Author: Schlegel, Christian
Date published: March 1, 2012
(ProQuest: ... denotes formulae omitted.)
The In the last few decades very precise static force measurements were developed and are now routinely used for calibration services in many national metrology institutes (NMFs) around the world. The force scale which is covered nowadays reaches from µ?-?? [1-2]. Thereby, relative measurement uncertainties down to 2-10"5 are obtained using deadweight machines, which are the best standard to realize a traceable force. The force, F, is just the product of the SI base unit mass, m, and the gravitational acceleration, g, following Newton's law, F=m-a, with the acceleration, a=g.
Besides the precise realization of a force in a standard machine, there must be selected force transducer available which can be used as a transfer standard to give the primary calibration to the secondarily calibration laboratories and industry. The crucial fact is now that often these static calibrated force transducers are used in dynamic applications. That is the reason why more and more NMFs have established procedures for a dynamic calibration of force transducers and also other sensors.
Currently in the European Metrology Research Programme (EMRP) one promoted research topic is the: "Traceable Dynamic Measurement of Mechanical Quantities", which includes, apart from a work package about dynamic force, also work packages about dynamic pressure, dynamic torque, the electrical characterization of measuring amplifiers and mathematical and statistical methods and modelling .
Similar to the static calibration philosophy primary calibrations have to be provided which guarantee traceability to the SI base units and also transfer transducers (reference standards) to transfer these calibrations e.g. to an industrial application. This transfer turned out to be the most complicated task because of the crucial influence of environmental conditions present in certain applications. Mostly the transducers are clamped from both sides which lead to sensitivity losses due to the dynamics of these connections which are more or less not infinitely stiff. On the other hand the resonant frequency often shifts down to lower frequencies which can also drastically change the sensitivity. The problem can be solved to a certain extent by modelling the whole construction including all relevant parameters. For that reason it is also important to determine the force transducer parameters like stiffness and damping which can be obtained during a dynamic calibration. This article describes one possibility for a primary dynamic force calibration using sinusoidal excitations. The whole procedure as well as most of the set-ups where developed over two decades and are extensively described in . Other methods as well as analysis procedures for dynamic force calibration are described in [5-9].
2. Mathematical Description
To obtain an analytical "handle" for the description of the dynamic behaviour of a dynamically excited force transducer, the well-known spring-mass-damper model can be applied. In Fig. 1 one can see a simplified picture of a force transducer which is equipped with a test mass, m^sub t^. The connection of that mass to the transducer is modelled by a certain stiffness, k^sub c^, and a damping constant, bc. The transducer itself consists of a bottom mass, mæ,, and a head mass, /w,·. Both masses are also connected by a spring with stiffness, &/, and a corresponding dumping constant, b/. The coordinates in space of all three masses are then given by the vector (xt9 xi9 x^sub b^ if only a vertical movement is considered. A periodical force acts from the bottom on the mass, m^sub b^,, (see Fig.l). This force is generated by an electrodynamic shaker system. The acceleration of the top mass, x^sub t^, the acceleration on the shaker table, x^sub b^ , and the force transducer electrical signal are measured during the calibration procedure. This transducer signal is directly proportional to the material tension/compression and can be described in the model by the difference of the spring coordinates x^sub i^-X^sub b^.
It should be noted that the system can be simplified if the coupling of the top mass has practically no influence on the dynamic process. This would correspond to the special case, k^sub c^ [arrow right] ∞, ^sub b^c [arrow right] 0, and the top mass as well as the head mass of the transducer can be summarized as one mass body. In the calibration process the dynamic sensitivity, which is the ratio between the measured force transducer signal and the acting dynamic force, is measured as follows:
In equation 2 the reduced mass μ=(m^sub t^*m^sub t^)/(m^sub t^+m^sub 1^) was introduced. This equation can be drastically simplified if the top mass coupling is neglected and one applies a Taylor series development of the second order for the frequency ω:
This equation is very convenient for fitting purposes to approximate the measured sensitivities just by the two parameters, p^sub 1^ and p^sub 2^. The measured sensitivity is calculated from the ratio of the transducer signal, U^sub f^, and the acting dynamic force:
The parameter, K^sub corr^, takes into account the vertical acceleration grathent over the mass body. Finite element simulations have shown that the individual mass points of the mass body have slightly different accelerations in the vertical direction . This correction factor can be neglected, if quite small masses are used (only a few centimetres in height). The factor S^sub f^ is the static sensitivity obtained for the limiting case ω=0, whereby p=pi. As one can see from approximation (4), the sensitivity drops down quadratically with increasing the frequency ω.
