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Design methodology of a six-component balance for measuring forces and moments in water tunnel tests
Measurement 58 (2014) 544–555
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Design methodology of a six-component balance for measuring forces and moments in water tunnel tests N.M. Nouri ⇑, Karim Mostafapour, Maryam Kamran, Robab Bohadori School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Narmak, Iran
a r t i c l e
i n f o
Article history: Received 8 April 2014 Received in revised form 31 July 2014 Accepted 5 September 2014 Available online 16 September 2014 Keywords: Six-component Force–moment balance Strain gauge Water tunnel Design evaluation
a b s t r a c t This article describes the methodology used in the design and evaluation of a new, sixcomponent balance. This balance is used to measure the forces and moments in model tests conducted in the water tunnel at the Iran University of Science & Technology (IUST). To design the structural parameters of the balance, were used derived equations as well as the finite element method in an iterative process. At every step in this process, the dimensions obtained from the derived equations were determined by considering the nominal strain. The finite element method was used to demonstrate the manner in which the strain was distributed and the reliability of the quantities obtained from the derived equations. To evaluate the designed balance, the strain distribution, linearity, stiffness, and the balance’s ability to produce separate component outputs were investigated. The results that were obtained indicating that the designed structure satisfied all the design criteria. A first-order model was used to calibrate the balance. The evaluation of the sensitivity matrix showed that the error that resulted from the effects of the non-linearity associated with the applied loads was less than 0.05%. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Measurement techniques are needed in water tunnel tests in order to estimate different operating parameters of hydrodynamic devices. Multi-component, strain-gauge balances normally are used to measure the hydrodynamic forces and moments in water tunnels. The force balance is a complex, elastic structure with a number of flexural components, and the forces exerted on the model cause strain in the flexural elements. The strains produced at specific locations on the elastic components are converted to variation of electrical signals by strain gauges that have been wired together as a Wheatstone bridge circuit. The electrical signals are proportional to the forces applied on the model. By considering the relationship between the ⇑ Corresponding author. Tel.: +98 2177240540x2982; fax: +98 2177240488. E-mail address: [email protected] (N.M. Nouri). http://dx.doi.org/10.1016/j.measurement.2014.09.011 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.
applied force and the balance’s output signal and by using the calibration models, the forces and moments exerted on the model in the water tunnel can be measured directly. Based on the design of their measuring elements, these balances can measure from one to six components. The separation of components from one another is achieved through an appropriate design and wiring scheme of elastic elements. Each element should be capable of independently measuring a specific component of force and moment with minimum interactions with the influence of other components. The ability to measure separate components and the linearity of the balance are two important features of its performance [1]. Each of these factors is influenced by structural design, fabrication, and calibration. At the structural design stage, the balance’s ability to measure separate components and the linearity of the balance depend on the selected dimensions, structural form, and materials. These parameters are obtained by considering the design-related requirements and issues.
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Past research in this area focused on the performance characteristics of the elastic components of a force transducer in order to obtain satisfactory performance. Elastic materials may exhibit different hysteresis responses, and the selection of the type of material for the elastic components causes different hysteresis errors in force transducers [2–4]. The sensing element of the sensor should be designed in such a way as to minimize interference errors and to provide the proper distribution of strain at various strain gauge locations. Some supplementary information given in [5] and the finite element studies were used to analyze the strain distribution on similar types of force transducers [6–8]. The proper structural design of a balance requires an accurate knowledge of the design criteria and adherence to these criteria. No explanation is presented concerning the extent to which each of these criteria exerts its influence, because their relative importance depends on the type of balance and the specific objective for which it will be used. In addition, these criteria are not independent, and many interactions exist between them. Therefore, the method used for the structural design can affect the cost and accuracy of the design. Different balances have been designed specifically based on the requirements of water [9–12] and wind [13–15] tunnels. This type of the balance is used in water tunnel where the frequency test is less than 10 Hz. Also, force balances were developed to measure aerodynamic forces and moments on hypersonic models in ground-based test facilities [16,17]. This measurement technique overcome the limitation with short time test and can be used for measuring force and moment on the cavitation test model in a water tunnel. The principles that govern the design of these balances were outlined in [18,19]. However, there has been no mention in existing documents of the process of determining the parameters of the structural design. There is a lack of information regarding the process-performance relationships of transducers due to highly-competitive market. This article describes the methodology that was used to design a new, six-component balance. This balance is used for measuring the forces and moments in testing models in the water tunnel at the Iran University of Science & Technology. The innovative methodology applied for the design and evaluation of a new, six-component balance that it is used for measuring the forces and moments in water tunnel tests is the novel contribution of this article.
2. Design requirements and considerations The proper structural design of a balance requires accurate knowledge of the design criteria and adherence to these criteria. No explanation is presented concerning the extent to which each of these criteria exerts its influence, because their relative importance depends on the type of balance and the specific objective for which it will be used. In addition, these criteria are not independent, and many interactions exist between them. The requirements and considerations for the design of the balance are listed below:
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1. In view of the limitations on the size of the model, the diameter of the balance cannot exceed 20 mm. 2. The ranges of forces and moments (nominal capacities) that the balance can measure were defined as follows:
Drag force : F D ¼ 0 to 60 N Lift force : F L ¼ 50 to 50 N Side force : F Y ¼ 50 to 50 N Pitching moment : M Y ¼ 1 to 1 N m Rolling moment : M X ¼ 1 to 1 N m Yawing moment : M Z ¼ 1 to 1 N m 3. Large signal strains with an appropriate safety factor (acceptable sensitivity of the Wheatstone bridge): to raise the sensitivity of the Wheatstone bridge, four active strain gauges are used in the circuit. Bendingtype strains are produced in the locations where the strain gauges were installed. The maximum output of the Wheatstone bridge was assumed to be about 1.5 mV/V ± 10% [19]. 4. High stiffness: As the stiffness of the flexural elements increases, the interference error decreases. 5. The deflection of the balance with respect to its longitudinal axis should be minimized because it causes the solutions to become non-linear. 6. The uniform distribution of strain where it is measured: since the electrical output of each measuring element is limited by the maximum allowable strain at the location of the strain gauge, this level of strain should exist uniformly throughout the entire measurement network so that the signal is maximized and the performance of the balance is improved. To properly distribute the strain, the maximum difference between the strains produced at the measurement locations is considered to be less than 15% of the maximum strain value [20]. 7. Design for the ease of machining and installation of strain gauges: one of the most important design considerations is the ease of installation of strain gauges and the ease of the machining operation [21]. If the flexural elements are designed in such a way that the installation of the strain gauges and the machining process are difficult to perform, high costs will be imposed on the system. 8. High strength and low hysteresis [19]: loading in excess of the defined design specifications may cause internal stresses in the force-measuring system. By selecting a material with high strength and low hysteresis, there will be less deviation from the linear state. 3. Structural design of the balance The structural design of the balance allows it to measure the applied forces and moments along the coordinates attached to the axes of the model. Flexural elements and strain gauges were used for the design of the balance, and each force or moment component was proportional
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to the strain produced on a specific elastic element. Four measurement sections were designed to measure the six components of force and moment (Fig. 1). The separation of forces and moments was made possible by the designs of the sections of the balance and by the way the strain gauges were positioned. Fig. 1 shows the balance that we designed, and the measurement sections included: The drag-measurement section: Since the balance was designed so that it was positioned along the axes attached to the model, the drag-measurement section was designed in such a way that the strain produced as a result of axial force at the locations of the strain gauges was bending strain type. The section contains four gauges on each of the two sides of the flexure member, as shown in Fig. 1(c). Gauges D1, D2, D3 and D4, were installed on beams at the beginning and end of the section. The four gauges are connected together to form one Wheatstone bridges, as shown. These beams were able to withstand the loads of five other components, but they were relatively flexible in the direction of axial load. Rolling moment section: The rolling moment section is a cross-shaped surface made up of four rectangular beams (two horizontal and two vertical). The section contains four gauges on each of the two sides of the flexure member, as shown in Fig. 1(b). Gauges R 1, R2, R3 and R4, were installed on each of the four beams at the end of the section. The four gauges are connected together to form one Wheatstone bridges, as shown. The application of load resulting from the rolling
moment created compressive strain at two of the strain-gauge locations and tensile strain at the other two locations. This cross-shaped section was very sensitive to moment changes, but it had relatively high stiffness against the other components. For this crossshaped section, the deformation created by the rolling moment was directly converted to pure bending in the beam. Pitching section: This section consisted of three rectangular beams. The lift force and the pitching moment were measured by this section. The strain gauges were installed on the side beam, and they were symmetrical with respect to the central beam. The eight gauges are connected together to form two Wheatstone bridges, as shown Fig. 1(a). The pitching moment was almost completely converted to tension or compression in the side beams. By appropriately wiring and arranging the strain gauges on the section, the lift force and bending moment can be separated and measured independently. The beams were made sufficiently thin so they would have the required sensitivity, and they provided the stiffness that was required and the minimum amount of deviation from the centerline. Yawing section: The yawing section resembled the pitching section with the exception that the yawing section was rotated 90° about the middle axis relative to the pitching section (Fig. 1(d)). This balance is used for measuring the hydrodynamic forces acting on model autonomous underwater vehicles (AUVs) such as, submarines, torpedoes in a water tunnel.
