Serif Gozen
email: serif.gozen@mcgill.ca

Brian J Olson
email: brian.olson@jhuapl.edu

Steven W Shaw
email: shawsw@egr.msu.edu

Christophe Pierre
email: christophe.pierre@mcgill.ca

Resonance Suppression in Multi-Degree-of-Freedom Rotating Flexible Structures Using Order-Tuned Absorbers

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Many rotating flexible structures consist of an array of interconnected constituent parts whose geometry and structural properties are rotationally periodic (i.e., cyclically symmetric). In a bladed disk assembly, for example, the fundamental substructure is one blade plus the corresponding segment of the disk, which is collectively referred to as a sector. During steady operation these systems rotate at a constant speed and are subjected to TW dynamic loading, or engine order (e.o.) excitation, which is characterized by excitation frequencies that are proportional to the mean rotational speed of the rotor. Such excitations result in component vibrations that can lead to high cycle fatigue failure, noise, reduced performance, and other undesirable effects. Order-tuned vibration absorbers exploit the centrifugal field from rotation of the primary system and are thus ideally-suited to address component vibration in rotating flexible structures. They essentially consist of masses that ride along designer-specified paths relative to the primary system. When properly tuned, these absorbers effectively counteract fluctuating loads applied to the primary system over all rotation speeds [START_REF] Alsuwaiyan | Performance stability and localization of systems of vibration absorbers[END_REF].

The dynamic performance, characteristics and features of order-tuned absorbers are well-understood in typical situations, and there are numerous examples of their implementation. They have been investigated by Shaw et al. [START_REF] Haddow | Centrifugal Pendulum Vibration Absorbers: An Experimental and Theoretical Investigation[END_REF][START_REF] Alsuwaiyan | Performance and Dynamic Stability of General-Path Centrifugal Pendulum Vibration Absorbers[END_REF][START_REF] Alsuwaiyan | Localization of Free Vibration Modes in Systems of Nearly-Identical Vibration Absorbers[END_REF][START_REF] Lee | On the Counteraction of Periodic Torques in Rotating Systems Using Centrifu-gally Driven Vibration Absorbers[END_REF][START_REF] Alsuwaiyan | Steady-State Response of Systems of Nearly-Identical Torsional Vibration Absorbers[END_REF] and others for torsional vibration reduction in rotating systems. Applications include light aircraft engines [START_REF] Ker Wilson | Practical Solutions of Torsional Vibration Problems[END_REF], helicopter rotors [START_REF] Wachs | The Main Rotor Bifilar Absorber and its Effect on Helicopter Reliability/Maintainability[END_REF], diesel camshafts [START_REF] Cook | Absolute Absorbtion[END_REF], and advanced technology automotive engines [START_REF] Nester | Vibration Reduction in Variable Displacement Engines Using Pendulum Absorbers[END_REF].

In previous works Olson et al. [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures[END_REF][START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Nonlinear path design[END_REF][START_REF] Olson | Vibration absorbers for cyclic rotating flexible structures: Linear and nonlinear tuning[END_REF] investigated the performance of order-tuned vibration absorbers applied to lumpedparameter models of a cyclically-coupled bladed disk assembly under engine order excitation. In these models, the sector dynamics are captured by two DOFs: one for the blade and one for the absorber. A key finding from the linearized coupled system is the existence of the so-called "no-resonance zone" in linear tuning parameter space. It was shown that there exists a range of absorber tuning values, close to the ideal tuning that exactly matches the excitation order, for which there are no resonances over the entire range of rotation speeds. This result was first reported by Shaw et al. [START_REF] Shaw | The Dynamic Response of Tuned Impact Absorbers for Rotating Flexible Structures[END_REF] for a model consisting of a single, isolated sector (i.e., one blade and one absorber attached to a rotating disk). Olson et al. showed that the no-resonance zone persists in the (coupled) multi-sector linear system [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures[END_REF] and that it qualitatively persists when weak nonlinearities are taken into account via the absorber paths [START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Nonlinear path design[END_REF][START_REF] Olson | Vibration absorbers for cyclic rotating flexible structures: Linear and nonlinear tuning[END_REF]. Existence of the no-resonance zone allows for the possibility of robustly-tuned absorbers that can function effectively in the presence of model and parameter uncertainties. This paper investigates if and how the no-resonance zone persists when higher-fidelity models are used to describe the bladed disk system. It will suffice to employ a 3-DOF sector model that consists of two DOFs for each blade/disk and one for the attendant absorber. The no-resonance zone is shown to persist in a restricted sense: one structural mode can be eliminated, but higher mode resonances still exist at some rotor speed for any absorber tuning. These results generically hold for models with many blade/disk DOFs because of the nature of dynamic coupling between the absorber and blade dynamics, which results in characteristic eigenvalue veering as the rotor speed is varied [START_REF] Pierre | Mode Localization and Eigenvalue Loci Veering Phenomena in Disordered Structures[END_REF][START_REF] Happawana | A singular perturbation analysis of eigenvalue veering and modal sensitivity in perturbed linear periodic systems[END_REF]. Suppression of multiple resonances may be achieved, for example, by implementing multiple sets of absorbers. This is left for future work.

The paper is organized as follows. A higher-fidelity model of a rotating bladed disk assembly under engine order excitation is developed in Section 2, where each blade is fitted with an order-tuned vibration absorber. Section 3 introduces a tuning strategy in which the absorber tuning order is set relative to the order of the TW excitation. Features of the forced response are discussed in Section 4, which contains the main results of this work. The paper closes in Section 5 with a discussion and suggestions for future work.

