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Abstract

The size of the load pad had a significant effect on the output of the column-type force transducer. It was important to eliminate the influence when the transducer was loaded in a force transducer build-up system which worked as the primary force transfer standard. A common model of the output about the column-type force transducer affected by the shape of load pad was analysed. Seven load pads with various dimensions were designed and fixed on a 300-kN force transducer to investigate the influence of the dimension on the indication error of the transducer. The results showed that the load pad of the sphere–plane contact design would cause the smallest indication error but the largest contact stress. It was better to use the load pad of the sphere–sphere contact design with the spherical radius ratio of ball to cup equalled to 0.2. The spherical radius of the ball should be small under the permission of material strength. In addition, it was important to enhance the stiffness of the build-up system to eliminate the effect of the load pad size on the indication error of the transducer.

Keywords

Force transducer column type load pad spherical radius indication error

Introduction

The demand of high-accuracy measurement with a nominal force more than 20 MN is increasing in various industries such as the aerospace, huge bridge construction, nuclear, building, and wind industries.^1–5 Since the late 1970s, several build-up force calibration machines (BUFCM) with capacities over 20 MN have been set up to satisfy the dissemination of the force in the meganewton range, which includes a 30 MN BUFCM at the National Physical Laboratory (NPL) in the United Kingdom,⁶ with its twin machine at ‘Ukrmetrteststandard’ in Ukraine;⁷ a 53 MN BUFCM at National Institute of Standards and Technology (NIST) in the United States;^8,9 and a 60 MN BUFCM at the Fujian Institute of Metrology (FJIM) in China.^10–12

Since unable to be traced directly to the base SI units of mass, length, and time, the BUFCM with such a large capacity should be calibrated by a force transducer build-up system (BUS), which works as the force transfer standard.¹³ As shown in Figure 1, the BUS consists several force transducers with the same capacities and build-up structure, such as the upper load pad, upper platen, and ground platen.⁶ To calibrate the BUFCM, the force transducers should be calibrated separately in a force standard machine (FSM) at first and then assembled to be a BUS. Therefore, the BUS works as the primary standard in the ultra-high force range, and it is significantly important to enhance its accuracy.¹⁴

Figure 1.

Schematic diagram of the force transducer build-up system (BUS): (a) components of the BUS and (b) loaded BUS with deformation.

An indication error exists in the BUS.^14,15 As shown in Figure 1, the upper platen will deform when the BUS is loaded. For being fixed on the platen, the ball of the load pad will rotate on the cup. Therefore, the contact point between the ball and the cup will move away from the axial line of the force transducer. The parasitic components, which include a side force and an additional moment, are induced on the force transducer. In this case, the loading condition of the force transducer is quite different from that when the transducer is measured individually in an FSM.

The indication error, which is considered as a systematic error,^14,15 should be eliminated to improve the accuracy of the BUS. The influence of the parasitic components should be diminished. The magnitude of the parasitic components is determined mainly by the rigidity of the upper platen and the dimension of the load pad, especially the spherical radius of the ball and cup. It is important to enhance the stiffness of the upper platen and also to improve the design of the load pad.

To date, there are few reports about the study of the load pad in the BUS. The load pad of the sphere–plane contact design, where the ball contacts with a flat pad, was often used in the BUS.^1,3,14,16 But for the force transducer with a capacity over 10 MN or up to 20 MN even, the contact stresses of the load pad are much large. The strength of the material should be checked carefully, and the suitable spherical radius is necessary to confine the stresses at the contact point within the limit of the material.⁷ Using a very small radius will result in failure of the material at the contact point or even fracture.^17,18

In 2015, Wagner et al.⁷ designed a 15 MN BUS to calibrate a 30 MN force calibration machine (FCM) in Ukraine. A concave-shaped load pad was designed for the transducer in the BUS to reduce contact stress. It is a new and beneficial idea in designing load pads for transducers in BUSs. More research should be carried out to investigate the performance of the concave-shaped load pads.