Fig. 1. Six-component balance for the measurement of three force components and three moment components by means of strain gauges.
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The above balance are placed inside the test model. The model shape shown in Fig. 2 is the length to diameter (L/D) ratio of 9.5. The total length and maximum diameter of the model was 0.333 and 0.035 m, respectively. The model consists of three anodized aluminum pieces which are connected to each other in the longitudinal direction. At the aft end of the model, some clearance from the sting was allowed to provide for model deflections. Thus, there is no contact between sting and the model and the endpoint of balance is fixed to sting. 4. Design methodology The measurement sections are the regions in the balance where the strains are measured. These sections were designed in such a way that the applied loads produced a bending type of stress at the locations of the strain gauges. In the given design, the design variables were the locations at which the strain gauges were installed and the dimensions of the flexural elements. These dimensions were determined from the design considerations, and they were based on the nominal capacity and the nominal strain. To design each of the measuring element, the structural parameters were extracted using the equations and the finite element method in an iterative process. In this process, at every step, the optimal dimensions obtained from the derived equations were calculated by considering the nominal strain. The finite element method demonstrated the manner in which the strain was distributed and the reliability of the dimensions obtained from the derived equations. 4.1. Fundamental equations The ultimate goal was to fabricate a balance in which the selected section was sensitive in the direction of the considered component and insensitive in the other directions. This objective can be accomplished through the proper design of the sections and appropriate wiring techniques. Four active strain gauges were used to measure each component of the balance in the Wheatstone bridge circuit. Eq. (1) shows the relationship between the strain that was produced and the output of the Wheatstone bridge [19].
V 1 ¼ kðe1 e2 þ e3 e4 Þ Ve 4
ð1Þ
where V is the output voltage, Ve is the input voltage, k is the gauge factor, and e1,2,3,4 are the amounts of strain obtained from the gauges. The measured strain is equal to:
e¼
r E
ð2Þ
where r is the stress measured by the gauge, and E is the Young’s modulus of the material used to construct the
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balance. The bending stress at the location of the strain gauge location is defined as:
r¼
M I C
ð3Þ
where CI and M are the section modulus and the moment produced at the location of the gauge, respectively. Considering the design of the balance in which the strains generated at the locations of the strain gauges are equal, the ratio of the input signal to output signal for the complete bridge can be simplified as follows:
V ¼ ke Ve
ð4Þ
4.2. Finite element method A finite-element model can be thought of as a system of solid springs. When a load is applied to the structure, all elements deform until all forces balance. In this study the finite-element analysis were carried out in order to analyze the stress and strain distribution on the measurement sections. FEM was used to evaluate the strain values in the area where the strain gauges were bonded. These values were compared with those obtained from the equation. For an analysis of the measurement sections, the following assumptions were made. (1) The elastic properties of the measurement sections were independent of direction. (2) The model was assumed to be perfectly elastic. A perfectly elastic model was obeyed of Hook’s law. (3) There were no body forces on the measurement sections. (4) The load distributions on locations of contact of the measurement sections were uniform. The FE meshes consisted of linear tetrahedral elements, the tetrahedral elements can adjust more easily to the curves and spline surfaces of the model, preserving its proportion form. The solid structure is meshed using NETGEN [22]. It is a powerful 3D tetrahedral mesh generator that can handle complex geometries and a great variety of meshing options. Grid independence was proved by taking a coarse, medium and a fine grid. All further analysis was done using the fine structures grid. In order to obtain the distribution of the strain, first, the measurement sections was simulated by some constraints. In simulation, the sections were isolated from the balance, and their locations of contact are replaced by different constrains. Strain probes were placed at relevant locations where strain gauges need to be mounted. Strain distribution was analyzed based on maximum loads throughout this study.
Fig. 2. Test model and balance assembly.
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5. Design of the measuring elements Selecting the types of materials for the balance required a lot of attention and forethought, because these decisions can have a significant effect on the cost and performance of the balance. The process of selecting the materials included three major categories that had to be considered, i.e. (1) mechanical characteristics, (2) thermal characteristics, and (3) a number of other characteristics that can be generally referred to as ‘fabrication considerations’ [20]. Two of the most important and effective factors in the selection of a material is the amount of stresses it will incur and the accessible space based on the considerations and requirements of the design. Considering the said factors and also the fact that the balance will be placed inside a water tunnel, Ti–Al6–V4 was the material of choice. Based on the design considerations, the maximum output of the Wheatstone bridge for the specified design can be considered to be about 1.5 mV/V Since the exact value of the gauge factor is not known in the design stage, we assumed that it was 2, i.e. k = 2. Using Eq. (1), the amount of nominal strain at the strain gauge location was calculated as 750 microstrain ðmicrostrain ¼ 750 106 m=mÞ. By inserting the amount of strain obtained and the Young’s modulus of the selected material ðE ¼ 110; 000 N=mm2 Þ into Eq. (2), the design stress was calculated at the location of the strain gauge based on the maximum load of 82.5 MPa. Based on this amount of calculated strain and the design considerations, the dimensions of the measuring sections were determined from the relation (3). 5.1. Drag section Fig. 3(a) shows the structure of the drag section. In the figure, h is the thickness of the beam, b is the width of the beam, and L is the distance between the two strain gauges on the beam. This section was designed in such a way that it acts like a beam that is fixed at both ends in response to the applied axial load; the amounts of strain produced at the two locations of the strain gauges due to the axial load are equal. By considering each beam as a beam with two fixed ends and substituting M ¼ F2D l, l ¼ 2L, C ¼ 2h and 3 1 I ¼ 12 bh into Eq. (3), the amount of bending stress at the locations of the strain gauges can be obtained as:
r¼
3F D L 2bh
2
ð5Þ
where r ¼ 82:5 MPa and F D ¼ 60 N. The initial values of dimensions L, h, and b were approximated by Eq. (5). The manner in which the strain of this section could be distributed in the longitudinal direction was investigated by applying the load of F D ¼ 60 N. In order to obtain the distribution of the strain, first, the drag section was simulated by some constrains. In this simulation, the section was isolated from the balance, and their locations of contact are replaced by different constrains. After the convergence of the solution, the uniformity of the strain distribution and the interference effects were examined and, if necessary, the dimensions were corrected again using Eq. (5). This procedure was continued until the dimensions that were identified satisfied the design criteria. The distribution of the strain along the installation line of the strain gauges for the finalized dimensions is shown in Fig. 3(b). The difference between the values of strain obtained from Eq. (5) and from the finite element method at the locations of the strain gauges was 5%. Therefore, the results provided by the FEM were considered to be reasonable considering the sensitivity criterion of ea ¼ 750 ls 10% for nominal strain. 5.2. Rolling moment section Fig. 4(a) shows the structure of the rolling section. In this figure, h, b, D and L are the thickness of the beam, the width of the beam, the diameter of balance, the distance of the installed strain gauge from the middle of the rolling section, along the axial direction, respectively. The governing equations for the analysis of the rolling section are similar to the equations associated with the drag section. The cross-shaped rolling section converts the rolling moment to the bending moment in rectangular beams. The designed section acts similar to a beam with two fixed ends. The amount of stress generated as a result of the rolling moment at the locations of the strain gauges is obtained from the following relationship:
r¼
3FL 2
bh
:
Fig. 3. Structure and strain distribution of the drag-measurement section.
ð6Þ
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Fig. 4. Structure and strain distribution of the rolling section.
