branches: Why and how do we classify languages?

p. 14

The classification of languages and living organisms in terms of shared ancestry is an important part of respectively linguistics and biology, and the two have a long history of mutual influence [START_REF] Atkinson | Curious parallels and curious connections: Phylogenetic thinking in biology and historical linguistics[END_REF]. Classifications most often take the shape of the well known tree diagram, but tree diagrams are not a goal in themselves. They only represent in a convenient way information about history, and what is important is just that information. Classifications, by summarizing their evolutionary history, or phylogeny, help us understand why languages or organisms are the way they are.

Phylogenetic classifications group together species that share the same innovations inherited from a recent common ancestor. They are better suited for reconstructing evolutionary history than traditional phenetic classifications based on the quantification of surface similarities, since those can be archaisms or pure coincidences. One linguistic analogue of the phenetic approaches in biology is lexicostatistics, a method which measures the number of words shared between languages in order to infer subgroups. But lexicostatistics does not distinguish between innovations and retentions, and it thus tends to group together conservative languages rather than languages with shared innovations. Lexicostatistical estimates of the relative proximity between the Ryukyuan languages and their relative distance to Japanese [START_REF] Hattori | Nihongo no keitō[END_REF](Hattori : 228, Ōshiro 1972) should thus not be taken at face value.

Among the various types of linguistic innovations that can be used to establish subgroups, some are more reliable than others. We should thus draw evidence from linguistic domains less prone to parallel development and borrowing, in order to minimize the effects of chance and contact. Sound changes are an often used criterion, but this should be done only cautiously for two reasons.2 First, some sound changes are due to natural tendencies and occur independently in many unrelated languages, like the palatalization of consonants near front vowels or the spirantization of k into h or of p into f or ɸ. Second, sound changes tend to diffuse geographically, even across language boundaries.3 This is why classifications based on sound changes are often inconclusive and conflict with other criteria. Innovations in the basic lexicon and the morphology constitute far more reliable evidence for subgrouping, especially when they involve irregular changes.

Listing and fully justifying every innovation supporting the different branches of the Japonic tree in Figure 1.1 is beyond the scope of this chapter, and the following sections will thus give only a limited number of the most convincing ones.

2. See [START_REF] Sagart | The higher phylogeny of Austronesian and the position of Tai-Kadai[END_REF] for a criticism of classifications of Austronesian based on sound changes. 3. As in the well known example of the spread of uvular r across Europe under the influence of Parisian French [START_REF] Trudgill | Linguistic change and diffusion: Description and explanation in sociolinguistic dialect geography[END_REF]. The Japonic language family tree 1.2 The Ryukyuan languages and their relationship with Japanese Though it is generally assumed that the Japonic family has two main branches [START_REF] Katō | Hōgen kukakuron[END_REF]), a Japanese one and a Ryukyuan one, some scholars have challenged the very idea that the Ryukyuan languages form a valid subgroup (Unger 2009: 94-106). Shuri -tɕaː, -taː, Ōgami -kɛː, -taː, Ishigaki -kjaː, -taː, Dunan -nta) are also unique to Ryukyuan, but these could also be retentions.

Japonic

On the other hand, all Japanese dialects seem to share the forms otoko for 'man' (< 'young man'; Iwate òdòɡó, Tokyo òtókó(ˋ), Kyoto ótòkò, Izumo òtókó(ˋ), Nagasaki òtòkó, Ōita òtókó(ˋ)) and kami for 'hair (of the head)' (< 'top, chief'; Iwate kàm (ˋ), Tokyo kàmí(ˋ), Kyoto kámì, Izumo kàm (ˋ), Nagasaki kàmí, Ōita kàmí(ˋ)) (Pellard 2009: 249-275).

p. 16 The position of the Hachijō language is unclear4 and deserves more investigation, but since it does not share any of the above innovations, it may constitute a third primary branch. However, no evidence is found to support [START_REF] Hattori | Ryūkyū hōgen to hondo hōgen[END_REF] idea that Hachijō might have been the first variety to branch off Proto-Japonic, and that Japanese and Ryukyuan thus might form a subgroup. Similarly, the hypothesis by Thorpe (1983: 236-238) that Ryukyuan is most closely related 4. Hachijō preserves many archaic lexical and grammatical features, but it also shows an important number of borrowings from Mainland Japanese. Any study on the phylogenetic position of Hachijō will need to first disentangle its different strata.

to Eastern Old Japanese (the probable ancestor of Hachijō) is not supported linguistically, and his idea Eastern Japan was settled from Kyushu by leapfrogging over Central Japan is not backed up by any evidence.

Since Ryukyuan is a sister and not a daughter language of Japanese, it follows that the Ryukyuan data is at least as important as the Old Japanese texts, and that any feature reconstructible at the Proto-Ryukyuan level potentially goes back to Proto-Japonic, even if there is no trace whatsoever of it in Japanese.5

Northern Ryukyuan

A Northern Ryukyuan branch can be established on the basis of an important grammatical irregularity, namely the voicing of the consonant in the converb/ medial stem and related forms of the verb 'go' .6 The irregular voicing *t > d of the initial consonant of 'bamboo' also defines Northern Ryukyuan as a branch (Table 1.1).

p. 17 Within Northern Ryukyuan, an Amami (Table 1.2) and an Okinawan (Table 1.3) subgroup can be identified on the basis of several mutually exclusive innovations, and there is no support for the Southern Amami-Northern Okinawan, or "Kunigami", hypothesis often seen in the literature (e.g. [START_REF] Uemura | Ryūkyū rettō no gengo: Sōsetsu[END_REF][START_REF] Karimata | Onsei no men kara mita Ryūkyū shohōgen[END_REF]. That hypothesis groups together dialects on the basis of their vowel system and of the patterns of lenition of *p and/or *k, but these are probably areal features or parallel developments. 5. Conversely, the absence of a feature from Ryukyuan does not obligatorily imply it cannot be reconstructed in Proto-Japonic. 6. According to [START_REF] Hattori | Ryūkyūgo V Bunpō[END_REF], this is a case of suppletism, with the converb/medial verb formed not on *ik-'go' but on the distinct verb *in-'leave' . The Miyako (Nishihara) and Yaeyama (Ishigaki) verb forms do not strictly correspond to the Northern Ryukyuan forms, but nevertheless, neither the old converb nor the newer form involves any irregularity. 1.3) is defined by the irregular lengthening of the first vowel in PR *kame B < PJ *kamai 2.3a 'turtle' , the irregular change *a > oː in PR *pato B < PJ *pato 2.3b 'pigeon' [START_REF] Lawrence | Okinawa hōgengun no kai kubun ni tsuite[END_REF], and a semantic shift 'disgusting, distasteful' > 'dirty' for PR *pago B (Pellard 2009: 249-275). The Southern Ryukyuan branch encompasses all varieties spoken in the Sakishima Islands, namely Miyako, Yaeyama, and Dunan. It is defined by innovations such as the irregular shift from tone class B to A for 'how many' [START_REF] Lawrence | Yonaguni hōgen no keitōteki ichi[END_REF], a special form for 'garden' (Pellard 2009: 249-275), and an irregular vowel shift *i > u (> ∅) in PR *pitae A < PJ pitape 3.1 'forehead'7 (Table 1.4). 1.5), such as the inclusive marker, 'head' (Pellard 2009: 249-275), 'round' , and 'get wet' [START_REF] Lawrence | Tarama hōgen no keitōteki ichi. In Daiyonkai Okinawa kenkyū shinpojiamu jikkō iinkai[END_REF]. On this basis, it is clear the Tarama dialect belongs to the Miyako subgroup and not the (Macro-)Yaeyama one, contra [START_REF] Karimata | Tarama hōgen no keifu: Tarama hōgen wo rekishi hōgengakuteki kanten kara miru[END_REF].8 ' , etc.) shows Tarama constitutes a primary branch on its own within the Miyako subgroup (Pellard 2009: 280-283).

