GB2537365A - Method and device for efficient determination of the vibro-acoustic properties of sound insulation materials - Google Patents

Abstract

A method and device for experimental characterisation of vibration and acoustic properties of sound insulation materials comprises; two or more loud speakers 3a and 3b for air-borne sound excitation on one side of a specimen material 1; two or more pistons 2a and 2b for exciting the specimen on the other side via vibration from force shaker exciters 6a and 6b; two or more accelerators 5a and 5b, two or more force transducers 7a and 7b, and at least one pressure-particle velocity (PU) sensor 4 for measuring relevant data; a control unit 9 to control the shaker exciters; and a data acquisition unit 10. Alternatively, the device may comprise only one force shaker exciter [fig. 4, 6], a solid support [fig. 4, 11] and one or more flexible springs [fig. 4, 12] located between the support and piston 2b. The device measures sound pressures, normal particle velocities, accelerations and forces, and uses a mathematical method to computer a matrix which defines the excitation and response pressures/velocities of patches of the material. A calibration procedure and method for correction of an air gap between the PU sensor and the specimen.

Description

Method and device for efficient determination of the vibro-acoustic properties of sound insulation materials

Background of the invention

Driven by both the ever-increasing tightening of legal regulations and the growing customers expectations, the noise, vibration and harshness (NVH) is becoming a crucial aspect in the vehicle development process. To achieve the N V H targets set for modern vehicles, sound insulation materials became an indispensable instrument to improve the vibro-acoustic behaviour. Typically, the sound insulation materials take advantage of so-called poroelastic materials, which exhibit favourable properties when it comes to structural damping as well as transmission and absorption of sound. However, due to the highly complex material micro-structure and the sound propagation mechanisms involved the computational modelling of poroelastic materials is a fairly challenging topic. An efficient yet accurate prediction of the NVH attributes of sound insulation materials therefore remains an unresolved issue and is addressed by this invention.

State-of-the-art Sound insulation materials materials are widely applied as dissipative treatments in vibro-acoustic systems. Whenever a vibrating structure radiates sound into an acoustic fluid, the insertion of sound insulation materials has three main effects: (i) structural loading and damping, (ii) decoupling the acoustic fluid from the structure (mass-spring systems) and (iii) adding absorption to the acoustic fluid. The sound insulation components are typically assembled by two or more material layers, from which so-called fluid-saturated poroelastic materials constitute a substantial part. The poroelastic materials consist of two phases -the solid one, which forms the skeleton, and the interstitial fluid phase, which is contained within the pores formed by the solid phase. Since both the transversal and longitudinal waves can exist in an isotropic solid, and since a longitudinal wave occurs in a fluid, three types of waves can propagate through a poroelastie domain. As the frame and the fluid exhibit a strong mutual interaction, visco-thermal dissipation mechanisms take place.

Numerical modelling of the vibro-acoustic behaviour of sound insulation materials Over the last decades, various mathematical models ranging from simple concepts to sophisticated methods (Allard and Atalla, 2009) have been developed to represent the vibro-acoustic behaviour of sound insulation materials. In traditional numerical schemes (Kropp and Hciserer, 2003), the influence on the structure is usually described by additional mass and damping. The damping of acoustic fluid is captured by solving a system with impedance boundary conditions. Required parameters may be estimated by material models, ranging from simple equivalent mechanical systems to phenomenological impedance models (Delany and Bailey, 1970).

The current state-of-the-art in the numerical modelling of the vibro-acoustic behaviour of poroelastic materials is represented by material micro-model based on the Biot theory (Biot, 1956), which is implemented in a finite element method (FEM) or in a reduced transfer matrix scheme (Allard and Alan, 2009; Al imonti et al., 2014). Material parameters are obtained experimentally on material samples (Jaouen et al., 2008). Although this approach allows for highly detailed material description, its practical implementation leads to very large computational burden, which limit its practical application for low frequency range. This becomes even more pronounced as far as industry-sized problems are considered. Moreover, a proper estimation of the material parameters required by the Biot model is not at all straightforward and is hence the reason, why are these parameters often not available in practice.

