Non-contact experimental methods to characterise the response of a hyper-elastic membrane
- M. Kamper^{1} and
- A. Bekker^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s40712-017-0082-6
© The Author(s) 2017
Received: 1 July 2017
Accepted: 12 July 2017
Published: 24 July 2017
Abstract
Background
Membranes often feature in dynamic structures. The design of such structures generally includes the evaluation of their dynamic characteristics, such as natural frequecies and mode shapes.
Methods
The quasi-statics ad dyamic responses of thin rubber sheeting were investigated through non-contact experimental techniques. The rubber sheeting was modelled as a membrane structure and the material was assumed to be hyper-elastic, isotopic and incompressible. Two hyper-elastic material models were considered, namely the Mooney-Rivlin model and the Neo-Hookean model. The natural frequencies and mode shapes of the hyprt-elastic membrane were anatically and numerically calculated by assuming small linear vibrations and an equi-bi-axial stress state in the membrane. To validate the mathematical analyses, experimental modal analysis was performed where the vibration response was measured with a laser Doppler vibrometer.
Results and conclusions
The analytical model, shows that the natural frequencies of the membrane depend on the initial stretch. Mathematical and experimental results agree well at the lower modes. However, measurement resolution is found to be a vital factor which limits the extraction of closely spaced modes due to difficulties with the accurate identification of nodal line in a purely experimental approach.
Keywords
Hyper-elasticity Non-contact techniques Membrane Mooney-Rivlin Neo-Hookean Natural frequency Mode shape Digital image correlationBackground
The design of dynamic structures often includes continuous mechanical systems which require the evaluation of vibration responses (Hagedorn and DasGupta 2007). Membrane structures are examples of such continuous systems and are utilised in a variety of applications, including space, civil and bio-engineering. Jenkins and Korde (2006) and Jenkins and Leonard (1991) provide reviews of membrane literature, including practical applications and the static and dynamic analysis of membranes. Advantages of membranes such as low mass and stowed volume have renewed interest in deployable structures and their utilisation of membranes for terrestrial use. Space applications utilise membranes in radars, antennas, telescopes, solar concentrators and shields (Young et al. 2005; Salama et al. 2000). Further applications are found in robotics and biomedical prosthesis, such as artificial organs, sensors, actuators and transducers (Jenkins and Korde 2006; Goulbourne et al. 2004).
Natural frequencies and their corresponding mode shapes are fundamental to understanding the dynamic response of mechanical structures. A natural frequency is a frequency at which the system will vibrate if initially disturbed from rest and not subjected to any external loads (De Silva 2007). A mode shape describes the displacement pattern of the system when it is vibrating or excited at a natural frequency (He and Fu 2001). Natural frequencies and mode shapes are inherent properties of a system and form part of the dynamic characteristics termed modal parameters. The dynamic characterisation of mechanical structures is typically performed through modal analysis techniques.
The aim of the instrument was twofold: firstly, to facilitate deeper learning and to encourage student engagement by introducing abstract mathematical concepts through concrete, tactile, audible and visual experiences and, secondly, to show how the deeper engineering abstractions of experimental, analytical and numerical modal analysis can be used to obtain the natural frequencies and mode shapes of the tensioned rubber sheeting. As such, the question is whether the modal properties of a thin rubber material can be predicted through a carefully planned experimental investigation and how this compares to mathematically derived modal parameters.
As the rubber sheet is thin and two-dimensional and does not support bending moments, it was modelled as a membrane (Jenkins and Korde 2006). Because of the ability to respond elastically at large deformations, the rubber was assumed to be hyper-elastic (Gent 2012). 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.
