IOMAC'17

7th International Operational Modal Analysis Conference

2017 May10-12 Ingolstadt - Germany

MODAL ANALYSIS OF A HYPERELASTIC

MEMBRANE FOR THE DEVELOPMENT OF A

MUSICAL INSTRUMENT

M. Kamper1 and A. Bekker

2

1 Miss, Department of Mechanical and Mechatronic Engineering, Stellenbosch University, South Africa,

mkamper94@gmail.com 2 Dr., Sound and Vibration Research Group, Department of Mechanical and Mechatronic Engineering, Stellenbosch

University, South Africa, annieb@sun.ac.za

ABSTRACT

A musical instrument was designed with the aim to visually and audibly demonstrate the natural

frequencies and mode shapes of a continuous system to under-graduate students. The concept

instrument comprised of a rubber membrane which was stretched over a cylindrical tube with a

tensioning device. Excitation was provided by a speaker, connected to an amplifier and signal

generator, which was placed centrally inside the tube. Salt was sprinkled on the membrane surface to

visualise the mode shapes of the instrument. The material properties of the rubber, required for the

Neo-Hookean hyperelastic model, were determined by uni-axial tensile testing and digital image

correlation. Digital image correlation was further utilised to determine the level of membrane tension

as an input to the analytical solution of the two-dimensional wave equation. The natural frequencies

and mode shapes of the instrument were determined analytically, by finite element methods and

through experimental modal analysis. In the experiment, a free-field microphone was used to capture

excitation measurements and a laser vibrometer was utilized to record membrane responses. The

discrepancy between analytical and experimental natural frequencies is attributed to the lack of strain

rate sensitivity in the material model. Measurement resolution is also a vital factor limiting the

identification of closely spaced modes and the accurate identification of nodal lines.

Keywords: Hyperelasticity, experimental modal analysis, membrane, non-contact methods

1. INTRODUCTION

The evaluation of the dynamic characteristics of continuous systems is important in the design of

engineering structures. Membranes are an example of such structures and are used in a variety of

applications, including space, civil and bio-engineering [1], actuators, transducers and sensors.

Jenkins and Korde [1] and Jenkins and Leonard [2] provide reviews of membrane literature with

emphasis on its applications as well as the static and dynamic analysis of membrane structures.

Natural frequencies and their corresponding mode shapes are two fundamental concepts when

considering the dynamic response of mechanical structures. A natural frequency is a frequency at

which a system will vibrate if it is disturbed from rest and not subjected to any external loads [3]. If

excited at its natural frequency, the system experiences a significant increase in its amplitude

response, even if the level of excitation is fairly low. This phenomenon is known as resonance. A

mode shape describes the displacement pattern of the system when it is vibrating at a natural

frequency [4]. Natural frequencies and mode shapes are inherent properties of a system and form part

of the dynamic characteristics termed modal parameters which are obtained by utilising modal

analysis techniques [4].

A musical instrument was developed to facilitate an active learning experience [5] to convey the

concepts of natural frequencies and mode shapes to under-graduate engineering students in the

instructional laboratory. The aim was to facilitate deeper learning and to encourage student

engagement by introducing abstract mathematical concepts through concrete tactile, audible and

visual experiences. It was reasoned that access to a simple practical rig would encourage students to

participate in active experimentation. Kolb [6] describes this as a necessary step in a cycle to

facilitate deeper learning and metacognition, which are desired traits of educated engineers.

As shown in Figure 1(a), the proposed instrument is based on a combination of the Chladni plate [7]

and an acoustic tonoscope [8]. It comprises rubber sheeting stretched over a circular tube using a

specially designed tensioning rig. A speaker is placed inside the tube and is connected to a frequency

generator to excite the rubber sheeting. Excitation at a single frequency is experienced as an audible,

single-pitched tone. Salt is sprinkled onto the rubber membrane (Figure 1(b)) and forms a distinct

pattern when the rubber sheet is excited at one of its natural frequencies. This visual experience is

associated with an increase in the sound intensity of the audible tone at the excitation frequency.