Experimental characterisation of the vibro-acoustic properties of sound insulation materials Similar to numerical approaches described above, different experimental techniques have been developed to characterise the vibro-acoustic properties of sound insulation materials. Mechanical parameters of sound insulation materials may be obtained in dynamic stiffness tests and impedances are measured in a standing wave tube or in-situ on the material surface (Lanoye et al., 2006). An impedance tube is typically used for estimation of the normal impedance, which allows then the derivation of sound absorption coefficient and transmission loss (ISO, 1996). Reverberation chambers and transmission suites are used for the measurement of, respectively, sound absorption coefficient and transmission loss under diffuse sound field conditions according to (ISO, 2003). As these testing procedures involve large, special-purpose environments, alternative, non-standardised measurement procedures based on small reverberation cabins (Bertolini and Falk, 2013) have been developed over the past years. In order to determine the dynamic properties of damping layers, Oberst test method can be applied (Oberst, 1952).

As far as Biot model is considered, a set of nine material parameters must be experimentally determined in order to provide the numerical model with corresponding data. Hence, highly dedicated, laboratory apparatuses need to be utilised in order to assess all required physical quantities (Allard and Atalla, 2009).

The patch transfer function (PTF) approach The patch transfer function (PTF) coupling scheme (Ouisse et al., 2005; Aucejo et al., 2010; Pavia, 2010) has been introduced as a simple yet effective method to reduce the calculation time in coupled fluid/fluid and fluid/structure simulations. While having been developed for numerical applications, the relatively small number of discrete surface elements (patches) makes this approach also applicable to experimental characterisation of physical systems (Rejlek et al., 2013; Veronesi et al., 2014). For structure-bourne sound similar approaches have been introduced to couple sub-systems by mobility matrices (Petersson and Gibbs, 2000).

The range of validity is roughly limited by the frequency, where the wavelength in any of the coupled systems reaches a spatial aliasing limit. High dynamic range of sensors and highly accurate calibration and characterisation measurements are required to avoid random and systematic errors masking the results.

Here. PTF coupling equations for a built-up system consisting of a structural 13, sound insulation 1 and acoustic fluid domain 14 are derived, while discretising the interfaces G and,43 between the respective subsystems into patches, see figure 1. A linear complex average of field variables is taken over each patch surface. Thereby, integral equations for infinitesimal elements and Green's functions are approximated by matrix equations for discrete patches. Detailed derivations of the patch transfer function method can be found in (Ouisse et al., 2005; Pavia, 2010; Bobrovnitskii, 2001).

In the following, a slightly alternative matrix formulation is used to describe coupling between subsystems. While structure and fluid arc, as usual, characterised by, respectively, a mobility matrix Y and an impedance matrix Z, the sound insulation will now he characterised by a hybrid matrix H instead of a conventional impedance matrix. A similar technique has been described by Atalla et al. (2001) to integrate poroelastic materials into finite ekment models of structural and acoustic domains.

On the structural side, a velocity response on surface a due to a pressure excitation is given by a mobility relation Yp" = v". (1) The n-th row of Y are die velocities of the free structure due to an excitation of p = 1 on position n. The fluid surface impedance on (3 relates the pressures to the excitation velocities on the upper sound insulation-fluid coupling surface, (2) The n-th row of Z are the blocked pressures due to an excitation of)273, = 1 on position 11.

A relationship between pressure and velocity on the two surfaces of the sound insulation are given by the matrix equation [ [ [ (3) L P h8" h8"3 [ p8 V describes a velocity (kinematic) excitation from the bottom side and p3 a pressure excitation from the top side.

For the one-dimensional case (one patch on each side), bottom and top quantities p AT' and sub-matrices h&j are given by scalars pi, vi, 11,* that may be interpreted in the following way: * the impedance as seen from the bottom side while keeping the top side free (the inverse of the bottom mobility) (4) pg =0 * the transmission ratio from a pressure excitation on the top to the blocked bottom.

hoi3 = Pa Pfi * the transmission ratio from a velocity excitation on the bottom to the free top 12$ v Cl * the top surface mobility with a blocked bottom (the inverse of the surface impedance) 8 12513 -

PS

jinn. P (1 v." =0 p2 =0 (5) (6) (7) The H-matrix elements can in principle be obtained by straightforward measurement of the state variables provided well-defined boundary conditions are imposed (velocity blocked at the bottom interface, pressure release at the top interface). These boundary conditions approximate the boundary conditions, which are realised in the proposed experimental layout.

