Modelling of train floor sound transmission using coupled FE-SEA analysis

Ulf Orrenius 1 Akustikdoktorn Sweden AB Drakens Gränd 6 II 111 30 Stockholm, Sweden Torsten Kohrs 2 Alstom Am Rathenaupark 16761 Hennigsdorf, Germany

ABSTRACT A hybrid FE-SEA numerical method is applied to calculate transmission and radiation properties of periodic structures typical for railway design. Train floor structures made from extruded aluminium profiles with and without interiors are analysed. Such structures are common in rail vehicle design, in particular for high-speed vehicles, and are typically subject to various noise control measures. Accurate and effective simulation models are key to reaching the acoustic design targets while respecting weight and cost budgets. The simulation concept applied, rests on that an accurate Statistical Energy Analysis (SEA) representation of the structure is determined from a small FE model consisting of a few cells of the periodic structure. The FE model can be kept small and computationally efficient, and the system model can for this reason be used for parametric studies of the effect of design changes. Calculated transmission loss results are found to compare well to measured data. Also, radiation efficiencies are calculated and compared to measured data.

1. BACKGROUND

The design of rail vehicles is driven by a number of functional requirements. One of these is the internal acoustic comfort of the vehicle. As sound pressure levels below the carbody may be above 120 dB(A), very effective noise barriers are needed for floors above bogies and Diesel engines, as well as for the roof below pantographs of high-speed trains. In addition, structural sound transmission should be kept to a minimum, e.g. for Diesel vehicles for which the engines may be strong sources of structure-borne sound. Solutions have to be sought in consideration of several other design criteria, such as static and dynamic stiffness, thermal insulation, partition thickness as well as weight and production costs. An overview of railway sources, and the challenges associated with sound transmission into the carbody, is given in [1].

1.1 Rail carbodies in aluminium

Rail carbodies are made in aluminum, steel or in some cases, composites, predominantly GRP but also metal sandwich designs. Aluminium carbodies, although more expensive than corresponding steel designs, can be made lighter and to a high degree of pressure tightness, making

1 ulf@akustikdoktorn.se 2 torsten.kohrs@alstomgroup.com

them the primary candidate for high-speed trains . For aluminum carbodies, wall, roof and floors are normally made from extruded hollow profiles that are welded together, cf. Figure 1.

Figure 1: Structural layout of extruded body-shell section.

1.2 The need for prediction tools

Due to their poor acoustic insulation properties, train structures in extruded profiles are subject to extensive noise control treatments and there is a strong interest from the industry to be able to validate and optimize such designs using computer models. The principal application is to support decision- making in the engineering process, preferably in its early stages when the design still can be influenced without major cost and project time losses. For this reason, the model should be able to correctly resolve the effects of design alterations that have an impact on noise transmission, including dedicated noise control treatments such as introduction of damping layers and stiffness alterations.

The ability to model the complex radiation and transmission mechanisms associated with the modes of built-up structures is critical to the accuracy of vibro-acoustic predictions. The transmission properties of built-up structures is, to a large extent, governed by near field radiation in the vicinity of exciting forces and at the member junctions making it necessary to resolve the detailed vibration field to correctly determine sound radiation.

1.3 Vibroacoustic modelling

Traditionally, methodologies applied for industrial acoustic predictions are based on either Statistical Energy Analysis (SEA) [2] or Finite Elements (FE) combined with Boundary Elements (BE) [3]. For large structures at audio frequencies, the fluid-structure FE-BEM models become numerically expensive. The SEA methodology is suitable for vibro-acoustic analysis of large structures, like those of aircraft and trains which mostly exhibit sufficient modal density throughout the audio frequency range. Early work applying SEA for extruded profiles in train structures are reported in [4], [5] and in [6]. Also, analytical models based on smearing of the stiffness and mass have been reported in [7] and waveguide Finite Elements (WFE) in [8] and in [10].

For structures made from extruded profiles, SEA models should account for the fact that the vibrational modes relevant for sound radiation can be divided into two groups: global and local modes [5] and [9]. This characteristic also applies to many other stiffened engineering structures like those of airplanes and ships but the cut on frequencies for the local modes are typically higher for rail structures making it relatively more important for the modeler to include both mode types.

A promising concept for periodic engineering structures is to determine the parameters needed for an SEA model from the solution of a small FE model. This FE model is then built up from one or a few cells of the periodic structure and periodic boundary conditions are applied. The FE model can be kept small and computationally efficient, allowing for parametric studies of the effect of design changes. This concept has been applied for train structures in [10], [11] and [14], and for aircraft structures in [14], [11] and [12]. In reference [13] the concept is applied to minimize the weight of an extruded train floor, subject to acoustical and structural constraints.