(a)

(b)

Figure 1: (a) Rubber sheets are tensioned over circular tubes and excited with speakers which are placed inside

the tubes. (b) The sound frequency is varied and salt is used to visualise the mode shapes.

The deeper engineering abstractions of experimental, analytical and numerical modal analysis are

subsequently demonstrated to obtain the natural frequencies and mode shapes of the tensioned rubber

sheeting. Whereas experimental modal analysis requires the excitation of the membrane and

measurements of the dynamic response, both analytical and numerical modal analyses require

information on the material response of the rubber sheeting.

The experimental evaluation of dynamic properties of thin structures often utilises non-contact

methods [1], since conventional devices, such as mechanical shakers, impact hammers and contact

accelerometers, have drawbacks attributed to mass loading affects [9]. Recent studies that employ

non-contact excitation and response measurement techniques ([9]–[13]), indicate that laser

vibrometers are often used to obtain the dynamic properties of membranes.

As the rubber sheet is thin, two-dimensional and does not support bending moments it was modelled

as a membrane [1]. Because of the ability to respond elastically at large deformations, hyperelastic

material models were considered to represent the material response [14]. To this end hyperelastic

materials are presented first, along with the hyperelastic material models which were selected to

analyse the rubber. The natural frequencies and mode shapes of a circular hyperelastic membrane are

analytically computed by solving the two-dimensional equation of motion. The analytical model is

validated by experimental testing. For the experimental investigation, a rubber membrane is excited

through acoustic means and the response is measured with a Laser Doppler Vibrometer (LDV).

Finally, the analytical and experimental results are compared to numerical results obtained from finite

element analysis (FEA).

2. MODELLING OF A HYPERELASTIC MATERIAL

2.1. Hyperelastic Material Models

Numerous hyperelastic material models are available in literature as discussed in overviews by Ali et

al. [15]. In the present work, the Neo-Hookean and Mooney-Rivlin hyperelastic material models were

considered to represent the dynamic response of the rubber sheeting. Constitutive models for

hyperelastic materials are based on the selection of an appropriate strain energy density function.

Comprehensive texts such as those by Treloar [16] and Mooney [17] contain great detail on the

fundamental concepts related to hyperelastic materials. The strain energy is a function of the

deformation tensor and can be expressed in terms of the strain invariants. For an isotropic,

incompressible body undergoing homogenous deformation, the strain energy function, W, is a

function of the first two strain invariants, 𝐼1 and 𝐼2. The strain energy function for the Mooney-Rivlin

material model is

𝑊 = 𝐶10(𝐼1 − 3) + 𝐶20(𝐼2 − 3). (1)

The Neo-Hookean strain energy function can be obtained by setting 𝐶20 in Eq. (1) equal to zero,

thereby obtaining

𝑊 = 𝐶1(𝐼1 − 3). (2)

In Eq. (1) and Eq. (2), 𝐶1, 𝐶10 and 𝐶20 are experimentally determined material parameters. The strain

invariants, 𝐼1 and 𝐼2, can further be expressed in terms of the principle stretch ratios.

𝐼1 = 𝜆12 + 𝜆2

2 + 𝜆32 ; (3)

𝐼2 = 𝜆12𝜆2

2 + 𝜆12𝜆3

2 + 𝜆22𝜆3

2. (4)

In Eq. (3) and Eq. (4) 𝜆1, 𝜆2 and 𝜆3 are the stretch ratios defining the amount of stretch in the

principle directions denoted by 1,2,3. The principle stretches are given by:

𝜆𝑖 =

𝑑𝑥𝑖

𝑑𝑋𝑖 (𝑖 = 1, 2, 3), (5)

where 𝑑𝑋𝑖 and 𝑑𝑥𝑖 respectively refer to the undeformed and deformed lengths of an infinitesimal

element. From the strain energy density functions, Eqs. (1) and (2), stress-strain relations can be

obtained for special cases of stress, such as uni-axial and bi-axial stress states ([14], [18]). The

material parameters, 𝐶1, 𝐶10 and 𝐶20, are then determined by curve-fitting the stress-strain

relationships to material testing data.

