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Strain measurement under the minimal controller synthesis algorithm and an extensometer design
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 194–201
journal homepage: www.elsevier.com/locate/jmatprotec
Strain measurement under the minimal controller synthesis algorithm and an extensometer design S. Bulut ∗ Cumhuriyet University, Mechanical Engineering Department, 58140 Sivas, Turkey
a r t i c l e
i n f o
a b s t r a c t
Article history:
In this paper, minimal controller synthesis (MCS) algorithm is used in the case of strain
Received 6 March 2006
measurement for the first time. The MCS control is an adaptive control method and it can
Received in revised form
be recommended as a robust controller for the servohydraulic materials testing machine.
4 October 2007
The algorithm coped with various specimens, which have different materials and diameters.
Accepted 23 January 2008
The control accuracy is important in materials testing due to the fact that even smaller overshoots or undershoots can cause undesirable results in cyclic loading. For this reason, the controller parameters needs to adjusted according to the changes in the plant parameters
Keywords: The MCS control
for acceptable plant output responses. In order to measure strain signal a simple LVDT extensometer was designed. In this set
Extensometer design
tests, two different specimens were used: aluminium alloy specimens of diameter 10 mm
Stress
and EN24T steel specimens of diameter 7 mm. The MCS control was implemented in two
Strain measurement
degrees of freedom form and produced very satisfactory plant output responses owing to
Load control
the fact that the results produced by the control are in the range of actual values. This indicates that the MCS control can be used in strain measurement very effectively. It may also indicate the possibility of using the MCS control other materials testing applications like strain control and temperature cycle tests. © 2008 Elsevier B.V. All rights reserved.
1.
Introduction
Adaptive control techniques are often used for a plant with unknown and time varying dynamics. Especially a model reference adaptive control (MRAC) technique can make the plant output coincide with a reference output. Since electrohydraulic servo systems are often used under varying conditions, the application of this technique to a servohydraulic materials testing machine is expected to be very powerful and useful as studied (Edge and Figuerodo, 1987). The main purpose of this paper is to present the results of the first known implementation of the minimal controller synthesis (MCS) algorithm in the case of strain measurement. The MCS algorithm was originally developed by Stoten and
∗
Tel.: +90 346 2191010 1698. E-mail address: [email protected]. 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.01.055
Benchoubane (1990a) as an extension to the model reference adaptive control (MRAC) algorithm of Landau (1979). The algorithm has a simple structure with relatively few computational requirements per time step. MCS requires no prior knowledge of the plant parameters for implementation, and yet is guaranteed to provide global asymptotic stability of the closed-loop system, unlike linear controller strategies. Additionally, the designer is not require to synthesize the MCS controller gains, since this done automatically by the algorithm, given arbitrary (often zero) initial conditions (Stoten and Benchoubane, 1990b). The MCS control has been shown to be robust in the presence of unknown external disturbances and unmodelled dynamics in the plant. The algorithm has been shown to be effective in a number of areas and it
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 194–201
Nomenclature A Am Ap Ar Bm Br Ce d(xr ,t) D e(s) E Gp h k K Kr L n P Pa Q r(s) s t ts u(s) x xe xm xr y(s) ye (s)
(n × n) nominal plant parameter matrix (h × h) reference model matrix cross-sectional area of the specimen (h × h) reduced order plant parameter matrix (h × 1) reference model matrix (h × 1) reduced order plant parameter matrix output error matrix (h × 1) disturbance vector diameter of the specimen tracking error the modulus of elasticity plant transfer function (scalar) reduced order plant state dimension, h < n instant of discrete-time integer (integer) MCS state feedback gain; (typically K(0) = 0) MCS forward loop gain; (typically Kr (0) = 0) unloaded length of the specimen nominal plant state dimension (h × h) symmetric positive definite matrix solution of the Lyapunov function equation applied load on the specimen (h × h) symmetric positive definite matrix associated with the Lyapunov equation reference vector Laplace variable time settling time control signal (n × 1) plant state vector (h × 1) state error vector (h × 1) reference model state vector (h × 1) the reduced order plant state vector plant output signal output error signal
Greek symbols ˛ MCS integral adaption gain (scalar), ˛ > 0 ˇ MCS proportional adaption gain (scalar), ˇ ≥ 0 ı elongation of the specimen sampling interval ε strain longitudinal stress time ˝ =ˇ − ˛
was applied to a large class of servohydraulic, pneumatic, electro mechanical systems and it was produced satisfactory plant output responses in work by Stoten (1990), Bulut (2000), and Stoten and Hodgson (1991). The algorithm was implemented to electrohydraulic servo systems in a simplified reduced order form in work by Bulut (2000), Stoten (1992) and Stoten and Bulut (1994) and produced satisfactory responses. It is very necessary to know the localized stress–strain history of aircraft components subjected to complex multiaxial stress conditions. However, a detailed finite element anal-
195