The force exerted on each beam in the rolling section, F, is equal to:
F¼
TAðD bÞ : 2J
ð7Þ
In the above relationship T, J, and A are the rolling moment, the second moment of area about the longitudinal axis of balance, and the beam’s cross-sectional area, respectively. 2 2 2 , T = Mx into By substituting A = bh, J ¼ bh ðb þh Þþ3ðDbÞ 3 Eq. (7) and substituting Eq. (7) into Eq. (6), the amount of bending stress at the locations of the strain gauges is obtained as:
r¼
3M x L
2 2 þh2 Þ 2 bh ðbDbÞ 3
þ ðD bÞ
ð8Þ
where r ¼ 82:5 MPa and M x ¼ 1 N m. The procedure for calculating the values of the dimensions (using Eq. (8) and applying the finite element method) is similar to the procedure used for the drag section. Fig. 4(b) shows the distribution of the strain for the rolling section along the installation line of strain gauges for the finalized dimensions. The difference between the values obtained from Eq. (8) and from the finite element method at the location of the strain gauge was 5%. Therefore, the results provided by the FEM were reasonable considering the sensitivity criterion of ea ¼ 750 ls 10% for nominal strain. 5.3. Pitching moment section Fig. 5(a) shows the structure of the pitching section. In this figure, h is the thickness of the beam, b is the width of the beam, d is the distance between the balance’s two side beams, and L is the distance between the two strain gauges used to measure lift force along the axial direction. The design of the pitching section allowed it separate the pitching moment from the lift force. This section consisted of three rectangular beams. To design the pitching section, the dimensions L, h, and b were calculated by applying pure force and not accounting for the pure effects of the pitching moment. In response to pure force FL, each of the beams acts as a beam with two fixed ends. The bending
stress at the locations of the strain gauges was obtained from:
r¼
3FL bh
2
:
ð9Þ
Based on Fig. 5(a) F ¼ F3L and the following values were used to determine the values of the dimensions, i.e. r ¼ 82:5 MPa and F L ¼ 50 N. The procedure for the calculation of the dimensions (using Eq. (9) and applying the finite element method) was similar to the procedure used for the drag section. Taking into account the design considerations for the manner of strain distribution and the application of the combined load (lift force and pitching moment), the distance between the two side beams (d) were determined. The effects of the combined load emerged in the form of compressive stress and pure tension in the two side beams. The stress due to the combined load at the locations of the strain gauges, which was used for the measurement of the pitching moment, is equal to:
r¼
MY þ F L S : dhb
ð10Þ
In the above relationship, S denotes the distance between the point of application of the load and the location of the strain gauge for the measurement of the moment. To calculate d using Eq. (10), the following values were used: M Y ¼ 1 N m, F L ¼ 50 N, r ¼ 82:5 MPa and S ¼ 36 mm. The values of h and b have already been calculated. For the pitching section, the distribution of the strain along the installation line of the strain gauges, which was obtained through the finite element method for the final dimensions by applying the pure and combined loads, is shown in Fig. 5(b) and (c), respectively. The maximum difference between the values of strain obtained from Eqs. (9) and (10) and from the finite element method at the locations of the strain gauges was 1.5%. Therefore, the results provided by the FEM were reasonable, considering the sensitivity criterion of ea ¼ 750 ls 10% for nominal strain. 5.4. Yawing moment section Fig. 6(a) shows the structure of the yawing section. In this figure, h is the thickness of the beam, b is the width of the beam, d is the distance between the balance’s two
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Fig. 5. Structure and strain distribution of the pitching section.
side beams, and L is the distance between the two strain gauges used to measure the side force along the axial direction. The governing equations for the analysis and the procedures for calculating the dimensions of the yawing section are similar to those used in pitching section. To determine the values of the dimensions, the following relationships were used: M z ¼ 1 N m, F s ¼ 50 N, r ¼ 82:5 MPa and S ¼ 150 mm. For the yawing section, the distribution of strain along the installation line of the strain gauges, which was obtained using the finite element method for the final dimensions by applying the pure and combined loads, is shown in Fig. 6(b) and (c), respectively. The maximum difference between the strain values obtained from Eqs. (9) and (10) and from the finite element method at the locations of the strain gauges was 2%. Therefore, the results provided by the FEM were reasonable considering the sensitivity criterion of ea ¼ 750 ls 10% for nominal strain.
6. Strain analysis The electrical output of each measuring section depends on the strain produced at the locations of the strain gauges; therefore, the manner of strain distribution where the strain gauges were installed would be effective
in improving the balance’s performance. Following the calculation of the main dimensions of the balance, strain distribution was analyzed based on maximum loads. The analysis of the strain distribution was performed by FEM, and 3D tetrahedral mesh was applied on the model. For the simulation of the balance, 991,764 mesh elements were generated. Four types of loads were used to evaluate the distribution of the strain, and this distribution for the application of each load is shown in Fig. 7. The first load that was applied was the force of drag. The strain distribution resulting from the drag force was shown along the installation locations of the strain gauges. Fig. 7(a) shows that the difference between the maximum and minimum strains at the locations of the strain gauges, considering a gauge length of 1 mm, was less than 14% of the maximum strain. Also, the difference between maximum values of sensitivity obtained from the derived equations and from the FE analysis (using Eq. (1) at the strain gauges’ installation locations in the drag section) was 4%. The second applied load was the rolling moment about the x-axis. Fig. 7(b) shows the distribution of strain that was produced as a result of the rolling moment. Fig. 7(b) shows that the difference between the maximum and minimum strains at the strain gauges’ installation locations, considering a gauge length of 1 mm, was less than 15% of the maximum strain. Also the difference between the
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Fig. 6. Structure and strain distribution of the yawing section.
Fig. 7. Strain distribution for four types of loadings.
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Fig. 8. Von Mises stress for critical loading.
maximum values of sensitivity obtained from the derived equations and from the FE analysis (using Eq. (1) at the strain gauges’ installation locations in the rolling section) was 4%. The third applied load was a combination of the lift force and the pitching moment. Fig. 7(c) shows the distribution of strain produced by the application of the combined load. Fig. 7(c) shows that the difference between the maximum and minimum strains at the strain gauges’ locations, considering a gauge length of 1 mm, was less than 13% of the maximum strain. And the difference between the maximum values of sensitivity obtained from the derived equations and from the FE analysis (using Eq. (1) at the strain gauges’ installation locations at this section on) was 5%. The fourth applied load was a combination of the side force and the yawing moment. Fig. 7(d) shows the strain analysis along the installation locations of the strain gauges. This figure also illustrates the distribution of the strain produced as a result of the combined load. Fig. 7(d) shows that the difference between the maximum and minimum strains at the strain gauges’ installation locations, considering a gauge length of 1 mm, was less than 10% of the maximum strain. Also, the difference between the maximum values of sensitivity obtained from derived equations and from the FE analysis (using Eq. (1) at the strain gauges’ installation locations at this section) was 5%. By comparing the strains obtained at the locations of the strain gauges using the finite element method and the derived equations for four types of loadings and considering the sensitivity criterion ð1:5 mV=V 10%Þ, it was concluded that the analyses that were performed produced acceptable results. Considering the analysis of strain performed for four types of loadings, the difference between the maximum and minimum strains at strain gauges’ installation locations, considering a gauge length of 1 mm, was less than 15%, which is acceptable with regards to the design criterion for strain distribution. 7. Stress analysis A six-component balance is a complex structure with many different dimensions. In IUST’s water tunnel tests,
combined loads are normally applied on the experimental models. Such combined loads induce more complicated stresses in the balance and therefore require special attention. In these conditions, the stress analysis of the balance is a concern. For the analysis of stress, the real-case combined loads were evaluated by different models, and the critical loading cases were selected. In this loading, six components with nominal values were applied simultaneously on the balance tip. The type of mesh configuration was similar to that in the strain distribution analysis, which could also be valid for stress analysis considering the results obtained, as compared to the values determined from the derived equations. Fig. 8 shows the Von Mises stress for the applied loads. In view of the illustrated distribution, the maximum stress occurred at the side section. The maximum stress value was 583 MPa, which, in comparison with the allowed stress of 1000 MPa, provided an acceptable safety margin. 8. Review of interactions and interference effects To minimize interference effects, it is necessary to install the strain gauges at the proper locations. The positions of the installed strain gauges used for the detection of forces are shown in Fig. 1. For the installation and wiring of the strain gauges’ bridges, the following considerations were envisioned: For measuring each component, four active strain gauges in the form of a Wheatstone bridge were installed next to each other. The installation’s location and arrangement of the strain gauges as a Wheatstone bridge were planned in such a way that the strain gauge had maximum sensitivity in the direction of the considered component and the lowest reaction to other components. In order to evaluate the sensitivity and interference effects, after determining the installation locations of the strain gauges, the ratio of output voltage to input voltage of each channel was calculated using the values of strain obtained from the finite element method and formula (1). The VVOUT ratio of each bridge versus the applied load IN is shown in Fig. 9 for the six types of loadings. VOUT and
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Fig. 9. Voltage ratio versus applied load for six force and moment measuring channels.