Wt H O (a) Model of Bladed Disk Assembly i + 1 i -1 i K d K d K d k d k d k b k b K b K b K b h b (b) Sector Model W j i q i y i M b ,L M d ,d M a Q A R i J i B V r F i (t) F i (t)

SYSTEM MODEL

A lumped-parameter model of the rotating bladed disk assembly is shown in Fig. 1a. It consists of a cyclic array of N identical, identically-coupled sectors (Fig. 1b), each with one blade and one absorber plus a corresponding segment of the disk. The disk component of each sector is composed of a rigid part of radius H, which rotates at a constant speed Ω about a fixed axis through O. This component is referred to as the rotor. A second flexible portion vibrates about a point Q at the periphery of the rigid rotor, which is modeled by a pendulum of length d and mass M d and is referred to as the disk. The blade dynamics are captured by a second pendulum of length L and mass M b that is attached to the disk pendulum at the vertex A. There are N such double pendulum systems, which are uniformly distributed along the circumference of the rotor. The sector model shown in Fig. 1b could also represent a 2-DOF blade model attached to a rigid disk/rotor. The key feature is multiple DOFs per sector, in this case two, which is sufficient to capture the effects ordertuned absorbers when there are arbitrarily many disk/blade DOFs per sector.

The flexural stiffness associated with the i th blade and disk is modeled using linear torsional springs of stiffness K b and K d , respectively. Adjacent blades (resp. disk portions) are coupled via translational springs with stiffness k b (resp. k d ) at a distance b (resp. h) relative to their attachment points to the disk (resp. rotor). It is assumed that the springs are unstressed when the disks and blades are in a purely radial configuration, that is, when ϕ i = θ i = 0 for each i ∈ N = {1, 2, . . . , N}. 1 Each stiffness element is paired with a corresponding linear viscous damper (not shown in Fig. 1a) to capture the effects of dissipation. The disk and inter-disk (resp. blade and inter-blade) damping is modeled by linear torsional and translational dampers with constants C d and c d (resp. C b and c b ). It is convenient to model the absorber damping with an effective torsional damper with constant C a that acts at point B (attachment point of absorber pendulum).

The blades are fitted with identical vibration absorbers, which generally consist of particle masses M a that ride on userspecified paths. Each absorber path is described by a vector of length R i (S i ) = R i (-S i ) relative to the outermost blade pendulums, where S i is the path arc length relative to its vertex V . Polchi [START_REF] Polchi | Multi-scaling and averaging for higher fidelity lumped model of bladed disk with cpva[END_REF] derived the governing EOM for the system shown in Fig. 1a fitted with the general-path absorbers using the path formulation described by Olson et al. [START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Nonlinear path design[END_REF][START_REF] Olson | Vibration absorbers for cyclic rotating flexible structures: Linear and nonlinear tuning[END_REF]. However, for the small-amplitude linearized model to be developed, the absorbers can be regarded as simple pendulums with length r and mass M a that are attached a distance αL along the outermost blade pendulum relative to point A. Their linearized dynamics are described by the relative angular coordinates ψ i = S i /r. Thus, each sector consists of a double-pendulum disk/blade model and a simple-pendulum (circular path) absorber attached to the outermost double-pendulim link.

The i th disk/blade double pendulums are harmonically forced in the transverse sense by e.o. excitation of order n such that the cyclic system is circumferentially-forced by a TW. The e.o. excitation is modeled by [START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Ewins | Vibration Characteristics of Bladed Disc Assemblies[END_REF]

F i (t) = F 0 e jφ i e jnΩt , (1)
where

φ i = 2π n N (i -1)
is the inter-sector phase angle, F 0 is the excitation strength, and j = √ -1. The e.o. is restricted such that n ∈ N , which includes all practically relevant situations [START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF].

The equations of motion (EOM) are formulated using Lagrange's method, linearized for small disk/blade and absorber 1 Throughout the remainder of this work it is understood that variables and equations with subscripts i are defined for each i ∈ N .

f ϕ = F 0 (L + d)/K d Strength of e.o. excitation (ϕ i -dynamics) f θ = F 0 L/K d Strength of e.o. excitation (θ-dynamics) λ = K b K d
Square of the blade torsional stiffness

ν a = (k d h 2 +k b d 2 ) K d
Stiffness coupling between sectors

ν b = k b db K d
Stiffness coupling between sectors

ν c = k b b 2 K d
Stiffness coupling between sectors

ξ ac = 1 d 2 c d h 2 +c b d 2 √ K d M d /d 2
Damping coupling between sectors

ξ bc = 1 d 2 c b db √ K d M d /d 2
Damping coupling between sectors

ξ cc = 1 d 2 c b b 2 √ K d M d /d 2
Damping coupling between sectors

ξ a = 1 d 2 Ca √ K d M d /d 2
Absorber torsional damping constant

ξ b = 1 d 2 C b √ K d M d /d 2
Blade torsional damping constant

ξ d = 1 d 2 C d √ K d M d /d 2
Disk torsional damping constant σ = Ω/ω 0 Angular speed of the rotor motions, and made dimensionless according to the nondimensional parameters defined in Table 1. Time is rescaled by τ = ω 0 t, where ω 0 = K d /M d d 2 is the undamped natural frequency of a single isolated disk DOF. In matrix-vector form the sector EOM are given by

Mz ′′ i + Cz ′ i + Kz i + C c (-z ′ i-1 + 2z ′ i -z ′ i+1 ) + K c (-z i-1 + 2z i -z i+1 ) = fe jφ i e jnστ , (2)
where the elements of the 3 × 3 sector mass, damping, and stiffness matrices (M, C, K) are defined in Table 2, the vector z i = (ϕ i , θ i , ψ i ) T captures the sector dynamics, and ( • ) ′ = d( • )/dτ denotes differentiation with respect to dimensionless time. Intersector coupling is captured by the matrices

C c =   ξ ac ξ bc 0 ξ bc ξ cc 0 0 0 0   , K c =   ν 2 a ν 2 b 0 ν 2 b ν 2 c 0 0 0 0   , (3)
Table 2. Elements of the sector mass, damping, and stiffness matrices.