ysis is often very time consuming. Consequently, a simple new method developed in work by Knop et al. (2000) which combines modern constitutive theory with either Neuber’s, or Glinka’s, approach to calculate the localized notch strains. Peters and Heymsfield (2003) examined the use of meshes consisting of constant elements created from polygons having differing numbers of element nodes. New extensometers were developed which needs no attachment of line markers or mechanical tracers on a specimen in work by Yamaguchi et al. (2006). In this work, the displacement of the marker position was tracked by moving a head containing a laser diode, an imaging lens and an image sensor under the feedback control that compensates for the speckle displacement detected. In a similar manner, the local diametral strains of specimens in the tension Kolsky bar (or split Hopkinson pressure bar) experiment were measured using a laser occlusive radius detector (LORD) in work by Li and Ramesh (2007). In situ image capture together with processing technique adapted from particle image velocimetry were used by Abadi et al. (2007) in order to have more detail about the local strain fields. The Direct Measurements, Inc. Symbolic Strain Gage was used to measure the local plastic strains in work by Ranson et al. (2005). The gage technology was utilized the symbolic properties of a two-dimensional bar code (or compressed symbol) to make stress-analysis measurements. Experimental results demonstrated that fiber Bragg grating (FBG) sensors could measure strain with higher resolution. Lou et al. (2002) described a proportional integrated control theory which was to control the filter FBG’s Bragg wavelength. Mizutani et al. (2003) found that the small-diameter FBG could also detect transverse cracks in quasi-isotropic laminates quantitatively. FBG sensors were interrogated with the Pcbus eXtension for Instrumentation, a type of opto-electronic instrument and this advanced interrogated system was used to measure strains inside the metal or composite structures in work by Tsuda and Lee (2007). Shobu et al. (2007) measured the internal strain of a 5-mm thick austenitic stainless steel sample (JIS-SUS304L) by using high energy white X-rays from a synchrotron radiation source at SPring-8. A multimodule strain measuring and data processing system equipped with microprocessors and 4096 channels were controlled by a single computer in work by Ser’eznov et al. (2004). In this work, in order to increase the general productivity of the system a Host controller was employed. The plant dynamics changes enormously during materials testing due to the changes in specimens and the machine characteristics. However, closed-loop control accuracy is crucial in materials testing applications due to the fact that even smaller overshoots or undershoots can cause undesirable results therefore, controller tuning is also important in such tests. The suitable controller parameter values for any one test depend on the nature of the test, the specimen characteristics and the dynamics of the materials testing machine. Manual tuning of the controller parameters can be a serious problem especially many controller parameters need to be reset during the operation of the machine. Therefore, using adaptive controller in this field has many advantages, such as adaptivity to the changes in the working condition.
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1995). They have the disadvantage that they can demage the specimen at the contact point due to the clamping forces required and the weight of the device itself which can influence test specimens. Surface preparation of the testpiece is particularly important for the accuracy of the test. The strain is written as below ε=
ı L
(2)
where ε and ı are the strain and elongation of the specimen respectively and L is the unloaded length of the specimen. The test specimen was a cylinder of cross-sectional area Ap . The force per unit area is called the stress which is denoted by and written as follows =
The ESH materials testing machine
The ESH materials testing machine which is presented in this paper, has been developed in the Materials Laboratory, the Mechanical Engineering Department at Bristol University. The plant consists of a double-ended, balanced hydraulic actuator, a servovalve, a fixed-displacement pump, a load cell and a test specimen, with input u and output y. The loads are measured by a load cell in series with the actuator ram (Fig. 1). A series of system identification tests were conducted on the open-loop plant which yielded a second-order plant model for the nominal operating condition (the supply pressure 13.8 MPa, aluminium alloy specimens of diameter 10 mm). The test results were generated from a swept sinusoid input, with data analysed by the Matlab System Identification Toolbox macro output error (oe) method. The supply pressure was kept at its nominal value of 13.8 MPa during these tests and the amplitude was 0.8 V (a corresponding load of 4.5 kN). At the mid-frequency range, the average second-order transfer function was found to be GP (s) =
2900 s2 + 110s + 2000
(3)
where Pa is the applied load. The axial stress in the specimen is calculated by dividing the load Pa by the cross-sectional area, Ap . If the material is linearly elastic then, it follows Hooke’s law, so that the longitudinal stress and strain can be related by the equation = Eε, where E is the modulus of the elasticity. Then the elongation of the specimen can be written as follows:
Fig. 1 – ESH materials testing machine.
2.
Pa Ap
(1)
The compliance of the supporting structure on which the actuator, test specimen, and associated fixtures dynamics were negligible for the mid-frequency test considered, but could be significant if the test signal frequencies are increased. Therefore, in the mid-frequency range a second-order model was relevant for the ESH materials testing machine.
3. Strain measurement by contacting specimens Contact extensometers have been used for many years in the case of strain measurement. These devices were used for applications where extremely high precision was required over a relatively small extension (a few mm) (Dyson et al.,
ı=
Pa L EAp
(4)
In tension or compression, strain is defined as the elongation per unit of the gauge length, and described as follows: ε=
l − l0 l0
(5)
where l is the gauge length at any time and l0 is the original gauge length. This expression is satisfactory for elastic strains since l − l0 is small. For plastic deformation the gauge length will change considerably, therefore using natural strain, εn is more suitable in such cases which is given below εn = ln
l l0
(6)
εn = ln(ε + 1) Eqs. (5) and (6) give similar results for strains less than 0.1. If strain is below this value, it is reasonable to use the first equation to measure strain. For larger strains, and depending on the application, it is more appropriate to use (6) when converting output signals.
4.
The LVDT extensometer (5 mm)
Linear variable differential transformer (LVDT) device, with good linearity and low cost they have become very common in creep laboratories. Transducers convert displacement into an electrical voltage output and they have been widely used in the case of strain measurement. Under load control, LVDT devices produced an error when the direction of the loading changed. The internal spring of the devices was the source of this error.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 194–201
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Fig. 4 – Test specimens. Fig. 2 – The top plate of the LVDT extensometer.
