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Unsteady axial force measurement by the strain gauge balance
Measurement 152 (2020) 107381
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Unsteady axial force measurement by the strain gauge balance A.R. Gorbushin a,b,⇑, A.A. Bolshakova b a b
Moscow Institute of Physics and Technology (MIPT), 9 Institutskiy per., Dolgoprudny, Moscow Region 141701, Russia Central Aerohydrodynamic Institute (TsAGI), 1 Zhukovsky Str., Zhukovsky, Moscow Reg. 140180, Russia
a r t i c l e
i n f o
Article history: Received 21 July 2019 Received in revised form 13 September 2019 Accepted 10 December 2019 Available online 17 December 2019 Keywords: Dynamic force Strain gauge balance Wind tunnel
a b s t r a c t A new method was developed to measure the unsteady force measurement by means of six-component strain gauge balance which finds use in technological processes, e.g. in wind tunnel experiments. A mathematical model in the form of damping spring was used as a core of the method. It was verified by results of experiments involving applying harmonic and step force. Applicability of the balance matrix defined in static conditions was validated by an experiment with harmonic steady-state oscillations of the balance. The developed methodology provides the measurement of arbitrary non-stationary load in inertial or non-inertial system using a strain gauge balance with relative deviation at the level of 0.3–3% in the whole balance’ operating frequency range, including the natural frequency. A dynamic calibration method was developed for the axial force of a six-component strain gauge balance by applying a harmonic force to the ground part of the balance. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Measuring dynamic loads is an important problem in various fields of science and technology. One of its application is measuring the mass of goods transported by trucks and trains [1]. Another application of this problem is the development of high-frequency sensors for industrial robotic manipulators [2] and measurement of dynamic forces acting on alpine skiers [3]. The most difficult task is to measure the aerodynamic force and the moment during wind tunnel testing. The flow pulsations existent in the wind tunnel affect the aircraft or automotive vehicle models and cause them to oscillate on the balance and the supporting device. The rapid increase in the flow velocity in short-duration and shock wind tunnels is another source of excitation of the model’s oscillations. The resulting oscillations of the model cause non-stationary aerodynamic loads acting on the model. In addition, there are so-called inertial forces and moments associated with the oscillating masses of the model and the balance. These inertial loads account for systematic errors in determining the aerodynamic forces and moments in wind tunnel experiments. Measurement of unsteady aerodynamic forces and moments during wind tunnel testing is important for solving aerodynamic, flight dynamics and strength problems:
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (A.R. Gorbushin), [email protected] (A.A. Bolshakova). https://doi.org/10.1016/j.measurement.2019.107381 0263-2241/Ó 2019 Elsevier Ltd. All rights reserved.
1) determination of the averaged forces and moments obtained in case of a fixed position of the model support system and under quasi-steady flow conditions (velocity, pressure, temperature, etc.) [4]; 2) determination of non-stationary aerodynamic derivatives of aircraft models at their free or forced oscillations in the flow [5]; 3) determination of forces and moments at continuously varying angle of attack or yaw angle [6]; 4) measurement of aerodynamic loads of models with oscillating lifting surfaces [7]; 5) measurement of the non-stationary aerodynamic loads during the separation phase of flight, for example, during opening the parachute in the wind tunnel, in simulating an aircraft maneuver or the trajectory of an object separating from a carrier. Measurement of non-stationary aerodynamic derivatives [5], loads at continuously varying angle of attack [6], forces acting on a model with oscillating lifting surfaces [7] require a frequency range not exceeding 50 Hz, which is below natural frequencies of the balance. To measure the force and moment vectors, a sixcomponent balance is used. Accelerometers [4,8,9], magnetic systems [10], piezo films [4,11–13] resistors (so-called strain gauges) [4,9,14,15], piezo-optical transducers [16] and fiber Bragg grating force sensors [2] are used as sensitive elements of the balance. Sometimes accelerometers are used together with the strain gauge balance to separate inertial forces [4,15]. The relative standard
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uncertainty in measuring the averaged aerodynamic coefficients with an accelerometer-based force balance is ~ 4–5% [8,9], whereas in case of using stress-wave force balance with piezo films it equals ~ 3% [12]. Most widely used in wind tunnels are strain gauge balances because of their high accuracy. One-component dynamometers have a standard measurement uncertainty referred to dynamometer’s half of range 0.01–0.02%, and that of sixcomponent ones is 0.05–0.3%. Strain gauge balances consist of an elastic body, sensitive elements and strain gauges which transform the deformations of the sensitive elements into electrical signals. Due to the elasticity of the sensitive elements, the balance itself is a dynamic system. Elasticity of the balance complicates the problem being solved as dynamic loads are measured by a dynamic system. As a rule, strain gauge balances are calibrated, under static conditions. To measure the dynamic loads, additional dynamic calibration is required. Currently, three methods are used to apply a known non-stationary force in case of dynamic calibration of balances: 1) step load (the Heaviside function) is set by cutting the thread connecting the load and the balance [4,12,15,17]; 2) impulse load (the Dirac d function) is applied by special calibrated hammer [4,9,12,13,17]; 3) harmonic force applied to the metric part of the balance [7,10]. The most common method for determining the average aerodynamic loads for model oscillations in a wind tunnel is the use of different filters [4,18] for filtering inertial forces and moments. Another method for solving the problem of determining quasistatic aerodynamic loads in shock wind tunnels is based on the step-shaped form of aerodynamic load at the start of the wind tunnel. The model and the strain gauge balance are considered as a linear dynamical system – ‘‘black box” concept. The second-order non-homogeneous equation of motion can be solved by means of the Laplace transform. The relation between the applied load and the strain gauge balance signal is represented as a convolution integral [4,9,13,17,19]. The impulse response function is determined by applying a known dynamic load – the Heaviside step function and (or) its derivative – the Dirac d function. The disadvantage of this method is the absence of the physical model of the model-balance system, which restricts the application of this method. It may be used only in shock wind tunnels. In [7], a method is proposed to define a correction factor to compensate for the inertial force during measuring the periodic force by means of a one-component strain gauge balance. The applicability of the method is limited to the measurement of sinusoidal force. The review carried out reveals that existing methods to measure nonstationary aerodynamic forces are applicable only for a limited number of cases – shock wind tunnels, free [5] and forced sinusoidal oscillations [7]. All these methods do not apply adequate physical model of the model-balance system. They provide a relative standard uncertainty in measuring the quasi-stationary forces at a level of 3–5% and dynamic force at a level of 10%. In this paper, we use a physical model in the form of a damping spring [4] for the axial force of the six-component strain gauge balance. Application of an adequate physical model allows solving a wide range of problems related to the high-accuracy measurement of arbitrary dynamic loads with strain gauge balances in a wide range of frequencies with a relative deviation at the level of 0.3– 3%. Masses of the oscillating model and balance, as well as the stiffness and damping coefficients of the balance are used in this model. Section 2 describes the facilities and methods to determine these constants. The mathematical model based on the physical model of the strain gauge balance is developed in Section 3. Section 4 deals with verification of the mathematical model using
experimental data obtained at experimental facilities described in Section 2. In Section 5, method for measuring unsteady force is developed. Discussions and conclusions are drawn in Sections 6 and 7 respectively. 2. Experimental facilities 2.1. Ground vibration testing of the strain gauge balance The scheme and detailed description of ground vibration testing (GVT) of the internal six-component strain gauge balance 6F-540 was given in [20]. Fig. 1 illustrates the GVT rig. Balance 1 was attached vertically to the vibration unit 2 by its ground end. Vibration unit 2 sets harmonic steady-state oscillations within the frequency range 10–1000 Hz with a nominal force of 10 N in vertical direction. The direction of the axial force was parallel to that of the force applied by the vibration unit. The axial force range of the balance was ±100 N and the standard measurement uncertainty (standard deviation) of its calibration equaled 0.32 N. Load 3 was mounted on the live end of the balance to decrease the natural frequency of the whole system to 752 Hz whereas that of the balance exceeded 1 kHz. The total mass of load, balance, and balance adapter was 1.48 kg. Signals of the balance were acquired by the NI PXIe-4330 measuring unit. Piezoelectric accelerometers 4 and 5 (PCB 333B32 Piezotronics, range ±50 g) were attached to the load 3 and the vibrator 2 to measure acceleration of live and ground parts of the balance in the direction of its axial force. The standard measurement uncertainties of accelerometers equaled 5 m/s2. Accelerometer signals were converted to the required level by the amplifier and acquired by the ADC NI PXI-6255 module. The measuring channels were sampled synchronously with the frequency fs1 = 25 kHz.