M 11 = µ b (ρ 2 + 1) + µ a (α + γ + 1) 2 + 1 C 11 = ξ d K 11 = δ(µ b (ρ + 1) + µ a (α + γ + 1) + 1)σ 2 + 1 M 12 = µ b ρ(ρ + 1) + µ a (α + γ)(α + γ + 1) C 12 = -ξ b K 12 = δ(µ b ρ + µ a (α + γ))σ 2 M 13 = µ a γ(α + γ + 1) C 13 = -ξ a K 13 = µ a γδσ 2 M 21 = M 12 C 21 = 0 K 21 = K 12 M 22 = µ b ρ 2 + µ a (α + γ) 2 C 22 = ξ b K 22 = (δ + 1)(µ b ρ + µ a (α + γ))σ 2 + λ 2 M 23 = µ a γ(α + γ) C 23 = -ξ a K 23 = µ a γ(δ + 1)σ 2 M 31 = M 13 C 31 = 0 K 31 = K 13 M 32 = M 23 C 32 = 0 K 32 = K 23 M 33 = µ a γ 2 C 33 = ξ a K 33 = µ a γ(α + δ + 1)σ 2
and the sector forcing vector is given by f = ( f ϕ , f θ , 0) T , where the dimensionless entries are defined in Table 1. The 3N linearized EOM defined by Eq. ( 2) serve as the basis for the analysis that follows.

ABSORBER TUNING

The linear absorber tuning order follows from the special case when the disks and blades are locked in their zero positions relative to the spinning rotor [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF]. This leads to a system of identical, dynamically isolated and unforced single-DOF absorbers that oscillate freely under the influence of centrifugal effects. Their dynamics follow from Eq. ( 2) by setting

ϕ i = ϕ ′ i = ϕ ′′ i = 0 and θ i = θ ′ i = θ ′′ i = 0. Then µ a γ 2 ψ ′′ i + ξ a ψ ′ i + µ a γ(α + δ 1)σ 2 ψ i = 0, (4)
where the undamped natural frequencies are given by ω33 ≡

ω 33 ω 0 = α + δ + 1 γ σ. (5)
Physically, each absorber acts like a centrifugally-driven pendulum of length γ whose pivot point is located a distance α + δ + 1 from the rotor axis O. The base accelerates towards the center or rotation at a constant rate (α + δ + 1)σ 2 , which is the effective gravity of the pendulum. This results in a centrifugal restoring force that scales with σ 2 and an undamped natural frequency that scales with σ according to the constant

ñ = α + δ + 1 γ , (6)
which is defined to be the absorber tuning order. Because the isolated absorber eigenfrequencies ω33 and the e.o. excitation frequency scale directly with the rotor speed, the absorbers are tuned to a given order of the excitation as opposed to a particular frequency [START_REF] Hartog | Tuned Pendulums as Torsional Vibration Eliminators[END_REF].

Absorber tuning refers to a choice of absorber geometric parameters, which are selected to attenuate the blade responses over a range of operating speeds, particularly near resonance. This is done by choosing a value for ñ and then selecting the dimensionless parameters α, δ, and γ to satisfy Eq. ( 6). It is shown that there exists an ideal absorber tuning that results in suppression of one of the blade resonances, where the attendant absorber amplitudes depend primarily on their mass and placement along the blade lengths and the level of the applied loads. Ideal tuning is defined by ñ = n, [START_REF] Ker Wilson | Practical Solutions of Torsional Vibration Problems[END_REF] in which case the isolated absorber natural frequency ω33 identically matches that of the excitation. Exact tuning is effective for all rotation speeds, but it is susceptible to uncertainties of the model or absorber parameters due to in-service wear, environmental effects, or the effects of tolerances. To account for such effects, and to allow for intentionally detuned designs, we let

ñ = n(1 + β), (8)
where β is a detuning parameter. Ideal linear tuning corresponds to β = 0 while undertuning (resp. overtuning) corresponds to β < 0 (resp. β > 0). One of the main goals of absorber design is to select β to achieve robust vibration reduction of the primary system.

FEATURES OF THE FORCED RESPONSE

This section reports the main results of the paper. The linearized EOM are analyzed to determine the effects of absorber tuning on the steady-state TW response to TW e.o. excitation.

The special case for which the absorbers are locked at their vertices is analyzed in Section 4.1. These results, which are typical for a cyclic system with two DOF per sector, are used as a baseline against which the absorber performance is assessed. The full model, in which all three sector DOF are free to move, is considered in Section 4.2. Of particular interest are the effects of varying the absorber tuning and whether or not the no-resonance zone reported by Olson et al. [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures[END_REF] exists for this higher fidelity model.

The forced response can be obtained by assembling the sector models defined by Eq. ( 2) into a single matrix EOM and proceeding in the usual way [START_REF] Meirovitch | Principles and Techniques of Vibrations[END_REF]. For large N, however, it is desirable to simplify the analysis by exploiting the system (cyclic) symmetry, as it is done by Olson et al. [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures[END_REF] and others [START_REF] Óttarsson | Dynamic Modeling and Vibration Analysis of Mistuned Bladed Disks[END_REF]. If P is the number of DOF in each sector (P = 3 for the model shown in Fig. 1), the PN × PN system mass, damping, and stiffness matrices are block circulant with P × P blocks. The PN-DOF model can be block decoupled to a set of N, P-DOF modal systems via a unitary transformation involving the complex Fourier matrix [START_REF] Wagner | Forced Harmonic Response of Grouped Blade Systems: Part I-Discrete Theory[END_REF][START_REF] Cai | Uncoupling of Dynamic Equations for Periodic Structures[END_REF][START_REF] Shen | Vibration of Rotationally Periodic Structures[END_REF]. The PN natural frequencies are preserved under the transformation and can be obtained in sets of P from the N decoupled modal EOM. The forced response follows accordingly, and allows for the full system response to be determined by considering a set of uncoupled three DOF models.

Response with the Absorbers Locked

If the absorbers are locked relative to the rotating blades, then ψ i = ψ ′ i = ψ ′′ i = 0 and each sector model has two DOFs. Figure 2a shows the 2N natural frequency loci in a Campbell diagram and the corresponding disk/blade amplitude frequency response (|ϕ i |, |θ i |) for a representative model with N = 10 sectors, order n = 3 excitation, α = 1.4, δ = 1.117, ρ = 1.67, λ = 0.1, γ = (α + δ + 1)/ ñ2 , µ a = 0.01, µ b = 0.1, ν a = 0.5, ν b = ν c = 0.1, f φ = 0.0016, and f θ = 0.001. There are two groups of N natural frequencies whose spread depends primarily on the strength of the inter-sector coupling. For nonzero coupling they generally appear in repeated pairs, except for mode p = 1 and mode p = (N + 2)/2 if N is even. For zero coupling they collapse onto a single pair of values [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF]. The bottom-most group corresponds to in-phase motions of the two DOF in each sector and the top-most curves represent out-of-phase motions of the disk/blade DOFs. The natural frequencies increase slightly with increasing rotor speed due to centrifugal stiffening.