Additionally, LVDT devices are also sensitive to ambient changes and the presence of magnetic fields. The long standing LVDT’s are best used in their more traditional role in the low temperatures (below 250 ◦ C), but they are not effective at high temperatures. The LVDT extensometer, which is presented in this paper, has been made in the Mechanical Engineering Department Workshop at Bristol. The extensometer consists of two parallel rectangular plates each of width 39 mm and height 5 mm and two LVDT’s which were placed either side of the bottom plate, shown in Figs. 2 and 3. The bottom plate has two 8 mm diameter holes either side of each corner which is 5 mm away from the plate edges, so that two LVDT’s can be placed in these holes and fixed by two M3 screws at zero position. The two LVDT’s have a range of 5 mm which corresponds to ±10 V. Both plates have a 25.5 mm diameter hole in the middle of them which will help to drag these plates on the specimens. Two plates were fixed in the middle of the specimens with ∼15 mm gauge length. In order to get fixed gauge length two collets had been used during the installation of the plates on the specimens, which have 10 mm height. The specimens are made of aluminium alloy (the stress = 150 MPa and Young’s modulus E = 72 GPa), steel (the stress = 320 MPa and Young’s modulus E = 210 GPa), with diameters D1 = 10 mm and D2 = 7 mm respectively, length L = 120 mm and a gauge length of 22 mm (Fig. 4). The structure of the extensometer was rather simple and it was designed for only room temperature. Alignment and bonding of the strain gauge was required a high degree of skill and the process of installing a gauge was time consuming. After fixing the extensometer on the specimens, the whole construction was loaded to the ESH materials testing machine under the MCS load control. The rig, shown schematically in Fig. 5, was actuated by a standard servohydraulic system.
Fig. 3 – The bottom plate of the LVDT extensometer.
Fig. 5 – Position, load and strain control.
During materials testing, the control problem was to ensure that the measured force y closely tracks for a given reference signal r despite the changes in specimen and servohydraulic characteristics. The collets made installation easy, by providing complete support to the specimens, and preventing the slip of the plates and the specimens, therefore it made possible to have predetermined fixed gauge length. Contact between the plates and the specimens was made by three M3 screws, for this reason, three M3 tap were located on the neck of the top and bottom plates, as shown in Fig. 6. All together, there
Fig. 6 – Test specimen under the MCS load control.
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 194–201
a second-order MCS formulation would normally be necessary. However, the MCS control was implemented in a simplified reduced first order form. The fact that the plant was higher order than the MCS algorithm indicates that MCS appears to be quite insensitive to such mismatches. Consider a SISO or MIMO plant described by the following state-space equation ˙ = Ax(t) + Bu(t) x(t)
(7)
where, A ∈ nxn and B ∈ nxn . The variables x(t) and u(t) denote the plant state vector and control input respectively. The reduced first order plant model is described by the following state-space equation: x˙ r (t) = Ar (t) + Br (t) + d(xr , t)
(8)
where, Ar ∈ hxh and Br ∈ hxh are the reduced first order plant parameters, xr is the reduced order plant state, u(t) is the control input signal, and d(xr , t) is the disturbance term. Any unmodelled terms, unknown external disturbances, plant nonlinearities and parameter variations are included into the
Fig. 7 – The responses of MCS control for aluminium specimens of diameter 10 mm, elastic region.
were 6 M3 taps located as 120◦ from each other on the neck of the top and bottom plates in order to provide good contact between the specimens and the plates.
5. Using the MCS load control in the case of strain measurement The MCS control was implemented in two degrees of freedom form. However, there was no control action in the second degree of the controller and it was used as a measurement channel of the strain signal. In fact, this implementation was only for controlling the load signal. Although, the MCS control was implemented to the plant in multi-input multi-output (MIMO) form, still it can be treated as single-input single output (SISO) MCS due to the fact that only the first channel of the controller was active. While the specimen was controlled under the MCS load control the strain signal was read and logged by the controller. Controller hardware consisted of a PC machine equipped with 12-bit D/A and A/D converters. The ESH material testing machine has second-order transfer function, under the nominal operating condition therefore
Fig. 8 – The responses of MCS control for aluminium specimens of diameter 10 mm, plastic region.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 194–201
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where Ce is the output error matrix, Ce = ts /8 and xe is the state error vector which is written below xe (k) = xm (k) − x(k)
(13)
where xe and xm are the state error vector and the reference model state vector respectively. The first order MCS reference model is x˙ m (t) = Am x(t) + Bm r(t)
(14)
where Am = −4/ts and Bm = 4/ts . The hyperstable condition is guaranteed if: Ce = BT eP
(15)
In this equation, Be = [1] and P is the positive definite solution to the Lyapunov equation PAm + AT m P = −Q;
Q>0
(16)
and the ‘weighting’ matrix Q in the Lyapunov equation, which was chosen as Q = [1]. The values of adaptive weights were ˛ = 0.01 and ˇ = 0.001. The reference model parameters were chosen in order to elicit ideal state trajectories xm which the plant state x guaranteed to follow in a stable manner. The first order reference model parameters were
Fig. 9 – The responses of MCS control for steel specimens of diameter 7 mm, elastic region.
disturbance vector. The first order MCS control signal is given in discrete-time scalar form as below: u(k) = K(k)x(k) + Kr (k)r(k)
4 4 ts = 0.0438 = −11.4286, Bm = = 11.4286 and Ce = ts ts 8
6.
Stress–strain tests
Two different specimens were used in this set of tests as follows:
(a) aluminium alloy specimens with D1 = 10 mm; (b) EN24T steel specimens with D2 = 7 mm.