2. AIM

The overall objective is to develop a robust modeling method to used for parametric studies of train floors and ultimately be a part of structural optimization scheme with acoustic transmission constraints. The present work aims at benchmarking the wave6 technology for periodic structures for the application to extruded panels of rail carbodies. The concept is described below and in reference [15] and is applied both to a structural floor design as well as to a complete floor, for which a metal sandwich suspended on rubber elements functions as a walking floor.

3. MODELLING STRATEGY

3.1 Summary of present approach

The current work adopts the periodic theory developed by Langley for two-dimensional structures [16]. Consider a general structure with 2D periodicity. A periodic cell can be extracted from the structure and modeled using the Finite Element method. The degrees of freedom q of the cell can be partitioned into interior ( I ), edge ( L , R , B , T ) and corner ( LB , RB , LT , RT ) degrees of freedoms. The internal degrees of freedom are not connected to other periodic cells. The vibration response of the structure at a given frequency can be analyzed by specifying a phase lag between the displacement degrees of freedom at the left, right, bottom and top edges:

y T B  − = q q . (1) Similar transformations apply to the corner degrees of freedoms. The terms  x and  y are referred to as ‘phase constants’. For free wave propagation in an undamped structure these constants are real and can take values between -  and  .

i e

i e x R L  − = q q ,

To apply periodic boundary conditions it is necessary to ensure that the node locations on opposite edges and corners of the cell are identical. The complete vector of local degrees of freedom of the cell can be ordered so that q = [𝐪 𝑰 𝐪 𝑩 𝐪 𝑻 𝐪 𝑳 𝐪 𝑹 𝐪 𝑳𝑩 𝐪 𝑳𝑩 𝐪 𝑳𝑻 𝐪 𝑹𝑻 ] T . The undamped harmonic equation of motion for the unit cell is then given by [ K -  2 M ] q = F , where M and K are the mass and stiffness matrices for the cell and F is the generalized force vector. An eigenvalue problem can be derived from this that can be phrased in different ways depending on whether frequency is fixed or phase constants epsilon is fixed. This enables that also frequency dependent materials can be modeled accurately.

By solving the algebraic eigenvalue problem, the properties of the free waves propagating in the periodic structure are obtained. From these properties, the vibro-acoustic properties of the structure are calculated using the concepts of the ‘wave approach to SEA’ [2]. Expressions are derived for the modal density, damping loss factor, and distribution of squared velocity response per unit energy of the structure. Using a Fourier transform approach, the resonant radiation efficiency, resonant and non-resonant transmission losses, and acoustic input power are also obtained. See reference [16] for details.

This general SEA subsystem formulation addresses many problems encountered with traditional analytical SEA formulations and has been applied below to predict the transmission loss of a number of structures common in aerospace and railway designs. For all calculations, the periodic subsystem module within the Wave6 software has been used, see reference [15] and [19] .

4. STRUCTURES ANALYSED

Modelling of train floor panels has been conducted and the results were compared to experimental results. Two different structures were analyzed:

- A bare aluminum structural floor from extruded profiles.

- A complete floor assembly an inner floor panel suspended on rubber elements

4.1 Bare aluminum structural floor

The bare extruded profile structure is shown in Figure 2 and Figure 3. On the top of the panel, CLD damping material is attached to increase the loss factor. The CLD consists of a 0.5 mm aluminum sheet bonded to the base with a viscoelastic layer. The total surface weight is 28 kg/m 2 .

Tomm 510mm

Figure 2: Train floor structure in measurement setup (left) with CLD strips mounted (right)

foam —_vibration isolator aluminium sandwich = glue_—floor cover

Figure 3: Cross-section of the bare floor structure (top) and of the complete floor (bottom).

4.2 Complete floor assembly

The complete floor is based on the bare aluminum floor structure as described in Section 4.1 with an inner floor panel suspended on rubber elements on top. The assembly is shown in Figure 3. The inner floor panel including glued floor cover has an overall thickness of about 10 mm and consists as base structure of an aluminium sandwich made from thin sinusoidal core sandwiched between two 1mm flat aluminium sheets. Between the inner floor and the base structure, a 15 mm 9.5 kg/m 3 melamine.foam is inserted for acoustical and thermal insulation. The total surface weight is about 40 kg/m 2 .

5. MEASUREMENTS

For sound reduction measurements the structure was set up in a horizontal test-rig with a transmitting room below and a receiving room on top, as shown in Figure 4. The sound insulation was determined using ISO 140-3 using a dodecahedron loudspeaker in the transmitting room with pink noise in the frequency range from 50 Hz to 5 kHz. Three different loudspeaker positions were used.

