# Estimation of material functions using system identification techniques

When using materials in constructions it is very important to fully know the mechanical properties of the materials and how they will respond to dynamic loads. Some materials are described by the well known Hooke's law of linear elasticity in which the stress and strain are related through the modulus of elasticity. Some materials, however, deviate from Hooke's law by exhibiting viscous characteristics apart from the elastic behaviour. These materials are called viscoelastic and for them the relation between stress and strain is more complicated: the stress depends not only on the current strain but also on all previous strain values. Metals can usually be well described as elastic, whereas plastics of different kinds are better characterized as viscoelastic.

Some of the properties of an isotropic linearly viscoelastic material are characterised by the frequency dependent complex modulus which relates stress and strain. Laboratory tests with polypropylene and polymethyl methacrylate have been carried out by generating extensional waves in rod specimens [2], described by a second order partial differential equation, and flexural waves in beam specimens [3], described by a fourth order partial differential equation. For identification of the complex modulus, these tests require a minimum of three and five independent measurements, respectively. Parametric [19,20] and non-parameteric identification methods are used in conjunction with frequency as well as time domain analyses. The methods developed are independent of boundary conditions. Therefore they can be used for in situ tests of elements in structures with properties which need not be known.

The time domain method developed has the advantage of being applicable to materials with linear as well as non-linear viscoelastic behaviour. However, it is an open question how extensional waves with amplitude levels high enough for a significantly non-linear response from the materials tested are to be generated. Laboratory tests concerning torsion of beam specimens and generation of flexural waves in plates are also in progress.

Previous methods have had poor accuracy near critical frequencies for the complex modulus. These critical frequencies correspond to situations with an integral multiple of a half wave length between measurement sections. A substantial improvement of the accuracy has been achieved through increased numbers and better configurations of strain gauges, and improved identification techniques [3,21].

Simulation models which can be used to evaluate different kinds of experiments have been developed. A thorough statistical analysis of the accuracy of the estimated material parameters has been undertaken. Expressions for the standard deviation, and its frequency dependence, of the estimated complex modulus have been derived [16,20,21]. This has given us important tools for evaluation of the quality of the determined material functions or parameters and for the planning of various experimental set-ups.

The complex extension and shear moduli, and the complex Poisson's ratio, have been identified from redundant strain measurements on test bars made of polymethyl methacrylate (PMMA) and polypropylene (PP) traversed by extensional and torsional waves [22,23]. The responses of both materials were found to be highly linear and nearly isotropic. Tests have also been carried out with both axial and circumferential strain gauges with the aim to completely characterise an isotropic and linearly viscoelastic material in a single test.

The Split Hopkinson Pressure Bar (SHPB) method has been improved in order to allow identification of the complex moduli of materials available in the form of short cylindrical specimens [24,25,30]. The work includes the design of a new test rig in which the specimen is inserted between two bars with known properties. An incident wave is generated in the first bar through axial impact, and the complex modulus of the specimen is identified on the basis of measured strains associated with waves reflected into the first bar and transmitted into the second. Unlike the procedure in classical SHPB testing, wave phenomena within the specimen are taken fully into account. The improved method is presently used for mechanical characterisation of pharmaceutical materials.

The analysis of algorithms has led to a fundamental study of using variable projection algorithms in optimization problems, with emphasis on their statistical properties, [6]. On the more fundamental side, all the considered algorithms can be posed as a separable nonlinear least squares problem. A general analysis for such estimators was developed in [13], and as a spin-off a new and efficient algorithm for frequency estimation was derived, [12].

For the case of equidistant sensors, a new algorithm has been developed, [14,15] which is much faster than the ones we have employed previously. Furthermore, it is not particularly vulnerable in cases where almost rank-deficiency occurs. This was previously a major problem, leading to low accuracy at critical frequencies corresponding to a standing wave pattern. In some situations certain boundary conditions can be assumed to be known. This is described in [11] for a more general setting. The estimation of material functions is one example out of many others, how the use of such a priori knowledge can be taken into account in the estimation. This leads to improved statistical accuracy, which can be quantified. More detailed developments appear in [9].

Some effects of sensor locations are discussed in [1,21] where we have shown that a 'smart ad hoc' choice of the sensor positions can give considerably improved statistical behaviour. The role of sensor location is further treated in [18,27], where optimal experiment design methods are used in order to further improve the accuracy of the estimated material parameters. Optimal experiment design methods are also used in [28], [35], where the excitation of the experimental setup is studied.

The frequency contents of strain waves in an axially excited bar, where one end of the bar is subject to a free end boundary condition is analysed in [29]. Wave propagation in both elastic and viscoelastic materials is treated, and the validity of the analysis confirmed through simulated and experimental data. The analysis is then used in order to interpret the large frequency variations in previous studies concerning the accuracy of the estimate and optimal input signal, respectively.