A Macro-Yaeyama subgroup that comprises Yaeyama proper and Dunan is defined by the grammaticalization of 'know' as a potential auxiliary, special forms for 'bud' [START_REF] Lawrence | Yaeyama hōgen no kukaku ni tsuite[END_REF], 'happy' , 'fresh' , and 'dirt' , and the semantic extension of 'nephew' to 'nephew or niece' (Pellard 2009: 249-275). Though Dunan is sometimes classified as a primary branch of Ryukyuan (Kokuritsu kokugo kenkyūjo 1963[START_REF] Thorpe | Ryūkyūan language history[END_REF] or of Southern Ryukyuan (Hirayama 1992-1994[START_REF] Uemura | Ryūkyū rettō no gengo: Sōsetsu[END_REF][START_REF] Karimata | Onsei no men kara mita Ryūkyū shohōgen[END_REF][START_REF] Bentley | A linguistic history of the forgotten islands: A reconstruction of the protolanguage of the Southern Ryukyus[END_REF], its close relationship with Yaeyama is beyond doubt. p. 20 The Nuclear Yaeyama subgroup differs from Dunan in that it shares a causative form of 'buy' that replaced the original verb 'sell' , a special form for 'get wet' (Pellard 2009: 249-275), and an irregular shift *g > n in PR *pige A 'beard' < PJ *piNkai 2.1. 2 When did Ryukyuan and Japanese separate?

Estimating when Ryukyuan and Japanese separated poses methodological problems. While it is common practice to give dates in archaeology and biology, by using carbon dating or the molecular clock, absolute chronological estimates are still controversial in linguistics.

Dating methods

In the case of Ryukyuan and Japanese, previous scholarship traditionally relied on the very controversial9 method of glottochronology. This method assumes words change at a universal fixed rate and measures the number of basic words shared between related languages in order to estimate when they separated. Several different formulas exist, and each of them give a different date for the split of Ryukyuan and Japanese, such as 500 ce (Hattori 1959: 80-83), 587 ce (Ōshiro 1972), or 996 ce (Unger 2009: 100).

More recently, following the trend initiated by [START_REF] Gray | Language-tree divergence times support the Anatolian theory of Indo-European origin[END_REF] for Indo-European, [START_REF] Lee | Bayesian phylogenetic analysis supports an agricultural origin of Japonic languages[END_REF] applied a more sophisticated statistical method and estimated Ryukyuan and Japanese split around the 2nd century bce. However, this novel methodology is still subject to debate,10 and in this case p. 21 there are problems with both the data and the application of the method (cognacy assessment, chronological calibration, etc.).11 If the new Yayoi chronology is correct and the beginnings of agriculture must be pushed back to around 950 bce,12 the 800-year gap with Lee & Hasegawa's date then needs to be explained, and the correlation with archaeology does not hold anymore.13

Leaving aside these new promising methods until the dust settles, it is possible in the case of Ryukyuan and Japanese to use less controversial methods. For instance, if we can show that some changes attested in the Japanese written records of a certain period did not affect Ryukyuan, then Ryukyuan and Japanese must have split before that time. Though this method does not allow to determine a precise date or even a terminus post quem, it will give a solid estimate of the latest possible date for this split.

Archaic features in Ryukyuan

Though some scholars (e.g., [START_REF] Yanagida | Muromachi-jidaigo o tōshite mita Nihongo on'inshi[END_REF] argue that the separation of Ryukyuan and Japanese dates from the (early?) Late Middle Japanese (LMJ) period (13th-15th century ce), this hypothesis not only lacks support from archaeology 9. See [START_REF] Bergsland | On the validity of glottochronology[END_REF], [START_REF] Blust | Why lexicostatistics doesn't work: The 'universal constant' hypothesis and the Austronesian languages[END_REF] 10. See the discussions collected in Forster & Renfrew (2006). 11. In particular, the position assigned to Hachijō is dubious, as well as the subgroupings of the Japanese dialects. Moreover, the fact no time calibration at all is offered for the Ryukyuan branch is problematic. 12. See the discussion in [START_REF] Shōda | A comment on the Yayoi period dating controversy[END_REF]. 13. The conclusion by [START_REF] Lee | Bayesian phylogenetic analysis supports an agricultural origin of Japonic languages[END_REF] that Japonic did not enter Japan during the Jōmon period but with the spread of agriculture during the Yayoi period is not controversial in itself and seems more than likely (see [START_REF] Hattori | Nihongo no keitō[END_REF][START_REF] Hudson | The linguistic prehistory of Japan: Some archaeological speculations[END_REF][START_REF] Unger | The role of contact in the origins of the Japanese and Korean languages[END_REF][START_REF] Whitman | Northeast Asian linguistic ecology and the advent of rice agriculture in Korea and Japan[END_REF].

(subsection 5.1), it is also irreconcilable with many linguistic facts which clearly point to an earlier date.

First, the syllables e and je merged around the first half of the 10th century in Japanese [START_REF] Mabuchi | Kokugo on'inron[END_REF]: 48-53, Frellesvig 2010: 206), but for example the distinction between the initial syllable of Early Middle Japanese (EMJ, 9th-12th century ce) ebi 'shrimp' and ye 'handle' is preserved in Northern Ryukyuan: Shodon ʔîp vs. jîː, Kamikatetsu ìbí vs. jǐː, Izena ʔibi vs. jiː. The split between Ryukyuan and Japanese cannot thus have happened later than the 10th century.

