Finite Element Modelling and Analysis of Wooden Musical Instruments The recent field of research by the Institute for Structural Analysis

described in this chapter is focused on two kinds of structures with respect to external loads: 1) purely hygrically loaded structures, such as furniture and panel paintings [21], and 2) more complex stringed wooden musical instruments, such as pianofortes [22, 23, 24], violins or guitars. The latter class, additionally loaded by mechanical forces, is strongly susceptible to plasticisations and creep deformations, intensi-fied by moisture-dependent mechano-sorptive effects.

The focus of this paper is the presentation of structural analyses of heavily-loaded wooden instruments. The investigated structures are subjects of research in recent cooperations with the Stiftung Händel-Haus, Halle. To provide easy access to the opportunities offered by FE-modelling, several features of the complex structural behaviour with respect to mechanical and hygrical loading are discussed step by step using concrete simulation examples. The effects of elastic and failure properties of wood depending on moisture content and their consequences on the structural scale are described in Section 2.2. Time-dependent, transient analyses consider the diffusion-based consequences of unsteady moisture distributions in the structure in Section 2.3. As stated above, the numerical models and simulations presented show the current state of development in complex structural analysis at the In-stitute for Structural Analysis. Further analyses of moisture-dependent, long-term behaviour are planned.

2.1 Finite Element Method

The concept of the finite element method is to discretise a volume into finite sections, which is also called meshing. These sections or elements have at least one node in each corner. Commonly applied elements consist of tri- or tetragonal faces. In the three-dimensional case, the volume is often discretised into tetraeder or hexaeder solid elements.

Furthermore, surface elements for interaction with the environment 1. Introduction

Museums with collections of historical music instruments are in conflict between conserving the original substance and maintaining the original use. Playable stringed keyboard instruments, in particular, are complex wooden structures under heavy mechanical loading. Hygrical loadings as alternating climate conditions induce additional mechanical loadings and influence the physical properties enforcing damage to the structure like large deformations and cracks.

The preservation of wooden cultural heritage is of eminent impor-tance, as pointed out by this book and a large number of further, recent publications on scientific experimental and numerical investigations of the mechanics and hygroscopicity of panel paintings (e.g., [1, 2, 3, 4, 5]), wooden musical instruments (e.g., [6, 7, 8, 9, 10, 11, 12, 13, 14]) and even shipwrecks (e.g., [15, 16]). In recent years, the investigation of complex structures in general and historic wooden musical instru-ments in particular has been strengthened by a further powerful tool for structural analysis with respect to the load-bearing behaviour at several external loadings, such as mechanical forces or environmental climate conditions. Due to the development of powerful computers also on the level of common work stations, computational engineering and the numerical simulation has become a standard in the repertoire of struc-tural analysis. Complex object geometries and loadings, high nonlinear structural interactions and material behaviour as well as the comprehen-sive consideration of multi-physical aspects as mechanical and climate issues need to be considered. Therefore, the Finite Element Method (FEM) is predestined for the numerical solution of multi-physically coupled, nonlinear differential equation systems.

In principle, the FEM offers a broad spectrum of imaginable structural analyses: Dynamical FE-analyses enable wave propaga-tion and vibrapropaga-tion simulapropaga-tion or modal analysis, which is necessary for numerical characterisation of an instrument’s tone and acoustics.

Static analyses enable the simulation of the load-bearing behaviour at multi-physical conditions and in the short- and long-term range.

The analysis of stresses, deformations, risk of failure, stability and durability, etc. are feasible.

Together, experimental analysis tools and the FE-analyses comple-ment each other. On the one hand, expericomple-ments on materials and struc-tures, for example, are necessary to validate numerical models. On the other hand, numerical methods help to avoid destructive or invasive experimental tests. Moreover, it is possible to gain insight into the invis-ible internal load-bearing behaviour at current conditions or to predict future behaviour. Therefore, the material properties need to be known and the material models have to be adequate for the aimed issue, yet as simple as possible to reduce the numerical effort.

(surface forces or heat and moisture transfer, see Section 2.3) and in-terface-elements are needed for the inhere analysed three-dimensional structures. Interface-elements enable the consideration of cracks by opening the initially coincident nodes of the lower and upper sides if the material strength is exceeded (see Section 2.2.3). In Fig.1, the applied solid and interface elements for the spatial discretisation of the keyboard instruments in this chapter are illustrated. They have an additional node at each edge. The three nodes per edge enable a qua-dratic shape function of the structural response instead of a linear shape for 8-node solid or 8-node interface elements. Each node has a certain number of degrees of freedom (DOF), i.e. the unknown char-acteristics, which need to be determined within the numerical solu-tion. The number and kind of DOF depend on the targeted results.

Common DOF are the displacements in all spatial directions and, furthermore, properties like moisture content, temperature, electric or magnetic field quantities etc.