In this study, both the load pads of the sphere–plane contact design and the sphere–sphere contact design were studied. A common model of the output about the column-type force transducer affected by the shape of the load pad was analysed. Seven load pads with various dimensions were designed and fixed on the same force transducer to investigate their performances.

Model proposed

The side force and additional moment produced on the force transducer

As shown in Figure 2, an angular misalignment will exist after the deformation of the upper platen, and the contact point between the ball and cup will move sideways. Therefore, as shown in Figure 3, the force on the primary axis of the transducer, $F_{N}$ , is calculated by

F_{N} = F \cos α

(1)

where $F$ is the force applied to the transducer.

Figure 2.

Schematic diagram of the force transducer and load pad.

Figure 3.

Schematic diagram of the load pad: (a) before rotation and (b) after rotation.

The side force, $F_{t}$ , is

F_{t} = F \sin α

(2)

and an additional moment, $M$ , is introduced. The moment can be calculated by

M = e \cdot F_{N}

(3)

where e is the eccentricity of the load (Figure 3)

e = R_{2} \sin θ_{2}

(4)

From the geometrical relationship in Figure 3, we obtain

{\begin{matrix} α + θ_{2} = θ_{1} \\ R_{1} θ_{1} = R_{2} θ_{2} \end{matrix}

(5)

where $θ_{1}$ is the angle between the former and the latter contact points to the centre of the ball; $θ_{2}$ is the angle between the former and the latter contact points to the centre of the cup; $R_{1}$ and $R_{2}$ are the spherical radii of the ball and cup, respectively.

Simplifying equation (5), $θ_{2}$ can be calculated by

θ_{2} = \frac{R_{1} α}{R_{2} - R_{1}}

(6)

Substituting equations (1), (4), and (6) into equation (3), we obtain

M = F \cdot R_{2} \sin [\frac{η α}{1 - η}] \cos α

(7)

where $η$ is the spherical radius ratio of $R_{1}$ to $R_{2}$ .

The output of the force transducer influenced by the side force and additional moment

Figure 4(a) shows that the loads, including the primary load, $F_{N}$ , a side force, $F_{t}$ , and an additional moment, $M$ , are applied to the column-type force transducer. The elastic element of the transducer is simplified as a cylinder as shown in Figure 4(b).

Figure 4.

Mechanical model of the force transducer: (a) loading condition of the transducer and (b) simplified model of transducer under loading.

Four strain gauges are attached to the elastic element, as shown in Figure 5(a). But for the technical limitation of the strain gauges attachment, the angular misalignment will exist, such as $β$ and $γ$ in Figure 5(b).

Figure 5.

Schematic diagram of the strain gauges attached to the elastic element: (a) strain gauge distribution and (b) strain gauges attached with angular misalignment.

With the knowledge of solid mechanics,¹⁹ the strains of the strain gauges can be calculated by

\begin{matrix} ε_{1} = \frac{1}{E} [\frac{F_{N}}{S} + \frac{M - F_{t} h}{I} R \cos (0^{°} + ω)], \\ ε_{2} = - \frac{μ}{E} [\frac{F_{N}}{S} + \frac{M - F_{t} h}{I} R \cos (90^{°} + ω)] \\ ε_{3} = \frac{1}{E} [\frac{F_{N}}{S} + \frac{M - F_{t} h}{I} R \cos (180^{°} + β + ω)], \\ ε_{4} = - \frac{μ}{E} [\frac{F_{N}}{S} + \frac{M - F_{t} h}{I} R \cos (270^{°} + γ + ω)] \end{matrix}

(8)

where S is the section area of the elastic element, $S = π R^{2}$ ; $R$ is the radius of elastic element section; E is Young’s modulus of the materials of the elastic element; $μ$ is Poisson’s ratio of the elastic element; h is the distance from the location where strain gauges attached to the bottom of the elastic element, I is the inertia moment, $I = π R^{4} / 4$ . $ω$ is the angle in the cylindrical coordinate system, as shown in Figure 4(b) and nominates the position of the load acting on the transducer.

From the Wheatstone bridge circuit, as shown in Figure 6, the output $X$ is

X = \frac{K}{4} (ε_{1} - ε_{2} + ε_{3} - ε_{4})

(9)

where K is the sensitivity coefficient.