Let--Ni" and 15" be the source terms, i.e. the responses of the isolated systems without sound insulation to internal excitations. In other words, cr " is the response of the free (Pc' = 0) structure due to some internal structural excitation and PS the response of the blocked (irl = 0) fluid due to some internal acoustical excitation. Then, using the superposition principle (Bobrovnitskii, 2001), the response of the coupled system is given by the following set of equations v" = s<f" -Yp". (8) p" = h"v" + h"spu, v/3 = hv" + It"ps, (9) (10) = + (11) or in matrix form I Y 0 0 V' I 0 -Ifl3 136 0 -fr31 0 I -ha" VP 0 (12) 0 0 -Z I P' _ Due to the linear patch average, an aliasing limit is given when the wavelength approaches twice the patch dimension.

For the case without sound insulation where surfaces a and fi coincide, the corresponding reduced H-matrix is given by (13) and (12) is reduced to [ I Y [ v [ v [ -Z I i[pi [Pi' where v" = vi3 = v and p0 = p0 = p. Sub-system characterisation Y. Z and H may be obtained by either numerical or experimental means. In existing works. PTF matrices were derived by analytical (Ouisse et al., 2005; Pavia, 2010) or numerical (Aucejo et al., 2010) means. In the following sections, some techniques for the direct experimental characterisation of subsystems without the requirement of a numerical material model are proposed. In the reconstruction, these sub-systems are combined with each other or with their numerical counterparts to predict the response of the coupled system.

The principle of the proposed characterisation method consists in the measurement of the isolated subsystem response for a number of load cases equal to the number of patches, thereby forming a full set of linear equations. (14)

S

Structure A structure with a relatively high impedance is assumed to be freely vibrating in air (1-1 >> Zo). Therefore, it is possible to measure the terms yu of the mobility matrix in a direct way by exciting on position j and measuring the velocity on position i. In our case the excitation was applied by means of an impact hammer. An excitation similar to a uniform pressure excitation was realised by averaging over a sufficient number of hammer blows on an equidistant grid inside a patch. If grid point distance is small compared to the structural wavelength, the excitation corresponds to a uniform pressure excitation of the whole patch. Convergence to this condition can he proved theoretically by wavelength criteria and experimentally by reaching the level of grid refinement where the results stop changing significantly in the desired frequency range. The response can be measured, as in our case, by a grid of accelerometers, a scanning laser vibrometer or by an array of pressure-velocity sensors referred to as PU-probes.

Acoustic fluid For a direct characterisation of the fluid cavity a flat, square-shaped piston source should in principle be progressively positioned in the different patch locations. In practice, an acceptable approximation may he achieved by a roving point source ("tube" source). The blocked (the cavity must he equipped with high impedance wall) pressure response zji is measured using an array of pressure microphones or PUprobes. where PU-probes offer the advantage of directly measuring the particle velocity on the exciting patch. A realisation of an experimental cavity characterisation is currently under investigation.

Sound insulation materials Sound insulation materials feature a porous solid phase (also referred to as skeleton or frame) with interconnected interstitial cavities filled with a fluid phase (air). At the two coupling surfaces the area fraction occupied by the solid phase is very small. At the structure/sound insulation interface, however, the considerable pressure concentrations in the tiny contact areas between skeleton and adjacent structure cause fluid and skeleton to have the same normal velocity. The characterisation procedure must account for this feature and mechanical excitation with a piston is thus necessary at this interface in order to obtain representative data.

Moreover, at the structure/sound insulation interface both solid as well as fluid phase may significantly contribute to the average patch pressure, and microphones can consequently not be used but the average patch pressure must be obtained indirectly through piston force measurements. Ai the sound insulation/fluid interface, on the other hand, the fluid of the cavity adjacent to the sound insulation material is unable to support shear stresses and no pressure concentration will occur. At this interface direct excitation of the solid phase of the sound insulation material will consequently be negligible. Normal velocities of fluid and solid phase do not necessarily coincide and the governing state variables arc fluid pressure and fluid particle velocities. At this interface the characterisation procedure should consequently use microphones and particle velocity sensors, while mechanical excitation shall be avoided.