Besides the amplitude of the sensitivity according to equation (4), also the phase shift between the acceleration x^sub t^ and the force signal U^sub f^ can be derived by the model:
The quite complicated equation (5) contains functions f^sub 1^-f^sub 3^ which are all proportional to l/kc , so that these terms can be neglected for the limiting case of infinite coupling stiffness of the top mass. In addition the arcus-tangent function can be approximated by a Taylor series of the first order for the frequency ?, which leads to a linear phase shift between the acceleration- and force transducer signal.
Last but not least, the parameters, k^sub f^ and b^sub f^ of the force transducer can be determined via a dynamic measurement. So far, the parameter determination was practically accomplished only in a quite simple approach. The model according to Fig. 1 can be simplified in such a way, that it has only one mass, m=m^sub 1^+m^sub 2^, connected with a spring, with spring constant, k, and damping factor b, which means that the whole coupling of the top mass is neglected. The bottom mass, m3i of the force transducer can be seen as a fixed part connected with the shaker table. In such a figure, the equation of motion is just given by
The complex output of the force transducer as a function of the frequency, Xff), is then obtained by the product of a complex transfer function, H(f) and the acting dynamic force:
Thereby, the complex transfer function can be obtained by solving eq. 6 in the frequency domain. If one draws the power of the transfer function, H(f/f^sub 0^)2, as a function of frequency one obtains a typical resonance peak as shown in Fig. 2. Thereby, the peak maximum is at f^sub 0^ and the Q value is approximately the ratio between the resonance frequency and the full width at half maximum, Δf^sub 1/2^.
The parameters, k and/ can be calculated from the resonance frequency, f^sub 0^ and the Q value:
If the Q value is large, one can use the bfm for its determination. The relative deviation between the exact Q value and that, which is derived just from the full width at half maximum, is lower than 0.1 % for the case of Q=20. The exact Q value is obtained by solving the equation: H(f^sub 1/2^)2=1/2*H(f^sub 0^)2.
3. Measurement Setup
The essential prerequisites for a primary sinusoidal force calibration are seen in Fig. 3. First of all one needs an exciter. At PTB we have three electromagnetic shaker systems, a small one for forces up to 100 N and 10 Hz until 2 kHz, a medium one up to 800 N for 10 Hz till 3 kHz and a large shaker up to forces of 10 kN and frequencies of 10 Hz to 2 kHz. The shakers consist of two parts, the vibration exciter itself and a power amplifier. The kind of excitation is determined by the chosen signal created by a function generator. This signal directly modulates the current signal which drives the coil of the shaker armature. The acceleration of the top mass can be measured principally in two different ways, either by a primary method using a laser Doppler vibrometer or by accelerometers.
Usually the vibrometer is used, which consists of a laser head providing a 632.8 nm red laser beam and a certain controller. There are two kinds of laser heads, one with a fixed beam and the other one with a scanning opportunity. The scanning vibrometer is able to scan surfaces in an angle region of ±25° in the x- and y- directions. This offers the possibility to investigate surface vibrations. The modular controller consists of different digital processing units, two velocity decoders, a displacement decoder and a digital quadrature decoder. In summary a frequency range from 0-2.5 MHz with a maximum velocity of 10 m/s and a resolution of 0.02 µ?t/s can be realized. The signal processing inside the decoders is fully digital, the output is provided as an analogue signal. For precise calibrations the digital quadrature encoder is used in connection with certain software which calculates the displacement according to the arcus-tangent procedure. The analogue IQ output signals are bundled together with the transducer output signal and the acceleration signal from the shaker armature in a junction box which is then cabled to a 5 MHz PC sampling card.
4. Tractability and Uncertainty Consideration
The sinusoidal calibration of force transducers is a primary calibration method which means that all measured quantities are traceable to the SI base units and all measuring equipment used is calibrated using certain standards, which are well established procedures. The calibration of the weights used as top masses is done according to the international recommendation OIML R 111-1 . According to this document the top masses can be classified at least as Class Mi, which leads to a maximum error for a 1 kg weight of 50 mg or a relative standard measurement uncertainty of 5-1 0'5. Apart from the mass determination, the acceleration measurement is the most important part of the calibration. In Fig. 4 the traceability chains are shown for 2 different ways of acceleration measurement.