Also the velocity levels at the receiving side of the test panel were recorded at 16 positions and energetically averaged. In parallel, the sound level was recorded in the receiving room using a mobile microphone at three microphone tracks and logarithmically averaged. The radiation efficiency was determined as

𝑊 𝑟𝑎𝑑 𝑣 2 𝜌 0 𝑐 0 𝑆 , (2)

𝜎=

with 𝑊 𝑟𝑎𝑑 = 𝑊 𝑑𝑖𝑠 =< p̃ 2 > A/4ρc , where V and A are the volume and the equivalent absorption area of the receiver room and S the panel area.

Figure 4: Sketch of test set up.

6. SIMULATION MODELS

6.1 Bare aluminum floor

The model geometry includes bays with triangular extrusions that have been simplified to a constant plate thickness of 2.7 mm. Based on the modeling strategy mentioned in Section 3 only a small FE model of a unit cell was created, as shown in Figure 5. The length in x-direction was chosen such that one repeating element is represented. The expansion in the y-direction ensures that the aspect ratio of the unit cell does not extend above ~5, being an experience based upper limit.

The model comprises 708 CQUAD elements with the cell dimension 0.51 × 0.09 m. Periodic boundary conditions, as described in Section 3, are applied to represent the real panel size of 2.54 × 4.20 m. Aluminum material parameter as of Table 1 was assumed for the entire model. Loss factors from decay tests were applied .

Table 1: Aluminum material parameters

Young’s modulus [Pa] Shear modulus [Pa] Poisson ratio  Density [kg/m 2 ]

70 e9 26 e9 0.33 2700

Receiving Room || _— Test object Sample Source Room

Figure 5: FE unit cell of cross-section of floor structure: top Model A, bottom Model B.

In a second set-up, Model B, the material thickness was adapted to better represent the real structure. An increased thickness of t =5 mm is here assumed in the middle-colored and t =7 mm in the dark colored areas. whereas t = 2.7 mm is kept in the light-colored area. The reason for this model update was the conclusions made from the transmission results from Model A as reported in Section 7 below. Even if this is not an exact description of the geometry, it roughly accounts for that the plates are significantly thickened in the joint areas.

6.2 Complete floor assembly

A representation of the complete floor assembly has also been developed representing the layup described in Figure 3 . The inner floor was modeled with CQUAD elements whereas the vibration isolators were modeled as lossy linear springs, The spring stiffness were taken to arrive at measured values of stiffness per unit area (including the air spring stiffness). The foam was not modelled explicitly, although the damping of the inner floor was slightly increased to represent the effect of the foam.

Figure 6: FE unit cell of cross-section of floor structure (Model C).

7. RESULTS

7.1 Bare aluminium floor

For structures made from extruded profiles, the vibrational waves relevant for sound radiation can be divided into two groups representing global and local cross-section modes, see references [5] and [9]. Such vibration characteristic also applies to other stiffened engineering structures like those of aircraft and ships.

The frequency at which the sub-panels start to have resonances is here referred to as the local cut- on frequency, or transition region, typically at 0.4-0.6 kHz [5] . Also important are the coincidence frequencies, above which the structure radiates sound efficiently. For the global modes, for which the panel structure vibrates as a whole, the coincidence frequency is typically between 130 and 170 Hz while for the sub-panel waves coincidence occurs around 4 kHz. The combined width of the two coincidence regions, together with the trace matching in the transition region, is a major reason for poor sound reduction properties of these panels.

Figure 7: Global wave of the FE unit cell at 125 Hz (visualization of five unit cells in the x-direction and one in the y-direction).

Figure 8: Local wave of the FE unit cell at 630 Hz (visualization of a single unit cell in x direction, ten in y-direction).

Transmission loss (TL) for the Model A structure were first calculated as shown in Figure 11 together with measured results. In the calculation, the excitation was described as a diffuse field, with a limiting angle of incidence at 78 deg. To resemble the measurement conditions, aperture effects based on the real panel size were accounted for. Generally speaking, the model captures the trends of the measured TL but the increase due to the cut-on of local modes appears at a lower frequency in the model compared to the measured results. This result indicates that Model A with the constant thickness shells for webs and the horizontal sheets was too simplified, See also reference [8] regarding stiffening effects.

To better understand the transition from local to global cross-section modes and the associated overprediction of TL, wavenumbers of different propagating waves were analysed as illustrated in Figure 9. At low frequencies (grey area), the panel behaves like a flat plate and only three wave types propagate. The upper grey line represents a flexural wave for which the cross-sections deform very little, see the cross-section bending wave in Figure 7.

At higher frequencies the individual sub-panels start to vibrate independently, and the transmission is determined by the properties of these panels, see Figure 8.

Figure 9: Model B wavenumbers for structural waves in y-direction (grey) and for air (blue).