Moreover, Ryukyuan preserves the phonological distinction between Proto-Japonic *ui and *əi, which had already merged into i2 by the 7/8th century in Japanese. That this distinction goes back to Proto-Japonic is confirmed by the fact that these diphthongs follow different alternation patterns in the morphophonology: i2 < *ui alternates with u, and i2 < *əi with o(2) in Old Japanese. For example, tuki2 'moon' has an alternate stem tuku-(tuku-yo1 'moon night'), while ki2 'tree' has a variant stem ko2-(ko2-no2-pa 'tree leaf') in Old Japanese, and these two ki2 have distinct reflexes in Ryukyuan (Table 1 Comparative studies of Japanese and Ryukyuan have shown that Ryukyuan also preserves several other phonological distinctions lost in Old Japanese. Thus, while PJ *i and *e, as well as *o and *u, had already merged in many environments in Old Japanese, they are distinct in Proto-Ryukyuan (Hattori 1978-1979[START_REF] Thorpe | Ryūkyūan language history[END_REF][START_REF] Serafim | The uses of Ryukyuan in understanding Japanese language history[END_REF][START_REF] Pellard | Proto-Japonic *e and *o in Eastern Old Japanese[END_REF], 2009[START_REF] Pellard | Ryukyuan perspectives on the proto-Japonic vowel system[END_REF]; Table 1.9). Several mergers that occurred in Old and Middle Japanese are not found in Ryukyuan, so the only logical conclusion is that Ryukyuan is not a daughter language of Old Japanese but that the two are sister languages that separated before the 7th century.

Later Japanese-Ryukyuan contacts

Though Ryukyuan exhibits many archaic features that can only be explained if we assume it split from Japanese before the 7th century, it also shows traces of later contacts with Japanese. For instance, a fair number of Sino-Japanese loanwords are found in the Ryukyuan languages, but many of them exhibit irregular correspondences and exist only in Okinawan. They are thus probably late loans due to contacts between the Ryukyu kingdom and Japan. On the other hand, p. 23 some Sino-Japanese loans are found not only in Northern Ryukyuan, but also in Southern Ryukyuan, in an area traditionally more isolated, and they can be reconstructed for Proto-Ryukyuan. Moreover, these show clear Sino-Japanese features and are thus not direct borrowings from Chinese either.

Particularly telling is the case of words with *a(j/ɨ/w)ŋ-ending rhymes in Early Middle Chinese (EMC), which were borrowed with aũ-endings in Early Middle Japanese and have cognates with au-finals in Miyako (Table 1.10). These Sino-Japanese words must have been borrowed from Japanese sometime between the 8/9th century, when Sino-Japanese words started to be massively borrowed in Japanese, and the 13/15th century, when au underwent monophtongization to ɔː in Japanese [START_REF] Mabuchi | Kokugo on'inron[END_REF]: 115-118, Frellesvig 2010: 319-321). We can thus conclude that after it separated from Japanese sometime before the 7th century, (pre-)Proto-Ryukyuan was in contact with Early Middle Japanese for several centuries, until at least the 8/9th century and at most the late 13th century.

Diversity and change

Linguistic substratum and language/dialect mixing are popular theories in Japanese linguistics, and the cline of linguistic diversity that culminates in the Yaeyama region is often attributed to the effect of contact with preexisting languages [START_REF] Uemura | Ryūkyū rettō no gengo: Sōsetsu[END_REF][START_REF] Serafim | Linguistically, what is Ryukyuan? Synchronic and diachronic perspectives[END_REF]. In particular, [START_REF] Serafim | Linguistically, what is Ryukyuan? Synchronic and diachronic perspectives[END_REF] hypothesizes that there was a linear expansion through the Ryukyus, where each move added contact with the language(s) of preexisting inhabitants and thus a layer of linguistic changes due to a substratum effect. But this is actually contradicted by direct evidence from linguistics (section 4), archaeology, and anthropology (section 5).

p. 24 Moreover, purely internal explanations are able to account for the Ryukyuan diversity. First, insular geography favors isolation and thus divergence by triggering punctuational bursts [START_REF] Atkinson | Languages evolve in punctuational bursts[END_REF]): colonization of new locations by small groups constitutes founder effects that cause rapid changes. This leads in turn to the appearance of new linguistic varieties potentially highly divergent from their parent.14 In the case of archipelagos of small islands, these effects are of course multiplied.

Not only is there no need to suppose the existence of any substratum in Ryukyuan, there is actually no direct evidence for it. In particular, no convincing linguistic comparison between Austronesian and Ryukyuan has been presented as yet (Pellard 2009: 19-20;Lawrence, this volume).

Inferring prehistory from language

Considering the above phylogeny of the Japonic languages and the evidence about the timing of their interactions, it is possible to draw some hypotheses about the speakers, as well as about their society.

Homeland

A widespread idea in historical linguistics is that the zone of maximum diversity of a language family is its most likely homeland [START_REF] Sapir | Time perspective in aboriginal American culture, a study in method[END_REF]: 79-83, Nichols 1997). In the case of Japonic, it would imply the Southern Ryukyus are the homeland from which both Ryukyuan and Japanese expanded. However, such a hypothesis is in total contradiction with what we know about archaeology (section 5), and more crucially, it is at odds with the linguistic facts. The evidence from Sino-Japanese loanwords in Proto-Ryukyuan implies prolonged contacts with Japanese, and it thus seems likely the two of them were at that time located in geographically close areas. The idea that the early Sino-Japanese loans were borrowed after the settlement of the Ryukyus implies that close contacts could have been maintained in spite of the distance and that those were borrowed all at once during the short time frame between the arrival of Proto-Ryukyuan in 14. Compare with the observation of accelerated evolution in organisms living on islands [START_REF] Millien | Morphological evolution is accelerated among island mammals[END_REF].

the islands and its expansion and diversification. Such a scenario seems rather unlikely.

Kyushu, the closest Japanese large island, immediately comes to mind as a likely homeland, and [START_REF] Serafim | When and from where did the Japonic language enter the Ryukyus[END_REF] pointed out some similarities between Ryukyuan and some Japanese dialects of Kyushu, which could suggest the existence of a Ryukyuan substratum there. However, the evidence remains rather scarce, p. 25 and it remains unknown whether these are not just shared retentions or parallel developments.

Migration and diversification

The phylogenetic tree of the Ryukyuan languages (Figure 1.1) represents a sequence of evolutionary changes, and its shape gives us a relative chronology of the separation of the different languages. If we lay out this tree on a geographical map, it is possible to associate the nodes and branches of the tree with migration moves.

First of all, since all Ryukyuan languages form a single branch, we can infer the settlement of the Ryukyus was not achieved by different waves of migration at different periods and from different places, but probably by one or a few moves from a single area, and within a rather narrow time frame.