There are, in principle, two ways, the primary method using interferometers/vibrometers and the secondary method based on a certain electrical chain. The vibrometer measurement can be classified according to the involved overlap of certain laser beams in the homodyne or heterodyne interferometers. Both instruments are based on modified Mach-Zehnder interferometers. In the heterodyne interferometer, as seen in Fig. 5, the measuring laser beam is spitted whereby one part is additionally mixed with a high frequency using a Bragg cell, usually 40 MHz, to provide the Doppler encoding. The homodyne interferometers are used for larger displacements which can be determined by counting the interference maxima, which is also known as the fringe counting method. The displacement is, thereby, only a function of the laser wave length and the number of fringes. Fringe counting can be performed with very precise instruments, like the high performance counter Fluke PM6681. According to the fringe counting equation, see Fig. 4, the uncertainty is very small. The relative counting error of the Fluke counter is at 1 kHz, ACounts/Counts -5 10" and the relative wavelength error is in the order of Δλ/λ[approximate]10^sup -6^. The main error is made if the displacement comes in the order of λ/2, because this is the resolution limit of the fringe counting. Keeping in mind a lower limit for the displacement of 400 µ??, one can obtain uncertainties of 0.1% in the range from 10 Hz to 1.5 kHz.
As mentioned above, the heterodyne technique is based on the arcus-tangent calculation of the quadrature signal. If one plots the IQ signals against each other, as seen in Fig. 4, one obtains a circle in the most perfect case. The fully digital quadrature encoding avoids all errors made in former times by analogue filters and mixing devices. By default the whole electrical chain of the vibrometer controller is calibrated by the manufacturer through coupling of very precise known artificial displacement signals directly in the controller. These measurements result in uncertainties which are below 0.1 % [H]. On the other hand, the vibrometer used for the sinusoidal force calibration was calibrated against the national acceleration standard. Thereby, the acceleration values obtained by the measuring program had deviations from the standard set-up of 0.02-0.04 %. With a clear conscience one can obtain an uncertainty at least of 0.1% for the frequency range of 10 Hz- 1.5 kHz with the heterodyne method.
The right-hand side of Fig. 4 shows the conventional method of acceleration measurement using accelerometers in combination with certain conditioning amplifiers. Normally a charge amplifier is used which can be calibrated with a very precise reference capacity and a high accurate voltmeter. The relative standard measurement uncertainty of both devices is a few 10"4 %, according to the calibration certificate which was obtained by a standard calibration procedure at PTB. For the use in a calibration set-up, one has to consider the whole measuring chain consisting of the accelerometer and its conditioning amplifier. The measuring chain can be included by a calibrated sensitivity factor, S^sub qa^, which commonly has an uncertainty around 0.2 %, as illustrated by example in Fig. 4. According to the charge amplifier calibration for the accelerometer measuring chain also the force transducer can be handled, if a piezoelectric transducer is used.
For the case of transducers based on the strain gauge technique, a special calibration device, also called bridge standard, was developed . The bridge standard simulates, in principle, a force transducer and is based on a Wheatstone bridge, whose bridge voltage is, as in the real case, provided by the amplifier, see Fig. 6. In place of a force transducer the bridge standard is connected to the conditioning amplifier. The output to the amplifier is a dynamic bridge detuning, which can be steered through an analogue input signal from an arbitrary function generator with voltage amplitudes. Inside the device the input signal is transformed to an mV voltage. The signal which is seen from the amplifier can be measured in addition on an auxiliary output channel, U^sub ref^.
Finally, the frequency dependent amplification factor can be given by the ratio of Uout /Uref The uncertainty for the amplification factor which can be achieved with this devise is normally around 0.05 %. The total (combined) measurement uncertainty, uCi of the sinusoidal calibration can then be separated into two main parts, a set-up dependent part, us and a part which is obtained by the actual calibration measurement, u^sub m^:
The part, us, is, in principle, a constant given for a certain set up and reflects the smallest achievable measurement uncertainty. This part depends - according to Fig. 7 on three parts, the uncertainty of the acceleration measurement, the uncertainty of the conditioning amplifier calibration and the uncertainty of the mass determination. Note that the uncertainty of the head mass, /W1, of the transducer is not included in this part, because this mass has to be determined with the aid of the actual measurement.