(9)
where K is MCS state feedback gain, Kr is MCS forward loop gain, r is the reference signal and k is instant of discrete-time (integer). The MCS adaptive gains are written as K(k) = K(k − 1) + ˇye xT (k) − ˝ye (k − 1)xT (k − 1)
(10)
Kr (k) = Kr (k − 1) + ˇye rT (k) − ˝ye (k − 1)rT (k − 1)
(11)
where ˛ is the MCS integral adaption gain (scalar); ˛ > 0, ˇ is the MCS proportional adaption gain (scalar), ˇ ≥ 0, ˝ = ˇ − ˛ and is the sampling interval of the discrete-time process. The output error signal is ye (k) = Ce xe (k)
Am = −
(12)
The reference signal was chosen as a sine wave of frequency 0.4 Hz, and amplitude 2 V (10 kN) in the case of aluminium alloy specimens of diameter 10 mm. This amplitude corresponds to the elastic region ( y = 150 MPa) for aluminium alloy specimens. The desired settling time was ts = 0.2 s, a reasonable choice of was = 10 ms. The adaptive rates were chosen empirically, the values were ˛ = 0.1 and ˇ = 0.01. In plastic region the amplitude of the reference signal was 2.7 V (13.5 kN). In the case of steel specimens of diameter 7 mm, the reference signal was a sine wave of amplitude 2.5 V and frequency 0.4 Hz. The desired settling time was ts = 0.4 s, a suitable choice of was = 20 ms. The adaptive rates were ˛ = 0.01 and ˇ = 0.001. Two different amplitudes, 2.5 V (12.5 kN) and 3.4 V (17 kN) were used for steel specimens which correspond to the applied load in elastic and plastic region respectively.
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 194–201
y = 300–670 MPa. Following the yielding point, MCS produced, y = 320 MPa as shown in Fig. 9a.
8.
Fig. 10 – The responses of MCS control for steel specimens of diameter 7 mm, plastic region.
7.
Results and discussion
The stress–strain diagram for aluminium alloy specimens in elastic region is shown in Fig. 7a. In elastic region, the modulus of elasticity of the specimens can be computed from Hooke’s Law ( = Eε). The corresponding, the elasticity modulus, MCS gains and elongation signals are shown in Fig. 7b–d respectively. The stress–strain diagram of aluminium alloy specimens in plastic region is shown in Fig. 8a, together with control, gains and elongation signals in Fig. 8b–d respectively. EN24T steel specimens of diameter 7 mm produced the stress–strain signal in elastic region which is shown in Fig. 9a. Corresponding the modulus of elasticity, MCS gains, and elongation signals are shown in Fig. 9b–d. The stress–strain signal of steel specimens in plastic region is shown in Fig. 10a, together with the control input, MCS gains, and elongation signals are shown in Fig. 10b–d. The standard yielding point for aluminium alloy specimens given as y = 120–200 MPa. The corresponding yielding point measured by MCS was y = 150 MPa (see in Fig. 6a). In the case of steel specimens the standard yielding point is given as
Conclusions
During most materials tests, the plant dynamics change enormously due to the variations in the specimens (for example crack growths, effects of high temperatures and modulus variations) and the test machine characteristics. The electrohydraulic materials testing machine exhibits significant nonlinearities, therefore a linear controller is not good enough to give satisfactory results due to the fact that the linear controller can only be optimised for one operating point. In this work, it has been shown that the MCS control performed very well in electrohydraulic system load control. Therefore, using the MCS control in materials testing applications has many advantages, such as adaptivity to the changes in the working condition (changes in load, supply pressure and specimens characteristics). The MCS algorithm was implemented in a simplified reduced order form. The fact that the MCS control reference model was one degree lower than the nominal plant model. The reduced order control produced satisfactory plant output responses, indicating that MCS appears to be robust and quite insensitive to model order reduction. In particular, a first order MCS algorithm produced excellent closed-loop responses. The results produced by the MCS control are in the range of the actual values, indicating that the MCS control can be used in strain measurement very effectively. The MCS control can be recommended as a robust controller for the servohydraulic materials testing machine. The algorithm coped with plant nonlinearities (which are high pressure, servovalve dynamics, the nonlinear effects of the hydraulic fluids, temperature changes, aeration and cavitation problems), parameter variations in specimens (small parameter variations introduced when the amplitude of the input signal is changed) and external disturbances. The MCS control coped with plant parameters variation, the changes in the plant working condition and disturbances. The algorithm can be recommended as a robust controller for the plants which are subject to internal and external disturbances. The MCS control of temperature cycles is perfectly viable.
references
Abadi, M., de Haan, F., Hahner, P., 2007. Tensile strain measurement using mechanical and in situ optical techniques in bi-metallic Ni-base superalloys. Meas.: J. Int. Meas. Confederation 40, 383–391. Bulut, S., 2000. The robustness of reduced order minimal controller synthesis control in the plastic region. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 214, 185–195. Dyson, B.F., Loveday, M.S., Gee, M.G., 1995. Materials Metrology and Standards for Structural Performance, first ed. Chapman & Hall. Edge, K.A., Figuerodo, K.R.A., 1987. Adaptively controlled electrohydraulic servo-mechanism, Part 2: implementation. Proc. I. Mech. E 201, 181–189.
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Knop, M., Jones, R., Molent, L., Wang, C., 2000. On the Glinka and Neuber methods for calculating notch tip strains under cyclic load spectra. Int. J. Fatigue 22, 743–755. Landau, I.D., 1979. Adaptive Control, the Model Reference Approach. Marcel Dekker, New York. Li, Y., Ramesh, K.T., 2007. An optical technique for measurement of material properties in the tension Kolsky bar. Int. J. Impact Eng. 34, 784–798. Lou, S., Song, I.-C., Jeong, S.-H., Lee, B.H., Lee, S.-K., 2002. Signal processing for strain measurement system with fiber Bragg grating. In: Proceedings of the Second International Symposium on Instrumentation Science and Technology, pp. 332–337. Mizutani, T., Okabe, Y., Takeda, N., 2003. Quantitative evaluation of transverse cracks in carbon fiber reinforced plastic quasi-isotropic laminates with embedded small-diameter fiber Bragg grating sensors. Smart Mater. Struct. 12, 898–903. Peters, J.F., Heymsfield, E., 2003. Application of the 2-D constant strain assumption to FEM elements consisting of an arbitrary number of nodes. Int. J. Solids Struct. 40, 143–159. Ranson, W.F., Hovls, G.L., Wood, M., Hickson, A.W., 2005. Low cycle fatigue plastic measurements in 2024-T3 aluminum. In: Proceedings of the 2005 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, pp. 1663–1668. Ser’eznov, A.N., Stepanova, L.N., Kozhemyakin, V.L., Lebedev, E.Yu., 2004. Multimodule strain-gauging system equipped with microprocessors. Russ. J. Nondestructive Test. 40, 66–70.