Fig. 1. Photo of the GVT rig for strain gauge balance.
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Experiment control was performed from the operator’s computer with specially developed software installed on it allowing customizing the initial test parameters, monitoring the readings of measuring channels during the experiment, recording and analyzing the data. It should be noted that this experimental setup did not provide steady-state oscillations at its natural frequency due to the fact that the maximum force generated by the setup was comparable with the weight of the balance and the load mounted on it.
2.4. Strain gauge balance matrix To calculate the loads acting on the balance, a balance matrix was used which correlated the electrical signals from the strain gauge balance and the loads applied to it. For the six-component balance, the following measurement equations, which are a second order polynomial [24], were used:
X i ¼ ai Ni þ
6 X
aij X j þ
j¼1j–i
2.2. Measurement of the masses of the load and the live end of the balance In problems of dynamics it is essential to know the masses of moving bodies. In the case where the model set on the strain gauge balance moves, it is required to determine the masses of the model and the live end of the balance. That of the live end of the balance is the mass of its part located in front of the sensitive element of the balance. The masses of the load and the balance metric part were determined in static conditions by the method described in [21] using the readings of all balance components and pendulous accelerometer [22] at several pitch angles:
AF ¼ Mg sin h;
ð1Þ
where AF – axial force; M – mass of the live end of the balance or the total mass of the load and the live end; g – acceleration of gravity; h – pitch angle measured by accelerometer [22]. The measuring system described in [22] was used in this experiment. The mass of the metric end of the balance described in Section 2.1 equaled 0.12 kg, and that of the load equaled 0.56 kg.
2.3. Experiment on step input to the axial force of the strain gauge balance The experiment was performed in the test section No. 1 of the TsAGI transonic wind tunnel T-128 [23]. The scheme and detailed description of the experimental setup was given in [20]. A sixcomponent internal strain gauge balance 6-EB-128-1 [22] with the axial force range ±2000 N was used. The standard deviation obtained during the calibration was 2 N. The balance was attached through its ground end to the rear sting in test section horizontally (h = 0). The aircraft model was fastened to the balance live end. Two PCB (333B32 Piezotronics) accelerometers were attached to the live and ground ends of the balance to measure accelerations of both ends in longitudinal direction and to control balance acceleration in normal and lateral directions. A negative axial force was applied to the model by the dead weight. The dead weight was connected to the model by a cable through the pulley. The cable was burned by a burner to form a step axial force. The LMS data acquisition system with the sampling frequency fs2 = 1600 Hz was used for recording balance and accelerometers’ readings. The standard measurement uncertainty in measuring the loads by this system was 10 N, and that of accelerations was 5 m/s2. The masses of the balance metric part and the model were also determined using (1) in accordance with the method described in [21]. The pitch angle of the balance and the model varied by means of the pitch sector of test section. The measurements were carried out under static conditions using the measuring system [22]. The mass of the live end of the balance was 8.33 kg, that of the model – 169.54 kg. Acceleration of the balance ground part along its longitudinal, lateral, and normal axes turned out to be negligible when the step load was applied.
6 X 6 X r¼1
airs X r X s
ð2Þ
s¼r
where Xi – load of i-th component of the balance; Ni – output signal of the i-th component of the balance; ai – sensitivity coefficient of the i-th component; aij – coefficient of the i-th row of the matrix of linear mutual influences of components; airs - coefficient of the i-th row of the matrix of nonlinear mutual influences of components. The balance matrix is determined by calibrating the strain gauge balance on a calibration rig. During the calibration, known static loads are applied to the balance, the output signals of the strain gauge balance are measured and the balance matrix coefficients are determined by the least square method. Standard deviation of the strain gauge balance calibration typically is of order 0.05–0.3% of balance’s half of range [24]. The loads applied to the balance components cause an interference of electrical signals of the strain gauge balance. Application of the balance matrix (2) exclude coupling effects of the load components. The resulting loads calculated by the balance matrix for all components become independent. 3. Dynamic model of the balance axial force We shall derive the second-order differential equation corresponding to the mathematical model of the balance axial force. Fig. 2 represents the balance dynamic model, including the model with mass M1, the balance with its live part mass M2, and the spring with the stiffness coefficient kx simulating the sensing element of the axial force component. The figure illustrates two coordinate systems – the balance coordinate system OXY and the normal one OXgYg. The balance coordinate system deviates from the normal one by a pitch angle h. The positive direction of the axial force is opposite to the direction of the OX axis of the balance system, as is customary in aerodynamics. Desired external force FX acts on the model along the balance’s longitudinal axis. The projection of total weight of the model and the balance live end on the OX axis Mg sinh is another force acting on balance’s axial force component. The coordinate of the ground part of the balance with respect to the origin O of the normal (inertial) coordinate system is denoted as x1 in Fig. 2, whereas the coordinate of the live end is denoted as x2. The external force applied to the balance by the sting causes the balance ground and correspondingly the balance coordinate system to move. In this case, the balance coordinate system is noninertial. According to Hooke’s law, the relation between the axial force (elastic force) and the balance (or spring) extension is as follows:
AF ¼ kx ðx2 x1 Þ:
ð3Þ
When the live end of the balance is moving relative to the ground end, the damping force D appears due to internal friction inside the balance. In accordance with Newton’s second law, the absolute motion equation for the model together with the live end of the balance projected on the OX axis of the balance will be as follows:
M€x2 ¼ Mgsinh kx ðx2 x1 Þ þ D þ F x ;
ð4Þ
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Fig. 2. Dynamic model of the balance axial force component.