According to the numbers for the certain uncertainties given above, this part results in an uncertainty contribution of 0.1-0.25 %. Fig. 7 shows as an example the uncertainty evaluation for us, which is obtained at 400 Hz.
The part, u^sub m^, includes the uncertainty of the internal mass determination and depends further mainly on the standard deviations of the performed measured sensitivity points. By using the scanning vibrometer for the acceleration measurement on the top mass, one can measure up to 100 points, depending on the actual geometry of the weight. Through this opportunity, special disturbing influences like rocking modes or mechanical adaptation influences of the transducer can be taken into account. It should be noted, that these influences contribute more than other errors made, by e.g., the sine approximation of the measured data or the uncertainties caused by special filter techniques applied in the analysis procedure. Experience has shown that the uncertainty part, um, is on average below 1 kHz between 0.4- 1 % and above 1 kHz around 1 -2%.
4. Measurement of a Strain Gauge Transducer
In Fig. 8 the measurement configuration of the force transducer can be seen. The transducer with a nominal load of 25 kN is mounted on the shaker table and equipped on the top with 10 kg loading mass. During the measurement also four other loading masses where mounted on the transducer to change the dynamic force acting on the transducer. The acceleration was measured in this case with an accelerometer, which is mounted inside the loading mass. A second accelerometer is on the shaker table, as seen in Fig. 8. The transducer was measured in a frequency range from 10 Hz- 1.6 kHz. The mean acceleration was around 100 m/s2. The signals of the accelerometers where amplified with a B&K- and a Kistler charge amplifier respectively. For the amplification of the force transducer signal a lock in amplifier of the type KWS 3073 was used. The signal where recorded with a 24 bit Analog to Digital Converter (ADC) which is a component of a PXI-System used for the data acquisition and analysis. PXI is a modular instrumentation platform where modules with different tasks can be combined on a common bus and trigger system.
4.1. Determination of the Internal Mass of the Force Transducer
For the determination of the internal mass of the force transducer equation 4 can be applied. Practically one determines the ratio between the signal of the amplifier of the force channel and the acceleration measured on the top mass at a few frequencies, mostly at very low frequencies. This ratio is the in a second step extrapolated e.g. via a linear fit to the frequency f=0, which gives then the ratio, (U^sub f^/a)^sub f=o^. As seen from the equation 10 this ratio is a linear function, thereby the slope is the sensitivity, S/=o, at f=0 and the offset is the product between the sensitivity, Sp, and the internal mass, m^sub i^. The task to determine w, consists first; in the measurement of the ratio of (U^sub f^ /a) at different frequencies and with different loading masses. Secondly, from each data set obtained by a certain loading mass the (U^sub f^/a)^sub f=o^ value can be determined. Third, the obtained data pairs of [(Uf/a)f=o , f ] can be fitted with a linear fit to obtain Sp and mi according equation 10.
For the calibrated force transducer a value of the internal mass of 479.7±25g was obtained.
4.2. The Influence of the Total Harmonic Distortion
If one is doing a sinusoidal calibration one has to be aware of the harmonic distortion of the shaker system. Thereby the total harmonic distortion, THD, is defined as:
The, A^sub n^, are the amplitudes of the harmonics occurring in the frequency spectrum of the certain signal.
To illustrate the spectral content of an excitation spectrum obtained with the electromagnetic shaker, two cases are shown in Fig. 9. The first spectrum was obtained by a sinusoidal excitation with 100 Hz. This frequency was for the used mechanical set up far from the resonance, which is around 750 Hz. As one can see, only one pronounced peak at 100 Hz are to seen. The small tables in Fig. 9 give the amplitudes for the certain harmonics. Also from this it can be seen, that the higher harmonics contribute at maximum only with 1-2 %. The spectrum is changing drastically, if the excitation is done around the resonance of the mechanical set up, which is shown in the second spectrum obtained with a 750 Hz excitation. Here, also higher harmonics contribute substantially to the spectrum. From the values shown in the table it is also seen that there is not a linear behaviour in the amplitude content of the harmonics as a function of the number of the harmonics, e.g. the third harmonic has a higher probability then the second one, which might be surprisingly. The harmonic distortion will be also a big influence of the sinusoidal time signal. If the amplitude of the sine wave is obtained e.g. via a fit algorithm using a sinusoidal function, one must taken into account also higher harmonics, see below. On the other hand, this is often problematic for the fit procedures. Because ofthat reason, it is better to make use of a FFT analysis in cases where the harmonic distortion is large.