A statistically-based structure testing method is described in [34]. The idea is to have a number of 'identical' experiments, which differ by random phases in the initial conditions. From the recorded data from the experiements one can construct a certain data matrix, that ideally has low rank, if the underlying model parameterization holds true. Due to the presence of measurement noise, the low-rank property is lost, but the noisy data matrix has a number of small singular values. The statistical test takes into account what is a reasonable level of these small singular values.

An estimation method with somewhat improved statistically behaviour for parametric models, is described and analysed in [37].

Overviews of the work can be found in [1,6,17,26,31-33,36,38,39].

A related study on estimating certain material parameters in a tube perforated by helical slots appeared in [40].

## Current (and previous) participants

Systems and Control Division:

(Bengt Carlsson), (Kaushik Mahata), (Magnus Mossberg), (Agnes Rensfelt), Torsten Söderström

Solid Mechanics Group:

(Lars Hillström), Bengt Lundberg, (Saed Mousavi), Urmas Valdek

(Scientific Computing Division: Leif Abrahamsson)

(Department of Mathematics, Linköping University: Lars-Erik Andersson)

The project is supported by the Swedish Research Council (and had previously support by the Swedish Research Council for Engineering Sciences).

A previous progress report for this project[1]

Some photos of the project group:

In Jpeg format:

In postscript format:Photo 1 Photo 2 Photo 3 Photo 4 Photo 5

## Publications

1- L. Hillström. Estimation of State and Identification of Complex Modulus Based on Measurements on a Bar Traversed by Waves. PhD thesis, Deptartment of Materials Science, Uppsala Univiversity, January 2001.

2- L. Hillström and B. Lundberg. Analysis of elastic flexural waves in non-uniform beams based on measurement of strains and accelerations. Journal of Sound and Vibration, 247(2):227-242, October 2001.

3- L. Hillström, M. Mossberg, and B. Lundberg. Identification of complex modulus from measured strains on an axially impacted bar using least squares. Journal of Sound and Vibration, 230(3):689-707, February 2000.

4- L. Hillström, U. Valdek and B. Lundberg. Estimering av tillstånd och komplex modul med hjälp av uppmätta töjningar på en stötbelastad balk (in Swedish). Svenska Mekanikdagar, Linköping, 2001.

5- L. Hillström, U. Valdek and B. Lundberg. Estimation of the state vector and identification of the complex modulus of a beam. Journal of Sound and Vibration, 261(4), 653-673, 2003.

6- K. Mahata. Estimation Using Low Rank Signal Models. PhD thesis, Department of Information Technology, Uppsala University, December 2003.

7- K. Mahata, S. Mousavi and T. Söderström. On the estimation of complex modulus and Poisson's ratio using longitudinal wave experiments. Mechanical Systems and Signal Processing, 20(8):2080-2094, November 2006.

8- K. Mahata, S. Mousavi, T. Söderström, M. Mossberg, U. Valdek and L. Hillström. On the use of flexural wave propagation experiments for identification of complex modulus. IEEE Transactions on Control Systems Technology, 11(6):863-874, November 2003.

9- K. Mahata, S. Mousavi, T. Söderström, and U. Valdek. Using boundary conditions for estimation of complex modulus from flexural wave experiments. IEEE Transactions on Control Systems Technology, 13(6),1093-1099, November 2005.

10- K. Mahata and T. Söderström. Bayesian approaches for identification of the complex modulus of viscoelastic materials. Automatica, 43(8):1369-1376, August 2007.

11- K. Mahata and T. Söderström. Improved estimation performance using known linear constraints. Automatica, 40(8):1307-1318, August 2004.

12- K. Mahata and T. Söderström. ESPRIT like estimation of real-valued sinusoidal frequencies. IEEE Transactions on Signal Processing, 52(5):1161-1170, May 2004.

13- K. Mahata and T. Söderström. Large sample properties of separable nonlinear least squares estimators. IEEE Transactions on Signal Processing, 52(6):1650-1658, June 2004.

14- K. Mahata, T. Söderström and L. Hillström. Computationally efficient estimation of wave propagation functions of viscoelastic materials. 13th IFAC Sympoisum on System Identification, SYSID 2003, August 27-29, 2003, Rotterdam, The Netherlands.

15- K. Mahata, T. Söderström, and L. Hillström. Computationally efficient estimation of wave propagation functions from 1-D wave experiments on viscoelastic materials. Automatica, 40(5):713-727, May 2004.

16- K. Mahata, T. Söderström, M. Mossberg, L. Hillström, S. Mousavi, and U. Valdek. Identification of complex elastic modulus from flexural wave experiments. In Proc. 3rd Int. Conf. on Identification in Engineering Systems, Swansea, Wales, UK, April 15-17 2002.