Figure 6.

Wheatstone bridge circuit.

Substituting equation (8) into equation (9), we obtain

\begin{matrix} X = \frac{F_{N} K}{2 ES} (1 + μ) + \frac{(M - F_{t} h) KR}{2 EI} [\sin (ω + \frac{β}{2}) \sin (\frac{β}{2}) + μ \sin (90 + ω + \frac{γ}{2}) \sin (\frac{γ}{2})] \\ = \frac{FK}{2 π R^{2} E} {(1 + μ) \cos α + 4 [\frac{R_{2}}{R} \sin (\frac{η α}{1 - η}) \cos α - \frac{h}{R} \sin α] [\sin (ω + \frac{β}{2}) \sin (\frac{β}{2}) + μ \sin (90 + ω + \frac{γ}{2}) \sin (\frac{γ}{2})]} \end{matrix}

(10)

Define a geometrical factor $δ$

δ = \sin (ω + \frac{β}{2}) \sin (\frac{β}{2}) + μ \sin (90 + ω + \frac{γ}{2}) \sin (\frac{γ}{2})

(11)

If the transducer is measured in several uniformly distributed positions over 360°, such as 0°, 120°, and 240°, the outputs, $X_{0}, X_{120}, and X_{240}$ , can be calculated by equation (10) with $ω = 0^{°}, 120^{°}, and 240^{°}$ , respectively. The average output, $\bar{X}$ , over the three positions is calculated by

\bar{X} = \frac{1}{3} (X_{0} + X_{120} + X_{240})

(12)

In formula (12), the geometrical factor $δ \equiv 0$ , and

\bar{X} = \frac{F_{N} K}{2 ES} (1 + μ) = \frac{FK}{2 π R^{2} E} (1 + μ) \cos α

(13)

From equation (13), it is verified that the average output is not sensitive to the additional moment when the column-type force transducer is calibrated in several uniformly distributed positions.

If the force, F, is applied to the axis of the force transducer, in which case $M = 0$ and $F_{t} = 0$ , the output of the circuit is

X' = \frac{FK}{2 π R^{2} E} (1 + μ)

(14)

The relative indication error of the force transducer, $E$ , can be calculated by

E = \frac{X - X'}{X'}

(15)

Substituting equations (10) and (14) into equation (15), we obtain

E = (\cos α - 1) + \frac{4 [\frac{R_{2}}{R} \sin (\frac{η α}{1 - η}) \cos α - \frac{h}{R} \sin α] δ}{1 + μ}

(16)

In this paper, the effects of the spherical radiuses of the ball and cup on the indication error are investigated.

The contact stress

According to the contact mechanics,¹⁹ the contact stress of the load pad is calculated by the following equation

σ_{\max} = \sqrt[3]{\frac{3 F}{2 π^{3}} {(\frac{R_{1} - R_{2}}{R_{1} R_{2}})}^{2} {(\frac{E}{1 - μ^{2}})}^{2}}

(17)

where $σ_{\max}$ is the maximum stress on the contact area.

The radius of the contact area, $a$ , is calculated by

a = \sqrt[3]{\frac{3}{2} F \cdot \frac{R_{1} R_{2}}{R_{1} - R_{2}} \cdot \frac{1 - μ^{2}}{E}}

(18)

Experiment

Figure 7 shows the experimental setup, which consists a column-type force transducer with a capacity of 300 kN, a load pad, a wedge-shape platen, and a 300-kN dead weight force standard machine (DWFSM) with a relative expanded uncertainty of 0.005% (k = 2). A load pad was fixed on the 300-kN column-type force transducer. The transducer was ground to make the top and bottom planes parallel. A wedge-shaped platen was placed under the transducer. A DMP41 (made by HBM, Germany) was used as the amplifier and indicator of the force transducer.

Figure 7.

Experiment setup.