Because the knowledge of the total harmonic distortion is a very crucial prerequisite for a sinusoidal calibration, normally before a calibration is done, it should be checked. Fig. 10 shows the total harmonic distortion of the three sensors, accelerometer on the shaker table, accelerometer on the top mass and the output of the conditioning amplifier of the force transducer. Beside the frequency, where a resonance occurs for this configuration, it can be seen, that the total harmonic distortion is below 5 %.
Nevertheless, it is interesting, that the enhancement of the total harmonic distortion around 0.8-IkHz, seen in the signal from the accelerometer on the shaker table, is drastically reduced in the other signals. One reason for that might be the fact that, the force transducer damps down the higher harmonics.
According to the modeling of the force transducer, as a damped spring mass system, several modes can be occur in a coupled mass spring system. The model introduced in chapter 1 is very simplified; in reality one has some more coupling elements, including the mechanical parts of the shaker system itself. If one consider, a chain of several coupled damped springs, then very complex oscillating schemes can be happen, e.g., that one spring is oscillating very fast and a another one is just at rest. According this picture, it is not surprisingly, that e.g. higher harmonics will be cancelled. A similar picture like Fig. 10, left, can be obtained also with other loading masses, as shown in Fig. 10, right. Always in the vicinity of resonance frequencies, one can see a drastically enhanced total harmonic distortion. Normally, one avoids during a calibration measurements in the resonance region.
If one averages the total harmonic distortion, excluding the values obtained at resonances, one gets values below 10 %, which is acceptable. It should be noted, that the shown behaviour is very dependent from the kind of shaker system as well as the certain transducer under test, which will be applied.
Normally during a dynamic calibration the time signal, will be recorded. From this signals then all the amplitudes have to be determined. This can be either done by a Fast Fourier Transformation (FFT) or by fitting a sinusoidal function on the data points. If a sinusoidal fit is applied one has to choose a certain fit function. It worked satisfactorily to make use of the following approach:
Thereby, A, is the amplitude, the real part of the signal and f, the phase, the imaginary part of the signal. Because of the fact, that the frequency is well known, it is just the excitation frequency of the shaker, this parameter can be often fixed. It is worthwhile to notice, that the fit can be performed more easily the more the number of free parameter can be reduced. On the other hand, one advantage of using a sinusoidal fit procedure compared with the FFT analysis is the fact, that the frequency can also be determined. Especially if a window for the FFT analysis must be applied, the frequency can be also have a certain uncertainty. Another advantage of the fit procedure is that one can also obtain an error of the fit parameter from the covariance matrix. Fig. 1 1 illustrates 2 examples of fitting the time signal for the frequencies, 100 Hz and 2 kHz, respectively. These examples should demonstrate the limitation of using the sinusoidal fit procedure. In the case of the 100 Hz data a very good convergence of the fit in respect to the measured data points can be obtained. The relative error of the fit parameter is on a level of 10"4. Also if one compares the fit procedure with the FFT analysis, there is only a relative deviation of 0.06 % in the amplitude between both algorithms. On the other hand, if one looks on the right part of Fig. 1 1, one can just see by eye, that there is not a pure sinusoidal signal. For that reason, the function shown in equation 12 cannot good describe the time signal. The signal was measured in the vicinity of a resonance and has a quite large harmonic distortion. In this case, the error of the parameters is on average a factor ten higher and the deviation to the FFT analysis is 1.6 %. From this example one can draw the conclusion, that depending from the content of higher harmonics in the time signal, one hast to choose the right analysis procedure. Only if the total harmonic distortion is quite low (e.g. below 10%), the sinusoidal fit procedure will provide the right values for amplitude and phase.