It is well-known that e.o. excitation excites only one mode in each eigenfrequency group due to special modal orthogonality properties of cyclic systems [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF]. The resonant rotor speeds can be identified in Fig. 2a by the intersections of the natural frequency loci corresponding to mode p = n + 1 with the e.o. line defined by nσ. 2 The corresponding resonances are shown in the frequency response diagrams of Fig. 2b. Of course, mis- 2 The excited mode is p = n mod N + 1 when considering n ∈ Z + [START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF]. tuning that arises due to imperfections among the sectors results in resonance with some degree of intensity at all of the intersections [START_REF] Wei | Localization Phenomena in Mistuned Assemblies with Cyclic Symmetry Part I: Free Vibrations[END_REF][START_REF] Wei | Localization Phenomena in Mistuned Assemblies with Cyclic Symmetry Part II: Forced Vibrations[END_REF].

Response with the Absorbers Free

When the absorbers are free to move there are three natural frequencies corresponding to each index p ∈ N . The three groups of N frequency loci ω(p) 1,2,3 are plotted in Fig. 3 for increasing inter-sector coupling and increasing absorber mass while holding all other parameters fixed. Each natural frequency corresponds to a system mode, which, together with fixed intersector phase relationships, form a forward TW, backward TW, or standing wave response among the sectors, depending on the value of n relative to N [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Óttarsson | Dynamic Modeling and Vibration Analysis of Mistuned Bladed Disks[END_REF]. The natural frequencies are tightly-spaced for weak inter-sector coupling (Fig. 3a) and they spread out as the coupling is increased (Fig. 3c). Each group exhibits the eigenvalue veering phenomenon, or mutual repulsion of the eigenfrequencies, which results from small dynamic coupling between the disks and blades and between the blades and absorbers [START_REF] Pierre | Mode Localization and Eigenvalue Loci Veering Phenomena in Disordered Structures[END_REF][START_REF] Happawana | A singular perturbation analysis of eigenvalue veering and modal sensitivity in perturbed linear periodic systems[END_REF][START_REF] Perkins | Comments on Curve Veering in Eigenvalue Problems[END_REF]. The ratio of absorber mass to the disk and blade masses is essential for the curve veering phenomenon. When the mass ratio is very small (Fig. 3d), the natural frequency curves are close to each other in the veering regions. (In Figs. 3d-f only one frequency locus ω(1) 1,2,3 is shown in each group, which corresponds to zero inter-sector coupling.) A larger difference between the groups of frequency loci can be obtained by increasing the mass ratio (Fig. 3 f ), although in most applications this mass ratio is small. Possible resonances can be identified by the intersections of ω(p) 1,2,3 (σ) with the e.o. line nσ. However, there is a system resonance in the perfectly symmetric case only when

nσ = ω(n+1) 1,2,3 (σ) (9)
is satisfied because only mode n + 1 is excited in the steadystate [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF]. The principle objectives of this work are to determine if any resonances persist for near-ideal absorber tuning (i.e., for β close to zero); whether or not any new resonances are introduced for multi-DOF disk/blade models; and, if so, whether the absorbers can be tuned to avoid these resonances over a range of rotor speeds.

The results are summarized in Fig. 4, which shows the eigenfrequency loci in Campbell diagrams and the corresponding disk, blade, and absorber amplitude frequency responses

(|ϕ i |, |θ i |, |ψ i |)
for a model with N = 10 sectors, order n = 3 excitation, various absorber tuning values that range from under-tuning (β < 0) to overtuning (β > 0), and for the same parameter values used in Fig. 2. Inter-sector coupling is assumed to be quite strong to more clearly show the effects of detuning on the excited modes; this does not affect the approach nor the conclusions. In many situations the eigenfrequency loci ω(p) 1,2,3 cross the e.o. line twice. For absorber overtuning (Fig. 4a) or sufficiently large undertuning (Fig. 4b) there are two system resonances, which are indicated by circles in the Campbell diagrams. Insets 1 and 2 show close-up views of the veering regions in Fig. 4a, including the resonances corresponding to ω(n+1) 1,2 . Figure 4c shows that the fundamental resonance disappears for ideal tuning (β = 0), but the resonance corresponding to ω(n+1) 2 persists. In this case, there is only one system resonance over the full range of possible rotor speeds.

One of the main findings of Olson et al. is the existence of a no-resonance zone in the 2-DOF sector model dynamics, that is, a range of absorber tuning values which results in complete resonance suppression over the full range of possible rotor speeds [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF]. The no-resonance gap corresponds to a finite range of absorber undertuning values, where the critical lower bound β cr < 0 depends on system and absorber parameters and the upper bound corresponds to ideal tuning with β = 0. Robust absorber design involves intentional undertuning within this (generally small) range, which allows for small variations of the model or parameter values without introducing resonance. For the 3-DOF sector model shown in Fig. 1b the no-resonance zone is replaced by a resonance suppression zone defined by

β cr < β ≤ 0, ( 10
)
where one of two resonances is avoided but the other resonance persists. It is clear from the eigenfrequency structures shown in Fig. 4 that there will be P -1 excited modes for the general case of P ≥ 2 disk/blade DOFs per sector. The resonance suppression zone can be clearly identified in Fig. 5, which is a plot of the resonant rotor speed σ res (defined implicitly by Eq. ( 9)) in terms of the detuning parameter β for the same parameter values used in Fig. 2. For the case of P = 2 DOFs per sector considered by Olson et al. [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures[END_REF][START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Nonlinear path design[END_REF][START_REF] Olson | Vibration absorbers for cyclic rotating flexible structures: Linear and nonlinear tuning[END_REF][START_REF] Shaw | The Dynamic Response of Tuned Impact Absorbers for Rotating Flexible Structures[END_REF] there exists a range of undertuning values for which there are no system resonances (i.e., the no-resonance zone). Figure 5 shows that no such range exists for P = 3 DOFs per sector, and it is clear that the same can be said for the general case of P ≥ 3. Instead, there is a finite range defined by Eq. [START_REF] Nester | Vibration Reduction in Variable Displacement Engines Using Pendulum Absorbers[END_REF] for which one resonance is robustly avoided and one resonance persists. For the parameter values used in Fig. 4 the critical undertuning value is β cr = -0.0554. Fig. 4d shows example disk, blade, and absorber frequency response plots and the corresponding Campbell diagram for tuning within the resonance suppression zone.

CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK

The performance of centrifugally-driven, order-tuned vibration absorbers has been investigated using a higher-fidelity lumped-parameter model of a bladed disk assembly under e.o. excitation. The model consists of a cyclic array of N identical, identically-coupled sectors, each with three DOFs that capture the effects of the disk, blades, and absorbers. The principle finding of this work is the existence of an absorber undertuning region in which one resonance is robustly avoided. The range is bounded from above by ideal tuning and from below by a critical linear undertuning, which forms a resonance-suppression zone. Thus, while the absolute no-resonance zone reported by Olson et al. [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures[END_REF] does not extend completely to higher-fidelity sector models, the absorbers do provide a range of speeds over which one resonance can be suppressed.

It is important to note that adding more DOFs per sector does not qualitatively alter this picture. That is, groups of higherorder modes will still experience resonance, albeit at higher rotor speeds. This is due to the structure of the natural frequency curves shown in Fig. 4, which veer only once and then flatten out (asymptotically approach the absorbers-locked frequencies [START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Linear order tuning[END_REF][START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF]) as the rotor speed is increased. Therefore, the e.o. line generically crosses all but one natural frequency group. Adding more DOFs per sector simply adds more such curves at higher frequencies, thereby introducing more resonances at higher rotor speeds. The main conclusion is that an identical set of ordertuned absorbers can suppress only one resonance.

There remains much work to be done on this class of problems, which can be divided into four general categories. First, very few systematic experiments have been performed. The work by Duffy and co-workers on self-tuned impact dampers makes use of an impacting mode of motion for order-tuned absorbers, and demonstrates that they are effective in attenuating vibrations in a rotating plate [START_REF] Duffy | On a Self-Tuning Impact Vibration Damper for Rotating Turbomachinery[END_REF][START_REF] Duffy | Self-Tuning Impact Dampers for Fan and Turbine Blades[END_REF].

The second group of problems address nonlinear effects. Shaw and Pierre [START_REF] Shaw | The Dynamic Response of Tuned Impact Absorbers for Rotating Flexible Structures[END_REF] have investigated a single-sector model with an order-tuned absorber that undergoes impacts when it reaches a certain amplitude. Olson and Shaw [START_REF] Olson | Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry with Applications to Turbomachinery[END_REF][START_REF] Olson | Vibration absorbers for a rotating flexible structure with cyclic symmetry: Nonlinear path design[END_REF][START_REF] Olson | Vibration absorbers for cyclic rotating flexible structures: Linear and nonlinear tuning[END_REF] analyzed the response of a two-DOF-per-sector model in which absorber path nonlinearity is taken into account, and they make some recommendations about nonlinear absorber tuning. Specifically, the absorber path, which is generalized to be non-circular, should be slightly softening for large amplitudes. Polchi [START_REF] Polchi | Multi-scaling and averaging for higher fidelity lumped model of bladed disk with cpva[END_REF] has some preliminary results along the same lines for the three-DOF-per-sector model considered here, which show that the linear resonance structure essentially persists for weakly nonlinear absorber motions. However, a detailed analysis with recommendations for path design remains to be completed.

The third category deals with systems of multiple absorbers, for which there are many possible studies. For example, is it necessary to have an absorber in every sector? The answer to this question likely depends on the level of intersector coupling. Also, can one use sets of absorbers to address multiple resonances? An affirmative answer to this question is given for the case of a single sector by Wang et. al [START_REF] Wang | Design of Pendulum Vibration Absorbers for the Attenuation of Transverse Vibrations in Rotating Beams[END_REF], but the multi-sector case has yet to be considered. The question of whether or not sets of absorbers can be arranged so that not every sector has absorbers tuned to every targeted order should also be considered.

The final, and probably most practical, line of work should consider the effects of mistuning and intentional detuning of the primary system and absorbers. It is well known that slight mistuning of cyclic systems leads to localization of vibration modes and of the forced steady-state response [START_REF] Castanier | Using Intentional Mistuning in the Design of Turbomachinery Rotors[END_REF][START_REF] Castanier | Consideration on the Benefits of Intentional Blade Mistuning for the Forced Response of Turbomachinery Rotors[END_REF][START_REF] Whitehead | Effect of Mistuning on the Vibration of Turbomachine Blades Induced by Wakes[END_REF][START_REF] Ewins | The Effect of Detuning upon the Forced Vibrations of Bladed Disks[END_REF]. All of the issues introduced by mistuning exist, and indeed are compounded, by the presence of absorbers, which raises a number questions. If the absorbers and/or sectors have mistuning, what is the degree of localization in the overall response? Of course, localized absorber motions are highly undesirable because the absorbers have only limited rattle space in which to operate. So, can one intentionally detune the system (sectors and/or absorbers) to mitigate the localization? Will such detuning be random, or just of some order of symmetry less than that of the ideal cyclic system? These questions, and no doubt others, should be addressed to more fully understand the effectiveness of order-tuned absorbers in practical systems.

Figure 1 .

 1 Figure 1. Lumped-Parameter Model.

Table 1 .

 1 Selected list of dimensionless parameters. Parameters Description δ = H/d Radius of the rigid rotor disk ρ = L/d Length of blade pendulum α Distance between points A and B γ = r/d Length of absorber pendulum µ a = M a /M d Absorber mass µ b = M b /M d Blade mass

Figure 2 .