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Shobu, T., Kaneko, H., Mizuki, J., Konishi, H., Shibano, J., Hirata, T., Suzuki, K., 2007. Strain measurements using high energy white X-rays at SPring-8. In: AIP Conference Proceedings, 1581–1588, 879. Stoten, D.P., 1990. Model Reference Adaptive Control of Manipulators. Research Studies Press, Taunton, U.K. Stoten, D.P., 1992. Implementation of minimal control synthesis on a servo-hydraulic testing machine. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 206, 189–194. Stoten, D.P., Benchoubane, H., 1990a. Robustness of a minimal controller synthesis algorithm. Int. J. Control 51, 851– 861. Stoten, D.P., Benchoubane, H., 1990b. Empirical studies of an MRAC algorithm with minimal controller synthesis. Int. J. Control 51, 823–849. Stoten,D.P., Bulut, S., 1994. Application of the MCS algorithm to the control of an electrohydraulic systems. Euriscon’94, 1861–1871. Malaga, Spain. Stoten, D.P., Hodgson, S.P., 1991. Comparative implementation studies of five adaptive control algorithms on a class 1 manipulator. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 205, 263–273. Tsuda, H., Lee, J.-R., 2007. Strain and damage monitoring of CFRP in impact loading using a fiber Bragg grating sensor system. Composites Sci. Technol. 67, 1353–1361. Yamaguchi, I., Kobayashi, K., Ida, T., Yokota, M., 2006. Noncontacting extensometer using digital speckle correlation. Mater. Eval. 64, 724–730.
journal homepage: www.elsevier.com/locate/jmatprotec
Strain measurement under the minimal controller synthesis algorithm and an extensometer design S. Bulut ∗ Cumhuriyet University, Mechanical Engineering Department, 58140 Sivas, Turkey
a r t i c l e
i n f o
a b s t r a c t
Article history:
In this paper, minimal controller synthesis (MCS) algorithm is used in the case of strain
Received 6 March 2006
measurement for the first time. The MCS control is an adaptive control method and it can
Received in revised form
be recommended as a robust controller for the servohydraulic materials testing machine.
4 October 2007
The algorithm coped with various specimens, which have different materials and diameters.
Accepted 23 January 2008
The control accuracy is important in materials testing due to the fact that even smaller overshoots or undershoots can cause undesirable results in cyclic loading. For this reason, the controller parameters needs to adjusted according to the changes in the plant parameters
Keywords: The MCS control
for acceptable plant output responses. In order to measure strain signal a simple LVDT extensometer was designed. In this set
Extensometer design
tests, two different specimens were used: aluminium alloy specimens of diameter 10 mm
Stress
and EN24T steel specimens of diameter 7 mm. The MCS control was implemented in two
Strain measurement
degrees of freedom form and produced very satisfactory plant output responses owing to
Load control
the fact that the results produced by the control are in the range of actual values. This indicates that the MCS control can be used in strain measurement very effectively. It may also indicate the possibility of using the MCS control other materials testing applications like strain control and temperature cycle tests. © 2008 Elsevier B.V. All rights reserved.
1.
Introduction
Adaptive control techniques are often used for a plant with unknown and time varying dynamics. Especially a model reference adaptive control (MRAC) technique can make the plant output coincide with a reference output. Since electrohydraulic servo systems are often used under varying conditions, the application of this technique to a servohydraulic materials testing machine is expected to be very powerful and useful as studied (Edge and Figuerodo, 1987). The main purpose of this paper is to present the results of the first known implementation of the minimal controller synthesis (MCS) algorithm in the case of strain measurement. The MCS algorithm was originally developed by Stoten and
∗
Tel.: +90 346 2191010 1698. E-mail address: [email protected]. 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.01.055
Benchoubane (1990a) as an extension to the model reference adaptive control (MRAC) algorithm of Landau (1979). The algorithm has a simple structure with relatively few computational requirements per time step. MCS requires no prior knowledge of the plant parameters for implementation, and yet is guaranteed to provide global asymptotic stability of the closed-loop system, unlike linear controller strategies. Additionally, the designer is not require to synthesize the MCS controller gains, since this done automatically by the algorithm, given arbitrary (often zero) initial conditions (Stoten and Benchoubane, 1990b). The MCS control has been shown to be robust in the presence of unknown external disturbances and unmodelled dynamics in the plant. The algorithm has been shown to be effective in a number of areas and it
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 194–201
Nomenclature A Am Ap Ar Bm Br Ce d(xr ,t) D e(s) E Gp h k K Kr L n P Pa Q r(s) s t ts u(s) x xe xm xr y(s) ye (s)
(n × n) nominal plant parameter matrix (h × h) reference model matrix cross-sectional area of the specimen (h × h) reduced order plant parameter matrix (h × 1) reference model matrix (h × 1) reduced order plant parameter matrix output error matrix (h × 1) disturbance vector diameter of the specimen tracking error the modulus of elasticity plant transfer function (scalar) reduced order plant state dimension, h < n instant of discrete-time integer (integer) MCS state feedback gain; (typically K(0) = 0) MCS forward loop gain; (typically Kr (0) = 0) unloaded length of the specimen nominal plant state dimension (h × h) symmetric positive definite matrix solution of the Lyapunov function equation applied load on the specimen (h × h) symmetric positive definite matrix associated with the Lyapunov equation reference vector Laplace variable time settling time control signal (n × 1) plant state vector (h × 1) state error vector (h × 1) reference model state vector (h × 1) the reduced order plant state vector plant output signal output error signal
Greek symbols ˛ MCS integral adaption gain (scalar), ˛ > 0 ˇ MCS proportional adaption gain (scalar), ˇ ≥ 0 ı elongation of the specimen sampling interval ε strain longitudinal stress time ˝ =ˇ − ˛
