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An improved high precision measuring method for shaft bending deflection
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An improved high precision measuring method for shaft bending deflection Cong-Hui Wang, Yong-Chen Pei∗, Qing-Chang Tan, Jia-Wei Wang School of Mechanical Science and Engineering, Jilin University, Nanling Campus, Changchun 130025, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 6 September 2016 Revised 4 January 2017 Accepted 22 February 2017 Available online xxx Keywords: Shaft bending deflection Eccentricity measuring High precision measuring Measuring algorithm Analytic model Experimental verification
a b s t r a c t The circular-section columnar parts, such as shaft, rod and pipe etc., are widely used in all kinds of machines. The bending deformation of such parts will seriously affect the assembly precision and working performance of the machines. A high efficiency and precision measuring method for bending deflection is needed. With a classic contact-detection mechanism at low cost composed of a displacement sensor, a lever and a shaft supporting device, a high precision measuring algorithm is proposed and deduced after strict mathematical analysis in this paper, and which is a closed analytic algorithm and will not produce any principle error. The parameters sensitivity auditing is carried out and the result shows the measuring method has high adaptability. Furthermore, the corresponding experimental calibration method of the current measurement technology is provided, and an experimental validation on the bending deformation of a particular shaft is carried out. The result shows in depth the proposed method has high accuracy. © 2017 Elsevier Inc. All rights reserved.
Notation A, B A0 , B0 C, r, E e, e0 k0 KA , KM R zc zA zM zAe e zM
x and y coordinates of the rotating center x and y coordinates of the geometric center structural size of the lever mechanism shaft section eccentricity lever proportional coefficient conversion coefficients for calibration shaft section radius actual measured value of the sensor fluctuating amplitude of the sensor indicating value sensor mean indicating value fluctuating amplitude in experiment mean value in experiment
zAL
theoretical fluctuating amplitude
L zM
theoretical mean value eccentricity deviation tilt angle between the lever and x axis
ε ϑ
∗
Corresponding author. E-mail address: [email protected] (Y.-C. Pei).
http://dx.doi.org/10.1016/j.apm.2017.02.050 0307-904X/© 2017 Elsevier Inc. All rights reserved.
Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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ϕ
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tilt angle for the equilibrium position eccentric angle of shaft section
1. Introduction As widely-used important components in machines, shaft, rod, pipe and other circular-section columnar parts have many functions and applications, such as transferring movement and power, supporting rotating parts, bearing loads, and so on. The bending deflection of such parts will severely affect the assembly precision and working performances of the machine in the way of centrifugal vibration, operational accuracy, even reliability and security [1]. In order to test, control or straighten the bending deformation of shaft parts [2] induced in the process of manufacturing or using, the bending deflection measuring is very important and necessary [3,4]. And in order to meet the requirements for automatic mass production, a high efficiency, high precision and automatic measuring method for bending deflection is needed. Manual measuring bending deflection of a shaft with micrometer and V type block is clearly unable to meet requirement of modern high efficiency production. So Minoru et al. [5] has developed a transmission-type position sensor for the straightness measurement, but it could only been applied for a large structure. Rana et al. [6] has proposed a non-contact method for rod straightness measurement based on transmitting the laser light from a laser source and detecting on the other end with quadrant laser sensor, but the ranges of the length and the bent of the measured rod were limited. Feng et al. [7] has used a single-mode fiber-coupled laser module for straightness measurement to enlarge the measuring range. In fact, bending deflection of shaft, pipe and rod parts can be evaluated by the cross-section eccentricity. Schalk et al. [8] has presented a pipe eccentricity measurement system based on laser triangulation. However, the laser collimator technique has been difficult to guarantee the measurement precision because of the influence produced by laser beam drift, reflection of light and air turbulence so