where € x2 – model’s absolute acceleration along the OX axis. We rewrite Eq. (4) for the model displacement with respect to the balance ground part:
M€x ¼ Mgsinh kx x þ D þ F x M€x1 ;
ð5Þ
where x ¼ x2 x1 . For the final formulation of the equation, it is necessary to determine the type of frictional damping force D. Solutions for free damping oscillations for different types of resistance are well known (e.g., see [25]). Using the installation described in Section 2.3, the function of model’s damped oscillations on the strain gauge balance along their longitudinal axis was obtained in time. The results revealed that damping of the balance oscillation amplitude was an exponential function. In accordance with the well known solution of (5), this decay law corresponds to a damping force proportional to the velocity (e.g., see [25]). The solution for free damped oscillations with a damping force proportional to the velocity is derived in [25] in the following form:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ x02 exp bt=2MÞcosð kx =M ðb=2MÞ2 t ;
ð6Þ
where x02 – initial oscillation amplitude of the balance live end; b – damping coefficient. Solution (6) for the amplitude of damped oscillations takes the following form (t ¼ nT=2):
xa2
¼
x02 expðbTn=4MÞ;
ð7Þ
where t ¼ nT=2; n = 0, 1, 2,. . . The initial value n = 0 (t = 0) corresponds to the first maximum or minimum of free oscillations. We define the damping coefficient b using (7) and results of the experiment described in Section 2.3. In our case, the damping force of the balance is proportional to the displacement velocity of the _ Eq. (5) live end of the balance relative to their ground D ¼ bx. takes the final form:
€x þ b=M x_ þ kx =M x ¼ gsinh þ F x =M €x1 :
ð8Þ
described in Section 2.1. The same data will allow validating the applicability of the balance matrix derived under steady conditions to dynamic force measurements. In the experiment described in Section 2.1 (h = 90°, FX = 0), steady-state oscillations were applied by the vibration exciter to the balance ground part. In this case, Eq. (8) takes the following form:
€x þ b=M x_ þ kx =M x ¼ x01 x2 sinxt
ð9Þ
where x1 ¼ x01 sin xt; x = 2pf – angular frequency; f – oscillation 0 frequency; x01 ¼ €x1 =x2 – oscillation amplitude of the balance ground part. Eq. (9) is the second-order ordinary differential equation. Solution of this equation is well known from the theory of oscillations (e.g., see [25]). In the case of low damping coefficient, the solution for steady-state oscillations is as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x ¼ x01 x2 sinðxt DuÞ= ðx20 x2 Þ þ ðbx=MÞ2 ;
ð10Þ
pffiffiffiffiffiffiffiffiffiffiffiffi where x0 ¼ kx =M – natural oscillation frequency without damping (resonant frequency) of the balance with the load; Du – phase difference of the oscillations of the live end of the balance with the model with respect to the balance ground end:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosDu ¼ ðx20 x2 Þ= ðx20 x2 Þ þ ðbx=MÞ2 :
ð11Þ
Solution (10) may be rewritten with allowance for (3):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 AF ¼ kx €x1 = ðx20 x2 Þ þ ðbx=MÞ2 :
ð12Þ
The force measured by the balance is determined by the acceleration amplitude of the balance ground, the forced oscillation frequency, the mass of the live end of the balance with the load, as well as the damping and stiffness coefficients of the balance. We determine x0 and b from (11) and (12):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x0 ¼ x AF=ðAF M€x01 cosDuÞ;
ð13Þ
b ¼ xM2 €x1 sinu=ðAF M€x1 cosDuÞ:
ð14Þ
0
0
4. Verification of mathematical model and validation of the static balance matrix application to the problem of unsteady force measurement
We use the least squares method to determine x0 from the available experimental data obtained at the setup described in Section 2.1. Expression (13) may be reduced to the following form:
Eq. (8) solving the problem of unsteady force measurement may be verified using experimental results obtained in the experiment
f ¼ f 0 ð1 M€x1 cosDu=AF V Þ; 2
2
0
ð15Þ
A.R. Gorbushin, A.A. Bolshakova / Measurement 152 (2020) 107381
where AFV – axial force measured by the balance; M – total mass of the load and balance metric part; €x1 – amplitude of accelerometer 5 (Fig. 1) readings (oscillation amplitude of the balance ground end); Du – phase difference of readings of accelerometers 4 and 5 (Fig. 1), representing the phase difference of oscillations of the metric and ground parts of the balance. Fig. 3 illustrates typical experimental data in the variables of expression (15). The slope coefficient corresponds to the square of the natural oscillation frequency without damping. The resonant frequency turned out to be equal to f0 = 652.5 Hz. Considering that pffiffiffiffiffiffiffiffiffiffiffiffi x0 ¼ kx =M, we obtain the value of the stiffness coefficient kx = 11.43106N/m. Substitution of the expression for cos Du from (11) in (13) gives the following expression: 0
2 2 0 2 ðM€x1 Þ 1 ðx=x0 Þ2 AF 2V ¼ b2 xAF V =Mx20 :
ð16Þ
We also use the least squares method to determine b using (16) and the available data obtained in experiment described in Section 2.1 as it was done above. The damping coefficient turned out to be equal b = 371.98 Ns/m. To verify the mathematical model, let us compare the amplitude of the balance readings under steady-state oscillations on the vibration rig (Fig. 1) with the calculated data using the mathematical model (12). Fig. 4 compares the axial force amplitude mea-
5
sured by the strain gauge balance AFv and the calculated one AFc by (12) depending on the oscillation frequency f of the vibration exciter. The AFv force measured by the balance was calculated following the readings of all six components and using the calibration matrix (2) derived under static conditions. The absolute value of difference between the two set of data does not exceed 0.3 N in the whole frequency range, except for the natural frequencies. The difference is of order of the balance axial force standard deviation 0.32 N (Section 2.1). It increases up to 3.5 N in the range of natural oscillation frequencies of 652.5 and 752 Hz. This effect can be explained as follows. As mentioned in Section 2.1, the experimental setup did not provide fully steady-state oscillations at its natural frequency because the maximum force generated by the setup was comparable to the total weight of the balance and the load mounted on it. Solution (12) becomes invalid for unsteady oscillations. Thus, applying the mathematical model (8) and the balance calibration matrix (2) obtained under static conditions is valid to measure unsteady loads within the accuracy of the experimental equipment used. The research carried out confirmed that the vibration rig described in Section 2.1 may be used for dynamic calibration of balances. The vibration setup determines the steady-state harmonic force for the axial force component of the strain gauge balance the amplitude and frequency of which can vary. 5. Measurement of the unsteady axial force by the strain gauge balance The dynamic force applied to the balance can be determined using the proposed dynamic model of the axial balance component (8). Let us rewrite (8) to express the unknown force FX and take into account (3):
_ M=kx AF € Mg sin h þ M€x1 : F X ¼ AF b=kx AF
Fig. 3. Square of the oscillation frequency as a function of the combination of oscillation parameters to determine the stiffness coefficient of the balance.