VIN are the output and input voltages of each channel, respectively. The range of loading for the balance was from zero to the nominal capacity of each component, which is indicated in the plotted diagrams. The diagrams demonstrate the linear performance of the balance against the applied loads. By choosing suitable installation locations for the strain gauges and wiring them properly as a Wheatstone bridge, it was possible for this balance to measure the forces and moments separately. In response to the load of the corresponding plane, every measuring section provided the desired sensitivity and the required stiffness against the other components. Fig. 9 shows that the maximum sensitivity of the Wheatstone bridge for the four measuring sections, considering the nominal load, was in the range of 1.36–1.63 mV/V, which constitutes an
acceptable sensitivity range for the balance if we consider the design sensitivity criterion of VVOUT ¼ 1:5 mV=V 10% . IN 9. Error analysis result For error analysis, the designed balance must be calibrated. Generally, there are different models for the calibration of the balance, i.e. different orders of equations (first, second, or third order) can be used depending on the types of equations selected for data processing and the desired degree of precision. In this article, after comparing various calibration models and considering different parameters with respect to the requirements of the water tunnel tests under investigation and the balance’s design, the [R] = [C][H] model was used [23]. In this model,
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Table 1 Errors for each component for a sample loading case. Type of applied load
Magnitude of applied load
Magnitude of load calculated from sensitivities ([H] = [C1][R])
Magnitude of load calculated from the first-order coefficients ([H] = [C1][R])
Interference error percentage
Nonlinear error percentage
Lift Pitch Roll Drag Side Yaw
19 0.4940 0.2 26 14 0.8760
19.5128 0.4822 0.1917 26.0286 14.1313 0.8894
19.0082 0.4939 0.0001 26.0044 14.0056 0.8756
2.70 2.39 1.00 0.11 0.94 1.53
0.043 0.020 0 0.017 0.04 0.046
the output voltage (R) is a function of calibration coefficients (C) and applied loads on the balance (H). To obtain the calibration coefficients, the loading of the balance was conducted by taking the real conditions of tests into account. The loads were applied according to Fig. 9. The electrical output of each measuring section depends on the strain produced installation locations of the strain gauges. Therefore, the strain at the locations of the strain gauges (according to Fig. 7) was calculated using the finite element method. After determining the strain at installation locations of the strain gauges, the VVOUT ratio of each IN bridge was calculated by substituting the values of strain obtained from the finite element method into the formula (1). The first-order coefficients were determined from the data of discrete loads applied on the balance and the ratio of the output voltage to the input voltage using the least squares regression method proposed by Ramaswamy et al. [24]. In this method, the calibration coefficients are determined based on the assumption that the sum of the differences of the squares between the measured voltage ratio and the voltage ratio obtained from calibration coefficients is a minimum value. For the six-component balance, calibration is always represented by six different equations; but the number of terms in each equation can vary depending on the order of the equation. In the designed balance, the range of stresses, in comparison with the yield strength Ti–Al6–V4, is relatively low, so the second-order interactions were disregarded against the first-order interactions. The sensitivity matrix is the inverse form of matrix [C], with the following array values: 2
0:6756 6 0:0090 6 6 6 0:0005 C 1 ¼ 6 6 0:0001 6 6 4 0:0374 0:0000
3 0:0048 0:0453 0:0012 0:0234 0:0004 0:7240 0:0011 0:0017 0:0008 0:0176 7 7 7 0:0000 0:6365 0:0003 0:0001 0:0057 7 7: 0:0002 0:0001 0:6235 0:0003 0:0001 7 7 7 0:0003 0:0453 0:0001 0:6930 0:0016 5 0:0305 0:0405 0:0001 0:0007 0:5982
The coefficients that were obtained indicate the sensitivities and interactions without considering the effects of fabrication, assembly, and wiring. The percentage error is defined as:
Percentage error ¼
ðApplied loadÞ ðComputed loadÞ 100: Applied load ð11Þ
Table 1 shows the data obtained from the calibration equations and also the percentage of error for a sample loading case. The interference errors were determined
from the sensitivities. The maximum error that resulted from the interference effects of the applied loads was less than 3%. Non-linear interaction terms omitted in this calibration and the process of fitting curves to obtain the calibration constants introduce non-linear errors in the calculated forces and moments. The results show that the six-component balance developed in this study has good performance, with the errors of non-linearity and repeatability less than 0.05%. 10. Conclusions The innovative methodology was applied to the design and evaluation of a new, six-component balance that it is used for measuring the forces and moments in water tunnel tests. The use of derived equations as well as the finite element method for determination of structural parameters of the designed balance resulted in the reduction of the time required for the design and increased the reliability of the solutions. The six-component balance developed in this study has good performance, with the errors of nonlinearity and repeatability less than 0.05%, and interference effects less than 3%. The results of our evaluations indicated that the dimensions obtained in an iterative process using analytical and numerical methods satisfied all the design criteria. Therefore, the design and evaluation process used for the new, six-component balance could provide a pattern for the design of similar, multi-component balances. References [1] A. Bray, G. Barbato, R. Levi, Theory and Practice of Force Measurement, Academic Press, London, 1990. pp. 42–168. [2] T. Allgeier, W.T. Evans, Mechanical hysteresis in force transducers manufactured from precipitation-hardened stainless steel, J. Mech. Eng. Sci. 209 (1995) 125–132. Part C. [3] P.S. Alexopoulos, C.W. Cho, C.P. Hu, Li Che-Yu, Determination of the anelastic modulus for several metals, Acta Metall. (1981) 549–577. [4] D.R. Chichili, K.T. Ramesh, K.J. Hemker, The high-strain-rate response of alpha titanium: experiments, deformation mechanisms and modeling, Acta Mater. (1998) 46–1025. [5] F.R. Ewald, Multi-component force balances for conventional and cryogenic wind tunnels, Meas. Sci. Technol. 11 (2000) R81–R94. 20th Conference, Albuquerque, NM, June 15–18. [6] M.C. Lindell, Finite element analysis of a NASA national transonic facility wind tunnel balance, in: Proceedings of the 1st International Symposium on Strain Gauge Balances, NASA CP1999-20901, Hampton, USA, 1999, pp. 595–606. [7] R. Karkehabadi, R.D. Rhew, Linear and nonlinear analysis of a windtunnel balance, in: Proceedings of the 4th International Symposium on Strain Gauge Balances, San Diego, USA, 2004. [8] V.I. Lagutin, V.I. Lapygin, S.I. Devyatkin, S.S. Trusov, Finite element analysis of a combined type strain-gauge balance, in: Proceedings of
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the 6th International Symposium on Strain Gauge Balances, Zwolle, the Netherlands, 2008. D.C. Johnson, A Coning Motion Apparatus for Hydrodynamic Model Testing in Non-Planar Cross-Flow, Department of Ocean Engineering, MIT, 1982. C.J. Suárez, G.N. Malcolm, B.R. Kramer, B.C. Smith, B.F. Ayers, Development of a multicomponent force and moment balance for water tunnel applications, NASA Contractor Report 4642, vols. I and II, 1994. C.S. Lee, N.L. Wong, S. Srigrarom, N.T. Nguyen, Development of 3component force–moment balance for low speed water tunnel, World Sci. Publ. Company Mod. Phys. Lett. B 19 (2005) 1575–1578. L.P Erm, Development of a Two-Component Strain-Gauge-Balance Load – Measurement System for the DSTO Water Tunnel, Defence Science and Technology Organisation, Melbourne, Australia, 2006. DSTO-TR-1835. M. Samardzic´, Z. Anastasijevic´, D. Marinkovski, J. Isakovic´, L.J. Tancic´, Measurement of pitch- and roll-damping derivatives using semiconductor five-component strain gauge balance, Proc. IMechE. Part G: J. Aerospace Eng. 226 (2012) 1401–1411. M. Samardzic´, Dj. Vukovic´, D. Marinkovski, Experiments in VTI with semiconductor strain gauges on monoblock wind tunnel balances, in: Proceedings of the 8th International Symposium on Strain-Gauge Balances, RUAG, Lucerne, Switzerland, 2012. pp. 1–8. M. Samardzic´, Ðj. Vukovic´, D. Marinkovski, Apparatus for measurement of pitch and yaw damping derivatives in high
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Reynolds number blowdown wind tunnel, Measurement 46 (2013) 2457–2466. A.L. Smith, D.J. Mee, W.J.T. Daniel, T. Shimoda, Design, modelling and analysis of a six component force balance for hypervelocity wind tunnel testing, Comput. Struct. 79 (11) (2001) 1077–1088. S. Trivedi, V. Menezes, Measurement of yaw, pitch and side-force on a lifting model in a hypersonic shock tunnel, Measurement 45 (2012) 1755–1764. R.D. Rhew, NASA Larc Strain Gage Balance Design Concepts, NASA Langley Research Center, Hampton, Virginia, 1998. Tropea, A.L. Yarin, J.F. Foss, Springer Handbook of Experimental Fluid Mechanics, 2007. Technical Note, Strain Gage Installation Procedures for Transducers, Measurement Group Inc., USA, 1978. Technical Note, Strain Gage Based Transducers, their Design and Construction, Measurement Group Inc., USA, 1988. J. Schäberl, J. Kepler, NETGEN. An Automatic Three-Dimensional Tetrahedral Mesh Generator, University, Linz Austria, 2004. S.Y.F. Leung, Y.Y. Link, Comparison and Analysis of Strain Gauge Balance Calibration Matrix Mathematical Models, Aeronautical and Maritime Research Laboratory, Defence Science and Technology Organisation, Melbourne, Australia, 1999. DSTO-TR-0857. M.A. Ramaswamy, T. Srinivas, V.S. Holla, A simple method for wind tunnel balance calibration including non-linear interaction terms, in: Proceedings of the ICIASF ‘87 RECORD, 1987.