on; and its larger size and more expensive price have prevented it from being widely used as a testing system. Lu et al. [9] and Derganc et al. [10] have tried to use machine vision technique to measure bending deflection of rotating parts, but the measuring capability is determined by the development of image processing ability. In present work, basing upon a classic measuring system at low cost composed of a displacement sensor, a lever and a shaft supporting device, a contact measurement technique for shaft bending deflection was made, which can avoid the concentric alignment error in some optical measurement techniques. And a high precision algorithm to obtain the deflection was presented, which was investigated by analyzing the accurate analytic relationship between the shaft cross-section eccentricity and the sensor indicating value. Sensitivity of the parameters in the algorithm was audited and a verification experiment was done on the effectiveness and high accuracy of the measuring method proposed. 2. Mechanism and model analysis 2.1. A contact-detection mechanism of bending deflection A contact-detection mechanism of bending deflection is simplified as shown in Fig. 1. It is composed of a displacement sensor, a lever and a shaft supporting device. Since the bending deformation of the measured shaft can be described by the eccentricities at several cross-sections, this mechanism focuses on the measurement of section eccentricity. When the shaft rotates, the circumference of a measured shaft section will always contact with the measuring lever tightly. The eccentricity of the measured section can be transformed into the fluctuation of the indication value of the displacement sensor by the swing of the lever. If the mathematical relationship between the value of the sensor and the eccentricity is established, the eccentricity of the shaft section can be obtained by tracking the sensor value. Comparing with some no-contact methods using laser or machine vision technique, the current contact detection with the lever mechanism cannot been interfered by light and air and can better control the influence factors of measurement accuracy. Moreover, a small range displacement sensor can be selected because of the proportional reduction of the displacement signal, thereby the measuring mechanism cost is low. As shown in Fig. 1, the rotating center of the lever mechanism is defined as the origin of Cartesian coordinates, recording as O(0,0). And C, r, E are the basic structural dimensions of the lever mechanism. The radius of a measured cross-section can be noted as R. The rotating center of the tip can be noted as O (A,B), the geometric center of the section is noted as O (A0 ,B0 ). Obviously, A0 =A +ecos(ϕ ), B0 =B +esin(ϕ ), where e and ϕ are the eccentricity and phase angle respectively. θ is the title angle between the lever and x axis, z is the sensor indicating value. If the measured cross-section is not eccentric and the radius R =B−E, it is easy to deduce that the lever mechanism is always in horizontal position, i.e. θ is always equal to 0, and the indicating value z remains unchanged as the workpiece rotates, as shown in Fig. 1(a). If the measured section is eccentric, the indicating value z will fluctuate with the lever mechanism swinging during the workpiece rotating. But when the section radius R =B−E, the equilibrium position of the lever mechanism is in horizontal line; when R=B-E, the equilibrium position of the lever mechanism will deviate from horizontal line, as shown in Fig. 1(b) and (c). According to the principle of plane geometry and trigonometric function, line L0 , L1 and L2 presented in the Fig. 1(b) can respectively be expressed as
L0 : x = −r
(1)
Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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Fig. 1. A lever mechanism for shaft bending deflection measuring. (a): R =B-E; (b): R >B-E; (c): R
L1 : y = −xtan(θ ) − C/cos(θ )
(2)
L2 : y = −xtan(θ ).
(3)
Due to the section being always tangent to the lever, the distance between O (A0 ,B0 ) and L2 is equal to R +E, which can also be obtained from the distance equation from dot to straight line. So, the following equation can be got
A0 sin(θ ) + B0 cos(θ ) = R + E.
(4)
From Eq. (4), θ can be solved as Eq. (5)
θ = sin−1
R+E A0 2 + B0 2
− tan−1
B 0
A0
.
(5)
Substituting Eq. (1) into Eq. (2), and removing the sensor initial position value -C, the sensor absolute indicating value z can be expressed as
z = r tan(θ ) − C/cos(θ ) + C.
(6)
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a
b
c
d
Fig. 2. Fluctuations of lever tilt angle and sensor indicating value. (a): R = 0.5(B-E), e = 0.5 mm; (b): R =B-E, e = 0.5 mm; (c): R = 2(B-E), e = 0.5 mm; (d): R = 2(B-E), e = 2 mm.