ð17Þ
Non-zero value of the last term on the right-hand side of (17) corresponds to the non-inertial balance coordinate system. For the inertial system this term is equal to zero. According to Eq. (17), two derivatives with respect to time of the AF force measured by the balance should be numerically calculated. The total mass of the model and the live end of the balance M, as well as the damping and stiffness coefficients b and kx of the balance can be determined by various methods. One of the possible methods is described in Section 4. In case of a non-inertial coordinate system, one should use an accelerometer to measure the balance ground acceleration. The method can be verified by applying a known dynamic force, e.g., a step one, to the balance. The experimental setup for this case is described in Section 2.3. For the conditions of this experiment h = 0°; € x1 ¼ 0 (17) reduces to the following:
_ M=kx AF: € F X ¼ AF b=kx AF
ð18Þ
A step force FX = 1.02 kN was applied to the model which vanished after the moment of rupture of the cable. As already mentioned in Section 2.3, the overall mass of the model and the live end of the balance was M = 177.87 kg. Fig. 5 illustrates the typical axial force of the balance as a function of time. The frequency of free oscillations was calculated using the least-squares method f = 66.4 Hz. The damping coefficient b was calculated from the obtained damping amplitude of free oscillations using (7). The stiffness coefficient of the balance kx was obtained following (6). Taking (6) into account, the frequency of free oscillations may be written in the following form:
x2 ¼ x20 ðb=2MÞ2 : Fig. 4. Axial force amplitude as a function of the oscillation frequency.
ð19Þ
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A.R. Gorbushin, A.A. Bolshakova / Measurement 152 (2020) 107381
Fig. 5. Axial force of the balance AF as a function of time.
Taking into account that x0 ¼ following:
pffiffiffiffiffiffiffiffiffiffiffiffi kx =M , (19) reduces to the
kx ¼ 4p2 f M þ b2 =4M: 2
ð20Þ
The coefficients equaled: kx = 3.093107N/m; b = 828.21 N∙s/m. The axial force illustrated in Fig. 5, was twice numerically differentiated. Then the desired force was calculated using (18). Fig. 6 illustrates the calculated applied force as a function of time. The standard deviation in measuring the averaged force was comparable to that of the balance and amounted to 2 N. The deviation of the calculated external force from the applied one for free oscillations does not exceed 3% of the balance’s axial force half of the range. The dynamic standard measurement uncertainty of the measuring system, as described in Section 2.3, was 10 N (0.5% of the half of range of the balance’s axial component). The resulted uncertainty is comparable with the uncertainties of the measuring system. The residual nonstationary force may be related to the second mode of oscillations, which was not considered in the mathematical model. The residuals of the remaining five balance components were less than 0.1% of the components’ half of the range for free oscillations. We may conclude that the developed method was verified with an uncertainty (deviation) of 3% of the half of range of the axial force. 6. Discussion A method is proposed to determine arbitrary unsteady forces using the strain gauge balance. The method consists of four stages.
1) First, one determines the masses of the live end of the axial component of the balance and the load (or aerodynamic model) using the method described in [21]. 2) Then one determines the damping and stiffness coefficients b and kx of the strain gauge balance. There are two options: a) using experimental results of steady-state oscillations of the strain gauge balance at the vibration rig, (13) and (14), as was done in this study; b) damping coefficient b is derived from the damping of the free oscillation amplitude of the axial component of the strain gauge balance (7); stiffness coefficient kx is defined by expression (20) for free damped oscillations. 3) Measurement of the acceleration of the balance ground part is required in case the coordinate system of the balance is non-inertial. Additional accelerometer should be installed on the balance ground end to measure acceleration. 4) Arbitrary unsteady force, including aerodynamic, is determined using expression (17). The derivatives of the axial force are calculated using the balance readings and its matrix. In stage 2) of the developed method, two methods for determining the stiffness and damping coefficients are proposed. When using a vibration rig, these coefficients are determined based on the readings of two accelerometers installed on the ground and the metric parts of the balance. The second method, which includes free damped oscillations, uses only the readings of the balance and its calibration matrix. The uncertainty of modern accelerometers is not less than 1% of their range, and that of balances obtained under static conditions is 0.05–0.3% of the balance’s half of range. The uncertainty in determining the coefficients in case of using a vibration rig is determined by that of the accelerometers. Therefore, one can make a preliminary conclusion that the method for determining the stiffness and damping coefficients of the balances based on the results of free damped oscillations is more accurate than when using the vibration rig. Table 1 represents the comparison of the uncertainty in measuring the average force by the method developed with the results described in the papers given in the Introduction. The relative deviation is considered as an estimate of uncertainty. The developed method reveals noticeably higher accuracy and its uncertainty is comparable with that of the balance calibration at static conditions. The nonstationary force uncertainty is compared in Table 2 with the results described by other authors. We consider the relative deviation of momentary force from the given one as an estimate of uncertainty. The developed method reveals noticeably higher accuracy for the measurement of a non-stationary force as well. Novelty of the study:
Table 1 Comparison of relative deviation of average axial force. Source
Method of uncertainty estimation
Relative deviation
[8]
comparison of shock wind tunnel results with short duration wind tunnel test data application of step force comparison of shock wind tunnel results with short duration wind tunnel test data comparison of shock wind tunnel results with short duration wind tunnel test data application of impulse force application of step force
±4.57%
[12] [14] [15] [17] present paper Fig. 6. Calculated external load FX as a function of time.
±3% ±4.69% ±3.5% ±0.53% ±0.2%
A.R. Gorbushin, A.A. Bolshakova / Measurement 152 (2020) 107381 Table 2 Comparison of relative deviation of the amplitude of axial force.
Acknowledgements
Source
Method of deviation estimation
Relative deviation
[7]
application of harmonic force (comparison of measurement results from two balances) application of step force application of impulse force application of harmonic force (deviation is refer to the balance’s half of range) application of step force (deviation is refer to the balance’s half of range)
±10.3%
[12] [12] present paper present paper
7
±9.8% ±11% ±0.3% ±3%
The authors are grateful to O. Pliuta for the translation of this paper. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References
1) the developed method provides the measurement of nonstationary load using a strain gauge balance in the whole operating frequency range, including the natural one; the upper frequency limit of the balance is ~30 kHz [4]; 2) we solve the dynamic problem for the general case where the strain gauge balance is a non-inertial measuring system; 3) the mass of the live end of the balance is taken into account; masses of live ends differ for different balance components in general case [21]; 4) the method provides high accuracy of dynamic load measurement, since only the readings of the balance itself and its matrix with a standard deviation of 0.05–0.3% of the balance half of range are used in (17); the coefficients in the mathematical model (17) are also determined only by the readings of the balance for free damped oscillations; 5) two methods are developed to determine the stiffness and damping coefficients of the balances; 6) the required arbitrary load can be determined for any arbitrary moment (17); only the results of measurements in the vicinity of the given time point are needed to calculate the first and second derivatives of the loads measured by the balance with respect to time. 7. Conclusion The research carried out allowed developing a method for measuring the dynamic force by means of a strain gauge balance in inertial or non-inertial coordinate systems. In addition, a dynamic calibration method was developed for the axial force of a six-component strain gauge balance using a vibration rig. The method allows balance dynamic calibration in a wide range of frequencies by applying a given harmonic force not to the model, but to the ground part of the balance. The developed approach to determine the unsteady force by the longitudinal component of the balances will allow solving the problem of measuring all three components of the dynamic force and the three components of the dynamic moment acting on an aircraft or automotive vehicle model in a wind tunnel using a six-component strain gauge balance. The approach to solving the problem for the other two force components and three moment components remains the same. The problem is the subject of a separate study, since new experimental studies will be required to verify the mathematical models of other balance components. The developed methodology laid the foundation for solving a wide range of problems related to the high-accuracy measurement of dynamic loads with strain gauge balances. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Unsteady axial force measurement by the strain gauge balance A.R. Gorbushin a,b,⇑, A.A. Bolshakova b a b