Seven load pads, were designed with various dimensions and numbered by group, as shown in Table 1 and Figure 8, to investigate the effect of the load pad shapes on the output of column-type transducer. The load pads were made from the same material (100Gr6), with the same heat treatment. The material with high strength and hardness was commonly used to manufacture bearings. According to the International Organization for Standardization (ISO) 376:2011,²⁰ the hardness of the load pads was controlled in the range of 45–50 HRC (Rockwell hardness). The machining errors of the load pads were smaller than 0.1%.

Table 1.

Dimensions of the load pads.

Experiment no.	Group	Factor assignment			Contact design
Experiment no.	Group	$R_{1} - R_{2}$ (mm)	$η$	$α$ (°)	Contact design
1	1	60–∞	0	0	Sphere– plane contact
2		60–∞	0	0.5
3		60–∞	0	1
4	2	120–∞	0	0
5		120–∞	0	0.5
6		120–∞	0	1
7	3	180–∞	0	0
8		180–∞	0	0.5
9		180–∞	0	1
10	4	60–180	1/3	0	Sphere– sphere contact
11		60–180	1/3	0.5
12		60–180	1/3	1
13	5	60–90	2/3	0
14		60–90	2/3	0.5
15		60–90	2/3	1
16	6	120–180	2/3	0
17		120–180	2/3	0.5
18		120–180	2/3	1
19	7	200–300	2/3	0
20		200–300	2/3	0.5
21		200–300	2/3	1

Figure 8.

The photograph of the load pads: (a) before being assembled and (b) after being assembled.

As shown in Table 1, 21 experimental setups, with various load pads and a wedge-shaped platen, were measured according to the ISO 376:2011²⁰ separately. Two series of increasing force with five uniformly distributed force steps were applied to the transducer after three times preloads in the first load position. And then, the force transducer and the load pad were rotated by 120° around the axis of the transducer. The wedge-shaped platen was fixed always in the same position. After one-time preload, one series of increasing and decreasing force with the same force steps was loaded to the transducer. Finally, the same measurements, which were the same as those in the second loading position, were carried out in the third loading position after rotating the load pad and the transducer by 120°.

The indication error of the experiment was calculated by

e_{α, j} = \frac{x_{α, j} - x_{0, j}}{x_{0, j}} \times 100 %

(19)

where $e_{α, j}$ and $x_{α, j}$ is the indication error and the output of the transducer tested using the wedge-shaped platen with an angular misalignment of $α$ ( $α$ = 0.5° or 1°) in the jth loading position, respectively; $x_{0, j}$ is the output of the transducer tested using the wedge-shaped platen with an angular misalignment of 0° in the jth loading position.

The indication error of the transducer under the rated load, which was measured with a relatively lower expanded uncertainty, was used in the following discussion.

Results and discussion

Sphere–plane contact

In the particular case of $R_{2} = \infty$ , the contact between the ball and cup is sphere–plane contact. The output of the force transducer is influenced by the spherical radius of the ball.

The scatter plot in Figure 9 shows the experiment results of the indication error of the transducer measured with the load pads of Groups 1, 2, and 3, respectively. It is obvious that larger spherical radius of the ball, $R_{1}$ , caused larger indication error of the transducer. The indication errors of all the load pads increased with the increment of the angular misalignment of the wedge-shaped platen. The indication error of Group 1, 2, and 3 increased by 0.02%, 0.07%, and 0.08%, respectively, which indicated that the indication error of the force transducer using the load pad of larger spherical radius increases more quickly than that of smaller spherical radius.

Figure 9.

Comparison of the indication errors from the experiment (Exp.) with the theoretical computation (Comp.). The load pads of Groups 1, 2, and 3, which were fixed on the 300-kN force transducer, were tested separately using the wedge-shaped platen with the angular misalignment of 0.5° and 1° in the first load position. The indication error was calculated using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ .

The theoretical computation of the indication errors plotted according to equation (16) with $β = 1^{°} and γ {= 1}^{°}$ were in good agreement with the experiment results (Figure 9). The indication error curve of each load pad grew linearly as the misalignment increased. But the slopes of curves were related to the spherical radius of the ball. The indication error of $R_{1} = 180 mm$ increased much sharper than that of $R_{1} = 60 mm$ , which was in agreement with the experiment results.