was applied to a large class of servohydraulic, pneumatic, electro mechanical systems and it was produced satisfactory plant output responses in work by Stoten (1990), Bulut (2000), and Stoten and Hodgson (1991). The algorithm was implemented to electrohydraulic servo systems in a simplified reduced order form in work by Bulut (2000), Stoten (1992) and Stoten and Bulut (1994) and produced satisfactory responses. It is very necessary to know the localized stress–strain history of aircraft components subjected to complex multiaxial stress conditions. However, a detailed finite element anal-
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ysis is often very time consuming. Consequently, a simple new method developed in work by Knop et al. (2000) which combines modern constitutive theory with either Neuber’s, or Glinka’s, approach to calculate the localized notch strains. Peters and Heymsfield (2003) examined the use of meshes consisting of constant elements created from polygons having differing numbers of element nodes. New extensometers were developed which needs no attachment of line markers or mechanical tracers on a specimen in work by Yamaguchi et al. (2006). In this work, the displacement of the marker position was tracked by moving a head containing a laser diode, an imaging lens and an image sensor under the feedback control that compensates for the speckle displacement detected. In a similar manner, the local diametral strains of specimens in the tension Kolsky bar (or split Hopkinson pressure bar) experiment were measured using a laser occlusive radius detector (LORD) in work by Li and Ramesh (2007). In situ image capture together with processing technique adapted from particle image velocimetry were used by Abadi et al. (2007) in order to have more detail about the local strain fields. The Direct Measurements, Inc. Symbolic Strain Gage was used to measure the local plastic strains in work by Ranson et al. (2005). The gage technology was utilized the symbolic properties of a two-dimensional bar code (or compressed symbol) to make stress-analysis measurements. Experimental results demonstrated that fiber Bragg grating (FBG) sensors could measure strain with higher resolution. Lou et al. (2002) described a proportional integrated control theory which was to control the filter FBG’s Bragg wavelength. Mizutani et al. (2003) found that the small-diameter FBG could also detect transverse cracks in quasi-isotropic laminates quantitatively. FBG sensors were interrogated with the Pcbus eXtension for Instrumentation, a type of opto-electronic instrument and this advanced interrogated system was used to measure strains inside the metal or composite structures in work by Tsuda and Lee (2007). Shobu et al. (2007) measured the internal strain of a 5-mm thick austenitic stainless steel sample (JIS-SUS304L) by using high energy white X-rays from a synchrotron radiation source at SPring-8. A multimodule strain measuring and data processing system equipped with microprocessors and 4096 channels were controlled by a single computer in work by Ser’eznov et al. (2004). In this work, in order to increase the general productivity of the system a Host controller was employed. The plant dynamics changes enormously during materials testing due to the changes in specimens and the machine characteristics. However, closed-loop control accuracy is crucial in materials testing applications due to the fact that even smaller overshoots or undershoots can cause undesirable results therefore, controller tuning is also important in such tests. The suitable controller parameter values for any one test depend on the nature of the test, the specimen characteristics and the dynamics of the materials testing machine. Manual tuning of the controller parameters can be a serious problem especially many controller parameters need to be reset during the operation of the machine. Therefore, using adaptive controller in this field has many advantages, such as adaptivity to the changes in the working condition.
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1995). They have the disadvantage that they can demage the specimen at the contact point due to the clamping forces required and the weight of the device itself which can influence test specimens. Surface preparation of the testpiece is particularly important for the accuracy of the test. The strain is written as below ε=
ı L
(2)
where ε and ı are the strain and elongation of the specimen respectively and L is the unloaded length of the specimen. The test specimen was a cylinder of cross-sectional area Ap . The force per unit area is called the stress which is denoted by and written as follows =
The ESH materials testing machine
The ESH materials testing machine which is presented in this paper, has been developed in the Materials Laboratory, the Mechanical Engineering Department at Bristol University. The plant consists of a double-ended, balanced hydraulic actuator, a servovalve, a fixed-displacement pump, a load cell and a test specimen, with input u and output y. The loads are measured by a load cell in series with the actuator ram (Fig. 1). A series of system identification tests were conducted on the open-loop plant which yielded a second-order plant model for the nominal operating condition (the supply pressure 13.8 MPa, aluminium alloy specimens of diameter 10 mm). The test results were generated from a swept sinusoid input, with data analysed by the Matlab System Identification Toolbox macro output error (oe) method. The supply pressure was kept at its nominal value of 13.8 MPa during these tests and the amplitude was 0.8 V (a corresponding load of 4.5 kN). At the mid-frequency range, the average second-order transfer function was found to be GP (s) =
2900 s2 + 110s + 2000
(3)
where Pa is the applied load. The axial stress in the specimen is calculated by dividing the load Pa by the cross-sectional area, Ap . If the material is linearly elastic then, it follows Hooke’s law, so that the longitudinal stress and strain can be related by the equation = Eε, where E is the modulus of the elasticity. Then the elongation of the specimen can be written as follows:
Fig. 1 – ESH materials testing machine.
2.
Pa Ap
(1)
The compliance of the supporting structure on which the actuator, test specimen, and associated fixtures dynamics were negligible for the mid-frequency test considered, but could be significant if the test signal frequencies are increased. Therefore, in the mid-frequency range a second-order model was relevant for the ESH materials testing machine.