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Design methodology of a six-component balance for measuring forces and moments in water tunnel tests N.M. Nouri ⇑, Karim Mostafapour, Maryam Kamran, Robab Bohadori School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Narmak, Iran
a r t i c l e
i n f o
Article history: Received 8 April 2014 Received in revised form 31 July 2014 Accepted 5 September 2014 Available online 16 September 2014 Keywords: Six-component Force–moment balance Strain gauge Water tunnel Design evaluation
a b s t r a c t This article describes the methodology used in the design and evaluation of a new, sixcomponent balance. This balance is used to measure the forces and moments in model tests conducted in the water tunnel at the Iran University of Science & Technology (IUST). To design the structural parameters of the balance, were used derived equations as well as the finite element method in an iterative process. At every step in this process, the dimensions obtained from the derived equations were determined by considering the nominal strain. The finite element method was used to demonstrate the manner in which the strain was distributed and the reliability of the quantities obtained from the derived equations. To evaluate the designed balance, the strain distribution, linearity, stiffness, and the balance’s ability to produce separate component outputs were investigated. The results that were obtained indicating that the designed structure satisfied all the design criteria. A first-order model was used to calibrate the balance. The evaluation of the sensitivity matrix showed that the error that resulted from the effects of the non-linearity associated with the applied loads was less than 0.05%. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Measurement techniques are needed in water tunnel tests in order to estimate different operating parameters of hydrodynamic devices. Multi-component, strain-gauge balances normally are used to measure the hydrodynamic forces and moments in water tunnels. The force balance is a complex, elastic structure with a number of flexural components, and the forces exerted on the model cause strain in the flexural elements. The strains produced at specific locations on the elastic components are converted to variation of electrical signals by strain gauges that have been wired together as a Wheatstone bridge circuit. The electrical signals are proportional to the forces applied on the model. By considering the relationship between the ⇑ Corresponding author. Tel.: +98 2177240540x2982; fax: +98 2177240488. E-mail address: [email protected] (N.M. Nouri). http://dx.doi.org/10.1016/j.measurement.2014.09.011 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.
applied force and the balance’s output signal and by using the calibration models, the forces and moments exerted on the model in the water tunnel can be measured directly. Based on the design of their measuring elements, these balances can measure from one to six components. The separation of components from one another is achieved through an appropriate design and wiring scheme of elastic elements. Each element should be capable of independently measuring a specific component of force and moment with minimum interactions with the influence of other components. The ability to measure separate components and the linearity of the balance are two important features of its performance [1]. Each of these factors is influenced by structural design, fabrication, and calibration. At the structural design stage, the balance’s ability to measure separate components and the linearity of the balance depend on the selected dimensions, structural form, and materials. These parameters are obtained by considering the design-related requirements and issues.
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Past research in this area focused on the performance characteristics of the elastic components of a force transducer in order to obtain satisfactory performance. Elastic materials may exhibit different hysteresis responses, and the selection of the type of material for the elastic components causes different hysteresis errors in force transducers [2–4]. The sensing element of the sensor should be designed in such a way as to minimize interference errors and to provide the proper distribution of strain at various strain gauge locations. Some supplementary information given in [5] and the finite element studies were used to analyze the strain distribution on similar types of force transducers [6–8]. The proper structural design of a balance requires an accurate knowledge of the design criteria and adherence to these criteria. No explanation is presented concerning the extent to which each of these criteria exerts its influence, because their relative importance depends on the type of balance and the specific objective for which it will be used. In addition, these criteria are not independent, and many interactions exist between them. Therefore, the method used for the structural design can affect the cost and accuracy of the design. Different balances have been designed specifically based on the requirements of water [9–12] and wind [13–15] tunnels. This type of the balance is used in water tunnel where the frequency test is less than 10 Hz. Also, force balances were developed to measure aerodynamic forces and moments on hypersonic models in ground-based test facilities [16,17]. This measurement technique overcome the limitation with short time test and can be used for measuring force and moment on the cavitation test model in a water tunnel. The principles that govern the design of these balances were outlined in [18,19]. However, there has been no mention in existing documents of the process of determining the parameters of the structural design. There is a lack of information regarding the process-performance relationships of transducers due to highly-competitive market. This article describes the methodology that was used to design a new, six-component balance. This balance is used for measuring the forces and moments in testing models in the water tunnel at the Iran University of Science & Technology. The innovative methodology applied for the design and evaluation of a new, six-component balance that it is used for measuring the forces and moments in water tunnel tests is the novel contribution of this article.