Considering the investigated hygro-mechanical interactions, the con-tribution of one node to the equation system contains the unknown three-dimensional vector of the displacements 𝒖 and the moisture 𝒎

�𝑲 ^{𝒖, 𝒖} 𝑲 𝒖, 𝒎 � � 𝒖 � = � 𝑭^𝒂𝒑𝒍 � – � 𝑭^𝒏𝒓 _�, (1) 𝑲 ^{𝒖, 𝒖} 𝑲 𝒖, 𝒎 𝒎 0 𝑭^𝒎

with the components of the compliance matrix 𝐾�^, � and the vectors of external (applied) minus internal loads on the right-hand side.

Depending on the applied material models, 𝒎 may consist of multiple field quantities (see Section 2.3). The total equation system of the ana-lysed structure now contains the sum of all equations of all nodes, i.e., the product of nodes and DOF per node in Eq. (1). Due to mutual non-linearities, the total load cannot be applied at once. Instead, sub-steps have to be introduced with a smaller percentage of the loadstep. In

- (average of ij and ji-components) - micro-model: E( ρ,�)

- constant Poisson’s ratios E,G

� Ductile

inelasticity Multi-surface plasticity (e.g., [17]) - limit of elasticity (“compression strength”) - quadratic approximation �(m)

- differential shrink number (e.g., [17])

- constant �

elasticity - not considered (missing reliable knowledge) Thermal

energy - not relevant for application range - not considered (so far)

sorption [28] - strain rate factor

- modified hygro-expansion 𝜅

����

Creep

failure [27] - strain rate energy

- critical strain rate energy 𝑒 𝑒����

Hygroscop

y Diffusion multi-Fickian diffusion (Frandsen) (e.g.[19])

- micro-model �� (�),

�� (�) Sorption - sorption rate (e.g., [19]) 𝑐  ̇ = ∂𝑐/∂𝑡 Surface

emissivity Boundary layer theory (e.g., [19]),

Varnish permeability (e.g., [20]) 𝑙, 𝑣

��

Thermal

energy - heat conduction, surface convection - not considered (so far)

2.2.1 Moisture dependent material models

The internal resistance to any applied loading state is defined via the specific material characteristics. Thus, in the scope of a realistic three-dimensional analysis by the FEM, the material behaviour needs to be known and appropriate material models formulated. Wood is an inhomogeneous, anisotropic and porous material with moisture-, temperature-, time- and stress-state-dependent behaviour (e.g., [25]).

Appropriate material models for the description of the macroscopical, mechanical response to multi-physical loadings are required. Structural analysis input parameters, such as Young’s moduli and Poisson’s ratios to describe the elastic characteristics or material strengths for failure behaviour, and diffusion parameters, such as diffusion coefficients for moisture transport inside the material are needed. Due to the hierar-chical structure of wood with the cells at micro-scale, early and late wood at meso-scale up to the wooden construction parts, sawn out of a tree with its cylindrical material coordinate system at the macro-scale, adequate modelling of the macroscopical behaviour with a minimum of numerical effort is a challenging task. When possible, the applied models in this chapter (see Table 1) are based on clear, detectable physi-cal material parameters, rather than fitted phenomenologiphysi-cal functions.

If necessary, multi-scale models help to consider smaller scales of the hierarchical material structure (e.g., coefficients of elasticity or tran-sient diffusion models). Detailed descriptions of the applied models are published in the cited references in Table 1.

Depending on the analysis purpose, the uncertainty of the material parameters, for example, due to natural variation as well as climate- and time-dependent input quantities can be considered. Compared to moisture, temperature-dependency plays a minor role within standard climate conditions. To visualise the massive influence of moisture content on wood material properties, selected applied material direc-tion and moisture-dependent material model parameters are shown in Fig.3. The diffusion coefficients are presented here for the two-phase

Fig.3 Selected material parameters for spruce (𝜌₀ = 0.38 kg/m³), depending on moisture and material direction 𝑖 ∈ {𝑟, 𝑡, 𝑙}: Young’s moduli ���, compression strengths 𝑓�, �� and diffusion coefficients of bound water 𝐷�, � and water vapour 𝐷�, � (log. scale)

Fig.3

enough and has fallen below a defined convergence tolerance. In terms of time-dependent simulations, the time will be discretised into inter-vals, too. That means, the number of solutions is again multiplied by the number of time steps.

2.2 Analysis of a Historic Pianoforte – Static/Stationary Wood Material Properties

In the first example, a pianoforte is investigated with respect to the in-fluence of tensioned strings and time-independent, i.e. stationary effects of the wood’s moisture content on the mechanics. The Hammer ügel MS-44 in Fig.2 by Conrad Graf (1835) is object of interest in previous publications (e.g., [24]). The figure shows the original structure in its current, deformed state, a simplified volume model reduced to the undeformed load-bearing structure, and finally the analysed, discretised (meshed) FE model.