3. Strain measurement by contacting specimens Contact extensometers have been used for many years in the case of strain measurement. These devices were used for applications where extremely high precision was required over a relatively small extension (a few mm) (Dyson et al.,
ı=
Pa L EAp
(4)
In tension or compression, strain is defined as the elongation per unit of the gauge length, and described as follows: ε=
l − l0 l0
(5)
where l is the gauge length at any time and l0 is the original gauge length. This expression is satisfactory for elastic strains since l − l0 is small. For plastic deformation the gauge length will change considerably, therefore using natural strain, εn is more suitable in such cases which is given below εn = ln
l l0
(6)
εn = ln(ε + 1) Eqs. (5) and (6) give similar results for strains less than 0.1. If strain is below this value, it is reasonable to use the first equation to measure strain. For larger strains, and depending on the application, it is more appropriate to use (6) when converting output signals.
4.
The LVDT extensometer (5 mm)
Linear variable differential transformer (LVDT) device, with good linearity and low cost they have become very common in creep laboratories. Transducers convert displacement into an electrical voltage output and they have been widely used in the case of strain measurement. Under load control, LVDT devices produced an error when the direction of the loading changed. The internal spring of the devices was the source of this error.
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Fig. 4 – Test specimens. Fig. 2 – The top plate of the LVDT extensometer.
Additionally, LVDT devices are also sensitive to ambient changes and the presence of magnetic fields. The long standing LVDT’s are best used in their more traditional role in the low temperatures (below 250 ◦ C), but they are not effective at high temperatures. The LVDT extensometer, which is presented in this paper, has been made in the Mechanical Engineering Department Workshop at Bristol. The extensometer consists of two parallel rectangular plates each of width 39 mm and height 5 mm and two LVDT’s which were placed either side of the bottom plate, shown in Figs. 2 and 3. The bottom plate has two 8 mm diameter holes either side of each corner which is 5 mm away from the plate edges, so that two LVDT’s can be placed in these holes and fixed by two M3 screws at zero position. The two LVDT’s have a range of 5 mm which corresponds to ±10 V. Both plates have a 25.5 mm diameter hole in the middle of them which will help to drag these plates on the specimens. Two plates were fixed in the middle of the specimens with ∼15 mm gauge length. In order to get fixed gauge length two collets had been used during the installation of the plates on the specimens, which have 10 mm height. The specimens are made of aluminium alloy (the stress = 150 MPa and Young’s modulus E = 72 GPa), steel (the stress = 320 MPa and Young’s modulus E = 210 GPa), with diameters D1 = 10 mm and D2 = 7 mm respectively, length L = 120 mm and a gauge length of 22 mm (Fig. 4). The structure of the extensometer was rather simple and it was designed for only room temperature. Alignment and bonding of the strain gauge was required a high degree of skill and the process of installing a gauge was time consuming. After fixing the extensometer on the specimens, the whole construction was loaded to the ESH materials testing machine under the MCS load control. The rig, shown schematically in Fig. 5, was actuated by a standard servohydraulic system.
Fig. 3 – The bottom plate of the LVDT extensometer.
Fig. 5 – Position, load and strain control.
During materials testing, the control problem was to ensure that the measured force y closely tracks for a given reference signal r despite the changes in specimen and servohydraulic characteristics. The collets made installation easy, by providing complete support to the specimens, and preventing the slip of the plates and the specimens, therefore it made possible to have predetermined fixed gauge length. Contact between the plates and the specimens was made by three M3 screws, for this reason, three M3 tap were located on the neck of the top and bottom plates, as shown in Fig. 6. All together, there
Fig. 6 – Test specimen under the MCS load control.
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a second-order MCS formulation would normally be necessary. However, the MCS control was implemented in a simplified reduced first order form. The fact that the plant was higher order than the MCS algorithm indicates that MCS appears to be quite insensitive to such mismatches. Consider a SISO or MIMO plant described by the following state-space equation ˙ = Ax(t) + Bu(t) x(t)
(7)
where, A ∈ nxn and B ∈ nxn . The variables x(t) and u(t) denote the plant state vector and control input respectively. The reduced first order plant model is described by the following state-space equation: x˙ r (t) = Ar (t) + Br (t) + d(xr , t)
(8)
where, Ar ∈ hxh and Br ∈ hxh are the reduced first order plant parameters, xr is the reduced order plant state, u(t) is the control input signal, and d(xr , t) is the disturbance term. Any unmodelled terms, unknown external disturbances, plant nonlinearities and parameter variations are included into the
Fig. 7 – The responses of MCS control for aluminium specimens of diameter 10 mm, elastic region.
were 6 M3 taps located as 120◦ from each other on the neck of the top and bottom plates in order to provide good contact between the specimens and the plates.
5. Using the MCS load control in the case of strain measurement The MCS control was implemented in two degrees of freedom form. However, there was no control action in the second degree of the controller and it was used as a measurement channel of the strain signal. In fact, this implementation was only for controlling the load signal. Although, the MCS control was implemented to the plant in multi-input multi-output (MIMO) form, still it can be treated as single-input single output (SISO) MCS due to the fact that only the first channel of the controller was active. While the specimen was controlled under the MCS load control the strain signal was read and logged by the controller. Controller hardware consisted of a PC machine equipped with 12-bit D/A and A/D converters. The ESH material testing machine has second-order transfer function, under the nominal operating condition therefore
Fig. 8 – The responses of MCS control for aluminium specimens of diameter 10 mm, plastic region.