"This invention is used to specify a method and a device for efficient determination of the vibroacoustic properties of sound insulation materials."

Description of the invention

The invention relates to a process of experimental characterisation intended to determine the relevant vibro-acoustic properties of sound insulation materials. In this invention, the PTF methodology is applied to a physical structure/sound insulation/acoustic fluid system, whose constituents are experimentally characterised. The method proposed for characterisation of sound insulation materials exhibits following distinctive features: * The material is characterised in conditions similar to the mounted situation. This is relevant because poroelastic materials consist of skeleton and fluid that are excited with different physical mechanisms depending on the boundary conditions.

* The measurement is non-destructive and can be performed on arbitrary samples. Mechanical separation of the layers, with the risk of modifying their characteristics, is hence not necessary.

* Non-local effects due to wave propagation in the lateral direction of the sound package are accounted for by transfer terms. Propagation across the thickness of the material is described by cross terms. Please refer to figure 6.

Device for experimental characterisation of sound insulation materials In order for the sound insulation materials to be characterised according to the methodology described above, following device is considered, see figure 3. The material specimen 1 is placed between pistons 2a and 2b and loudspeakers 3a and 3b. The pistons 2a and 2b are driven independently by two force shaker exciter 6a and 6b, which mechanically excite the material specimen 1 from the bottom side. From the top side, the material specimen 1 is acoustically excited by two loudspeakers 3a and 3b. The pistons 2a and 2b are separated by a rigid baffle 8. Accelerometers 5a and 5b are attached to pistons 2a and 2b, whereas force transducers 7a and 7b are mounted between, respectively, the shakers 6a and 6b and pistons 2a and 2b to allow for reading of the pressure and velocity at the bottom of the material specimen 1. An array of pressure-velocity sensors 4 is located in the air gap between the specimen 1 and loudspeakers 3a and 3b to measure the sound pressure and normal particle velocity on the top side of the specimen 1. The shakers 6a and 6b and loudspeakers 3a and 3b are controlled by a control unit 9, whereas a data acquisition unit 10 is used to record the measurement data from PU-probes 4, accelerometers 5a and 5b and force transducers 7a and 7b. To experimentally re-construct the H matrix on the 2N patches (top and bottom), 2N linearly independent load cases are needed while measuring pressure and velocity on all patches using the test rig described here.

It has to be stated that top side and bottom side are only possible representations of the invention. The invention also comprises devices in other orientations than top and bottom.

Method for experimental characterisation of sound insulation materials Bottom surface active patch The bottom surface of the sound insulation specimen 1 is excited by a shaker 6 on patch i. To excite the whole patch, a patch-sized lightweight piston 2 is mounted on top of the shaker 6. The velocity of the piston 2, v7, is measured using an accelerometer 5 and the reaction force below the piston is obtained by force transducers 7. The reaction pressure on the sound insulation surface is given by = (15) where Ztet is the acoustic piston impedance. Zr primarily depends on the piston mass and on the radiation loading below the piston.

The pressure p7 on the sound insulation surface has to be reconstructed by subtracting the reaction pressure of the piston, Zrit from the pressure measured by the force transducers, JJ op' (16) "teas Fa, meas/ where Sp is the surface area of the piston 2.

The acoustic piston impedance Zrrt is obtained by a calibration measurement consisting of driving the piston 2 without sound insulation 1 and measuring the piston force Fia' cal with a force transducer 7 and the pressure pft "1 flush to the piston surface as well as the piston velocity it7''81 using the PU-probe array 4. The PU particle velocity sensor sensitivities are adjusted in such a way as to match the piston accelerometer reading. Clearly, under the mentioned conditions the indirect pressure reading with the force transducer 7 must match the pressure measured directly with the PU array 4 flush to the surface. The acoustical piston impedance on patch i is consequently obtained as na, cal a. cal (17) r1. / c'n a, cal Bottom surface passive patches For load cases requiring a non-driven bottom patch j, the pressure on the piston surface is determined indirectly by the relation Here the acoustical impedance Zit's' of the passive piston is again obtained through a calibration mea-surement. The speaker 3 on patch.' is switched on and the calibration pressure is again determined from measurements with the PU array 4 flush to the surface. The impedance of the passive system (combination of piston and remaining setup below) is then given directly as 7yass ".1 a. cal Top surface Acoustical excitation is provided by loudspeakers 3a and 3b centred above each patch. Pressures and velocities are measured by averaging over an array of PU-probes 4 located between the loudspeaker and the sound insulation material 1 as close as possible to the surface of the latter.