For the further investigation, the indication error with various misalignments, $α$ , and spherical radius, $R_{1}$ , was plotted as a surface using equation (16), as shown in Figure 10. It is shown that the indication error was not sensitive to the spherical radius of the ball when the angular misalignment was smaller than 0.2°. However, as the misalignment became more and more larger, the indication error increased sharply with the increment of $R_{1}$ .

Figure 10.

The indication error plotted using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ and the angular misalignment in the range of 0°–1° and the spherical radius of the ball in the range of 60–180 mm, in the first loading position.

When the transducer was measured with a large angular misalignment, the contact point between the sphere and the plane could move farther away from the original contact point. In this case, a large eccentricity of the load was induced, and a large moment was introduced to the force transducer, which caused a large indication error.

In summary, it was important to enhance the stiffness of the upper platen of the BUS to eliminate the angular misalignment.

Similar experiment results were also received when the transducer was tested on the wedge-shaped platen with an angular misalignment of 0.5° in the other two loading positions. As shown in Figure 11, the indication error of the load pad with $R_{1} = 180 mm$ was higher than that with $R_{1} = 120 mm$ , and the one with $R_{1} = 60 mm$ was the smallest.

Figure 11.

Comparison of the indication errors from the experiment (Exp.) with the theoretical computation (Comp.). The load pads of Groups 1, 2, and 3, which were fixed on the 300-kN force transducer, were tested separately using the wedge-shaped platen with the angular misalignment of 0.5°. The indication error was calculated using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ .

Using equation (16), the theoretical computation curves of the indication errors of Group 1, 2, and 3 were plotted with the loading position ranging from 0° to 360°, as shown in Figure 11. The curves were characterized by a sinusoid with the change of the loading position, which is similar to the results by Kang and Hong.²¹ The phase positions of the curves were not dependent on the spherical radius of the ball. All the peaks and troughs of the three curves were located on 75° and 255°. The indication error equals 0 in the loading positions of 165° and 345°. The deviation between the peak and the trough of the sinusoid was the so-called ‘position error’ of the force transducer.

Because of the housing of the force transducer, it was hard to observe the exact locations of the strain gauges attached on elastic element. In this case, the coordination system in Figure 4 and the loading positions, $ω$ , were impossible to be confirmed at first. However, it was found that the testing results agreed very well with the cures when $ω = 2 0^{°}, 14 0^{°}, and 26 0^{°}$ (Figure 11).

It was obvious that the amplitudes of the indication error sinusoids in Figure 11 were related to $R_{1}$ for the load pad of the sphere–plane contact design. No matter in which loading position, the indication error and position error increased as the spherical radius was increased.

Figure 12 shows the indication error surface of the force transducer using the load pad with $R_{1} = 60 - 200 mm$ when the transducer was loaded in the range of 0°–360°. It was shown that the indication error and position error increased sharply with the increment of $R_{1}$ .

Figure 12.

The indication error plotted using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ and the loading position in the range of 0°–360°, the spherical radius of the ball in the range of 60–180 mm, and the angular misalignment $α = 0 . 5^{°}$ .

With the same misalignment, $α$ , the ball with a larger spherical radius rotated farther away from the original point than a smaller one did. So a larger moment was introduced to the force transducer, which caused a larger indication error.

The spherical radius of the ball had a significant effect on the indication error. It should not be too large when this type of load pad was designed.

Sphere–sphere contact

Effect of spherical radius ratio

The indication error of the force transducer tested using the load pads of Group 1, 4, and 5 with the spherical radius ratios of 0, 1/3, and 2/3 were compared and discussed.

Figure 13 shows that the indication error increased with the increment of the ratio when the load pads were tested using the same wedge-shaped platen with an angular misalignment of 0.5° or 1°. And when the angular misalignment increased, all the indication error increased. It is easy to find that the increase rate was related to the spherical radius ratio. The larger spherical radius ratio resulted in the quicker increase of the indication error. In Figure 13, the indication error of $η = 0$ increased from 0.05% to 0.07%; that of $η = 1 / 3$ increased from 0.11% to 0.19%; and that of $η = 2 / 3$ increased from 0.21% to 0.55% as $α$ increased from 0.5° to 1°. The indication error from the theoretical computation presented that same variation tendency with the experimental results.