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where Ce is the output error matrix, Ce = ts /8 and xe is the state error vector which is written below xe (k) = xm (k) − x(k)
(13)
where xe and xm are the state error vector and the reference model state vector respectively. The first order MCS reference model is x˙ m (t) = Am x(t) + Bm r(t)
(14)
where Am = −4/ts and Bm = 4/ts . The hyperstable condition is guaranteed if: Ce = BT eP
(15)
In this equation, Be = [1] and P is the positive definite solution to the Lyapunov equation PAm + AT m P = −Q;
Q>0
(16)
and the ‘weighting’ matrix Q in the Lyapunov equation, which was chosen as Q = [1]. The values of adaptive weights were ˛ = 0.01 and ˇ = 0.001. The reference model parameters were chosen in order to elicit ideal state trajectories xm which the plant state x guaranteed to follow in a stable manner. The first order reference model parameters were
Fig. 9 – The responses of MCS control for steel specimens of diameter 7 mm, elastic region.
disturbance vector. The first order MCS control signal is given in discrete-time scalar form as below: u(k) = K(k)x(k) + Kr (k)r(k)
4 4 ts = 0.0438 = −11.4286, Bm = = 11.4286 and Ce = ts ts 8
6.
Stress–strain tests
Two different specimens were used in this set of tests as follows:
(a) aluminium alloy specimens with D1 = 10 mm; (b) EN24T steel specimens with D2 = 7 mm.
(9)
where K is MCS state feedback gain, Kr is MCS forward loop gain, r is the reference signal and k is instant of discrete-time (integer). The MCS adaptive gains are written as K(k) = K(k − 1) + ˇye xT (k) − ˝ye (k − 1)xT (k − 1)
(10)
Kr (k) = Kr (k − 1) + ˇye rT (k) − ˝ye (k − 1)rT (k − 1)
(11)
where ˛ is the MCS integral adaption gain (scalar); ˛ > 0, ˇ is the MCS proportional adaption gain (scalar), ˇ ≥ 0, ˝ = ˇ − ˛ and is the sampling interval of the discrete-time process. The output error signal is ye (k) = Ce xe (k)
Am = −
(12)
The reference signal was chosen as a sine wave of frequency 0.4 Hz, and amplitude 2 V (10 kN) in the case of aluminium alloy specimens of diameter 10 mm. This amplitude corresponds to the elastic region ( y = 150 MPa) for aluminium alloy specimens. The desired settling time was ts = 0.2 s, a reasonable choice of was = 10 ms. The adaptive rates were chosen empirically, the values were ˛ = 0.1 and ˇ = 0.01. In plastic region the amplitude of the reference signal was 2.7 V (13.5 kN). In the case of steel specimens of diameter 7 mm, the reference signal was a sine wave of amplitude 2.5 V and frequency 0.4 Hz. The desired settling time was ts = 0.4 s, a suitable choice of was = 20 ms. The adaptive rates were ˛ = 0.01 and ˇ = 0.001. Two different amplitudes, 2.5 V (12.5 kN) and 3.4 V (17 kN) were used for steel specimens which correspond to the applied load in elastic and plastic region respectively.
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y = 300–670 MPa. Following the yielding point, MCS produced, y = 320 MPa as shown in Fig. 9a.
8.
Fig. 10 – The responses of MCS control for steel specimens of diameter 7 mm, plastic region.
7.
Results and discussion
The stress–strain diagram for aluminium alloy specimens in elastic region is shown in Fig. 7a. In elastic region, the modulus of elasticity of the specimens can be computed from Hooke’s Law ( = Eε). The corresponding, the elasticity modulus, MCS gains and elongation signals are shown in Fig. 7b–d respectively. The stress–strain diagram of aluminium alloy specimens in plastic region is shown in Fig. 8a, together with control, gains and elongation signals in Fig. 8b–d respectively. EN24T steel specimens of diameter 7 mm produced the stress–strain signal in elastic region which is shown in Fig. 9a. Corresponding the modulus of elasticity, MCS gains, and elongation signals are shown in Fig. 9b–d. The stress–strain signal of steel specimens in plastic region is shown in Fig. 10a, together with the control input, MCS gains, and elongation signals are shown in Fig. 10b–d. The standard yielding point for aluminium alloy specimens given as y = 120–200 MPa. The corresponding yielding point measured by MCS was y = 150 MPa (see in Fig. 6a). In the case of steel specimens the standard yielding point is given as
Conclusions
During most materials tests, the plant dynamics change enormously due to the variations in the specimens (for example crack growths, effects of high temperatures and modulus variations) and the test machine characteristics. The electrohydraulic materials testing machine exhibits significant nonlinearities, therefore a linear controller is not good enough to give satisfactory results due to the fact that the linear controller can only be optimised for one operating point. In this work, it has been shown that the MCS control performed very well in electrohydraulic system load control. Therefore, using the MCS control in materials testing applications has many advantages, such as adaptivity to the changes in the working condition (changes in load, supply pressure and specimens characteristics). The MCS algorithm was implemented in a simplified reduced order form. The fact that the MCS control reference model was one degree lower than the nominal plant model. The reduced order control produced satisfactory plant output responses, indicating that MCS appears to be robust and quite insensitive to model order reduction. In particular, a first order MCS algorithm produced excellent closed-loop responses. The results produced by the MCS control are in the range of the actual values, indicating that the MCS control can be used in strain measurement very effectively. The MCS control can be recommended as a robust controller for the servohydraulic materials testing machine. The algorithm coped with plant nonlinearities (which are high pressure, servovalve dynamics, the nonlinear effects of the hydraulic fluids, temperature changes, aeration and cavitation problems), parameter variations in specimens (small parameter variations introduced when the amplitude of the input signal is changed) and external disturbances. The MCS control coped with plant parameters variation, the changes in the plant working condition and disturbances. The algorithm can be recommended as a robust controller for the plants which are subject to internal and external disturbances. The MCS control of temperature cycles is perfectly viable.
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