Reconstruction of the H-matrix According to (3), the H-matrix of the sound insulation can be reconstructed from 2N load cases by H -[ VPa [ where the matrices P', P3, V. Vi3 contain the excitation and response pressures and velocities for each load case in one column, i.e. [ Va [ [ P 1:. j "8 il-P 2 P 2N1 In principle, any linearly independent set of excitation conditions is suited to reconstruct the matrix H. V' In practice, however, bad conditioning of the matrix FS may lead to substantial error amplification in the inversion process, especially when experimental data containing inevitable inaccuracies have to be [Ara processed. Ideal excitation conditions consist of an orthonormal set of vectors i, . Stable results were obtained using the following four excitation conditions: loudspeakers "off" and left shaker "on", loudspeakers "off' and right shaker "on", shakers "off' and speakers "on" in phase, shakers "off' and speakers "on" in anti-phase.

Reduced device for experimental characterisation of sound insulation materials Wave propagation inside the sound insulation material in the lateral (in-plane) direction is characterised by substantial spatial decay. In this particular case the size of the patches is such that only the driven patch itself and the patch adjacent to the driven patch need to be considered, whereas propagation to farther patches can be neglected. In practice H may consequently be approximated by a banded matrix and the test rig he limited to two patches.

For two adjacent patches 1 and 2, pressures and velocities are labelled p, 7)!2, v1 and v. The elements of the 2 x 2 sub-matrices of a banded H in (3) are / I = 2 1, I:13 122 with surface 1,7 E {a "3}. If the material is both homogeneous and isotropic in the lateral direction, these terms are equal and symmetric for any two adjacent patches. The single elements of are called input (in) and transfer (tr) terms, (18) The input element = h;31 = 42 describes the input iesponse and the transfer element Or = ht,'31. the response of the next neighbour.

Applying these symmetry properties of h11, the number of load cases is reduced to half of the original ones. Since the response on patch 1 due to excitation on patch 2, i.e. [ p7 hnn Jan " P2 2 [ rinr I [ I L 2)? 2 is identical to the response on patch 2 due to the same excitation on patch IL with swapped patch indices 1 and 2, [PP:21: ii lir hr 1' q 1 ± [ haa h aa tr HI L tt 1. . . Consequently, for symmetric patch configurations, virtual load cases may be introduced by swapping indices and a reduced version of the test rig with only one exciting patch is possible. This particular property of sound insulation materials exhibiting a symmetry in the lateral direction is made use of in the reduced device for experimental characterisation, see figure 4. Here, only one shaker exciter 6 and piston 2a are used, whereas the other unit is replaced by a solid support 9 and the flexible spring 10. The pressure and normal velocity below the material specimen 1 are acquired indirectly by means of calibration step conducted prior to actual measurement.

Calibration procedure It has to be noted that Zr in (15) is not necessarily the actual impedance of the free piston, but rather a calibration factor that compensates for the combination of the piston impedance and the calibration between force cell and microphones. In the final measurement, the desired pressure below patch IL is given by (15) with p(Pmeas = TicHmens /S,. For nearly blocked bottom and nearly open top, the sound insulation reaction pressure on the active patch i is given by and the measured pressure is 0. meam (ht-ena ztact) The resulting relative error is Apr' /72(3 zact. Aa. inens ineas * haa 13; The relative error in the reconstructed sound insulation pressure is thus amplified as compared to the relative error in the direct force measurements. The amplification factor is governed by the piston impedance, and thus, to a large extent, by the piston mass. The piston must consequently not only behave like a rigid body in the frequency range of interest but also be as lightweight as possible in order to minimise the measurement uncertainty.

For the measurement on a passive piston j, a lightweight piston ensures a higher velocity signal

I ur

Zi?"ss above the noise level.