Moscow Institute of Physics and Technology (MIPT), 9 Institutskiy per., Dolgoprudny, Moscow Region 141701, Russia Central Aerohydrodynamic Institute (TsAGI), 1 Zhukovsky Str., Zhukovsky, Moscow Reg. 140180, Russia
a r t i c l e
i n f o
Article history: Received 21 July 2019 Received in revised form 13 September 2019 Accepted 10 December 2019 Available online 17 December 2019 Keywords: Dynamic force Strain gauge balance Wind tunnel
a b s t r a c t A new method was developed to measure the unsteady force measurement by means of six-component strain gauge balance which finds use in technological processes, e.g. in wind tunnel experiments. A mathematical model in the form of damping spring was used as a core of the method. It was verified by results of experiments involving applying harmonic and step force. Applicability of the balance matrix defined in static conditions was validated by an experiment with harmonic steady-state oscillations of the balance. The developed methodology provides the measurement of arbitrary non-stationary load in inertial or non-inertial system using a strain gauge balance with relative deviation at the level of 0.3–3% in the whole balance’ operating frequency range, including the natural frequency. A dynamic calibration method was developed for the axial force of a six-component strain gauge balance by applying a harmonic force to the ground part of the balance. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Measuring dynamic loads is an important problem in various fields of science and technology. One of its application is measuring the mass of goods transported by trucks and trains [1]. Another application of this problem is the development of high-frequency sensors for industrial robotic manipulators [2] and measurement of dynamic forces acting on alpine skiers [3]. The most difficult task is to measure the aerodynamic force and the moment during wind tunnel testing. The flow pulsations existent in the wind tunnel affect the aircraft or automotive vehicle models and cause them to oscillate on the balance and the supporting device. The rapid increase in the flow velocity in short-duration and shock wind tunnels is another source of excitation of the model’s oscillations. The resulting oscillations of the model cause non-stationary aerodynamic loads acting on the model. In addition, there are so-called inertial forces and moments associated with the oscillating masses of the model and the balance. These inertial loads account for systematic errors in determining the aerodynamic forces and moments in wind tunnel experiments. Measurement of unsteady aerodynamic forces and moments during wind tunnel testing is important for solving aerodynamic, flight dynamics and strength problems:
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (A.R. Gorbushin), [email protected] (A.A. Bolshakova). https://doi.org/10.1016/j.measurement.2019.107381 0263-2241/Ó 2019 Elsevier Ltd. All rights reserved.
1) determination of the averaged forces and moments obtained in case of a fixed position of the model support system and under quasi-steady flow conditions (velocity, pressure, temperature, etc.) [4]; 2) determination of non-stationary aerodynamic derivatives of aircraft models at their free or forced oscillations in the flow [5]; 3) determination of forces and moments at continuously varying angle of attack or yaw angle [6]; 4) measurement of aerodynamic loads of models with oscillating lifting surfaces [7]; 5) measurement of the non-stationary aerodynamic loads during the separation phase of flight, for example, during opening the parachute in the wind tunnel, in simulating an aircraft maneuver or the trajectory of an object separating from a carrier. Measurement of non-stationary aerodynamic derivatives [5], loads at continuously varying angle of attack [6], forces acting on a model with oscillating lifting surfaces [7] require a frequency range not exceeding 50 Hz, which is below natural frequencies of the balance. To measure the force and moment vectors, a sixcomponent balance is used. Accelerometers [4,8,9], magnetic systems [10], piezo films [4,11–13] resistors (so-called strain gauges) [4,9,14,15], piezo-optical transducers [16] and fiber Bragg grating force sensors [2] are used as sensitive elements of the balance. Sometimes accelerometers are used together with the strain gauge balance to separate inertial forces [4,15]. The relative standard
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uncertainty in measuring the averaged aerodynamic coefficients with an accelerometer-based force balance is ~ 4–5% [8,9], whereas in case of using stress-wave force balance with piezo films it equals ~ 3% [12]. Most widely used in wind tunnels are strain gauge balances because of their high accuracy. One-component dynamometers have a standard measurement uncertainty referred to dynamometer’s half of range 0.01–0.02%, and that of sixcomponent ones is 0.05–0.3%. Strain gauge balances consist of an elastic body, sensitive elements and strain gauges which transform the deformations of the sensitive elements into electrical signals. Due to the elasticity of the sensitive elements, the balance itself is a dynamic system. Elasticity of the balance complicates the problem being solved as dynamic loads are measured by a dynamic system. As a rule, strain gauge balances are calibrated, under static conditions. To measure the dynamic loads, additional dynamic calibration is required. Currently, three methods are used to apply a known non-stationary force in case of dynamic calibration of balances: 1) step load (the Heaviside function) is set by cutting the thread connecting the load and the balance [4,12,15,17]; 2) impulse load (the Dirac d function) is applied by special calibrated hammer [4,9,12,13,17]; 3) harmonic force applied to the metric part of the balance [7,10]. The most common method for determining the average aerodynamic loads for model oscillations in a wind tunnel is the use of different filters [4,18] for filtering inertial forces and moments. Another method for solving the problem of determining quasistatic aerodynamic loads in shock wind tunnels is based on the step-shaped form of aerodynamic load at the start of the wind tunnel. The model and the strain gauge balance are considered as a linear dynamical system – ‘‘black box” concept. The second-order non-homogeneous equation of motion can be solved by means of the Laplace transform. The relation between the applied load and the strain gauge balance signal is represented as a convolution integral [4,9,13,17,19]. The impulse response function is determined by applying a known dynamic load – the Heaviside step function and (or) its derivative – the Dirac d function. The disadvantage of this method is the absence of the physical model of the model-balance system, which restricts the application of this method. It may be used only in shock wind tunnels. In [7], a method is proposed to define a correction factor to compensate for the inertial force during measuring the periodic force by means of a one-component strain gauge balance. The applicability of the method is limited to the measurement of sinusoidal force. The review carried out reveals that existing methods to measure nonstationary aerodynamic forces are applicable only for a limited number of cases – shock wind tunnels, free [5] and forced sinusoidal oscillations [7]. All these methods do not apply adequate physical model of the model-balance system. They provide a relative standard uncertainty in measuring the quasi-stationary forces at a level of 3–5% and dynamic force at a level of 10%. In this paper, we use a physical model in the form of a damping spring [4] for the axial force of the six-component strain gauge balance. Application of an adequate physical model allows solving a wide range of problems related to the high-accuracy measurement of arbitrary dynamic loads with strain gauge balances in a wide range of frequencies with a relative deviation at the level of 0.3– 3%. Masses of the oscillating model and balance, as well as the stiffness and damping coefficients of the balance are used in this model. Section 2 describes the facilities and methods to determine these constants. The mathematical model based on the physical model of the strain gauge balance is developed in Section 3. Section 4 deals with verification of the mathematical model using
experimental data obtained at experimental facilities described in Section 2. In Section 5, method for measuring unsteady force is developed. Discussions and conclusions are drawn in Sections 6 and 7 respectively. 2. Experimental facilities 2.1. Ground vibration testing of the strain gauge balance The scheme and detailed description of ground vibration testing (GVT) of the internal six-component strain gauge balance 6F-540 was given in [20]. Fig. 1 illustrates the GVT rig. Balance 1 was attached vertically to the vibration unit 2 by its ground end. Vibration unit 2 sets harmonic steady-state oscillations within the frequency range 10–1000 Hz with a nominal force of 10 N in vertical direction. The direction of the axial force was parallel to that of the force applied by the vibration unit. The axial force range of the balance was ±100 N and the standard measurement uncertainty (standard deviation) of its calibration equaled 0.32 N. Load 3 was mounted on the live end of the balance to decrease the natural frequency of the whole system to 752 Hz whereas that of the balance exceeded 1 kHz. The total mass of load, balance, and balance adapter was 1.48 kg. Signals of the balance were acquired by the NI PXIe-4330 measuring unit. Piezoelectric accelerometers 4 and 5 (PCB 333B32 Piezotronics, range ±50 g) were attached to the load 3 and the vibrator 2 to measure acceleration of live and ground parts of the balance in the direction of its axial force. The standard measurement uncertainties of accelerometers equaled 5 m/s2. Accelerometer signals were converted to the required level by the amplifier and acquired by the ADC NI PXI-6255 module. The measuring channels were sampled synchronously with the frequency fs1 = 25 kHz.