Hyper-elastic membrane models are presented in other applications including the behaviour of biological materials and tissues (Chagnon et al. 2015; Mihai et al. 2015; Rashid et al. 2012). Chakravarty (2013) modelled the wings of micro air vehicles as hyper-elastic membranes to investigate their modal parameters. Experimental modal analysis (EMA) was performed by Chakravarty and Albertani (2011) on a pre-stretched latex membrane to investigate the effects of added mass and damping on modal characteristics. Hyper-elastic membranes further model the dynamic response of dielectric elastomers (Chakravarty 2014; Mockensturm and Goulbourne 2004). Gonçalves et al. (2009) analysed the vibrations of a pre-stretched circular hyper-elastic membrane, both analytically and numerically. The linear and non-linear vibration of a hyper-elastic rectangular membrane was additionally investigated by Soares and Gonçalves (2014) using analytical and finite element (FE) models.
The experimental evaluation of the dynamic properties of thin structures often utilises non-contact methods (Jenkins and Korde 2006), since conventional devices, such as mechanical shakers, impact hammers and contact accelerometers, present drawbacks attributed to mass loading affects (Siringoringo and Fujino 2009). Recent studies that employ non-contact excitation and response measurement techniques (Chakravarty 2013; Siringoringo and Fujino 2009; Ameri et al. 2012; Xu and Zhu 2013) indicate that laser vibrometry is a popular tool to obtain the dynamic properties of membranes.
In light of this background, the present work documents the evaluation of experimental, analytical and numerical investigations to determine the dynamic response of a tensioned, circular, rubber membrane. To this end, hyper-elastic materials are presented first, along with the hyper-elastic material models which were considered to model rubber material. Uni-axial tensile testing was performed to determine the hyper-elastic material parameters. Digital image correlation (DIC), a non-contact technique, was used to obtain measurements of the material stretch ratio over a selected region of the rubber specimen. The natural frequencies and mode shapes of a circular hyper-elastic membrane are analytically computed by solving the two-dimensional equation of motion. The vibrations are assumed to be small, and consequently, the equation of motion in the transverse direction is linear. The analytical model is validated by experimental modal analysis. 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).
Methods
Constitutive modelling of rubber
Cauchy stress for special cases of stress states
Stress state | Stretch ratios | Neo-Hookean | Mooney-Rivlin |
---|---|---|---|
Equi-bi-axial | λ _{1}=λ _{2}=λ | \(\sigma _{1} = \sigma _{2} = 2C_{1}\left (\lambda ^{2}-\frac {1}{\lambda ^{4}}\right)\) | \(\sigma _{1} = \sigma _{2} = 2\left (\lambda ^{2} - \frac {1}{\lambda ^{4}} \right) \left (C_{1} + C_{2}\lambda ^{2} \right)\) |
\(\lambda _{3} = \frac {1}{\lambda ^{2}}\) | σ _{3}=0 | σ _{3}=0 | |
Uni-axial | λ _{3}=λ | σ _{1}=σ _{2}=0 | σ _{1}=σ _{2}=0 |
\(\lambda _{1} = \lambda _{2} = \frac {1}{\sqrt {\lambda }}\) | \(\sigma _{3} = 2C_{1}\left (\lambda ^{2} - \frac {1}{\lambda }\right)\) | \(\sigma _{3} = 2\left (\lambda - \frac {1}{\lambda ^{2}} \right) \left (C_{1} \lambda + C_{2} \right)\) |
Uni-axial material testing
Several test methods exist to determine the required hyper-elastic material parameters. These include uni-axial, bi-axial, shear and bulge tests (Kim et al. 2012; Liu et al. 2015; Sasso et al. 2008; Selvadurai 2006; Selvadurai and Shi 2012). 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 specimens (10 mm × 170 mm). The specimens were cut from 1-mm rubber sheeting with a density of 1350 kg/m^{3}. The same rubber sheeting was used for construction of the musical instrument. As done by Selvadurai (2006), additional pieces of rubber were glued to the grip-contact area of the rubber specimens to minimise pull out from the grips. Since the testing was performed within 30 min of applying the adhesive, the possibility of chemical reaction between the rubber specimen and adhesive can be neglected (Selvadurai and Shi 2012).