Method for correction of the effect of an air gap In principle the PU-probes 4 should be located in correspondence with the top surface of the sound insulation (3-surface), but in practice a slight air gap 15 between the sound insulation surface /3 and the plane Tv of the transducers 4 is unavoidable. This air gap results in two error sources. On one hand the air layer may exhibit radiation through the side walls of the gap, on the other hand, the stiffness of the layer itself is finite.

For a thin cylindrical air gap of height It and radius it can be shown that for a single patch 7' (19) Zgap = + Z-1 rad 2h 21ch.

In the high frequency range, the overall impedance is governed by the stiffness term. The radiative term Zrad will generally lead to a mass-like drop in the low frequency range (Robey, 1955). This leads to an underestimation of the material surface input impedance. Moreover, the power radiated through the side walls of the gap towards the neighbouring patches leads to an overestimation of transfer terms. In the high frequency range, the finite air gap stiffness dominates and may mask the material underneath. A lower height h, naturally leads to a higher air-gap impedance and to improved characterisation results.

Experimental air gap correction For an experimental correction procedure, the air gap 15 is discretised into the patch grid defined on the specimen surface. Since the thickness of the air gap is very small as compared to the wavelength in the frequency range of interest, the pressure may be considered uniformly distributed within the air gap of each patch. The pressure vector pi3 measured at the top surface of the sound insulation may now be approximated by = Z,"" -where v3 is the velocity vector containing the patch velocities measured with the particle velocity transducers 4 at the upper surface 7 of the air gap 15. Zga, represents the air gap impedance matrix. The desired patch velocity vector v3 at the sound insulation surface is obtained as - -ireinP (20) v = v The air gap impedance matrix Zgap may be assessed experimentally beforehand by replacing the sound insulation specimen I with a rigid dummy 16 (v3 = 0), see figure 5. By acquiring a sufficient number of independent load eases the air gap impedance Zgtip can again be obtained through a matrix inversion. Thus the particle velocity at the sound insulation specimen surface is reconstructed from the measurement with PU sensors 4 at a finite distance from the specimen 1.

Claims (4)

1. Claims 1. A device for experimental characterisation of vibro-acoustic properties of sound insulation materials, which comprises two or more loudspeakers 3a and 3b for air-home sound excitation on one side of the material specimen 1, two or more force shaker exciters 6a and 6b with pistons 2a and 2b for structure-borne excitation on the other side of the material specimen 1, one rigid baffle 8 which separates the pistons 2a and 2b, two or more accelerometers 5a and 5b attached to the pistons 2a and 2b, two or more force transducers 7a and 7b mounted between the shakers 6a and 6b and the pistons 2a and 21). respectively, at least one but preferably an array of pressure-particle velocity probes 4 located between the loudspeakers 3a and 3b and the material specimen 1, a control unit 9 for shakers fia and fib and loudspeaker 3a and 3b, respectively, and a data acquisition unit 10 for determining measured signals of PU-probe 4, accelerometers 5a and 5b and force transducers 7a and 7b.
2. 2. A device for experimental characterisation of vibro-acoustic properties of sound insulation materials according to claim 1, which comprises only one force shaker exciter 6, one solid support 11 and one or more flexible springs 12 mounted between the piston 2b and the solid support 11.
3. 3. A method for correction of the effect of an air gap 15 between the surface $ of the sound insulation material specimen 1 and the actual position a of the pressure-particle velocity sensor 4, which consists in replacement of the sound insulation material 1 with rigid block of the same thickness 16, assessment of sound pressures and normal particle velocities at sensor position "y in either numerical or experimental manner and determination of the resulting air gap impedance by solution of a system of linear equations.
4. 4. A method for determination of vibro-acoustic properties of sound insulation materials, which comprises a calibration procedure 17 used to adjust the readings of the pressure-particle velocity sensors with those of accelerometers and force transducers, a method 18 for correction of the effect of an air gap according to claim 3, the measurement 19 of sound pressures, normal particle velocities, accelerations and forces on N patches corresponding to 2N linearly independent load cases of the material specimen 1 according to device described in claim 1 and 2 and an assembly of the H-matrix 20, which is obtained by solution of a system of linear equations, where the matrix coefficients contain the excitation and response pressures and velocities for each load case on both sides a and 3 of each patch.