Fig. 1. Photo of the GVT rig for strain gauge balance.
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Experiment control was performed from the operator’s computer with specially developed software installed on it allowing customizing the initial test parameters, monitoring the readings of measuring channels during the experiment, recording and analyzing the data. It should be noted that this experimental setup did not provide steady-state oscillations at its natural frequency due to the fact that the maximum force generated by the setup was comparable with the weight of the balance and the load mounted on it.
2.4. Strain gauge balance matrix To calculate the loads acting on the balance, a balance matrix was used which correlated the electrical signals from the strain gauge balance and the loads applied to it. For the six-component balance, the following measurement equations, which are a second order polynomial [24], were used:
X i ¼ ai Ni þ
6 X
aij X j þ
j¼1j–i
2.2. Measurement of the masses of the load and the live end of the balance In problems of dynamics it is essential to know the masses of moving bodies. In the case where the model set on the strain gauge balance moves, it is required to determine the masses of the model and the live end of the balance. That of the live end of the balance is the mass of its part located in front of the sensitive element of the balance. The masses of the load and the balance metric part were determined in static conditions by the method described in [21] using the readings of all balance components and pendulous accelerometer [22] at several pitch angles:
AF ¼ Mg sin h;
ð1Þ
where AF – axial force; M – mass of the live end of the balance or the total mass of the load and the live end; g – acceleration of gravity; h – pitch angle measured by accelerometer [22]. The measuring system described in [22] was used in this experiment. The mass of the metric end of the balance described in Section 2.1 equaled 0.12 kg, and that of the load equaled 0.56 kg.
2.3. Experiment on step input to the axial force of the strain gauge balance The experiment was performed in the test section No. 1 of the TsAGI transonic wind tunnel T-128 [23]. The scheme and detailed description of the experimental setup was given in [20]. A sixcomponent internal strain gauge balance 6-EB-128-1 [22] with the axial force range ±2000 N was used. The standard deviation obtained during the calibration was 2 N. The balance was attached through its ground end to the rear sting in test section horizontally (h = 0). The aircraft model was fastened to the balance live end. Two PCB (333B32 Piezotronics) accelerometers were attached to the live and ground ends of the balance to measure accelerations of both ends in longitudinal direction and to control balance acceleration in normal and lateral directions. A negative axial force was applied to the model by the dead weight. The dead weight was connected to the model by a cable through the pulley. The cable was burned by a burner to form a step axial force. The LMS data acquisition system with the sampling frequency fs2 = 1600 Hz was used for recording balance and accelerometers’ readings. The standard measurement uncertainty in measuring the loads by this system was 10 N, and that of accelerations was 5 m/s2. The masses of the balance metric part and the model were also determined using (1) in accordance with the method described in [21]. The pitch angle of the balance and the model varied by means of the pitch sector of test section. The measurements were carried out under static conditions using the measuring system [22]. The mass of the live end of the balance was 8.33 kg, that of the model – 169.54 kg. Acceleration of the balance ground part along its longitudinal, lateral, and normal axes turned out to be negligible when the step load was applied.
6 X 6 X r¼1
airs X r X s
ð2Þ
s¼r
where Xi – load of i-th component of the balance; Ni – output signal of the i-th component of the balance; ai – sensitivity coefficient of the i-th component; aij – coefficient of the i-th row of the matrix of linear mutual influences of components; airs - coefficient of the i-th row of the matrix of nonlinear mutual influences of components. The balance matrix is determined by calibrating the strain gauge balance on a calibration rig. During the calibration, known static loads are applied to the balance, the output signals of the strain gauge balance are measured and the balance matrix coefficients are determined by the least square method. Standard deviation of the strain gauge balance calibration typically is of order 0.05–0.3% of balance’s half of range [24]. The loads applied to the balance components cause an interference of electrical signals of the strain gauge balance. Application of the balance matrix (2) exclude coupling effects of the load components. The resulting loads calculated by the balance matrix for all components become independent. 3. Dynamic model of the balance axial force We shall derive the second-order differential equation corresponding to the mathematical model of the balance axial force. Fig. 2 represents the balance dynamic model, including the model with mass M1, the balance with its live part mass M2, and the spring with the stiffness coefficient kx simulating the sensing element of the axial force component. The figure illustrates two coordinate systems – the balance coordinate system OXY and the normal one OXgYg. The balance coordinate system deviates from the normal one by a pitch angle h. The positive direction of the axial force is opposite to the direction of the OX axis of the balance system, as is customary in aerodynamics. Desired external force FX acts on the model along the balance’s longitudinal axis. The projection of total weight of the model and the balance live end on the OX axis Mg sinh is another force acting on balance’s axial force component. The coordinate of the ground part of the balance with respect to the origin O of the normal (inertial) coordinate system is denoted as x1 in Fig. 2, whereas the coordinate of the live end is denoted as x2. The external force applied to the balance by the sting causes the balance ground and correspondingly the balance coordinate system to move. In this case, the balance coordinate system is noninertial. According to Hooke’s law, the relation between the axial force (elastic force) and the balance (or spring) extension is as follows:
AF ¼ kx ðx2 x1 Þ:
ð3Þ
When the live end of the balance is moving relative to the ground end, the damping force D appears due to internal friction inside the balance. In accordance with Newton’s second law, the absolute motion equation for the model together with the live end of the balance projected on the OX axis of the balance will be as follows:
M€x2 ¼ Mgsinh kx ðx2 x1 Þ þ D þ F x ;
ð4Þ
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Fig. 2. Dynamic model of the balance axial force component.