The general topology of the tree also indicates the closest islands were not always colonized first.15 If every island encountered had been settled as the Proto-Ryukyuans advanced straight ahead in the archipelago, each island would be associated with a bifurcation on the main branch, and there would be no Northern Ryukyuan, Amami, nor Okinawan subgroup. However, the Southern Ryukyuan languages form a sister branch of Northern Ryukyuan as a whole, and not just Okinawan. If the settlement of the Ryukyus had taken place in a linear fashion from north to south, we would find innovations shared by Okinawan and Southern Ryukyuan but not by Amami, and no innovation shared by Amami and Okinawan but not by Southern Ryukyuan. This is the exact reverse of the actual situation. On the other hand, the inner branching of Southern Ryukyuan is compatible with a north-to-south expansion: Miyako is settled first, then Yaeyama, with the too small Tarama Island bypassed, and finally Yonaguni.

Several scenarios16 can be proposed to account for this tree topology. First, the Southern Ryukyus might have been settled independently of the north, directly from Mainland Japan (probably Kyushu). This implies two distinct migration waves, though there is no need to suppose any particular chronological order, or even that one wave happened after the other.17

Second, there could have been a single move out of Japan, that reached either Okinawa or Amami first. Then, a subset of these settlers would have left for 15. This is congruent with the typology of island colonization by [START_REF] Keegan | Colonization of islands by humans: A biogeographical perspective[END_REF]. 16. I won't consider the hypothesis that the Northern Ryukyus may have been settled from the Sakishima for the reasons exposed in section 5. 17. I leave out the possibility these two waves were just one that split into two on the way to the Ryukyus.

the south, separating from those who thereby became the Proto-Northern Ryukyuans. This supposes either the first wave bypassed Amami and went straight to Okinawa, or that the second wave toward the Southern Ryukyus started from Amami and bypassed Okinawa. This hypothesis implies either Amami or Oki-p. 26 nawa was not settled before the Sakishima.

Yet another hypothesis is possible, which supposes language replacement happened in either Amami or Okinawa. The Ryukyus could have been indeed settled progressively from north to south, but later either Amami or Okinawan intruded beyond its original area and replaced preexisting varieties of Northern Ryukyuan there. This language shift would have occurred before the diversification of the intruding language.

Linguistic palaeontology

Following the idea language reflects the culture of the people who use it, we can assume that if a word for a particular object can be reconstructed in a protolanguage, that object was probably part of the culture of the speakers. Applying such a method of linguistic palaeontology, it is possible to hypothesize that the Proto-Ryukyuan speakers were most probably farmers familiar with rice agriculture, pottery, and navigation. This is supported by the existence of a common set of vocabulary pertaining to agriculture,18 livestock, pottery, and sailing in Proto-Ryukyuan (Table 1.11).

Toward a new synthesis

The window opened on the Ryukyuan past by historical linguistics is only a partial one. In particular, the problem of absolute dating cannot be solved with confidence, though it is possible to provide a time frame. Other inferences made on the basis of linguistics also need to be complemented by and checked against the results of archaeology and anthropology, in order to write a holistic scenario. p. 27

Archaeology

The earliest human remains in the Japan-Ryukyu region are found in the Ryukyus, in the Yamashita-chō cave of Okinawa, and date from 32,000 years ago. However, these ancient populations do not seem to have survived the Palaeolithic, and they are thus not related to the modern Ryukyuan people [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF], Takamiya 2005: 95-100).

Following the Palaeolithic, the Shellmound period of the Northern Ryukyus (6400 bp-11th century ce) hosted a hunter-gatherer culture roughly similar to the Jōmon culture of Mainland Japan. From the Middle Shellmound period on, the Northern Ryukyus were in contact with Japan via Kyushu through a trade 18. There is however variation in the diversity of rice vocabulary within Ryukyuan. This suggests rice agriculture was not evenly practiced and had not the same importance everywhere. network that provided them with pottery, and perhaps rice, in exchange of seashells used in ornaments [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF][START_REF] Asato | Okinawa-ken no rekishi[END_REF]). However, neither the Yayoi culture (950/500 bce-300 ce) nor the following Kofun culture (3rd-6th century ce) expanded into the Ryukyus [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF][START_REF] Asato | Okinawa-ken no rekishi[END_REF][START_REF] Pearson | Archaeological perspectives on the rise of the Okinawan state[END_REF]. Despite being in contact with the Yayoi agriculturalists, the Shellmound foragers did not shift to agriculture, even though they underwent a food stress crisis. Their population was probably not very large, as islands cannot sustain a large number of foragers, and most of the islands were thus probably uninhabited (Takamiya 2005: 57-60).

In the meanwhile, the Southern Ryukyus formed a separate area isolated from the North until the 11/12th century No traces are found of the Jōmon and Yayoi cultures, but an Austronesian colonization from Taiwan around 4500 bp can be identified in the Shimotabaru pottery of Yaeyama [START_REF] Summerhayes | An Austronesian presence in southern Japan: Early occupation in the Yaeyama Islands[END_REF]. However, this Austronesian colony did not last and, after a blank of several centuries, it was followed until the 11/12th century ce by a culture which did not make use of pottery and did not practice agriculture [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF][START_REF] Asato | Okinawa-ken no rekishi[END_REF].

It is only at the end of the Late Shellmound period that agriculture developed in the Ryukyus, around the 10/11th c (Takamiya 2005: 144-172), with rice as the main crop and other cereals (wheat, barley, foxtail millet) as supplements. This mixed agriculture corresponds to that practiced in Mainland Japan at that time [START_REF] Kinoshita | Kai kōeki to kokka keisei: 9 seiki kara 13 seiki wo taishō ni[END_REF]. The transition from foraging to agriculture was abrupt, accompanied with a sudden demographic explosion in all the Ryukyus and particularly in Southern Okinawa, and it is not explainable by a simple scenario of p. 28 shift to agriculture by the existing foragers [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF]: 88-91, Takamiya 2005: 174-179). The exact status of the recently discovered archeological site of the Gusuku village on Kikai Island remains to be determined, but it seems to have served as a major trade center with Japan between the 9th and 13th ce.19 It thus could have played a major role in the settlement of the Ryukyus by Japonic speakers.

This Proto-Gusuku period [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF][START_REF] Asato | Okinawa-ken no rekishi[END_REF]) laid in a few centuries (10-12th century) the bases of a hierarchical society based on farming, metallurgy, and trade. It led to the development of competitive polities in Okinawa, and later to the rise of the Ryukyu kingdom. Moreover, for the first time the isolation of the Sakishima was broken, and roughly the same culture as in the North developed there. There is evidence that the Yamato State of Japan knew of the existence of the Ryukyus, and some of its embassies to China called at Ryukyuan islands on their way to the continent [START_REF] Kinoshita | Kai kōeki to kokka keisei: 9 seiki kara 13 seiki wo taishō ni[END_REF]). The contacts intensified with the start of the Proto-Gusuku period, and a trade network developed between the Ryukyus and Japan, ultimately extending to China. The main trade goods were soapstone cauldrons from Nagasaki in Kyushu, iron from Kyushu, Chinese porcelain, Ryukyuan sulfur, and Ryukyuan seashells used in lacquer-ware ornaments. In parallel, kamui-yaki stoneware from Tokunoshima in the Amami region diffused throughout all the Ryukyus, where it was probably used to stock grains, and a new type of "Gusuku" ceramics appeared [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF][START_REF] Asato | Okinawa-ken no rekishi[END_REF][START_REF] Pearson | Archaeological perspectives on the rise of the Okinawan state[END_REF].