The volume model already shows the single parts of the structure in different shades. Based on these parts, specific material properties are assigned to every element in the FE model. The elements at the location of the iron brace receive the characteristics of iron. The wooden parts have information on the wood species and material directions (longitu-dinal, radial, tangential). With the specific material models, the whole structural behaviour can be described.

The static analysis deals with the time-independent phenomena.

With respect to the equation of motion, all observed phenomena are balanced immediately without any movement. Velocity and acceleration terms disappear. Stationary moisture diffusion and heat conduction lead to constant moisture and temperature values over time for every mate-rial point with constant flux.

Fig.2 Hammer ügel MS-44 (Stiftung Händel-Haus, Halle): current geometry, three-dimensional volume model and FE model.

Fig.2

quantitatively, since there is an over-estimation by regarding a constant maximum 𝒎-distribution throughout the whole volume, which would never be reached during a thunderstorm. On the other hand, some effects due to brittle failure (see Section 2.2.3), as well as stress concentrations with 𝒎-gradient by transport processes (see Section 2.3) and creeping (long-term behaviour) are not captured in this model problem. Neverthe-less, the influence of moisture on the load-bearing behaviour becomes obvious. Further results are published in [24]. The application of this approach to critical construction details, such as the joints of the piano, or to the entire structure can identify the failure potential or, in terms of a long-term simulation, the development of creep deformations.

2.2.3 Plastic and brittle failure of a pianoforte

To illustrate the potential failure mechanisms at hygro-mechanical loading, a heavily-loaded detail of the pianoforte was investigated (e.g., [26]). The pin block of a keyboard instrument has to take up the strings’ tension and to transfer it to the framework of the corpus over three connections: an iron brace in the centre and the two side wall joints (see Fig.6). Material and element formulations are equal to those in Section 2.2.2. Additionally, the veneer layer existing between

Fig.4

Fig.5

u_z [mm]

-5 5

(a) (b) (c)

multi-Fickian diffusion model. While the bound water diffusion is re-lated to moisture content 𝑚 (mass of water per mass of wood), like all wood-specific material parameters, the water vapour diffusion coefficient is related to vapour pressure and relative humidity 𝑅𝐻. A recent review on material modelling of wood with special focus on musical instruments is presented in [17]. A survey (in German) on developed moisture-dependent wood material models within the scope of FEM, including a comprehensive survey of experimental data for most of European wood species commonly used in musical instruments, is given in [20].

2.2.2 Elastic deformations and hygro-expansion of a pianoforte with respect to moisture content

The above structure of a pianoforte is chosen to visualise the influence of moisture-dependent elasticity of wood on structural behaviour.

The pianoforte is mechanically loaded by the tensioned strings at the pins carrying the loads into the load-bearing wooden frame-system.

These forces are determined for an original, historic tuning of cham-ber pitch 𝑎₁ = 450 Hz. The climate conditions are chosen as the observed limit values in the exhibition room as a dry climate in winter (𝑅𝐻/𝑇: 40%/18°C) and maximum possible climate conditions (𝑅𝐻/𝑇:

70%/28°C), corresponding to a thunder-storm in summer.

Three cases are analysed, the first at a constant winter climate and the second at a constant summer climate. The third case considers a quasi-stationary increase of moisture from winter to summer load case.

This means that the influence of swelling is considered, but not any tran-sient phenomena due to the time-dependent inequilibrium of moisture over the structure’s volume (see Section 2.3). The reduced FE-model is described without the bondlines and veneer layers (see Section 2.3.3).

Fig.4 depicts the deformations in vertical direction of the whole load-bearing structure (excluding the soundboard), while Fig.5 shows the displacement along the path on top of the side walls. The differ-ences between the winter and summer case are marginal, but show an interesting shift of the flux of forces, combined with a different defor-mation figure. Although the wood’s stiffness is reduced with increasing moisture, some regions show smaller displacements. At the same tun-ing, but considering the swelling during adsorption, the distinct mag-nification of deformations becomes obvious. This result shows also the huge influence of choosing the right reference moisture conditions, i.e., the conditions belonging to the undeformed, unswollen volume model at the beginning. The second and third case are finally situated in the

be seen in Section 2.2.2, the 𝑚-dependent hygro-expansion leads to massive volume changes with different quantities in the three material directions, or may lead to external constraints due to connected struc-tural elements. Another issue now is the path from the initial state to a new equilibrium over time. Due to the non-equilibrium of  𝑚 over the volume, additional internal constraints occur, caused by the 𝑚-gradient and the anisotropic shrinkage values.