 2 Figure 2. Campbell Diagram and Frequency Response.

15 FrequencyFrequencyFigure 3 .

 153 Figure 3. Campbell diagrams showing the effects of increasing inter-sector coupling and absorber mass.

Figure 4 .Figure 5 .

 45 Figure 4. Campbell diagrams and the corresponding amplitude frequency response.

ACKNOWLEDGMENT

The authors are grateful to Mathias Legrand for many enlightening discussions related to this work. This work was supported by the National Science Foundation (NSF) under grant NSF-0408866. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.


Toril Eskeland Rangnes

Andrea Synnøve

Blomsø Eikset

Preservice teachers' reflections on language diversity

Keywords: Language as resource, language as problem, preservice teachers

In this paper, we investigate how preservice teachers (PTs) express their awareness about language diversity in mathematics classrooms. We use "language as resource" and "language as a problem" as theoretical constructs. The data is taken from PTs´ group assignments, including transcriptions of dialogues between the PTs and multilingual students, and from PTs´ written reflections about teaching in multilingual classrooms. We found that in the dialogues, PTs and multilingual students produced meanings together through negotiating a shared repertoire from out-of-school and in-school mathematics context. Nevertheless, the students' home languages were not evident as resources in PT´s dialogues and reflections. A deficit perspective was present in their reflections. These findings lead us to rethink our teacher education courses so that language as a resource becomes more prominent.

Background

In this paper, we explore preservice teachers (PTs) understandings of how to teach mathematics in multilingual classrooms. In particular, we consider if, and how, they used the construct of "language as a resource" in their interactions with multilingual students and in their written reflections about teaching in multilingual classrooms.

The construct of "language as resource" has been developed as a framework to understand how languages are used and perceived in multilingual mathematics classrooms [START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF][START_REF] Planas | On the process of gaining language as a resource in mathematics education[END_REF]. Planas and her co-authors build on [START_REF] Ruiz | Orientations in language planning[END_REF] policy planning concepts of language as a right, as a problem, and as a resource. Ruiz focused on the resource perspective. This inspired [START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF] to develop a theoretical framework for exploring and theorizing how language can be seen as a resource in mathematics education. The focus on "language as a resource" has the potential to contribute to a shift away from seeing home languages, which differ from the instruction language, from a deficit perspective. This shift is also identified in CERME'S 20-year history [START_REF] Planas | Mathematics education and language[END_REF].

Nevertheless, how to understand and operationalize "language as resource" in multilingual classrooms can be challenging in teacher education, schools and in research. One attempt to do this is to use ideas about translanguaging in mathematics lessons. [START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF] described the focus of translanguaging as being on what learners do to convey meaning, on how they use all their languages, and not on the normative performance of the language of instruction. This includes "situations in which multiple mathematical sign systems, iconic representations of mathematical objects, and narratives of mathematical ideas co-exist" (p. 218). According to Planas, the concept of translanguaging supports a move from a focus on correct language to a more deliberate use of language resources in the participants' meaning making so that they develop mathematical ideas together. This is seen as deeply connected to equality in the classroom, as multilingual students often find themselves being viewed as lacking because their home languages are suppressed [START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF].

Teacher education context in Norway and in former research

The "ordinary" mathematics classroom in Norway has changed in the last 20 years from being monolingual to multilingual. In Norway, there are three official languages, Norwegian, Sami and sign language. Norwegian dominates as the language of instruction in most municipalities. In some classrooms with language diversity, there can be many different languages, spoken by refugees and immigrants. Often there will be students who will not have any classmates or teachers who can understand their home language. The discourse around classrooms with language diversity seems to be dominated by a "Norwegian only" attitude [START_REF] Naustdal | Diskursar om bruk av språk og omgrepslaering i matematikklasserom med språkleg mangfald -eit laerarperspektiv[END_REF], similar to what has been identified in neighbouring Sweden, where the described discourse is "Swedish only" (Norén, 2016).

In teacher education in Norway, there has been little focus on language diversity in mathematics education. For example, in the former National Guidelines for Teacher Education, language diversity is mentioned once in the chapter about mathematics and then only as something the PTs should learn to take into consideration. In the new National Guidelines (Universities Norway, 2016), the resource aspect is added: "the students' different needs, where cultural, linguistic and social backgrounds must both be taken into consideration and seen as a resource in the teaching" (p. 23). Our study is situated in this change, as we refocus our teacher education to incorporate the view that all students' languages can be a resource in the learning and teaching of mathematics. Through a four years long research project Learning about teaching Argumentation for Critical mathematics Education in Multilingual classrooms [LATACME], our research group aims to develop both tools for teaching and theories about this development through an educational design research program. The aim in this paper is to identify our starting point, namely, how preservice teachers are currently aware of, and interpret, multilingual diversity in mathematics teaching and learning as resource, which will be the basis for subsequent interventions.

Internationally we find few studies on how teacher education can develop understandings of language diversity in mathematics classrooms. For instance, [START_REF] Mcleman | Regarding the mathematics education of English learners: Clustering the conceptions of preservice teachers[END_REF] study from the USA revealed that issues about mathematics education of English learners needed to have more prominence and to be integrated into teacher education to hinder the development of a deficit perspective among PTs. They explicitly suggested that PTs would need to be provided with experiences within the context of mathematics to help them better understand the interconnection between language and mathematics.

Based on three different contexts (South Africa, Malawi, and Spain/Catalonia), [START_REF] Essien | Language diversity in mathematics teacher education: Challenges across three countries[END_REF] found that even though the mathematics teacher educators are aware of the linguistically diverse contexts, the "teacher education institutions rarely attend to the complexity of teaching mathematics to linguistically diverse mathematics student teachers in a structured way in their programs" (p. 107). Yet even when there is such awareness, it is challenging to change discourses [START_REF] Eikset | When does a difference make a difference? Teaching about language diversity in mathematics teacher education[END_REF]. In their analysis of their conversations about preparing PTs for teaching in multilingual classrooms, Eikset and Meaney found that other competing discourses, such as the PTs´ expressed needs for learning more mathematics, gained priority.