2. Design requirements and considerations The proper structural design of a balance requires accurate knowledge of the design criteria and adherence to these criteria. No explanation is presented concerning the extent to which each of these criteria exerts its influence, because their relative importance depends on the type of balance and the specific objective for which it will be used. In addition, these criteria are not independent, and many interactions exist between them. The requirements and considerations for the design of the balance are listed below:
545
1. In view of the limitations on the size of the model, the diameter of the balance cannot exceed 20 mm. 2. The ranges of forces and moments (nominal capacities) that the balance can measure were defined as follows:
Drag force : F D ¼ 0 to 60 N Lift force : F L ¼ 50 to 50 N Side force : F Y ¼ 50 to 50 N Pitching moment : M Y ¼ 1 to 1 N m Rolling moment : M X ¼ 1 to 1 N m Yawing moment : M Z ¼ 1 to 1 N m 3. Large signal strains with an appropriate safety factor (acceptable sensitivity of the Wheatstone bridge): to raise the sensitivity of the Wheatstone bridge, four active strain gauges are used in the circuit. Bendingtype strains are produced in the locations where the strain gauges were installed. The maximum output of the Wheatstone bridge was assumed to be about 1.5 mV/V ± 10% [19]. 4. High stiffness: As the stiffness of the flexural elements increases, the interference error decreases. 5. The deflection of the balance with respect to its longitudinal axis should be minimized because it causes the solutions to become non-linear. 6. The uniform distribution of strain where it is measured: since the electrical output of each measuring element is limited by the maximum allowable strain at the location of the strain gauge, this level of strain should exist uniformly throughout the entire measurement network so that the signal is maximized and the performance of the balance is improved. To properly distribute the strain, the maximum difference between the strains produced at the measurement locations is considered to be less than 15% of the maximum strain value [20]. 7. Design for the ease of machining and installation of strain gauges: one of the most important design considerations is the ease of installation of strain gauges and the ease of the machining operation [21]. If the flexural elements are designed in such a way that the installation of the strain gauges and the machining process are difficult to perform, high costs will be imposed on the system. 8. High strength and low hysteresis [19]: loading in excess of the defined design specifications may cause internal stresses in the force-measuring system. By selecting a material with high strength and low hysteresis, there will be less deviation from the linear state. 3. Structural design of the balance The structural design of the balance allows it to measure the applied forces and moments along the coordinates attached to the axes of the model. Flexural elements and strain gauges were used for the design of the balance, and each force or moment component was proportional
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to the strain produced on a specific elastic element. Four measurement sections were designed to measure the six components of force and moment (Fig. 1). The separation of forces and moments was made possible by the designs of the sections of the balance and by the way the strain gauges were positioned. Fig. 1 shows the balance that we designed, and the measurement sections included: The drag-measurement section: Since the balance was designed so that it was positioned along the axes attached to the model, the drag-measurement section was designed in such a way that the strain produced as a result of axial force at the locations of the strain gauges was bending strain type. The section contains four gauges on each of the two sides of the flexure member, as shown in Fig. 1(c). Gauges D1, D2, D3 and D4, were installed on beams at the beginning and end of the section. The four gauges are connected together to form one Wheatstone bridges, as shown. These beams were able to withstand the loads of five other components, but they were relatively flexible in the direction of axial load. Rolling moment section: The rolling moment section is a cross-shaped surface made up of four rectangular beams (two horizontal and two vertical). The section contains four gauges on each of the two sides of the flexure member, as shown in Fig. 1(b). Gauges R 1, R2, R3 and R4, were installed on each of the four beams at the end of the section. The four gauges are connected together to form one Wheatstone bridges, as shown. The application of load resulting from the rolling
moment created compressive strain at two of the strain-gauge locations and tensile strain at the other two locations. This cross-shaped section was very sensitive to moment changes, but it had relatively high stiffness against the other components. For this crossshaped section, the deformation created by the rolling moment was directly converted to pure bending in the beam. Pitching section: This section consisted of three rectangular beams. The lift force and the pitching moment were measured by this section. The strain gauges were installed on the side beam, and they were symmetrical with respect to the central beam. The eight gauges are connected together to form two Wheatstone bridges, as shown Fig. 1(a). The pitching moment was almost completely converted to tension or compression in the side beams. By appropriately wiring and arranging the strain gauges on the section, the lift force and bending moment can be separated and measured independently. The beams were made sufficiently thin so they would have the required sensitivity, and they provided the stiffness that was required and the minimum amount of deviation from the centerline. Yawing section: The yawing section resembled the pitching section with the exception that the yawing section was rotated 90° about the middle axis relative to the pitching section (Fig. 1(d)). This balance is used for measuring the hydrodynamic forces acting on model autonomous underwater vehicles (AUVs) such as, submarines, torpedoes in a water tunnel.
Fig. 1. Six-component balance for the measurement of three force components and three moment components by means of strain gauges.
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The above balance are placed inside the test model. The model shape shown in Fig. 2 is the length to diameter (L/D) ratio of 9.5. The total length and maximum diameter of the model was 0.333 and 0.035 m, respectively. The model consists of three anodized aluminum pieces which are connected to each other in the longitudinal direction. At the aft end of the model, some clearance from the sting was allowed to provide for model deflections. Thus, there is no contact between sting and the model and the endpoint of balance is fixed to sting. 4. Design methodology The measurement sections are the regions in the balance where the strains are measured. These sections were designed in such a way that the applied loads produced a bending type of stress at the locations of the strain gauges. In the given design, the design variables were the locations at which the strain gauges were installed and the dimensions of the flexural elements. These dimensions were determined from the design considerations, and they were based on the nominal capacity and the nominal strain. To design each of the measuring element, the structural parameters were extracted using the equations and the finite element method in an iterative process. In this process, at every step, the optimal dimensions obtained from the derived equations were calculated by considering the nominal strain. The finite element method demonstrated the manner in which the strain was distributed and the reliability of the dimensions obtained from the derived equations. 4.1. Fundamental equations The ultimate goal was to fabricate a balance in which the selected section was sensitive in the direction of the considered component and insensitive in the other directions. This objective can be accomplished through the proper design of the sections and appropriate wiring techniques. Four active strain gauges were used to measure each component of the balance in the Wheatstone bridge circuit. Eq. (1) shows the relationship between the strain that was produced and the output of the Wheatstone bridge [19].
V 1 ¼ kðe1 e2 þ e3 e4 Þ Ve 4
ð1Þ
where V is the output voltage, Ve is the input voltage, k is the gauge factor, and e1,2,3,4 are the amounts of strain obtained from the gauges. The measured strain is equal to:
e¼
r E
ð2Þ
where r is the stress measured by the gauge, and E is the Young’s modulus of the material used to construct the
547
balance. The bending stress at the location of the strain gauge location is defined as:
r¼
M I C
ð3Þ
where CI and M are the section modulus and the moment produced at the location of the gauge, respectively. Considering the design of the balance in which the strains generated at the locations of the strain gauges are equal, the ratio of the input signal to output signal for the complete bridge can be simplified as follows:
V ¼ ke Ve
ð4Þ
4.2. Finite element method A finite-element model can be thought of as a system of solid springs. When a load is applied to the structure, all elements deform until all forces balance. In this study the finite-element analysis were carried out in order to analyze the stress and strain distribution on the measurement sections. FEM was used to evaluate the strain values in the area where the strain gauges were bonded. These values were compared with those obtained from the equation. For an analysis of the measurement sections, the following assumptions were made. (1) The elastic properties of the measurement sections were independent of direction. (2) The model was assumed to be perfectly elastic. A perfectly elastic model was obeyed of Hook’s law. (3) There were no body forces on the measurement sections. (4) The load distributions on locations of contact of the measurement sections were uniform. The FE meshes consisted of linear tetrahedral elements, the tetrahedral elements can adjust more easily to the curves and spline surfaces of the model, preserving its proportion form. The solid structure is meshed using NETGEN [22]. It is a powerful 3D tetrahedral mesh generator that can handle complex geometries and a great variety of meshing options. Grid independence was proved by taking a coarse, medium and a fine grid. All further analysis was done using the fine structures grid. In order to obtain the distribution of the strain, first, the measurement sections was simulated by some constraints. In simulation, the sections were isolated from the balance, and their locations of contact are replaced by different constrains. Strain probes were placed at relevant locations where strain gauges need to be mounted. Strain distribution was analyzed based on maximum loads throughout this study.
Fig. 2. Test model and balance assembly.
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5. Design of the measuring elements Selecting the types of materials for the balance required a lot of attention and forethought, because these decisions can have a significant effect on the cost and performance of the balance. The process of selecting the materials included three major categories that had to be considered, i.e. (1) mechanical characteristics, (2) thermal characteristics, and (3) a number of other characteristics that can be generally referred to as ‘fabrication considerations’ [20]. Two of the most important and effective factors in the selection of a material is the amount of stresses it will incur and the accessible space based on the considerations and requirements of the design. Considering the said factors and also the fact that the balance will be placed inside a water tunnel, Ti–Al6–V4 was the material of choice. Based on the design considerations, the maximum output of the Wheatstone bridge for the specified design can be considered to be about 1.5 mV/V Since the exact value of the gauge factor is not known in the design stage, we assumed that it was 2, i.e. k = 2. Using Eq. (1), the amount of nominal strain at the strain gauge location was calculated as 750 microstrain ðmicrostrain ¼ 750 106 m=mÞ. By inserting the amount of strain obtained and the Young’s modulus of the selected material ðE ¼ 110; 000 N=mm2 Þ into Eq. (2), the design stress was calculated at the location of the strain gauge based on the maximum load of 82.5 MPa. Based on this amount of calculated strain and the design considerations, the dimensions of the measuring sections were determined from the relation (3). 5.1. Drag section Fig. 3(a) shows the structure of the drag section. In the figure, h is the thickness of the beam, b is the width of the beam, and L is the distance between the two strain gauges on the beam. This section was designed in such a way that it acts like a beam that is fixed at both ends in response to the applied axial load; the amounts of strain produced at the two locations of the strain gauges due to the axial load are equal. By considering each beam as a beam with two fixed ends and substituting M ¼ F2D l, l ¼ 2L, C ¼ 2h and 3 1 I ¼ 12 bh into Eq. (3), the amount of bending stress at the locations of the strain gauges can be obtained as:
r¼
3F D L 2bh
2
ð5Þ
where r ¼ 82:5 MPa and F D ¼ 60 N. The initial values of dimensions L, h, and b were approximated by Eq. (5). The manner in which the strain of this section could be distributed in the longitudinal direction was investigated by applying the load of F D ¼ 60 N. In order to obtain the distribution of the strain, first, the drag section was simulated by some constrains. In this simulation, the section was isolated from the balance, and their locations of contact are replaced by different constrains. After the convergence of the solution, the uniformity of the strain distribution and the interference effects were examined and, if necessary, the dimensions were corrected again using Eq. (5). This procedure was continued until the dimensions that were identified satisfied the design criteria. The distribution of the strain along the installation line of the strain gauges for the finalized dimensions is shown in Fig. 3(b). The difference between the values of strain obtained from Eq. (5) and from the finite element method at the locations of the strain gauges was 5%. Therefore, the results provided by the FEM were considered to be reasonable considering the sensitivity criterion of ea ¼ 750 ls 10% for nominal strain. 5.2. Rolling moment section Fig. 4(a) shows the structure of the rolling section. In this figure, h, b, D and L are the thickness of the beam, the width of the beam, the diameter of balance, the distance of the installed strain gauge from the middle of the rolling section, along the axial direction, respectively. The governing equations for the analysis of the rolling section are similar to the equations associated with the drag section. The cross-shaped rolling section converts the rolling moment to the bending moment in rectangular beams. The designed section acts similar to a beam with two fixed ends. The amount of stress generated as a result of the rolling moment at the locations of the strain gauges is obtained from the following relationship:
r¼
3FL 2
bh
:
Fig. 3. Structure and strain distribution of the drag-measurement section.