where € x2 – model’s absolute acceleration along the OX axis. We rewrite Eq. (4) for the model displacement with respect to the balance ground part:
M€x ¼ Mgsinh kx x þ D þ F x M€x1 ;
ð5Þ
where x ¼ x2 x1 . For the final formulation of the equation, it is necessary to determine the type of frictional damping force D. Solutions for free damping oscillations for different types of resistance are well known (e.g., see [25]). Using the installation described in Section 2.3, the function of model’s damped oscillations on the strain gauge balance along their longitudinal axis was obtained in time. The results revealed that damping of the balance oscillation amplitude was an exponential function. In accordance with the well known solution of (5), this decay law corresponds to a damping force proportional to the velocity (e.g., see [25]). The solution for free damped oscillations with a damping force proportional to the velocity is derived in [25] in the following form:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ x02 exp bt=2MÞcosð kx =M ðb=2MÞ2 t ;
ð6Þ
where x02 – initial oscillation amplitude of the balance live end; b – damping coefficient. Solution (6) for the amplitude of damped oscillations takes the following form (t ¼ nT=2):
xa2
¼
x02 expðbTn=4MÞ;
ð7Þ
where t ¼ nT=2; n = 0, 1, 2,. . . The initial value n = 0 (t = 0) corresponds to the first maximum or minimum of free oscillations. We define the damping coefficient b using (7) and results of the experiment described in Section 2.3. In our case, the damping force of the balance is proportional to the displacement velocity of the _ Eq. (5) live end of the balance relative to their ground D ¼ bx. takes the final form:
€x þ b=M x_ þ kx =M x ¼ gsinh þ F x =M €x1 :
ð8Þ
described in Section 2.1. The same data will allow validating the applicability of the balance matrix derived under steady conditions to dynamic force measurements. In the experiment described in Section 2.1 (h = 90°, FX = 0), steady-state oscillations were applied by the vibration exciter to the balance ground part. In this case, Eq. (8) takes the following form:
€x þ b=M x_ þ kx =M x ¼ x01 x2 sinxt
ð9Þ
where x1 ¼ x01 sin xt; x = 2pf – angular frequency; f – oscillation 0 frequency; x01 ¼ €x1 =x2 – oscillation amplitude of the balance ground part. Eq. (9) is the second-order ordinary differential equation. Solution of this equation is well known from the theory of oscillations (e.g., see [25]). In the case of low damping coefficient, the solution for steady-state oscillations is as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x ¼ x01 x2 sinðxt DuÞ= ðx20 x2 Þ þ ðbx=MÞ2 ;
ð10Þ
pffiffiffiffiffiffiffiffiffiffiffiffi where x0 ¼ kx =M – natural oscillation frequency without damping (resonant frequency) of the balance with the load; Du – phase difference of the oscillations of the live end of the balance with the model with respect to the balance ground end:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosDu ¼ ðx20 x2 Þ= ðx20 x2 Þ þ ðbx=MÞ2 :
ð11Þ
Solution (10) may be rewritten with allowance for (3):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 AF ¼ kx €x1 = ðx20 x2 Þ þ ðbx=MÞ2 :
ð12Þ
The force measured by the balance is determined by the acceleration amplitude of the balance ground, the forced oscillation frequency, the mass of the live end of the balance with the load, as well as the damping and stiffness coefficients of the balance. We determine x0 and b from (11) and (12):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x0 ¼ x AF=ðAF M€x01 cosDuÞ;
ð13Þ
b ¼ xM2 €x1 sinu=ðAF M€x1 cosDuÞ:
ð14Þ
0
0
4. Verification of mathematical model and validation of the static balance matrix application to the problem of unsteady force measurement
We use the least squares method to determine x0 from the available experimental data obtained at the setup described in Section 2.1. Expression (13) may be reduced to the following form:
Eq. (8) solving the problem of unsteady force measurement may be verified using experimental results obtained in the experiment
f ¼ f 0 ð1 M€x1 cosDu=AF V Þ; 2
2
0
ð15Þ
A.R. Gorbushin, A.A. Bolshakova / Measurement 152 (2020) 107381
where AFV – axial force measured by the balance; M – total mass of the load and balance metric part; €x1 – amplitude of accelerometer 5 (Fig. 1) readings (oscillation amplitude of the balance ground end); Du – phase difference of readings of accelerometers 4 and 5 (Fig. 1), representing the phase difference of oscillations of the metric and ground parts of the balance. Fig. 3 illustrates typical experimental data in the variables of expression (15). The slope coefficient corresponds to the square of the natural oscillation frequency without damping. The resonant frequency turned out to be equal to f0 = 652.5 Hz. Considering that pffiffiffiffiffiffiffiffiffiffiffiffi x0 ¼ kx =M, we obtain the value of the stiffness coefficient kx = 11.43106N/m. Substitution of the expression for cos Du from (11) in (13) gives the following expression: 0
2 2 0 2 ðM€x1 Þ 1 ðx=x0 Þ2 AF 2V ¼ b2 xAF V =Mx20 :
ð16Þ
We also use the least squares method to determine b using (16) and the available data obtained in experiment described in Section 2.1 as it was done above. The damping coefficient turned out to be equal b = 371.98 Ns/m. To verify the mathematical model, let us compare the amplitude of the balance readings under steady-state oscillations on the vibration rig (Fig. 1) with the calculated data using the mathematical model (12). Fig. 4 compares the axial force amplitude mea-
5
sured by the strain gauge balance AFv and the calculated one AFc by (12) depending on the oscillation frequency f of the vibration exciter. The AFv force measured by the balance was calculated following the readings of all six components and using the calibration matrix (2) derived under static conditions. The absolute value of difference between the two set of data does not exceed 0.3 N in the whole frequency range, except for the natural frequencies. The difference is of order of the balance axial force standard deviation 0.32 N (Section 2.1). It increases up to 3.5 N in the range of natural oscillation frequencies of 652.5 and 752 Hz. This effect can be explained as follows. As mentioned in Section 2.1, the experimental setup did not provide fully steady-state oscillations at its natural frequency because the maximum force generated by the setup was comparable to the total weight of the balance and the load mounted on it. Solution (12) becomes invalid for unsteady oscillations. Thus, applying the mathematical model (8) and the balance calibration matrix (2) obtained under static conditions is valid to measure unsteady loads within the accuracy of the experimental equipment used. The research carried out confirmed that the vibration rig described in Section 2.1 may be used for dynamic calibration of balances. The vibration setup determines the steady-state harmonic force for the axial force component of the strain gauge balance the amplitude and frequency of which can vary. 5. Measurement of the unsteady axial force by the strain gauge balance The dynamic force applied to the balance can be determined using the proposed dynamic model of the axial balance component (8). Let us rewrite (8) to express the unknown force FX and take into account (3):
_ M=kx AF € Mg sin h þ M€x1 : F X ¼ AF b=kx AF
Fig. 3. Square of the oscillation frequency as a function of the combination of oscillation parameters to determine the stiffness coefficient of the balance.