Wavenumber (rad/m) 100 04 2 100 200 400 600 800.1000 Frequency (Hz) 2000 ‘4000

Figure 10: Frequency zoom of wavenumber plot for the transition range for Model A and B. In Model B the regions close to the web intersections are instead modeled with an increased thickness better representing the real geometry. With the updated geometry, the transition region is moved to higher frequencies, see Figure 10, and the match between calculated and measured TL is much better as displayed in Figure 11. However, in the low frequency range from 100 to 500 Hz TL is still under-predicted by 3 to 7 dB.

The modes shapes associated with the wave types shown in Figure 10 are presented in Figure 8 for 630 Hz, showing large deflection on the long side of the outer triangles.

Wavenumber (rad/m) 300 400~«500.—«600—«700 00 Frequency (Hz) Air Model A, Model B 900 1000 11001200,

Figure 11: Calculated and measured transmission losses for Model A and Model B In Figure 12, measured and calculated radiation efficiencies are shown. The match is quite good. Measured data above ~1 kHz are not reliable, possibly due to mass loading effects from the weight of the accelerometers. The radiation efficiency function displays three ranges of high radiation: 0.1- 0.4 kHz, 0.8-1.2 kHz and above 2.5 kHz; all corresponding to trace matching of flexural waves in the structure and the fluid waves, as illustrated by the wavenumber plot of Figure 9 .

Transmission loss (dB re: 1) ‘= Measurment without floor + Model A = Model 8 100 200 400600 800 1000 2000 +4000 Frequency (Hz)

Figure 12: Measured and calculated radiation efficiencies

7.2 Complete floor model with interior floor

Before anlysing the complete floor structure, a model of the inner floor aluminum sandwich panel was determined based on the stiffness properties provided by the manufacturer. As the stiffness properties did not differ much in the x and y directions, an isotropic representation was applied. The resulting TL matches measured results quite well as displayed in Figure 13.

Radiation efficiency () 200 00 800 1000 Frequency (Hz) 2000 Measurement + Mode! +4000

Figure 13: Calculated and measured transmission losses for the inner floor

In Figure 14, TL results determined for the full floor assembly according to Model C is shown together with measured results. The correlation is reasonable, although at high frequency the transmission loss is underpredicted. For the present models, fluid elements in the unit cell representation were not included (apart from the air-spring effect), but the Model C could be extended to also include the melamine foam using e.g. a Delaney and Bazley representation.

100 400 2000 Measurement

Figure 14: Calculated and measured transmission losses for Model B and Model C.

8. CONCLUDING REMARKS AND PERSPECTIVES

The unit cell method to feed SEA calculations as presented in [19], has been used here to analyse train floor structures. It’s been possible to correctly calculate the transmission and radiation of complex engineering structures typical of railway floors throughout the entire audio frequency range at moderate calculation costs.

The method as such is based on a deterministic (FE) model of the periodic structure surrounded by semi-infinite fluids. From the deterministic analysis of the wave propagation in the structure, the vibro-acoustic parameters are calculated with algorithms using the concepts of the ‘wave approach to SEA’. For this reason, the resulting parameters are by nature statistical (even though the structure is described using a deterministic approach) and can be imported into an SEA system model. This allows for an improved description of subsystems that cannot readily be described using the standard library of subsystems typically available in commercial SEA codes. The modeling concept is a powerful tool to analyse effects of design alterations, such as alternative stiffener lay-out or various noise control treatment which is useful in the early design phases.

100 200 ‘Measurement with inner floor Model 8 “Model C

Various loads can be applied, as a point force or a diffuse field (this paper), or also a turbulent boundary layer or a pressure field imported from CFD codes [18]. The fluid-structure coupling model allows for different fluids on the source and receiving sides of a structure. The calculation cost is a few minutes to hours on a modern workstation.

The unit cell can then be integrated into a more realistic model, which can be constructed within wave6, starting either from a 3D CAD model or from a Finite Element model, depending on what is available. Using modern geometry manipulation objects and meshers, a full train can be built within a few hours (or a couple of days depending on the exact inputs). It’s then straightforward to apply realistic loads to the floor and get the vibro-acoustic response of the full train. Figure 15 shows a typical example on a realistic train geometry.

The method theoretically requires the analyzed structures to be periodically repetitive, but since results from the deterministic unit cell model are used to determine parameters for SEA subsystems, an ensemble and frequency average is implicit, and it is therefore reasonable to deviate slightly from exact periodicity. Therefore, the method applies well to real train structures.

Figure 15: Integration of unit cell models in a carbody system model (top), and typical results using the SEA gradient functionality described in [19], for both the pressure within the interior cavity and the acceleration of the panels (bottom).

9. ACKNOWLEDGEMENTS

The authors would like to thank the wave6 support team at Dassault Systèmes, in particular Mark Teschner, for the support with reviewing of models.

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