Figure 13.

Comparison of indication error of the transducer from experiment (Exp.) with the theoretical computation (Comp.) measured in the first loading position with the angular misalignments of 0.5° and 1°, using the load pads of Groups 1, 4, and 5 in Table 1. The indication error was calculated using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ .

To deeply understand the effect of the ratio, the indication error surface was plotted according to equation (16) with $α = 0^{°} - 1^{°}$ and $η = 0 - 0.66$ (Figure 14). When $η \leq 0.2$ , the variation of indication error was very small with the change of the angular misalignment; but when $η > 0.2$ , the indication error increased rapidly along with the increase of the angular misalignment, especially when $α > 0 . 6^{°}$ . Besides that, it is also confirmed that the indication error of transducer using this type of pad load was not sensitive to the spherical radius ratio when the angular misalignment was small.

Figure 14.

The indication error plotted using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ and the angular misalignment in the range of 0°–1° and spherical radius ratio in the range of 0–0.66, in the first loading position.

The indication errors of the force transducer tested on a 0.5° wedge-shaped platen using the three load pads in three different load positions are shown in Figure 15. It obviously shows that a large ratio causes a large error no matter in which position the transducer was loaded.

Figure 15.

Comparison of the indication error of the transducer from experiment (Exp.) with the theoretical computation (Comp.) measured in various loading positions with an angular misalignment of 0.5°, using the load pads of Groups 1, 4, and 5 in Table 1. The indication error was calculated using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ .

The three sinusoids in Figure 15 were plotted using equation (16). The theoretical computation presented similar results as the experiment. The ratio had a significant effect on the amplitude. Large ratio caused a larger indication error and position error. In addition, all the curves reached the same peaks and troughs in 75° and 255°, respectively, as the same with the curves in Figure 11.

The surface in Figure 16 presents the indication error of the force transducer using the load pad with the ratio in the range of 0–0.66 loaded in any position. The biggest error existed when $η = 0.66$ . The error decreased rapidly with the decrement of the ratio. It tended to smooth when the ratio smaller than 0.2.

Figure 16.

The indication error plotted using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ and the loading pad in the range of 0°–360° and the spherical radius ratio in the range of 0–0.66, $α = 0 . 5^{°}$ .

In summary, the spherical radius ratio should not be larger than 0.2 in the sphere–sphere contact design load pads.

Effect of the spherical radius with the same radius ratio

The indication error of the force transducer tested using the load pads of Group 5, 6, and 7, of which the spherical radius ratios were 2/3, with the spherical ball radius of 60, 120, and 200 mm were compared and discussed.

As shown in Figure 17, the experiment results presented that the indication error increased as the spherical radius increased when the load pads were tested with the same spherical radius ratio. The indication errors increased as $α$ increased from 0.5° to 1°.

Figure 17.

Comparison of the indication error of the transducer from experiment (Exp.) with the theoretical computation (Comp.) measured in the first loading position with the angular misalignments of 0.5° and 1°, using the load pads of Groups 5, 6, and 7 in Table 1. The indication error was calculated using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ .

The indication error of the transducer tested using the load pad with the biggest radius of 200 mm increased by 0.68%, while the one with 120 mm increased by 0.38% and the one with 60 mm increased by 0.34%. It is possible to conclude that the large spherical radius is the main influencial factor of the large indication error. The phenomenon was verified by the experiment results in Figure 17.

Figure 18 shows the indication error surface of the transducer using the load pad with the ratio equal to 2/3 and the radius in the range from 60 to 200 mm. Being similar to the surface in Figure 10, with the increment of the radius, the indication error increased slowly when $α \leq 0 . 2^{°}$ but increased sharply when $α > 0 . 5^{°}$ . The indication error increased slowly along with the increment of $α$ when radius was small.

Figure 18.