2.2. Uni-Axial Material Testing

A number of test metods exist to determine the required hyperelastic material parameters, including

uni-axial, bi-axial and bulge tests ([19]–[23]). In the present work uni-axial tensile testing was

performed on a MTS Universal Testing Machine to investigate the response of five rectangular

neoprene rubber samples (10 × 170 mm). The specimens were cut from 1 mm rubber sheeting with a

density of 1350 kg/m3. Homogeneous deformation is not possible over the full sample, because the

grips prevent lateral deformation. For this reason, digital image correlation (DIC) was used to obtain

strain measurements over a smaller section of the specimen. Images were captured with two high

resolution cameras (Imager E-lite 2M) from the LaVision Strain Master DIC System and processed

with the DIC software, DaVis. The force, measured with a 1 kN load cell, was divided by the original

cross-sectional area to obtain the engineering stress. The specimens were tensioned at a nominal strain

rate of 1 × 10-3

s-1

up to a final strain of approximately 70%.

The material constants were determined by curve fitting the Neo-Hookean and Mooney-Rivlin uni-

axial stress relationships to the experimental data. The rubber is not expected to exceed 30% strain

during construction or use of the musical instrument and the models were therefore matched to the

test data of the first 30% strain region. Figure 2 shows the curve-fits of the material models to the

average experimental stress-strain curve. The computed material parameters as well as the root-mean-

square (RMS) errors between the model fittings and the averaged experimental data (for the strain

region of interest) are presented in Table 1. Although the Mooney-Rivlin model provides a superior

fit, the Neo-Hookean model was selected to describe the response of the rubber sheeting. It was

reasoned that this simpler model provides a good approximation in the desired strain range.

Figure 2: Curve-fitting of hyperelastic material models to uni-axial material testing data.

Table 1: Material parameters and RMS error for uni-axial material testing.

Material model Average material constants RMS error

Neo-Hookean C1 = 0.791 MPa 0.032 MPa

Mooney-Rivlin C10 = 0.152 MPa; C20 = 0.785 MPa 0.012 MPa

0.0

0.5

1.0

1.5

2.0

0 10 20 30 40 50 60 70 80

Ste

ss [

MP

a]

Strain [%]

Neo-Hookean model

Mooney-Rivlin model

Average experimental data

3. ANALYTICAL MODEL

To analytically obtain the natural frequencies and mode shapes, the rubber sheeting is modelled as a

flat circular membrane with an initial radius 𝑅0. The analysis is separated into two steps. First, the

membrane is stretched from its initial position (State 1 in Figure 3) in the radial direction to a radius

of 𝑅𝑓 (State 2) and fixed at its boundaries.

The equi-bi-axial stress, caused by the stretching, is obtained using the Neo-Hookean hyperelastic

material model. This is followed by an analytical linear modal analysis about the pre-stretched state.

Small vibrations (State 3) are assumed such that the equations of motion may be assumed linear. The

natural frequencies and mode shapes of a circular membrane fixed along its circumference, are

determined by solving the two-dimensional wave equation [24].

Figure 3: Different states of deformation of a membrane (Adapted from [24] and [25]).

Seeing that a circular membrane is considered, the two-dimensional wave equation for the transverse

displacement is given in terms of cylindrical coordinates as indicated in Figure 3:

𝜕2𝑤

𝜕𝑡2= 𝑐2 (

𝜕2𝑤

𝜕𝑟2+

1

𝑟

𝜕𝑤

𝜕𝑟+

1

𝑟2

𝜕2𝑤

𝜕𝜃2 ). (6)

The wave speed, 𝑐2, in Eq. (6) can be expressed as a function of the Neo-Hookean material constant,

the material density and the stretch ratio, 𝜆 = 𝑅𝑓/𝑅0, ([25], [26]) such that:

𝑐2 =

2𝐶1

𝜌(𝜆2 −

1

𝜆4). (7)

The method of separation of variables, together with the boundary condition, 𝑤(𝑅𝑓 , 𝜃, 𝑡) = 0, is used

to solve Eq. (6). The transverse displacement of the vibrating membrane at a natural frequency, 𝜔𝑛𝑚,

describes the mode shape and is given by:

𝑤(𝑟, 𝜃, 𝑡) = 𝑑𝑛𝑚𝐽𝑚 (

𝛼𝑛𝑚

𝑅𝑓𝑟) cos(𝑚𝜃) cos(𝜔𝑛𝑚𝑡), (8)

where 𝑑𝑛𝑚 is the vibration amplitude, 𝐽𝑚 is the Bessel function of the first kind order m and 𝛼𝑛𝑚 is

the nth root of the Bessel function. The order of the Bessel function, m, determines the number of

nodal lines, whereas n determines the number of nodal circles of the mode shape. The natural

frequency of the (m, n)-mode is given by:

𝜔𝑛𝑚 = 𝛼𝑛𝑚

𝑅𝑓√

2𝐶1

𝜌(𝜆2 −

1

𝜆4) . (9)

4. EXPERIMENTAL MODAL ANALYSIS

Experimental modal analysis (EMA) was performed to determine the natural frequencies and mode

shapes of the tensioned rubber sheeting. The tensioning rig was used to stretch the rubber membrane

over a 160 mm circular tube as shown in Figure 4. The membrane was cut from the same rubber

sheeting used for material testing and has a thickness of 1 mm, density of 1350 kg/m3

and a Neo-

Hookean material constant of 0.791 MPa. Two flanges, attached to a back panel with threaded rods,

secured the membrane. The membrane was excited acoustically and the response was measured with a

Vibromet 500V Laser Doppler Vibrometer (LDV). Classical EMA was adapted by using a free field

microphone to measure the sound pressure from the speaker, which served as the reference signal. A

circular hole was cut in the back panel to allow sound pressure waves to reach the membrane. As done

by Ameri et al [10], the back of the panel (side facing the speaker) was covered with acoustic foam to

minimise vibration of the wooden board. The sound source, a 10 W speaker, was placed roughly

400 mm from the tensioned rubber and the free field microphone was set to point in the direction of

the source. A LMS SCADAS data acquisition (DAQ) system acquired the measurements and

provided the input signal to the speaker, which was amplified to ensure sufficient excitation.

Figure 4: EMA setup with the rubber pre-tensioned using the tensioning rig.

Figure 5: Circular grid pattern and measurement points on the rubber sheeting

A grid was drawn on the membrane with a white paint marker (Figure 5) and comprised 41 points

arranged in a circular pattern. To identify the mode shapes, the LDV was moved using a roving sensor

approach ([10], [13]) for response measurements at the points on the grid. The excitation and response

data were processed with the LMS Test Lab software and the modal parameters were estimated based

on the Polymax algorithm. Five measurements were taken at each point and averaged to obtain the

response. The bandwidth was set to 1024 Hz with 2048 spectral lines and an acquisition time of 2 s

was used for each measurement. For excitation, a linear sine sweep, spanning the bandwidth from 0 to

1024 Hz was used.

As revealed by the analytical solution, the natural frequencies dependent on the initial stretch

(𝜆 = 𝑅𝑓/𝑅0) experienced by the membrane (Eq. (9)). This initial stretch ratio was measured by using

DIC techniques. The flange fixture, which secured membrane, was displaced in stepwise increments

towards the back panel in order to tension the membrane. Images were incrementally taken at the

different strain levels, with two high resolution cameras (Imager E-lite 2M) from the LaVision Strain

Master DIC System. To obtain the in-plane strain values, the images were processed by the DIC

software, DaVis. Figure 6 presents the computed in-plane strains, εxx and εyy, as a function of the

displacement of the flange fixture. In Figure 6, the inserts (a) to (e) represent the deformation of the

membrane with progressive displacement of the tensioning device. The final in-plane strains, εxx and

εyy, were respectively computed to be 4.0% and 4.1%. Ideally, to compare the analytical and

experimental results, it was required that the rubber sheeting must be subjected to equi-bi-axial strain.

This would require the two strain components to be equal. Even though exact equi-bi-axial strain was

not achieved, the values for εxx and εyy agree closely.

Figure 6: Strain-displacement curve of the rubber membrane stretched over a 160 mm pipe, together with

images of the deformation field at (a) 0 mm displacement (b) 5 mm displacement (c) 10 mm displacement (d) 15

mm displacement and (e) 17.5 mm displacement.