ð17Þ
Non-zero value of the last term on the right-hand side of (17) corresponds to the non-inertial balance coordinate system. For the inertial system this term is equal to zero. According to Eq. (17), two derivatives with respect to time of the AF force measured by the balance should be numerically calculated. The total mass of the model and the live end of the balance M, as well as the damping and stiffness coefficients b and kx of the balance can be determined by various methods. One of the possible methods is described in Section 4. In case of a non-inertial coordinate system, one should use an accelerometer to measure the balance ground acceleration. The method can be verified by applying a known dynamic force, e.g., a step one, to the balance. The experimental setup for this case is described in Section 2.3. For the conditions of this experiment h = 0°; € x1 ¼ 0 (17) reduces to the following:
_ M=kx AF: € F X ¼ AF b=kx AF
ð18Þ
A step force FX = 1.02 kN was applied to the model which vanished after the moment of rupture of the cable. As already mentioned in Section 2.3, the overall mass of the model and the live end of the balance was M = 177.87 kg. Fig. 5 illustrates the typical axial force of the balance as a function of time. The frequency of free oscillations was calculated using the least-squares method f = 66.4 Hz. The damping coefficient b was calculated from the obtained damping amplitude of free oscillations using (7). The stiffness coefficient of the balance kx was obtained following (6). Taking (6) into account, the frequency of free oscillations may be written in the following form:
x2 ¼ x20 ðb=2MÞ2 : Fig. 4. Axial force amplitude as a function of the oscillation frequency.
ð19Þ
6
A.R. Gorbushin, A.A. Bolshakova / Measurement 152 (2020) 107381
Fig. 5. Axial force of the balance AF as a function of time.
Taking into account that x0 ¼ following:
pffiffiffiffiffiffiffiffiffiffiffiffi kx =M , (19) reduces to the
kx ¼ 4p2 f M þ b2 =4M: 2
ð20Þ
The coefficients equaled: kx = 3.093107N/m; b = 828.21 N∙s/m. The axial force illustrated in Fig. 5, was twice numerically differentiated. Then the desired force was calculated using (18). Fig. 6 illustrates the calculated applied force as a function of time. The standard deviation in measuring the averaged force was comparable to that of the balance and amounted to 2 N. The deviation of the calculated external force from the applied one for free oscillations does not exceed 3% of the balance’s axial force half of the range. The dynamic standard measurement uncertainty of the measuring system, as described in Section 2.3, was 10 N (0.5% of the half of range of the balance’s axial component). The resulted uncertainty is comparable with the uncertainties of the measuring system. The residual nonstationary force may be related to the second mode of oscillations, which was not considered in the mathematical model. The residuals of the remaining five balance components were less than 0.1% of the components’ half of the range for free oscillations. We may conclude that the developed method was verified with an uncertainty (deviation) of 3% of the half of range of the axial force. 6. Discussion A method is proposed to determine arbitrary unsteady forces using the strain gauge balance. The method consists of four stages.
1) First, one determines the masses of the live end of the axial component of the balance and the load (or aerodynamic model) using the method described in [21]. 2) Then one determines the damping and stiffness coefficients b and kx of the strain gauge balance. There are two options: a) using experimental results of steady-state oscillations of the strain gauge balance at the vibration rig, (13) and (14), as was done in this study; b) damping coefficient b is derived from the damping of the free oscillation amplitude of the axial component of the strain gauge balance (7); stiffness coefficient kx is defined by expression (20) for free damped oscillations. 3) Measurement of the acceleration of the balance ground part is required in case the coordinate system of the balance is non-inertial. Additional accelerometer should be installed on the balance ground end to measure acceleration. 4) Arbitrary unsteady force, including aerodynamic, is determined using expression (17). The derivatives of the axial force are calculated using the balance readings and its matrix. In stage 2) of the developed method, two methods for determining the stiffness and damping coefficients are proposed. When using a vibration rig, these coefficients are determined based on the readings of two accelerometers installed on the ground and the metric parts of the balance. The second method, which includes free damped oscillations, uses only the readings of the balance and its calibration matrix. The uncertainty of modern accelerometers is not less than 1% of their range, and that of balances obtained under static conditions is 0.05–0.3% of the balance’s half of range. The uncertainty in determining the coefficients in case of using a vibration rig is determined by that of the accelerometers. Therefore, one can make a preliminary conclusion that the method for determining the stiffness and damping coefficients of the balances based on the results of free damped oscillations is more accurate than when using the vibration rig. Table 1 represents the comparison of the uncertainty in measuring the average force by the method developed with the results described in the papers given in the Introduction. The relative deviation is considered as an estimate of uncertainty. The developed method reveals noticeably higher accuracy and its uncertainty is comparable with that of the balance calibration at static conditions. The nonstationary force uncertainty is compared in Table 2 with the results described by other authors. We consider the relative deviation of momentary force from the given one as an estimate of uncertainty. The developed method reveals noticeably higher accuracy for the measurement of a non-stationary force as well. Novelty of the study:
Table 1 Comparison of relative deviation of average axial force. Source
Method of uncertainty estimation
Relative deviation
[8]
comparison of shock wind tunnel results with short duration wind tunnel test data application of step force comparison of shock wind tunnel results with short duration wind tunnel test data comparison of shock wind tunnel results with short duration wind tunnel test data application of impulse force application of step force
±4.57%
[12] [14] [15] [17] present paper Fig. 6. Calculated external load FX as a function of time.
±3% ±4.69% ±3.5% ±0.53% ±0.2%
A.R. Gorbushin, A.A. Bolshakova / Measurement 152 (2020) 107381 Table 2 Comparison of relative deviation of the amplitude of axial force.
Acknowledgements
Source
Method of deviation estimation
Relative deviation
[7]
application of harmonic force (comparison of measurement results from two balances) application of step force application of impulse force application of harmonic force (deviation is refer to the balance’s half of range) application of step force (deviation is refer to the balance’s half of range)
±10.3%
[12] [12] present paper present paper
7
±9.8% ±11% ±0.3% ±3%
The authors are grateful to O. Pliuta for the translation of this paper. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References
1) the developed method provides the measurement of nonstationary load using a strain gauge balance in the whole operating frequency range, including the natural one; the upper frequency limit of the balance is ~30 kHz [4]; 2) we solve the dynamic problem for the general case where the strain gauge balance is a non-inertial measuring system; 3) the mass of the live end of the balance is taken into account; masses of live ends differ for different balance components in general case [21]; 4) the method provides high accuracy of dynamic load measurement, since only the readings of the balance itself and its matrix with a standard deviation of 0.05–0.3% of the balance half of range are used in (17); the coefficients in the mathematical model (17) are also determined only by the readings of the balance for free damped oscillations; 5) two methods are developed to determine the stiffness and damping coefficients of the balances; 6) the required arbitrary load can be determined for any arbitrary moment (17); only the results of measurements in the vicinity of the given time point are needed to calculate the first and second derivatives of the loads measured by the balance with respect to time. 7. Conclusion The research carried out allowed developing a method for measuring the dynamic force by means of a strain gauge balance in inertial or non-inertial coordinate systems. In addition, a dynamic calibration method was developed for the axial force of a six-component strain gauge balance using a vibration rig. The method allows balance dynamic calibration in a wide range of frequencies by applying a given harmonic force not to the model, but to the ground part of the balance. The developed approach to determine the unsteady force by the longitudinal component of the balances will allow solving the problem of measuring all three components of the dynamic force and the three components of the dynamic moment acting on an aircraft or automotive vehicle model in a wind tunnel using a six-component strain gauge balance. The approach to solving the problem for the other two force components and three moment components remains the same. The problem is the subject of a separate study, since new experimental studies will be required to verify the mathematical models of other balance components. The developed methodology laid the foundation for solving a wide range of problems related to the high-accuracy measurement of dynamic loads with strain gauge balances. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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