The indication error plotted using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ , the angular misalignment in the range of 0°–1°, and the spherical radius of the ball in the range of 60–200 mm, in the first loading position.

As shown Figure 19, the experiment results of the indication errors of the transducer tested using the three load pads with the same radius ratio indicated that large radius was an important factor for the increment of the indication error wherever the load applied to the transducer. The curves plotted using equation (16) and the experiment results had a good agreement. The large radius caused the big amplitudes of the indication error sinusoid, which also was verified by the indication error surface in Figure 20. It was suggested that big radius should be avoided when the ratio was large, such as 2/3 in this study.

Figure 19.

Comparison of the indication error of the transducer from experiment (Exp.) with the theoretical computation (Comp.) measured in various loading positions with an angular misalignment of 0.5°, using the load pads of Groups 5, 6, and 7 in Table 1. The indication error was calculated using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ .

Figure 20.

The indication error plotted using equation (16) with the parameters of the 300-kN force transducer in the experiment that R = 19 mm, h = 37 mm, $β_{i} = γ_{i} = 1 . 5^{°}$ , and $μ = 0.3$ , the loading pad in the range of 0°–360°, and the spherical radius of the ball in the range of 60–200 mm, $α = 0 . 5^{°}$ .

Comparison of the contact stress

From the above analysis results, it is known that the spherical radius of the ball should be small enough to avoid large indication error. But a too small radius would induce large contact stress, which caused the plastic deformation and failure of the load pad. To evaluate the contact stresses of the load pads of the sphere–plane and the sphere–sphere contact designs, a contact stress ratio $λ$ is defined as follows

λ = \frac{σ_{\max, s - s}}{σ_{\max, s - p}}

(20)

where $σ_{\max, s - s}$ and $σ_{\max, s - p}$ are the maximum contact stress of the load pads of the sphere–sphere and the sphere–plane contact design, respectively. $σ_{\max, s - p}$ can be calculated by substituting $R_{2} = \infty$ into equation (17), and $λ$ is

λ = {[(1 - η) \cos (\frac{η}{1 - η} α)]}^{- \frac{2}{3}}

(21)

The surface of $λ$ with $α$ in the range of 0°–1° and $η$ in the range of 0–0.9 was plotted using equation (21), as shown in Figure 21. It is found that the contact stress ratio was not affected obviously by the angular misalignment ranging from 0° to 1°. It is clear that the contact stress ratio $λ$ decreased quickly with the increment of the spherical radius ratio. The contact stress of the load pad of the sphere–sphere contact design was just about only 80% of that of the sphere–plane contact design when the ratio increased to 0.2.

Figure 21.

The contact stress ratio of the maximum contact stress of the sphere–sphere contact design to that of the sphere–plane contact design.

Conclusion

From the theoretical analysis and experiment results, it is revealed that the dimension of the spherical radius of the load pad had a significant effect on the indication error of the transducer.

Compared with the load pad of the sphere–sphere contact design, the load pad of the sphere–plane contact design would cause the smallest indication error and the largest contact stress with the same spherical radius of the ball. It is suggested that the spherical radius ratio of the ball to the cup could be fixed at about 0.2 to get a relatively small indication error and contact stress.

The spherical radius of the ball should be small under the permission of material strength to avoid producing large side force and additional moment and causing a large indication error.

In addition, it is important to enhance the stiffness of the upper platen of the BUS to eliminate the angular misalignment because big misalignment would enlarge the effect of the load pad size on the indication error of the transducer in the BUS.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The research has been funded from the research budget for a research project 2011YQ090009 of the Chinese major national scientific instrument and equipment development projects.

ORCID iD

Wei Liang

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Effect of the load pad size on the output of the column-type force transducer

Abstract

Keywords

Introduction

Model proposed

The side force and additional moment produced on the force transducer

The output of the force transducer influenced by the side force and additional moment

The contact stress

Experiment

Results and discussion

Sphere–plane contact

Sphere–sphere contact

Effect of spherical radius ratio

Effect of the spherical radius with the same radius ratio

Comparison of the contact stress

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References