5. FINITE ELEMENT MODEL

To validate the results obtained from the experimental modal analysis, the finite element (FE)

software, Abaqus, was used to investigate the linear vibration of a pre-stretched hyperelastic

membrane. The membrane was modelled using 4-node M3D4 membrane elements and the Neo-

Hookean material model with a material constant of 0.791 MPa and mass density of 1350 kg/m3. The

circular membrane was specified to have an un-deformed radius of 76.9 mm and thickness of 1 mm.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 5 10 15 20

Str

ain

[%

]

Displacement [mm]

εxx data points

Average of εxx

εyy data points

Average of εyy

εxx data points

Average of εxx

εyy data points

Average of εyy (a)

(b)

(c)

(d)

(e)

Similar to the analytical model, the analysis was performed in two steps. The first step consisted of a

fully non-linear analysis where a radial displacement of 3.1 mm was specified along the

circumference of the membrane. The stretched radius was therefore 80 mm, with a strain of 4.03%,

thereby simulating similar conditions to that achieved in the experimental modal analysis. This was

followed by a linear perturbation step in which the natural frequencies and mode shapes were

computed.

6. RESULTS AND DISCUSSION

Frequency response functions (FRF) and coherence functions for several points on the membrane are

presented in Figure 7. Initial noise (in the frequency range of 0 to 50 Hz) can be attributed to the

limited excitation of the membrane by the speaker below 50 Hz.

(a)

(b)

Figure 7: EMA results for several locations on the membrane. (a) FRFs. (b) Coherence functions.

To measure the out-of-plane vibrations of the rubber sheeting, the LDV was to be aligned

perpendicular to the surface. It was found that the LDV measurements were affected by the

irregularity of the rubber surface, in particular near the edge of the membrane where the surface was

slightly curved. The noisy response measurements from the LDV affected the resulting FRF curves.

A stabilisation diagram (Figure 8) was calculated by LMS Polymax from the experimental modal

analysis results. The ‘Automatic Modal Parameter Selection’ feature was used to select poles from the

stabilisation diagram. Six modes were identified in the frequency range of 0 to 350 Hz. A high level

of noise in the response measurements at higher frequencies (above 350 Hz) was a result of poor

signal quality from the LDV measurements at the boundary.

1.E-06

1.E-04

1.E-02

1.E+00

0 50 100 150 200 250 300 350

Mag

nit

ud

e [(

m/s

)/P

a]

Frequency [Hz]

FRF point 1

FRF point 3

FRF point 25

FRF point 34

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200 250 300 350

Co

her

ence

mag

nit

ud

e

Frequency [Hz]

Coherence point 1

Coherence point 3

Coherence point 25

Coherence point 34

The modal assurance criteria (MAC) [27] matrix is reported in Figure 9. The AutoMAC matrix

presents the extent of consistency between a set of mode shapes and indicates a fair amount of

independency between the mode shapes. The high degree of similarity between mode pair 5 and 6

(indicated with yellow) is a result of the low resolution of the measurement grid which was not

adequate to represent the mode shape of the fifth mode. Since the acoustic source excited a finite

region of the membrane [28], no proper driving point was available for the experimental modal

analysis. Future investigation of modal parameters could therefore consider the implementation of

operational modal analysis (OMA) techniques from response-only measurements [29].

Figure 8: EMA stabilisation diagram, showing the sum FRF with 2048 Spectral lines, frequency resolution of

0.5 Hz and uniform windowing ( × indicates selected poles).

(a)

(a) (b)

Figure 9: AutoMAC matrix for EMA mode shapes. (a) AutoMAC tabular view. (b) AutoMAC isometric view.