The data for this paper is collected from a mandatory group assignment in the first compulsory mathematics course for those who will become teachers in Grades 1-7. The PTs were asked to write 8-10 pages about number sense connected to children's language and to link this to theory and experiences from their practicum. As part of the process, they had supervision with the second author. The PTs had to include a research question and transcriptions of dialogues between themselves and students. These PTs were in language diverse classrooms, and their interpretation of teaching in the multilingual classroom became an important part of their text. The PTs were encouraged to use their own home language as resource in their problem solving during their teacher education program and were aware of the concept of language as resource. Consequently, our research question is: How do PT express their awareness and understanding of the construct "language as resource" through written dialogues and reflections? Theoretical framework [START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF] described "language as a resource" in mathematics education research as being used more as a metaphor than as a defined concept in the last decade. She demonstrated that "language as resource is not about the unproblematic functioning of language as producer of meaning taken as mathematical and shared" (p. 226). From her perspective, language contributes to the realization of normative meaning in the culture of schooling and school mathematics and does so by reproducing discourses connected to these cultures. In addition, knowledge of language enables learners to influence the process of gaining meaning through interaction. Therefore, language can be a resource both for meaning making and for the relocation of culture in learning situations in mathematics classrooms. From this perspective, mathematical language cannot be separated from the learning gained from other languages or be given priority. Instead, there is a need for language from different contexts, both in and out of school, to be combined in meaning production. In Bakhtin's dialogism, learning can occur in the tension between different forces [START_REF] Bakhtin | The dialogic imagination: Four essays[END_REF]. Tension can occur between voices where different opinions, understanding and linguistics settings are expressed. For example, demands to use the correct language can be one force, while focus on solving a mathematical problem can be another. Tensions might occur in the mathematics classroom between the stratified and formal mathematical language and the language of instruction on one side and the more informal and diverse multivoicedness on the other side (see for example [START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF][START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF]. However, tensions provide opportunities to negotiate meaning and in multilingual classrooms to learn both language and mathematics.

In their investigations of what happens in classrooms, both Barwell and Planas have focused on translanguaging. [START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF] emphasized the view of students and teachers as languaging, where they use "a repertoire of language practices that draws on students' varied experiences of communication in multiple contexts to make meaning in mathematics" (p. 161), whereas Planas (2018) stated, "translanguaging refers to what people actually do with language to convey meaning" (p. 218). This is a change of focus, from the language systems and the mix between them, to what students and teachers do with language. This includes the use of different representations and mathematical signs to make sense and convey meaning.

Methodology

The data was collected during the autumn of 2017 when groups of PTs completed a written assignment about mathematics, in particular number sense, connected to children's language, conceptual knowledge and representations in mathematics. The PTs were in their second year of a course to become teachers of Grades 1-7 and were in their first of two semesters of mathematics education. During this semester, teaching mathematics in a multilingual classroom was one of several topics. In their assignment, they were asked to take into consideration how they could adapt their teaching. Two groups chose to include language diversity as an issue in their assignment, and one of these will be analysed in this paper. It was mandatory for the PTs to obtain data that included oral responses from students, where the PTs' role was not to evaluate or judge their students. They were asked not to collect any personal data about the students. The assessment was completed before the PTs received a written request about whether this material could be analysed for research purposes. The dialogues presented in the group's assignment were between a PT and a student in grade 6 in a classroom setting (18 students). The PTs had given the students some tasks, provided them with individual help and in some cases asked the students to explain their thinking.

We analyse two excerpts containing dialogues between a PT and a multilingual, primaryschool student and the PTs' reflections on the dialogues and their comments about language and multilingual students. We use a two-layered analysis. First, we sought to identify what PTs and students did to convey meaning, how they used languages, and which repertoire they used, in order to identify instances of translanguaging. Second, we sought to identify language as a resource or as a problem in PTs reflections. To do this, we searched for tensions between voices and the potential for learning that they can create [START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF][START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF].

The original dialogues were in Norwegian and have been translated into English as faithful as we could to retain the original sense. Where the PTs have transcribed incorrect words that the student used, we place the word [incorrect Norwegian] in brackets.

Result and discussions

The PTs' aim in the activity was for the students to discover the connection between multiplication and division. Although this was the headline on the student work sheet, the PTs claimed that the student, Laban, whose home language is not Norwegian, did not read it. The task was 98/7= PT:

Can you explain for us how you were thinking here?

Laban:

Here I took 7, but the 9 is bigger. We minus [minuse] only with the nine, nine minus 7 is 2. Then we take the eight and wait… 14… so we took only 1 here (points to the second digit for tens) with 2 which is 28, then we take 7 up to 28 so that is 4, so that is 28 minus 28, which is zero. You should all the time, when it's 7 just you should just insert 1 up here, then it becomes 7 and if you take up till 9. Then it's 14, it can't be that, so we just take 7 times 9 then minusing [minuse] the 9. I can do it all in my headed [hoden].

PT:

Oh yeah! That is so cool, have you worked a lot with the timetable [gangetabell] before?

Laban: No, division.

PT:

That is sort of times, multiplication. It's kind of, the twin of division.

Together with his written solution, Laban managed to explain to the PT how he was thinking.

Laban demonstrated that he saw the connection between multiplication and division when, for instance, he said "which is 28 then we take 7 up to 28 so that is 4 so that is 28 minus 28 which is zero". He was switching constantly between multiplying up (7 up to 28) and reversing by subtracting so he could check if there would have been zero left to divide.

In the PT's response to Laban's explanation, s/he amplified Laban's explanation by exclaiming "Oh yeah! That is so cool". The PT also asked Laban if he had worked a lot with times tables (gangetabell) which is an informal word for the multiplication table in Norwegian. Laban denied this by saying, "No, division". His "No, division" is an interesting change as division is a more formal expression, even if the PT uses an informal expression "times" (gange) instead of multiplication. The PT then followed up with "that is sort of times" before s/he changed the wording to "multiplication", "It´s kind of, the twin of division". The introduction of "multiplication" as "the twin of division" seems to be a new language construction, perhaps to support Laban to be more aware of the connection between multiplication and division, even though he already demonstrated this in his explanation. As the student and PT used resources from their informal and formal language to make meaning together, they were languaging [START_REF] Barwell | From language as a resource to sources of meaning in multilingual mathematics classrooms[END_REF]. The student's idiosyncratic language of "minuse" (subtract), and his conjugation of a noun into a verb as in "headed" (hoden) was in the transcription in the group assignment. However, the PTs did not highlight these examples.