Several scenarios have been proposed to account for these facts, which differ on details but tell more or less the same story. Asato [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF][START_REF] Asato | Okinawa-ken no rekishi[END_REF]) sets the start of the Proto-Gusuku period rather early (10th century) and thinks there was a temporal sequence of events, starting from the diffusion of Nagasaki cauldrons, followed by that of kamui-yaki ware, and then of agriculture. He also stresses the importance of a group of traders from the Ryukyus who diffused the kamui-yaki ware throughout the Ryukyus from the 11th century on. On the other hand, [START_REF] Kinoshita | Kai kōeki to kokka keisei: 9 seiki kara 13 seiki wo taishō ni[END_REF] thinks all happened within a short and later time frame and was the deed of Japanese merchants from Fukuoka motivated by a rise in price of the seashells from the Ryukyus. Both remain vague about who the Japanese merchants traded with, and who colonized and developed agriculture in the Ryukyus. We can infer the settlers must have been a group of Japanese, comprising traders, craftsmen and farmers, who were transported there along the road of the seashell trade.

Anthropology

Recent studies of both ancient and modern dna [START_REF] Li | Phylogenetic relationship of the populations within and around Japan using 105 short tandem repeat polymorphic loci[END_REF][START_REF] Shinoda | Mitochondrial DNA analysis of human skeletal remains obtained from the old tomb of Suubaru: Genetic characteristics of the westernmost island Japan[END_REF][START_REF] Matsukusa | A genetic analysis of the Sakishima islanders reveals no relationship with Taiwan aborigines but shared ancestry with Ainu and main-island Japanese[END_REF] tend to show that Ryukyuans form a group closely related to the Mainland Japanese, and only more distantly to the Jōmon and Ainu p. 29

19. See [START_REF] Takanashi | Okinawa bunka wa doko kara kita ka: Gusuku jidai to iu kakki[END_REF] and references therein. [START_REF] Tajima | Genetic origins of the Ainu inferred from combined DNA analyses of maternal and paternal lineages[END_REF]). Despite their geographical proximity, Southern Ryukyuans do not show any particular affinity with the Austronesian populations of Taiwan, and they form a clear subgroup with Northern Ryukyuans [START_REF] Li | Phylogenetic relationship of the populations within and around Japan using 105 short tandem repeat polymorphic loci[END_REF][START_REF] Shinoda | Mitochondrial DNA analysis of human skeletal remains obtained from the old tomb of Suubaru: Genetic characteristics of the westernmost island Japan[END_REF][START_REF] Matsukusa | A genetic analysis of the Sakishima islanders reveals no relationship with Taiwan aborigines but shared ancestry with Ainu and main-island Japanese[END_REF]). However, Southern Ryukyuans do not form a cluster within that subgroup, and the position of Amami Islanders remains to be determined. Genetic diversity is not particularly high in the Ryukyus, which suggests neither long-term isolation nor population size reduction. There is also evidence for sex-biased migrations, with a higher rate of geographic isolation in males than in females [START_REF] Matsukusa | A genetic analysis of the Sakishima islanders reveals no relationship with Taiwan aborigines but shared ancestry with Ainu and main-island Japanese[END_REF].

The results of different dental and cranial studies are contradictory, and there seems to be no consensus on the question of whether there is a connection between Ryukyuans and the Jōmon people or the Ainu. Ryukyuans appear however to be generally close to (near-)Modern Mainland Japanese. Dental analyses also reveal high inter-island diversity within the Ryukyus.20

All of this suggests that both Northern and Southern Ryukyuans descend from a population influx from Mainland Japan after the beginning of the Yayoi period that simply replaced any preexisting inhabitants without significant admixture with them. The fact Southern Ryukyuans are genetically close to Northern Ryukyuans, although these two regions previously hosted distinct populations, suggests Ryukyuan did not enter the area by pure diffusion to preexisting populations but was brought there by a migration wave that replaced earlier inhabitants, contra [START_REF] Serafim | Linguistically, what is Ryukyuan? Synchronic and diachronic perspectives[END_REF].

A unified scenario

At least three scenarios of the settlement of the Ryukyus and the spread of the Japonic civilization there have been proposed: the "Ocean Road", the "Hayato", and the "Proto-Gusuku" scenario.

According to the Ocean Road hypothesis, a scenario originally proposed by the Japanese ethnographer [START_REF] Yanagita | Kaijō no michi[END_REF], wet rice agriculture was introduced from China in the Southern Ryukyus first, and then was transmitted to Okinawa and finally to Japan. While not concerned by this issue in particular, it suggests an early settlement of the Ryukyus by Japonic agriculturalists before the Yayoi period. However, such a scenario is contradicted by archaeological evidence, as agriculture began earlier in Mainland Japan than in the Ryukyus [START_REF] Takamiya | Introductory routes of rice to Japan: An examination of the Southern route hypothesis[END_REF], and from a linguistic point of view, if Japonic had spread with agriculture from the south, the presence of Sino-Japanese loanwords in Southern Ryukyuan would not be expected.

The Hayato hypothesis associates the spread of Japonic in the Ryukyus with a migration before the 8th century ce of the Hayato "barbarians" of Kyushu depicted in the Old Japanese chronicles. The Hayato would have fled farther and p. 30 farther South as they were repelled by the extending forces of the Yamato state [START_REF] Uemura | Ryūkyū rettō no gengo: Sōsetsu[END_REF][START_REF] Serafim | Linguistically, what is Ryukyuan? Synchronic and diachronic perspectives[END_REF]. Glottochronological evidence played a major role 20. See [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF], [START_REF] Pietrusewsky | A multivariate craniometric study of the inhabitants of the Ryukyu Islands and comparisons with cranial series from Japan, Asia, and the Pacific[END_REF], [START_REF] Haneji | Non-metric dental variation of Sakishima Islanders, Okinawa, Japan: A comparative study among Sakishima and neighboring populations[END_REF], [START_REF] Toma | Metric dental diversity of Ryukyu Islanders: A comparative study among Ryukyu and other Asian populations[END_REF] and references therein. in this scenario, as it was accepted the Ryukyuan-Japanese split happened before or around the Kofun period (3rd-6th century ce) and that the settlement of the Ryukyus would have occurred around that time too. However, no evidence is found for such a migration at that time, and the evidence from loanwords suggests Ryukyuan stayed in contact with Japanese until a much later date.