Material parameter identification
Material parameters [MPa] for the Neo-Hookean and Mooney-Rivlin hyper-elastic models
Neo-Hookean model | Mooney-Rivlin model | ||
---|---|---|---|
Sample | C _{1} | C _{10} | C _{20} |
1 | 0.797 | 0.140 | 0.806 |
2 | 0.764 | 0.193 | 0.701 |
3 | 0.825 | 0.152 | 0.829 |
4 | 0.800 | 0.131 | 0.823 |
5 | 0.768 | 0.142 | 0.767 |
Average | 0.791 | 0.152 | 0.785 |
RMS errors for uni-axial material testing
Material mode | Average material constant | RMS error |
---|---|---|
Neo-Hookean | C _{1} = 0.791 MPa | 0.032 MPa |
Mooney-Rivlin | C _{10} = 0.152 MPa | 0.012 MPa |
C _{20} = 0.785 MPa |
Analytical analysis
Equation (18) was also obtained by Chakravarty (2013) and Gonçalves et al. (2009).
Experimental modal analysis
The natural frequencies depend on the initial deformation experienced by the membrane (Eqs. (17) and (18)). Non-contact methods were employed to measure the initial stretch ratio. The flange fixture was displaced in stepwise increments towards the back panel to a final displacement of 20 mm. The membrane was subjected to bi-axial loading, and images were incrementally taken at the different stretch levels with two high-resolution cameras (Imager E-lite 2M) from the LaVision Strain Master DIC System. To obtain the in-plane stretch values, the images were again processed by the DIC software, DaVis.
Finite element analysis
The finite element (FE) software, Abaqus, was used to investigate the linear vibration of a pre-stretched hyper-elastic membrane in order to validate the results obtained from the analytical and experimental modal analyses. A circular membrane was modelled using membrane elements with a thickness of 1 mm and a Neo-Hookean material model with a material constant of 0.791 MPa and a mass density of 1350 kg/m^{3}. Four-node M3D4 elements were selected to mesh the domain. (Hence, the dominant element type was a four-node quadrilateral elements.) Some transition three-node triangular elements were allowed in order to mesh the circular domain. Similar to the analytical model, the analysis was performed in two steps. The first step consisted of a fully non-linear analysis in which a radial displacement was specified along the circumference of the membrane. To simulate the dimensions of the pipe and achieve similar conditions to that of the experimental setup, the final radius was chosen to be 80 mm. This was followed by a linear perturbation step in which the natural frequencies and mode shapes were computed.
Results
Initial deformation of the rubber sheeting
Also shown in Fig. 12 are inserts (a) to (e), illustrating the in-plane displacement field of the selected region with the progressive displacement of the tensioning device. The initial uneven displacement seen in inserts (b) and (c) is caused by the rubber surface not being entirely flat. As the stretch is increased, the radial displacement is increasingly more evident. At the final flange-fixture displacement of 20 mm, the in-plane stretches, λ _{1} and λ _{2}, were respectively computed to be 1040 and 1041. Ideally, to compare the analytical and experimental results, the rubber sheeting has to be subjected to equi-bi-axial stretch. This requires the two stretch components to be equal. Even though exact equi-bi-axial stretch was not achieved, the values for λ _{1} and λ _{2} are still in close proximity.
Experimental modal analysis
Natural frequencies and mode shapes
To simulate similar conditions to that obtained during the experimental analysis, the circular membrane of the finite element model was specified to have an un-deformed radius of 76.9 mm. The tensioning of the membrane was simulated by specifying a displacement of 3.1 mm in the radial direction. The initial radius of 76.9 mm and radial displacement of 3.1 mm were chosen in order to achieve a similar stretch value as the experimentally computed in-plane stretches, λ _{1} and λ _{2}. The stretched radius was therefore 80 mm with a stretch ratio of 1.0403 (R _{ f }/R _{0}).