In this case the tension in language use was apparent in the juggling between formal and informal mathematical language, rather than between a home language and the language of instruction.

When Laban explained and requested time to think, indicated by his "wait", he used informal words and sentence construction. While he was answering the PT's question, he began to use formal mathematical words, such as division, which belong to the school mathematical vocabulary. The dialogue between PT and Laban is characterized as a mutual willingness to make sense together. It seems that the student affected the language choices of the PT.

The group of PTs reflected on the dialog in their assignment:

He explained his way of thinking and led us through his calculations. This gave us the impression that it was not his number sense that was the challenge for Laban. He is a student we can clearly see has gained much practice in mathematics.

By mentioning that Laban explained his thinking and led them through his calculation, they recognized his skills to communicate and his number sense. However, they indicated that there was still something which was a challenge for Laban, without specifying what this was.

By including his language errors, one interpretation is that the PTs saw language as a problem even as they recognized his communication skills. In the summary, the PTs repeated their belief in Laban's and another multilingual student´s number sense and their explanation skills:

The students could also explain their way of thinking, the different types of calculations and how they could use this in their daily lives without much difficulty. … Thus, we will say that it is not the numerical understanding or the number concept that hinders the students from dealing with mathematics, but rather that it is insufficient understanding of the concepts.

By stating that the students can "explain their way of thinking" and that "numerical understanding or number concept" is not the problem, they recognized the skills the students had, and implicitly that the students were able to use language in Norwegian as a resource to make others understand their thinking. However, they also mentioned what hinders the students from "dealing with mathematics"; the multilingual students had "insufficient understanding of concepts". In this way they saw the student in a deficit perspective and the lack of concepts in Norwegian as a problem.

As an example of the problem that a multilingual child had with understanding mathematical concepts, they included this conversation with another multilingual student, Ahmed: Ahmed: That it's fair. That there are equally many on each team.

Ahmed: What is "even" [
PT: YES! That is another way of explaining "even". The PT used an example from a familiar situation for Ahmed, soccer, to explain what "even" means in mathematics. By using language repertoires that they had in common from outside of school mathematics, they had made sense and gained a shared understanding of the mathematical concept. In their assignment, they explained how they used the example of soccer as a translational link to help the student understand the concept of "even". The PTs used the conversation as an example of where language can cause a problem, as they write "the student does not have an understanding of Norwegian term for the concept of even". At the same time, they demonstrated how they used language as a resource in the dialogue by drawing on the example from soccer as translation link. They used repertoires from different contexts, formal and informal school mathematics, and out of school activities as soccer, to make sense together.

A question arises about whether the students' and PTs' use of languages from different situations, can be considered translanguaging [START_REF] Planas | Language as resource: A key notion for understanding the complexity of mathematics learning[END_REF]. In the Norwegian context, it was repertoire from formal and informal Norwegian which were being transformed as meaning is being made. The PTs' understanding of language in multilingual classrooms also reflected a tension between how they made use of theirs and the students' language resources from different contexts, and seeing their student from a deficit perspective where language is a problem:

During the practicum, we learned that there is a connection between being able to master mathematics and to master the language. Minority-speaking students may have limited vocabulary and little understanding of Norwegian terms. This may be because the student speaks little Norwegian at home, in his immediate social environment, or never experiences situations where the terms are used.

The PTs recognized the connection between language and mathematics learning. It is clear from their perspective, that it could be a disadvantage for a student to not have Norwegian as their home language, which could cause limited understanding of Norwegian terms. The PTs only mentioned the Norwegian language. The home language and mathematical concepts that the students may have in this language were not discussed. There was no evidence that the PTs saw the students using their home languages in mathematics. This suggests that it may be difficult for PTs to know how to use the students' language and cultural background as resource and this pedagogical practice remains more an idea than an experienced reality in primary schools.

Concluding remarks

In this study, we identify tensions in more than one sense. There is a tension between "language as resource" captured in the transcribed dialogues and "language as problem" in the PTs' reflections. This tension could provide opportunities for learning if this tension become topics for discussion. In addition, we find a tension between students' and PTs' language use with informal and formal mathematical terms. Together they were languaging by using experiences of communication in multiple contexts to make meaning (Barwell, 2018, p. 161), as when they used soccer to explain even, and in the PTs' effort to connect concepts, when making the new language construction, multiplication as "twin to division".

Planas' (2018) conceptualisation of language as resource has given us the opportunity to see how PTs use a repertoire from different contexts in their meaning making. The opposite metaphor, language as problem, made us aware of where PTs see language diversity in the mathematics classroom as a problem and how the students' home languages are neglected.

If language as resource, which includes translanguaging, is going to be valuable in PTs' teaching in the future, they need to be exposed to broader understandings about multilingual teaching both in teacher education and in the practicum, as [START_REF] Mcleman | Regarding the mathematics education of English learners: Clustering the conceptions of preservice teachers[END_REF] suggested.

How can different languages be present in Norwegian schools where there can be so many different languages in one classroom and when the "Norwegian only" discourse is so strong? There is a need to develop a common understanding of what language as resource can be together with the teachers in the schools where students do their practicums. As the PTs' dialogues demonstrated, repertoires from different contexts, informal and formal, and the students' culture, such as soccer, can be resources in teachers' mathematical dialogues with students. Identifying these potential resources in shared repertoires of experience can be developed further to include students' home language. Teacher educators need to engage in open discussions with PTs about which discourses dominate and which discourses should be developed to avoid language being viewed from a deficit perspective. Our next step is to develop an intervention in which practicum teachers, PTs and teacher educators explicitly discuss tensions between language as resource and language as problem. Working together to change our discourses may help us to explore how different languages can be present.

Acknowledgment

This research project is supported by the Research Council of Norwayprogram FINNUT.