Fig.7 Clavichord MS-85 (C.G. Sauer, Dresden, 1807; Stiftung Händel-Haus, Halle) and simu-lation results of the vertical displacements with pure mechanical loading of the ten-sioned strings at the beginning (𝑡 = 0𝑑) and after climate changes (𝑡 = 18𝑑) (see Fig.9).

Fig.8 Comparison of the hygro-mechanical structural response of a clear spruce wood sample to increasing relative humidity, from 65% 𝑅𝐻 to 95% 𝑅𝐻, with two different transient diffusion models.

Fig.7

Fig.8

x y z

the upper and the lower part of the pin block is described by 16-node interface-elements and a hygro-mechanically coupled interface material model (see Table 1). In both element types, the four degrees of freedom, i.e., the displacements in the three material directions and the wood moisture, are considered. For the bulk material, a coupled multi-surface plasticity model is applied. The load of the tensioned strings on the supported pin block is characterised in Section 2.2.2. Concerning the hygric loading, the third load case with increasing moisture from 40%

to 70% RH and hygro-expansion (swelling), is applied.

In Fig.6, two details of the original damaged pin block are shown.

The simulated detail at point P1 shows the plasticised wood due to the iron brace, which provokes a strong pressure perpendicular to the fi-bre direction of the wood. The veneer layer of the pin block is cracked on the bass side, where the loading due to string tension is extremely high (P2). The damaged parts of the interface coincide with the areas of damage observed in the real instrument.

2.3 Analysis of a Historic Clavichord – Quasi-Static/Transient Wood Material Properties

Within this second example of a clavichord by C.G. Sauer (1807) [Fig.7]. the effects of additional time-dependent, i.e., transient moisture trans-port through the volume shall be considered and analysed. As could

Fig.6 Hammer ügel MS-44: Damage in the pin block caused by local overstraining with local plasticisations around the iron brace (P1) and cracks near lateral support on the bass side (P2).

Fig.6

uz [mm]

applied, with 78% RH and 30% RH for the wet and dry cycles, respec-tively, starting with an initial room climate of 50% RH.

The clavichord’s wooden load-bearing structure is very complex.

The structural parts include different species with different hygro-mechanical characteristics and fibre orientations, namely spruce (side-walls and bottom, dry density 𝜌0 = 0, 38kg/ m3), beech (wrestplank, 𝜌0 =  0, 65kg/ m3) and oak (hitchpin rail, 𝜌0 =  0, 65kg/ m3). Due to the unknown exact material directions, a Cartesian coordinate system is applied for the anisotropic material directions. The fibre direction (local longitudinal material direction) is assumed to be in the length direction of the structural parts, and the tangential material direction is simply assumed to be in the vertical direction for all parts. Further simplifica-tions are applied to the structural model. To reduce the numerical effort, the oak veneer and the balance rail have not been considered thus far.

The glued joints are assumed as planar and force-fitted wood to wood connections, without any further diffusion resistance (see Section 2.3.3).

The previously introduced material models for elastic and fail-ure behaviour with respect to moistfail-ure dependency are applied. Com-pressively loaded wood, especially perpendicular to the grain, leads to ductile failure with plastic deformations beyond the elastic range, modelled by a multi-surface plasticity model (see Table 1). The mois-ture transport plays an important role in the analysis of transient processes, i.e., climate changes. The transfer at the surface is captured by a boundary layer model, while the inner transport is characterised by a multi-Fickian diffusion approach. A simpler and commonly used single-phase diffusion model is found to be not accurate enough (see Section 2.3.1). The two phases of bound water in the cell walls and the water vapour in the lumens are coupled via a sorption isotherm, without considering hysteresis.

The simulated deformed structure for the pure mechanical string tension load and at the end of the three cycles (see Fig.9) is presented on the right-hand side in Fig.7. Next to the influences of changing 𝑚-dependent material parameters, such as stiffness and strength, and the external hygro-expansional constraints of swelling and shrinking in Section 2.2, stress peaks at every climate change due to internal constraints are influencing the structural behaviour (see Fig.8). Due to the transient simulation, it is now possible to investigate the continu-ous development versus time. The simulation can give an insight into the process of time-dependent stress peaks due to internal constraints, caused by the anisotropic swelling and shrinking behaviour of wood.

2.3.1 Time- and moisture-dependent material models

In contrast to dynamic behaviour, the quasi-static, transient behaviour is characterised by mass-independent and acceleration-free processes of the macroscopic structure with unsteady changes over large time scales. As listed in Table 1, there are the mechanical transient models, i.e., all creep issues, and the climate-dependent behaviour. Additional descriptions of creep features and viscous material models for wood can be found in [e.g., 17, 27, 28].

An example, which might be a cross-section of a clavichord wrest-plank, shall illustrate the influence of transient moisture transport

An example, which might be a cross-section of a clavichord wrest-plank, shall illustrate the influence of transient moisture transport