Comparison of the natural frequencies [Hz] from experimental modal analysis (EMA), finite element analysis (FEA) and analytical (AN) analysis for λ=1.0403 and R _{ f }=80 mm
Mode | (m, n) | α _{ nm } | EMA | FEA | AN | 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 |
Discussion
Non-contact techniques, such as DIC to measure strain and the use of a LDV to capture vibration responses, overcome problems associated with attachment of sensors, such as strain gauges and accelerometers. To measure the out-of-plane vibrations of the rubber sheet, the LDV had 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. This resulted in somewhat noisy response measurements from the LDV.
The roving sensor approach in EMA is time-consuming but can be improved with the use of a scanning laser Doppler vibrometer (SLDV). SLVDs have automatic scanning abilities and, hence, have the capability for a higher resolution of measurement points as well as faster response measurements. Because DIC requires the application of speckle patterns and the assembly of expensive equipment, this method is also relatively time-consuming. An alternative to measuring displacement of highly elastic materials, such as rubber, is the use of long-travel contact extensometers.
The AutoMAC matrix in Fig. 17 indicates a fair amount of independency between the mode shapes. The high degree of similarity between mode pairs 5 and 6 is a result of the low resolution of the response measurement grid which was not adequate to represent the mode shape of the fifth mode (Fig. 18e). As it was not possible to determine the number of nodal lines and nodal circles of the fifth EMA mode shape, it cannot be related to the FE or analytical results, and this mode is therefore not included in Table 4.
Rubber is expected to exhibit viscoelastic material behaviour (Gent 2012), and its response therefore depends on the rate of deformation. Hyper-elastic material models do not account for this strain-rate sensitivity, which would 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 hyper-elastic 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 4 are the deformation-amplitude dependency, strain-history dependency (Diercks et al. 2016) and material damping of rubber as well as the boundary conditions and the slippage of the sheeting between the flanges of the tensioning device. The latter results in a reduction of the actual stress experienced by the membrane. The analytical and numerical models do not account for the surrounding medium and are further limited by the assumption of small linear vibrations.
Conclusions
The analytical, numerical and experimental investigation of the natural frequencies and mode shapes of a tensioned rubber sheeting is presented. The rubber sheeting is a component of a musical instrument, which is used to explain the concepts of natural frequencies and mode shapes to under-graduate students.
The use of digital image correlation for uni-axial tensile testing was proven to be a beneficial technique in instances where slippage of test machine grips is a particular concern. For experimental modal analysis, non-contact excitation and measurement techniques were successfully employed to evaluate the modal properties of the tensioned rubber sheeting. To mathematically obtain the natural frequencies and mode shapes, the sheeting was modelled as a hyper-elastic membrane. From uni-axial material testing, it was found that the Neo-Hookean hyper-elastic model provided a good fit to material testing data for the desired strain range.
The mathematical and experimental results agree in their predictions of the natural frequency and mode shape at the lower modes. However, some challenges are encountered with the experimental identification of closely spaced modes as a result of the limited spatial resolution of response measurements. Although spatial resolution is always a concern in EMA, this is especially applicable to symmetric circular membranes where nodal lines could be un-detected.
Discrepancies between the mathematically and experimentally derived frequencies are attributed to simplification of the complex material response exhibited by rubber, which is not completely modelled by the hyper-elastic material models. A proposed extension of the present study is the extraction of material properties from the modal test data of tensioned membranes. The modal properties of a system depend on the system’s geometric and material characteristics (He and Fu 2001). Since there is good agreement between the mathematical and experimental results at the fundamental mode, it can possibly be used as a non-destructive testing method to estimate material parameters.
Declarations
Authors’ contributions
MK is responsible for the design of the constitutive modelling, experimental design, data acquisition and analysis presented in this work. AB initiated the project, contributed to the critical interpretation of results and experimental modal analysis and conceptualised the contribution of the manuscript, its documentation and revisions. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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