The Proto-Gusuku hypothesis [START_REF] Asato | Okinawa-jin wa doko kara kita ka? Ryūkyū=Okinawa-jin no kigen to seiritsu[END_REF][START_REF] Takamiya | Shima no senshigaku: Paradaisu dewanakatta Okinawa shotō no senshi jidai[END_REF]) convincingly argues that the only event which can be meaningfully associated with a Japonic expansion in the Ryukyus is the migration around the 10th century that led to the formation of the Gusuku culture. Linguistic evidence was seen as ir-reconcilable21 with the other data, due to the reliance on the glottochronological calculations by [START_REF] Hattori | Nihongo no keitō[END_REF]. But glottochronology, though very appealing to archaeologists, is a rather controversial tool, and the loanword evidence indicates that the ancestor of Ryukyuan was still spoken on the mainland until perhaps as late as the 12th century. There is thus no obstacle from linguistics to the Proto-Gusuku scenario, as already acknowledged by [START_REF] Serafim | When and from where did the Japonic language enter the Ryukyus[END_REF], and the loanword evidence is actually congruent with it.

Refining the Proto-Gusuku hypothesis, we can say that the Ryukyuan languages form a sister branch of Japanese and that their ancestor separated from Japanese probably during the first centuries ce. This is likely to have happened at the end of the Yayoi period, just before the rise of the Kofun culture and the emergence of strong polities in Japan that led to the formation of the Yamato state in Central Japan, the homeland of Old Japanese.22 If Ryukyuan and Japanese had split much earlier than that, such as around the beginning of the Yayoi period a millenium earlier, they would probably be much more divergent.

After that, (pre-)Proto-Ryukyuan was still spoken on the mainland, most likely in Kyushu, for several centuries. The socio-linguistic situation of Kyushu at that time is not known, but (pre-)Proto-Ryukyuan was influenced by Japanese, the extending language of the prominent Yamato state. The Ryukyus were known to the inhabitants of the mainland since Yayoi, thanks to a trade network. Then, around the 10/12th century, several millennia after the disappearance of the Austronesians from the Sakishima, a group of merchants from Kyushu, accompanied by craftsmen and farmers who settled there, entered the Ryukyus, down to the Sakishima, probably motivated by the rising market value of the Ryukyuan seashells.

p. 31 The settlers gave birth to and diffused the Gusuku culture, a package comprising agriculture, ceramics, and the Proto-Ryukyuan language. If the Gusuku culture can indeed be associated with the Proto-Ryukyuan speakers, the fact it is found in the Southern Ryukyus but not on Kyushu leads to the conclusion that the current population of the Ryukyus is probably not the product of several migration waves from Japan. The Northern Ryukyus must have been settled first, 21. For instance Takamiya (2005: 196) believes he will trigger violent opposition from linguists when he concludes the colonization of the Ryukyus is most likely to have happened around the 10/12th century. 22. See [START_REF] Hattori | Nihongo no keitō[END_REF] for a similar idea about pre-Kofun as a likely archaeological setting for the Ryukyuan-Japanese split. On the Kofun period and state formation, see [START_REF] Barnes | State formation in Japan: Emergence of a 4th-century elite[END_REF].

perhaps starting with Kikai Island, and the Gusuku culture evolved there before it was brought to the Southern Ryukus. The exact timing of this Southern expansion remains to be determined.

The bearers of the Gusuku culture expanded within the whole Ryukyu Archipelago, and preexisting foragers, who were few, simply died out or were assimilated without leaving a significant trace. The relative isolation of the different islands and the fact settlers probably formed small groups accelerated linguistic change, but contacts did exist between islands, and their populations intermarried. After Proto-Ryukyuan entered the Ryukyus, all its remaining relatives in Kyushu were slowly annihilated by the ongoing influence of Japanese, and no trace remains of them today.

Conclusions and future prospects

Trying to synthesize the results of historical linguistics, archaeology and anthropology has proven particularly fruitful in circumventing the lack of ancient written records and the low survival rate of organic remains in the Ryukyu Islands. Results all converge to tell the same story, that of a migration from Kyushu around the 10/12th century that replaced the preexisting inhabitants and founded the Gusuku culture.

Linguistic evidence plays here a key role in reconciling contradictory data, but the conclusion that the migration of Proto-Ryukyuan probably happened later than the split with Japanese itself was motivated by purely linguistic evidence, namely that of Sino-Japanese loanwords. The fact such an hypothesis independently formulated is congruent with other kinds of evidence only adds to its credibility. The proposed scenario also fits well with the Farming/Language Dispersal hypothesis [START_REF] Bellwood | Farmers, foragers, languages, genes: The genesis of agricultural societies[END_REF]: Proto-Ryukyuan managed to expand in the Ryukyus because its speakers were able to durably settle there, and this was possible only because they practiced agriculture.

Though the main line of the above scenario is rather clear, many details remain to be determined. For instance, when exactly did Proto-Ryukyuan split from Proto-Japanese and when did it move to the Ryukyus? Where in Kyushu did the settlers come from? Who were they actually? What role did the traders play in the colonization? When and how did language shift to Japanese happen in Kyushu? Such questions are still open for future research. Also, the exact chronology of the migration and of the spread of agriculture has not been sufficiently investigated.

p. 32 Future research will need to clarify such details, to thoroughly evaluate the different hypotheses and to test how well the data matches the emerging scenario.23 For the time being, the scenario of the expansion Japonic in the Ryukyus presented here is no more than a plausible narrative, though a rather convincing one.

23. See [START_REF] Greenhill | Testing population dispersal hypotheses: Pacific settlement, phylogenetic trees, and Austronesian languages[END_REF] for an interesting example of such an evaluation in the case of the Austronesian settlement of the Pacific.

Note on sources and transcriptions

All Ryukyuan linguistic forms are given in a unified broad phonetic transcription following the conventions of the ipa. Tense ("glottalized") consonants are transcribed with a superscript glottal stop. Tones are left unmarked when I have no reliable information. The tone category C in Dunan is marked as final-falling, though it is actually realized nowadays as a high tone in the case of light syllables. The letters A, B and C correspond to the historical tonal categories first established by [START_REF] Hattori | Amami guntō no shohōgen ni tsuite: Okinawa/Sakishima shohōgen to no hikaku[END_REF]Hattori ( , 1978Hattori ( -1979) ) and [START_REF] Matsumori | Ryūkyū akusento no rekishiteki keisei katei: Ruibetsu goi 2-hakugo no tokui na gōryū no shikata o tegakari ni[END_REF]Matsumori ( , 2000a,b),b). Proto-Japonic tone categories follow the notation of [START_REF] Martin | The Japanese language through time[END_REF], with some additions.