17- M. Mossberg. Identification of Viscoelastic Materials and Continuous-Time Stochastic Systems. PhD thesis, Deptartment of Systems and Control, Uppsala University, May 2000.

18- M. Mossberg. Optimal experimental design for identification of viscoelastic materials. IEEE Transactions on Control Systems Technology, 12(4):578-582, July 2004.

19- M. Mossberg, L. Hillström, and L. Abrahamsson. Parametric identification of viscoelastic materials from time and frequency domain data. Inverse Problems in Engineering, 9(6): 645-670, December 2001.

20- M. Mossberg, L. Hillström, and T. Söderström. Identification of viscoelastic materials. In Proc. 12th IFAC Symp. System Identification, volume 2, pp 663-668, Santa Barbara, CA, USA, June 21-23 2000.

21- M. Mossberg, L. Hillström, and T. Söderström. Non-parametric identification of viscoelastic materials from wave propagation experiments. Automatica, 37(4):511-521, April 2001.

22- S. Mousavi and B. Lundberg. Identification of complex moduli and Poisson's ratio from measured strains on an impacted bar. 5th Euromech Solid Mechanics Conference, Thessaloniki, 16-22 August 2003.

23- S. Mousavi, D. F. Nicolas, and B. Lundberg. Identification of complex moduli and Poisson's ratio from measured strains on an impacted bar. Journal of Sound and Vibration, 277(4-5):971-986, November 2004.

24- S. Mousavi, U. Valdek, K. Welch, and B. Lundberg. SHPB technique for identification of complex modulus under condition of non-uniform stress. 21th International Congress for Theoretical and Applied Mechanics, Warsaw, Poland, August 15-20 2004.

25- S. Mousavi, K. Welch, U. Valdek, and B. Lundberg. Non-equilibrium split Hopkinson pressure bar procedure for non-parametric identification of complex modulus. International Journal of Impact Engineering, 31(9):1133-1151. October 2005.

26- A. Rensfelt. Nonparametric Identification of Viscoelastic Materials. Lic. Thesis, Department of Information Technology, Uppsala University, October 2006.

27- A. Rensfelt, S. Mousavi, M. Mossberg, T. Söderström. Optimal sensor locations for nonparametric identification of viscoelastic materials. Automatica, 44(1):28-38, January 2008.

28- A. Rensfelt, T. Söderström. Optimal Excitation for Nonparametric Identification of Viscoelastic Materials. Technical Report 2006-034. Department of Information Technology, Uppsala University.

29- A. Rensfelt and T. Söderström. Frequency content in an axially impacted bar subject to boundary conditions. IFAC 17th World Congress, Seoul, Korea, July 06-11, 2008.

30- A. Runqvist, S. Mousavi, T. Söderström. Nonparametric identification of complex modulus using a non-equilibrium SHPB procedure. 14th IFAC Symposium on System Identification, March 29-31, 2006, Newcastle, Australia.

31- T. Söderström. Using system identification for estimating material functions from wave propagation experiments. Keynote lecture in Proc. 3rd Int. Conf. on Identification in Engineering Systems, Swansea, Wales, UK, April 15-17 2002.

32- T. Söderström. System identification techniques for estimating material functions from wave propagation experiments. Inverse Problems in Engineering, 10(5):413-439, October 2002.

33- T. Söderström: Identifying characteristics of viscoelastic materials from wave propagation experiments -- recent advances. In T. Glad and G. Hendeby, eds.: Forever Ljung in System Identification, Studentlitteratur, Lund, 2006.

34- A. Rensfelt and T. Söderström: Structure testing of wave propagation models used in identification of viscoelastic materials. Automatica, vol 46, no 4, pp 728-734, April 2010.

35- A. Rensfelt and T. Söderström: Optimal excitation for nonparametric identification of viscoelastic materials. IEEE Transactions on Control Systems Technology, vol 19, no 1, pp 238-244, January 2011.

36- A. Rensfelt: Agnes Rensfelt: Viscoelastic Materials: Identification and Experiment Design. PhD thesis, Uppsala University, 2010.

37- A. Rensfelt and T. Söderström: Parametric identification of complex modulus. Automatica, vol 47, no 4, pp 813-818, April 2011.

38- T. Söderström and A. Rensfelt: Estimation of material functions using system identification techniques. Invited plenary paper, 4th International Symposium on Advanced Control of Industrial Processes (Adconip), May 23-27, 2011, Hangzhou, China.

39- T. Söderström and A. Rensfelt: Estimation of material functions using system identification techniques. Control Engineering Practice, vol 20, no 10, pp 972-990, October 2012.

40- H. Norlander, U. Valdek, B. Lundberg and T. Söderström: Parameter estimation from wave propagation tests on a tube perforated by helical slots. Mechanical Systems and Signal Processing, vol 40, no 1, pp 385-399, October 2013.