ð6Þ
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Fig. 4. Structure and strain distribution of the rolling section.
The force exerted on each beam in the rolling section, F, is equal to:
F¼
TAðD bÞ : 2J
ð7Þ
In the above relationship T, J, and A are the rolling moment, the second moment of area about the longitudinal axis of balance, and the beam’s cross-sectional area, respectively. 2 2 2 , T = Mx into By substituting A = bh, J ¼ bh ðb þh Þþ3ðDbÞ 3 Eq. (7) and substituting Eq. (7) into Eq. (6), the amount of bending stress at the locations of the strain gauges is obtained as:
r¼
3M x L
2 2 þh2 Þ 2 bh ðbDbÞ 3
þ ðD bÞ
ð8Þ
where r ¼ 82:5 MPa and M x ¼ 1 N m. The procedure for calculating the values of the dimensions (using Eq. (8) and applying the finite element method) is similar to the procedure used for the drag section. Fig. 4(b) shows the distribution of the strain for the rolling section along the installation line of strain gauges for the finalized dimensions. The difference between the values obtained from Eq. (8) and from the finite element method at the location of the strain gauge was 5%. Therefore, the results provided by the FEM were reasonable considering the sensitivity criterion of ea ¼ 750 ls 10% for nominal strain. 5.3. Pitching moment section Fig. 5(a) shows the structure of the pitching section. In this figure, h is the thickness of the beam, b is the width of the beam, d is the distance between the balance’s two side beams, and L is the distance between the two strain gauges used to measure lift force along the axial direction. The design of the pitching section allowed it separate the pitching moment from the lift force. This section consisted of three rectangular beams. To design the pitching section, the dimensions L, h, and b were calculated by applying pure force and not accounting for the pure effects of the pitching moment. In response to pure force FL, each of the beams acts as a beam with two fixed ends. The bending
stress at the locations of the strain gauges was obtained from:
r¼
3FL bh
2
:
ð9Þ
Based on Fig. 5(a) F ¼ F3L and the following values were used to determine the values of the dimensions, i.e. r ¼ 82:5 MPa and F L ¼ 50 N. The procedure for the calculation of the dimensions (using Eq. (9) and applying the finite element method) was similar to the procedure used for the drag section. Taking into account the design considerations for the manner of strain distribution and the application of the combined load (lift force and pitching moment), the distance between the two side beams (d) were determined. The effects of the combined load emerged in the form of compressive stress and pure tension in the two side beams. The stress due to the combined load at the locations of the strain gauges, which was used for the measurement of the pitching moment, is equal to:
r¼
MY þ F L S : dhb
ð10Þ
In the above relationship, S denotes the distance between the point of application of the load and the location of the strain gauge for the measurement of the moment. To calculate d using Eq. (10), the following values were used: M Y ¼ 1 N m, F L ¼ 50 N, r ¼ 82:5 MPa and S ¼ 36 mm. The values of h and b have already been calculated. For the pitching section, the distribution of the strain along the installation line of the strain gauges, which was obtained through the finite element method for the final dimensions by applying the pure and combined loads, is shown in Fig. 5(b) and (c), respectively. The maximum difference between the values of strain obtained from Eqs. (9) and (10) and from the finite element method at the locations of the strain gauges was 1.5%. Therefore, the results provided by the FEM were reasonable, considering the sensitivity criterion of ea ¼ 750 ls 10% for nominal strain. 5.4. Yawing moment section Fig. 6(a) shows the structure of the yawing section. In this figure, h is the thickness of the beam, b is the width of the beam, d is the distance between the balance’s two
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Fig. 5. Structure and strain distribution of the pitching section.
side beams, and L is the distance between the two strain gauges used to measure the side force along the axial direction. The governing equations for the analysis and the procedures for calculating the dimensions of the yawing section are similar to those used in pitching section. To determine the values of the dimensions, the following relationships were used: M z ¼ 1 N m, F s ¼ 50 N, r ¼ 82:5 MPa and S ¼ 150 mm. For the yawing section, the distribution of strain along the installation line of the strain gauges, which was obtained using the finite element method for the final dimensions by applying the pure and combined loads, is shown in Fig. 6(b) and (c), respectively. The maximum difference between the strain values obtained from Eqs. (9) and (10) and from the finite element method at the locations of the strain gauges was 2%. Therefore, the results provided by the FEM were reasonable considering the sensitivity criterion of ea ¼ 750 ls 10% for nominal strain.
6. Strain analysis The electrical output of each measuring section depends on the strain produced at the locations of the strain gauges; therefore, the manner of strain distribution where the strain gauges were installed would be effective
in improving the balance’s performance. Following the calculation of the main dimensions of the balance, strain distribution was analyzed based on maximum loads. The analysis of the strain distribution was performed by FEM, and 3D tetrahedral mesh was applied on the model. For the simulation of the balance, 991,764 mesh elements were generated. Four types of loads were used to evaluate the distribution of the strain, and this distribution for the application of each load is shown in Fig. 7. The first load that was applied was the force of drag. The strain distribution resulting from the drag force was shown along the installation locations of the strain gauges. Fig. 7(a) shows that the difference between the maximum and minimum strains at the locations of the strain gauges, considering a gauge length of 1 mm, was less than 14% of the maximum strain. Also, the difference between maximum values of sensitivity obtained from the derived equations and from the FE analysis (using Eq. (1) at the strain gauges’ installation locations in the drag section) was 4%. The second applied load was the rolling moment about the x-axis. Fig. 7(b) shows the distribution of strain that was produced as a result of the rolling moment. Fig. 7(b) shows that the difference between the maximum and minimum strains at the strain gauges’ installation locations, considering a gauge length of 1 mm, was less than 15% of the maximum strain. Also the difference between the
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Fig. 6. Structure and strain distribution of the yawing section.
Fig. 7. Strain distribution for four types of loadings.
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Fig. 8. Von Mises stress for critical loading.