The Ryukyuan and Japanese linguistic forms are cited from the following sources. Yamatohama: [START_REF] Osada | Amami hōgen bunrui jiten[END_REF][START_REF] Osada | Amami hōgen bunrui jiten[END_REF]; Shodon: personal field notes, [START_REF] Martin | Shodon: A dialect of the Northern Ryukyus[END_REF], [START_REF] Karimata | Kagoshima-ken ōshima-gun Setouchi-chō Shodon hōgen no fonēmu (ge)[END_REF]; Kamikatetsu: personal field notes, [START_REF] Kibe | Shōmetsu kiki hōgen no chōsa/hozon no tame no sōgōteki kenkyū: Kikai-jima hōgen chōsa hōkokusho[END_REF][START_REF] Shirata | Ryūkyūgo Kikai-jima Kamikatetsu hōgen no danwa shiryō[END_REF]Kametsu and Wadomari: Hirayama (1986); Yoron: [START_REF] Kiku | Yoron hōgen jiten[END_REF]; Izena: [START_REF] Uchima | Okinawa hokubu/nanbu no kijutsuteki kenkyū[END_REF]; Iejima: [START_REF] Oshio | Okinawa Ie-jima hōgen jiten[END_REF]; Sesoko: Hirayama (1992Hirayama ( -1994)); Nakijin: [START_REF] Nakasone | Okinawa Nakijin hōgen jiten[END_REF]; Shuri: Kokuritsu kokugo kenkyūjo (1963); Kume: [START_REF] Uchima | Okinawa hokubu/nanbu no kijutsuteki kenkyū[END_REF]; Ōgami: personal field notes; Ikema, Nagahama and Tarama: Hirayama, Ōshima & Nakamoto (1967), Hirayama (1983); Ishigaki: Miyagi (2003); Taketomi [START_REF] Hirayama | Ryūkyū Sakishima hōgen no sōgōteki kenkyū[END_REF], [START_REF] Maeara | Taketomi hōgen jiten[END_REF]; Hoshidate: [START_REF] Maeō | Iriomote hōgen shū[END_REF]; Kuroshima: personal field notes; Hateruma: [START_REF] Hirayama | Ryūkyū Sakishima hōgen no sōgōteki kenkyū[END_REF], Hirayama (1988), [START_REF] Aso | Hateruma (Yaeyama Ryukyuan)[END_REF]; Dunan: personal field notes, [START_REF] Ikema | Yonagunigo jiten[END_REF], [START_REF] Uwano | Ryūkyū Yonaguni hōgen no akusento shiryō (1)[END_REF]; Japanese dialects: Hirayama (1992Hirayama ( -1994)). Proto-Ryukyuan and Proto-Japonic reconstructions are my own, and Middle Chinese reconstructions come from [START_REF] Pulleyblank | Lexicon of reconstructed pronunciation in Early Middle Chinese, Late Middle Chinese, and Early Mandarin[END_REF].

  Figure 1.1: The Japonic language family tree

  systematic comparison of the Ryukyuan and Japanese languages reveals all Ryukyuan varieties share a set of innovations absent from Japanese (Pellard 2009: 249-275). For instance, all Ryukyuan languages share a Proto-Ryukyuan (PR) etymon *do C 'body' (Shodon dǔː, Shuri dúː, Ōgami tuː, Ishigaki dúː, Dunan dûː). Proto-Ryukyuan further innovated by grammaticalizing this noun into a reflexive pronoun. Other Ryukyuan innovations include for example the semantic change 'intestines' > 'belly' for PR *wata B < Proto-Japonic (PJ) *wata 2.3b (Shodon wátʰǎ, Shuri wátá, Ōgami pata, Ishigaki bádá, Dunan bàtà). The interrogative pronoun *n[a|o][o|u] C 'what' (Shodon nǔː, Shuri núː, Ōgami nau, Ishigaki nóː, Dunan nûː) and the plural markers *-kjaa and *-taa (Shodon -kʰja,

Table 1

 1

		.1: Northern Ryukyuan innovations
			'go' (converb) 'bamboo'
		Yamatohama	ʔidʑi	dɘxɘ
		Shodon	ʔìdʑî	d xɘǨ
	A	amikatetsu Kametsu	ʔidʑi ʔɪdz ́dɘː děː
		Wadomari	ʔìdʑ-áŋ	déː
		Yoron	idʑi	dàì
		Iejima	ʔidʑi	dákí
	O	Nakijin	ʔìdʑí	dàkíː
		Shuri	nˀdʑi	dákì
	M Nishihara	itsɨ, ikiː	taki
	Y Ishigaki	ikɨ, ikitte	tákì
	Dunan	ititi	tʰaɡi

  The Amami subgroup includes Yoron and all varieties located to its north. It is defined by the irregular vowel shift *u > a in PR *kakozu C 'jaw' and a unique form for 'dust' instead of PJ *p[o|ə]k[o|ə]ri 3.1 (Table1.2;[START_REF] Lawrence | Okinawa hōgengun no kai kubun ni tsuite[END_REF] p.c. 2009).

		Table 1.2: Amami innovations
			'jaw'	'dust'
		Yamatohama kʰaxadzɨ	ɸuɸuŋ
		Shodon	kʰáxǎt	ɸúɸǔm
	A	Kamikatetsu kʰaːdu ('talkative') ɸûːmú
		Kametsu	kâːdz ̀hóːmûŋ
		Wadomari	kàːdʑí
		Yoron	káːdʑí	pùːmú
		Iejima	hákúzì	pʰu̥ kúì
	O	Nakijin	hàkùːdʑí	pu̥ kúí
		Shuri	kákúdʑí	ɸu̥ kúì
	M Ōgami	kamaks	puki
	Y Ishigaki	kḁkúdz ́ɸúkúì
	Dunan	kʰàɡúdî	kˀùŋ
	On the other hand, the Okinawan subgroup (Table	p. 18

Table 1

 1

			.3: Okinawan innovations
			'turtle'	'pigeon'	'dirty'
		Izena	haːmi	ɸuːtu	haɡusaŋ
		Iejima	háːmí	pʰóːtú	pʰáɡósà
		Sesoko	hàːmí	hòːtú	hòɡòɕéŋ
	O	Nakijin Shuri	hàːmíː pòːtúː káːmíː hóːtú	pàɡòːɕêŋ háɡóːsáŋ
		Kume	kaːmiː hoːtu	haɡoːsaː
	A	Shodon Wadomari hàmǐː kʰám ̌hátʰǒ ɸàːtǔː	janaɡɘsam nˀjáːɕâŋ
		Yoron	hàmí	pàtú	jànàɡísàŋ
	M Ōgami	kami	mːpatuɯ skatakam
	Y Ishigaki	kàmí	pátú	jàníɕɕáːŋ
	Dunan	kʰàmì hàtû	dìnâŋ