Using Eq. (9) with a stretch ratio of 1.0403 (λ = strain + 1), the natural frequencies were determined

analytically. The natural frequencies of five modes, determined using the different modal analysis

techniques, are presented in Table 2. Since the analytically and the numerically derived frequencies

are closely matched, the percentage differences between the numerical and experimental results are

given. The percentage difference is defined as |(ωn)EMA - (ωn)FEM|/(ωn)EMA. It is normalised with respect to

the experimental modal analysis results, as these results are based on the data collected from the

Mode [Hz]

1 2 3 4 5 6

80.87 132.66 180.33 193.59 278.74 316.39

1 80.87 100 3.90 9.89 21.20 2.27 4.10

2 132.66 3.90 100 2.59 5.08 3.03 2.64

3 180.33 9.89 2.59 100 14.68 0.54 0.79

4 193.59 21.20 5.08 14.68 100 18.17 14.82

5 278.74 2.27 3.03 0.54 18.17 100 65.88

6 316.39 4.10 2.64 0.79 14.82 65.88 100

actual system response. The mode shapes obtained numerically and experimentally, are presented in

Figure 10.

Table 2: Comparison of natural frequencies [Hz] from different modal analysis techniques. m refers to the

number of nodal lines and n to the number of nodal circles.

Mode # (m, n) 𝜶𝒏𝒎

(Eq. 9)

Experimental

(EMA)

Numerical

(FEA) Analytical Diff [%]

1 (0, 1) 2.405 80.87 78.28 78.28 3.20

2 (1, 1) 3.832 132.66 124.73 124.72 5.98

3 (2, 1) 5.136 180.33 167.15 167.16 7.31

4 (0, 2) 5.520 193.59 179.65 179.66 7.20

6 (0, 3) 8.654 316.39 281.54 281.67 11.01

(a) (b)

(c) (d)

(e) (f)

Figure 10: Comparison of mode shapes using FEA (left) and EMA (right) for (a) mode (0, 1), (b) mode (1,1),

(c) mode (2,1), (d) mode (0,2) and (f) mode (0,3).

From the analytical and FE models, several closely spaced modes were identified between the fourth

and sixth EMA modes. The FE computed mode shapes of the aforementioned modes are presented in

Figure 11. Difficulty is especially encountered for mode shapes with nodal lines, as a result of the

symmetric structure and limited resolution of response measurements. For example, it was not

possible to determine the number of nodal lines and nodal circles for the fifth EMA mode shape

(Figure 10(e)). As such it cannot be related to the FE or analytical results and is therefore not included

in Table 2.

The results in Table 2 show that the mathematical and experimental results agree well at the four

lower modes. Rubber exhibits viscoelastic material behaviour [3], and is therefore dependant on the

rate of deformation, which the hyperelastic material model does not account for. Strain-rate sensitivity

is expected to be increasingly dominant at higher modes with increased rates of vibration. Uni-axial

material testing was used to obtain the Neo-Hookean material parameter. However, the modal

analyses were performed with the assumption of a bi-axial stress state. The hyperelastic material

model can be improved by way of bi-axial material testing. Other factors that contribute to the

discrepancy between the frequency values in Table 2 are the deformation-amplitude dependency,

strain-history dependency [30] and boundary conditions s slippage of the sheeting between the flanges

of the tensioning device.

(a) (b) (c) (d)

Figure 11: Additional mode shapes that were identified with FEA: (a) mode (3,1) at 207.6 Hz, (b) mode (1,2) at

228.3 Hz (c) mode (4,1) at 246.9 Hz and (d) mode (2,2) at 273.8 Hz.

7. CONCLUSIONS

The analytical, numerical and experimental investigation of the natural frequencies and mode shapes

of tensioned rubber sheeting is presented. The rubber sheeting is a sub-component of a musical

instrument, which is used to explain the concepts of natural frequencies and mode shapes to under-

graduate students. To mathematically obtain the natural frequencies and mode shapes, the sheeting

was modelled as a hyperelastic membrane. From uni-axial material testing it was found that the Neo-

Hookean hyperelastic model provided a good fit to material testing data for small strains experienced

by the rubber. The analytically and numerically derived modes are in agreement, although some

challenges are encountered with the identification of closely spaced modes as a result of the limited

resolution of response measurements. The discrepancy between the mathematical and numerical

derived frequencies is a result of the complex material response exhibited by rubber, which is not

completely modelled by the hyperelastic Neo-Hookean material model.

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