maximum values of sensitivity obtained from the derived equations and from the FE analysis (using Eq. (1) at the strain gauges’ installation locations in the rolling section) was 4%. The third applied load was a combination of the lift force and the pitching moment. Fig. 7(c) shows the distribution of strain produced by the application of the combined load. Fig. 7(c) shows that the difference between the maximum and minimum strains at the strain gauges’ locations, considering a gauge length of 1 mm, was less than 13% of the maximum strain. And the difference between the maximum values of sensitivity obtained from the derived equations and from the FE analysis (using Eq. (1) at the strain gauges’ installation locations at this section on) was 5%. The fourth applied load was a combination of the side force and the yawing moment. Fig. 7(d) shows the strain analysis along the installation locations of the strain gauges. This figure also illustrates the distribution of the strain produced as a result of the combined load. Fig. 7(d) shows that the difference between the maximum and minimum strains at the strain gauges’ installation locations, considering a gauge length of 1 mm, was less than 10% of the maximum strain. Also, the difference between the maximum values of sensitivity obtained from derived equations and from the FE analysis (using Eq. (1) at the strain gauges’ installation locations at this section) was 5%. By comparing the strains obtained at the locations of the strain gauges using the finite element method and the derived equations for four types of loadings and considering the sensitivity criterion ð1:5 mV=V 10%Þ, it was concluded that the analyses that were performed produced acceptable results. Considering the analysis of strain performed for four types of loadings, the difference between the maximum and minimum strains at strain gauges’ installation locations, considering a gauge length of 1 mm, was less than 15%, which is acceptable with regards to the design criterion for strain distribution. 7. Stress analysis A six-component balance is a complex structure with many different dimensions. In IUST’s water tunnel tests,
combined loads are normally applied on the experimental models. Such combined loads induce more complicated stresses in the balance and therefore require special attention. In these conditions, the stress analysis of the balance is a concern. For the analysis of stress, the real-case combined loads were evaluated by different models, and the critical loading cases were selected. In this loading, six components with nominal values were applied simultaneously on the balance tip. The type of mesh configuration was similar to that in the strain distribution analysis, which could also be valid for stress analysis considering the results obtained, as compared to the values determined from the derived equations. Fig. 8 shows the Von Mises stress for the applied loads. In view of the illustrated distribution, the maximum stress occurred at the side section. The maximum stress value was 583 MPa, which, in comparison with the allowed stress of 1000 MPa, provided an acceptable safety margin. 8. Review of interactions and interference effects To minimize interference effects, it is necessary to install the strain gauges at the proper locations. The positions of the installed strain gauges used for the detection of forces are shown in Fig. 1. For the installation and wiring of the strain gauges’ bridges, the following considerations were envisioned: For measuring each component, four active strain gauges in the form of a Wheatstone bridge were installed next to each other. The installation’s location and arrangement of the strain gauges as a Wheatstone bridge were planned in such a way that the strain gauge had maximum sensitivity in the direction of the considered component and the lowest reaction to other components. In order to evaluate the sensitivity and interference effects, after determining the installation locations of the strain gauges, the ratio of output voltage to input voltage of each channel was calculated using the values of strain obtained from the finite element method and formula (1). The VVOUT ratio of each bridge versus the applied load IN is shown in Fig. 9 for the six types of loadings. VOUT and
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Fig. 9. Voltage ratio versus applied load for six force and moment measuring channels.
VIN are the output and input voltages of each channel, respectively. The range of loading for the balance was from zero to the nominal capacity of each component, which is indicated in the plotted diagrams. The diagrams demonstrate the linear performance of the balance against the applied loads. By choosing suitable installation locations for the strain gauges and wiring them properly as a Wheatstone bridge, it was possible for this balance to measure the forces and moments separately. In response to the load of the corresponding plane, every measuring section provided the desired sensitivity and the required stiffness against the other components. Fig. 9 shows that the maximum sensitivity of the Wheatstone bridge for the four measuring sections, considering the nominal load, was in the range of 1.36–1.63 mV/V, which constitutes an
acceptable sensitivity range for the balance if we consider the design sensitivity criterion of VVOUT ¼ 1:5 mV=V 10% . IN 9. Error analysis result For error analysis, the designed balance must be calibrated. Generally, there are different models for the calibration of the balance, i.e. different orders of equations (first, second, or third order) can be used depending on the types of equations selected for data processing and the desired degree of precision. In this article, after comparing various calibration models and considering different parameters with respect to the requirements of the water tunnel tests under investigation and the balance’s design, the [R] = [C][H] model was used [23]. In this model,
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Table 1 Errors for each component for a sample loading case. Type of applied load
Magnitude of applied load
Magnitude of load calculated from sensitivities ([H] = [C1][R])
Magnitude of load calculated from the first-order coefficients ([H] = [C1][R])
Interference error percentage
Nonlinear error percentage
Lift Pitch Roll Drag Side Yaw
19 0.4940 0.2 26 14 0.8760
19.5128 0.4822 0.1917 26.0286 14.1313 0.8894
19.0082 0.4939 0.0001 26.0044 14.0056 0.8756
2.70 2.39 1.00 0.11 0.94 1.53
0.043 0.020 0 0.017 0.04 0.046
the output voltage (R) is a function of calibration coefficients (C) and applied loads on the balance (H). To obtain the calibration coefficients, the loading of the balance was conducted by taking the real conditions of tests into account. The loads were applied according to Fig. 9. The electrical output of each measuring section depends on the strain produced installation locations of the strain gauges. Therefore, the strain at the locations of the strain gauges (according to Fig. 7) was calculated using the finite element method. After determining the strain at installation locations of the strain gauges, the VVOUT ratio of each IN bridge was calculated by substituting the values of strain obtained from the finite element method into the formula (1). The first-order coefficients were determined from the data of discrete loads applied on the balance and the ratio of the output voltage to the input voltage using the least squares regression method proposed by Ramaswamy et al. [24]. In this method, the calibration coefficients are determined based on the assumption that the sum of the differences of the squares between the measured voltage ratio and the voltage ratio obtained from calibration coefficients is a minimum value. For the six-component balance, calibration is always represented by six different equations; but the number of terms in each equation can vary depending on the order of the equation. In the designed balance, the range of stresses, in comparison with the yield strength Ti–Al6–V4, is relatively low, so the second-order interactions were disregarded against the first-order interactions. The sensitivity matrix is the inverse form of matrix [C], with the following array values: 2
0:6756 6 0:0090 6 6 6 0:0005 C 1 ¼ 6 6 0:0001 6 6 4 0:0374 0:0000
3 0:0048 0:0453 0:0012 0:0234 0:0004 0:7240 0:0011 0:0017 0:0008 0:0176 7 7 7 0:0000 0:6365 0:0003 0:0001 0:0057 7 7: 0:0002 0:0001 0:6235 0:0003 0:0001 7 7 7 0:0003 0:0453 0:0001 0:6930 0:0016 5 0:0305 0:0405 0:0001 0:0007 0:5982
The coefficients that were obtained indicate the sensitivities and interactions without considering the effects of fabrication, assembly, and wiring. The percentage error is defined as:
Percentage error ¼
ðApplied loadÞ ðComputed loadÞ 100: Applied load ð11Þ
Table 1 shows the data obtained from the calibration equations and also the percentage of error for a sample loading case. The interference errors were determined
from the sensitivities. The maximum error that resulted from the interference effects of the applied loads was less than 3%. Non-linear interaction terms omitted in this calibration and the process of fitting curves to obtain the calibration constants introduce non-linear errors in the calculated forces and moments. The results show that the six-component balance developed in this study has good performance, with the errors of non-linearity and repeatability less than 0.05%. 10. Conclusions The innovative methodology was applied to the design and evaluation of a new, six-component balance that it is used for measuring the forces and moments in water tunnel tests. The use of derived equations as well as the finite element method for determination of structural parameters of the designed balance resulted in the reduction of the time required for the design and increased the reliability of the solutions. The six-component balance developed in this study has good performance, with the errors of nonlinearity and repeatability less than 0.05%, and interference effects less than 3%. The results of our evaluations indicated that the dimensions obtained in an iterative process using analytical and numerical methods satisfied all the design criteria. Therefore, the design and evaluation process used for the new, six-component balance could provide a pattern for the design of similar, multi-component balances. References [1] A. Bray, G. Barbato, R. Levi, Theory and Practice of Force Measurement, Academic Press, London, 1990. pp. 42–168. [2] T. Allgeier, W.T. Evans, Mechanical hysteresis in force transducers manufactured from precipitation-hardened stainless steel, J. Mech. Eng. Sci. 209 (1995) 125–132. Part C. [3] P.S. Alexopoulos, C.W. Cho, C.P. Hu, Li Che-Yu, Determination of the anelastic modulus for several metals, Acta Metall. (1981) 549–577. [4] D.R. Chichili, K.T. Ramesh, K.J. Hemker, The high-strain-rate response of alpha titanium: experiments, deformation mechanisms and modeling, Acta Mater. (1998) 46–1025. [5] F.R. Ewald, Multi-component force balances for conventional and cryogenic wind tunnels, Meas. Sci. Technol. 11 (2000) R81–R94. 20th Conference, Albuquerque, NM, June 15–18. [6] M.C. Lindell, Finite element analysis of a NASA national transonic facility wind tunnel balance, in: Proceedings of the 1st International Symposium on Strain Gauge Balances, NASA CP1999-20901, Hampton, USA, 1999, pp. 595–606. [7] R. Karkehabadi, R.D. Rhew, Linear and nonlinear analysis of a windtunnel balance, in: Proceedings of the 4th International Symposium on Strain Gauge Balances, San Diego, USA, 2004. [8] V.I. Lagutin, V.I. Lapygin, S.I. Devyatkin, S.S. Trusov, Finite element analysis of a combined type strain-gauge balance, in: Proceedings of
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