Table 1

 1

			.5: Miyako innovations
			'incl.'	'head'	'round'	'get wet'
		Ikema	mai kanamai	maːku	mmiː
	M	Nagahama mai kanamaɭ Ōgami mai kanamaɯ maːku maːku	mmiɭ mmi
		Tarama	mai kanamaɭ	maːku	mmiɭ
	Y Ishigaki	ŋ	ts búɾ ́múɾúsàːŋ dzóɸɸìŋ
	Dunan	ŋ	mìmbùɾù màɾúŋkâ ŋɡáɾúŋ
	A Shodon	m	kʰámǎtɕ	maɾsa	nùɾ tʰ	Ó
		Nakijin	ŋ	tɕˀìmbú	màɾùɕêŋ díːɾûŋ
	7. The Dunan form could come from either *putae or *pitae.
	8. Though Lawrence (2003) considers Tarama to belong to the main Miyako branch, the fact it
	does not share several innovations seen in all other Miyako dialects (e.g., *-(s)amar-'honorific' ,
	*b > *g in 'knee

Table 1

 1

				.6: Macro-Yaeyama innovations
			'potential' 'bud'	'happy'	'niece' 'dirt'	'fresh'
		Ishigaki	ɕɕiŋ	báí	sánìɕàːŋ	búì(-ɸàː) ɡábá píɾáɡíɕáːŋ
	Y	Taketomi Hatoma	ɕɕuŋ ɕɕeːŋ	bəi bàì	sənisəŋ sàníjàŋ	bùì-ɸàː ɡəbə́píːɾèsàŋ bùì ɡàbàː pìɾákèŋ
		Hateruma ɕɕaŋ	bëː	sḁn̥ iɕahaŋ bui	ɡaba piɾiːsahaŋ
	Dunan	tsˀúŋ	bai	ɕànáŋ	búíhá	ɡaba çiɾaɡjaŋ
		Ikema		miː	hu̥ kaɾasɨ̥ kai mjuːi	fu̥ sɨ çiɡuɾukai
	M	Ōgami	juːs	miː	pukaɾaskam miuɯ	napa pukuɾukam
		Tarama		fukɨ	pu̥ kaɾaɕaːɭ mjuːɨ	naba piɡuɭɭaɭ
	A Shodon		mɘː	ʔuhoɾaɕa	m kkˀwá çîɡ.ɾú sɨdaɕa
	O Nakijin	ʔúːsûŋ mìdùɾíː ɸu̥ kùɾáɕèŋ mìːúì	píŋɡù ɕìdàːɕêŋ

Table 1

 1

			.7: Yaeyama innovations
			'sell'	'get wet' 'beard'
		Ishigaki	kâːs ŋ	dzóːɾìŋ	p ńì
		Taketomi	kàːsùŋ	zuɾiɾuŋ pìnì
	Y	Kuroshima haːsuŋ Hoshidate kaːsu	zoːɾiɾu	pini pi̥ ni
		Hatoma	káːsúŋ	dzóːɾúŋ pìní
		Hateruma	kasɨ̥ m̥ iɾuŋ duɾiɾuŋ pi̥ n̥ i
	Dunan	ùɾúŋ	ŋɡáɾúŋ ŋɡí
	M Ōgami	ʋː	mmi	pɯki
	A Shodon	ʔùɾjûm	nùɾ tʰ ́çìɡî
	O Nakijin	ʔúŋ	díːɾûŋ	pˀìdʑíː

Table 1

 1

			.9: PJ *i vs. *e and *u vs *o
		'daytime'	'garlic'	'mortar'	'medicine'
	PJ	*piru 2.2	*peru 2.1	*[u/o]su 2.5a *kusori 3.7a
	OJ	pi1ru	pi1ru	usu	kusuri
	PR	*piru A	*peru A	*[u/o]su C	*kusori C
	Shodon	çîɾ	ɸ ɾ	ʔús ̌kˀu̥ sǔɾ
	Nakijin	pˀìɾúː	pʰìɾúː	ʔúɕì	ku̥ sùí
	Ōgami	psː-ma	piɯ	us	ffuɯ
	Ishigaki	p ːɾ ̀pîŋ	ús ́ɸu̥ ɕíɾ	Dunan
		tsˀúː	çiɾu	ùtɕî	tsˀùɾî

Table 1 .

 1 10: Sino-Japanese loans in Ryukyuan

		'stick'	'1st month'	'skillful'
		棒	正月	上手
	EMC	baɨwŋʰ tɕiajŋ ŋuat	dʑɨaŋ ɕuw'
	EMJ	baũ	syaũgwat	zyaũzu
	LMJ	bɔː	syɔːgwat	zyɔːzu
	A Shodon bôː	ɕóːɡwàdd k	dʑǒt
	O Shuri	bóː	ɕóːɡwátsí	dʑóːdʑí
	M Ōgami pau	saukaks	taukɯ
	Y Ishigaki bóː	ɕóŋɡwàdz ̀dʑóːdz	Dunan
		bûː	sùŋátî	dudi
	PR	*bau C *sjaugwatu C *zjauzu B/C

Table 1

 1

			.11: Ryukyuan cultural lexicon
		PR	Shodon Shuri Ōgami Ishigaki Dunan
	'rice'	*kome B kʰúm ̌kúmí
	'rice'	*mai A		mêː	maɯ	máì	màí
	'rice plant'	*ine B	ʔínǐ	ńˀní		íní	ǹnì
	'unhulled rice'	*momi A mûm múmì			mumi
	'wheat'	*mogi B	múɡǐ múʑí	mukɯ múŋ	mùŋ
	'foxtail millet'	*awa B	ʔǒː	ʔáwá	aː	áː	àː
	'broomcorn millet' *kimi B	kˀímǐ mâː-ʑìŋ kɯm	k ŋ	tɕˀin-ti
	'taro, yam'	*umo B	ʔúmǔ ḿˀmú mː	úŋ	ùn-tî
	'field'	*patake C xàtɘxɘ́hátákí		pàtáɡí hàtáɡî
	'rice paddy'	*ta B	thǎː	táː	taː	táː	tʰàː
	'cow'	*usi A	ʔûɕ	ʔúɕì	us	ús ̀ùtɕí
	'pig'	*uwa C	wˀǎː	wˀáː	ʋaː	óː	wàː
	'horse'	*uma B	ʔúmǎ ḿˀmá nuːma ḿmá	m̀mà
	'pot'	*tubo A	tˀ bû	tsíbù	kɯpu	ts bù	tɕˀìbú
	'jar'	*kame C kʰám ̌káːmí	kami	kàmí	kʰami
	'boat'	*pune C	ɸún ̌ɸúní	funi	ɸúní	ǹnî
	'sail'	*po A	ɸûː	ɸûː		pûː	húː
	'paddle'	*ijako B	júhǒ	ʔéːkú	ɯaku	jákú	dàŋù

Acknowledgement

I would like to thank Sven Osterkamp, Wayne Lawrence, Laurent Sagart, Anton Antonov, and J. Marshall Unger for providing useful comments and suggestions on earlier drafts. All remaining errors are mine. p. 33