In Fig. 12 quantitatively the uncertainty are shown of the two parameters, amplitude and phase. Each data point corresponds to a certain top mass and frequency. According to the behaviour of the total harmonic distortion, as it shown in Fig. 10, also the uncertainty of the amplitude and phase is correlated with the resonance frequency region. If a resonance occurs, the total harmonic distortion is higher and the signal cannot longer described only by a simple sinusoidal signal. Beside the resonance region, the sinus fit is a good tool to determine amplitude and phase with the corresponding uncertainty. From Fig. 12 it can be seen, that by averaging the uncertainties for a certain top mass, all in all, quite small values for the uncertainty can be obtained. The small tables inside Fig. 12 show the overall averaged values, which result for an amplitude uncertainty in -0.05 % and for the phase in 0.0024 %.
It should be noted, that this uncertainties can also be neglected, because other sources of errors e.g. from mechanical influences will be more pronounced in such a kind of dynamic measurement.
4.3. Discussion of the Calibration Result
Beside the internal mass of the force transducer, which is a necessary result to determine the dynamic sensitivity, a second outcome of a dynamic calibration is the dynamic sensitivity of the whole measuring chain as shown in the upper row of Fig. 13. It is obvious to notice, that from this data nothing can be concluded concerning the behaviour of the force transducer itself, for that the frequency response of the used amplifier has to be unfolded. As shown in Fig. 13, there is to be seen a quite pronounced frequency dependence of the sensitivity.
Connected with the sensitivity is the phase shift of the force channel in respect to the acceleration measured on the top mass, which can be seen in the upper right panel of Fig. 13. In this figure two data sets are plotted over each other, the dots represent the phase shift of the whole measuring chain and the squares the phase shift only from the used conditioning amplifier. Both data sets show almost the same characteristics, from which one can conclude, that the phase shift is only dominated from the amplifier. This special example shows how important the knowledge of the dynamic behaviour of the used amplifier is.
To get the pure sensitivity of the force transducer, the frequency response of the amplifier was unfolded from the data points, which results in the diagram shown in the lower left panel of Fig. 13.
The curves, obtained by measuring with different top masses, are all in good agreement below a frequency of 1 kHz. This is also confirmed by the combined relative standard measurement uncertainty, given in %, which can be seen in the lower right panel of Fig. 13. The different top masses cause different resonance frequencies, which lie in the range of 700-1400 Hz. Measuring points near the resonance and also beyond naturally, have a bigger uncertainty.
To acquire a figure of merit, all the sensitivity curves obtained with the different top masses can be fitted with a function according to equation 4 and the mean value for the obtained parameters can be calculated. Including the uncertainties of the individual points in the fit procedure moreover, leads to a realistic error also for the fit parameters. The averaged fit results, together with the obtained uncertainty range are shown in Fig. 14. The sensitivity at frequency, f=0, was scaled to 100 % to illustrate the sensitivity drop as a function of frequency in an easy readable way. Thus, it can immediately be seen, that the transducer shows only -96 % of its sensitivity at 1.6 kHz.
The traceable sinusoidal calibration of force transducers was demonstrated. The calibration mainly depends on the acceleration measurement and the calibration of the electrical chain of the used conditioning amplifier. For the most accurate acceleration measurement, laser Doppler vibrometers can be used, which are traced back to the laser wavelength. In the case of piezoelectric force transducers, the charge amplifiers can be very precisely calibrated using a reference capacity and a primary calibrated multimeter. For strain gage transducers, a special calibration bridge standard was developed to dynamically calibrate voltage ratios.
In chapters 4 it was shown, that one can perform the calibration depending on the involved top masses with relative standard measurement uncertainties of -0.4-2.0 %. The main uncertainty contributions are not caused by the setup but rather by the mechanical influences, like adaptations and the rocking modes of the transducer. These disturbing influences can be detected during a calibration measurement by applying a scanning vibrometer or additional sensors like, e.g., triaxial accelerometers. If a certain threshold of transverse acceleration is exceeded, e.g., caused by rocking modes or side resonances of the transducer, the corresponding data will be no further considered. In addition special adapters can be developed to suppress these effects.
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Christian Schlegel, Gabriela Kiekenap, Bernd Glöckner, Rolf Kumme
Bundesallee 100, D-38116 Braunschweig, Germany
Tel.: +49 5315921230
Received: 15 November 201 1 /Accepted: 20 December 2011 /Published: 12 March 2012