According to Eq. (6), when θ = 0 and z = 0, as shown in Fig. 1(a); when θ > 0 and z > 0, as shown in Fig. 1(b); when θ < 0 and z < 0, as shown in Fig. 1(c). For example, the basic structural dimensions of the lever mechanism are followed as C = 15 mm, r = 65 mm, E = 20 mm, A = 130 mm and B = 32 mm, the tilt angle θ of the detection mechanism and the sensor indicating value z can be separately calculated based on the above equations. And the fluctuations of both with the phase angle ϕ are illustrated as Fig. 2 theoretically. It can be seen that both the tilt angle θ and the indicating value z vary periodically when the workpiece rotates, that the larger difference between R and B −E, the larger tilt angle θ 0 for the equilibrium position of the lever mechanism, and that the larger eccentricity e of the workpiece section, the larger swinging range of the lever mechanism. A mathematical model must be set up to find the relationship between the cross-section eccentricity and the sensor indicating value. 2.2. A direct proportional algorithm Intuitionally, if a direct proportional algorithm is used to calculate the section eccentricity, the eccentricity e is the production of zA and k0 , which can be expressed as
e = k0 zA ,
(7)
where zA =[max(z)−min(z)]/2, which is the fluctuating amplitude of the sensor indicating value z; and k0 =A/r, which is the lever proportional coefficient. Practice shows that when the section radius R is close to the value of (B-E) and the eccentricity is small, e calculated by Eq. (7) is less deviation from the actual eccentricity. Whereas, when the difference between the radius R and the value of (B-E) is larger, or when the section eccentricity is larger, e obtained from Eq. (7) will show a larger deviation from the actual value, and the deviation is even much larger than the mechanical system precision and the measurement accuracy of the sensor. By assuming that the actual eccentricity of the shaft cross section is e0 , the eccentricity obtained from the Eq. (7) is e, the deviation ε between e and e0 is expressed as ε =e −e0. When the following parameters are used: C = 15 mm, r = 65 mm, E = 20 mm, A = 130 mm, B = 32 mm, the variation of the deviation ε with the eccentricity e0 and the section radius R is shown as Figs. 3 and 4. It is evident that the deviation caused by the proportional algorithm is sharply increasing with the eccentricity e0 increasing or the difference between R and (B−E) being larger. Although it decreases to some extent when R =B−E, the deviation ε still increases rapidly with the eccentricity e0 increasing. Thus, the system detection accuracy is covered by the deviation ε caused by proportional algorithm. In addition, ε > 0 (i.e. e >e0 ) shows that proportion algorithm Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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Fig. 3. Deviation ε varying with the eccentricity e0 and the radius R of shaft section.
Fig. 4. Deviation contour map with the change of eccentricity e0 and the radius R of shaft section.
is overestimating the section eccentricity in fact. The analysis results are consistent with the practice situation perfectly. Therefore, it is particularly significant to develop a high precision algorithm for calculating the eccentricity of shaft section. 2.3. An improved closed analytic algorithm In order to solve the problem of large error produced in the direct proportional algorithm, an investigation on the accurate analytic relationship between the section eccentricity e and the sensor indicating value z is carried out. Since the measured indicating value z has maximum value max(z) and minimum value min(z) at əz/əϕ = 0, the partial derivation of z with respect to ϕ can be written as
∂z ∂ z ∂θ r − C sin(θ ) ∂θ = = = 0, ∂ϕ ∂θ ∂ϕ cos2 (θ ) ∂ϕ
(8)
where [r-Csin(θ )]/cos(θ )=0 cannot be always true, thus there is
∂θ −e cos(θ + ϕ ) = = 0. ∂ϕ A cos(θ ) − B sin(θ ) + e cos(θ + ϕ )
(9)
And Eq. (9) can be established by
θ + ϕ = nπ + π / 2 .
(10)
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Substituting Eq. (10) into Eq. (4), Eq. (11) will be yielded.
A sin(θ ) + B cos(θ ) = R + E ± e,
(11)
when tan(θ /2) is used to express all the trigonometric functions, Eq. (11) can be resolved as shown in Eq. (12). The maximum value and minimum value for tan(θ /2) are denoted as x+ and x− respectively as followed
tan
θ 2
=
A−
A2 + B2 − ( R + E ± e ) = x± , B + (R + E ± e ) 2
(12)
where A2 +B2 -(R +E +e)2 must be greater than 0, the measurable condition of the lever mechanism is
R+e<
A2 + B2 − E.
(13)
Substituting Eq. (12) into Eq. (6), the following Eq. (14) can be obtained.
r tan(θ ) − C/ cos(θ ) + C = 2
r x± − C x± 2 = z± , 1 − x± 2
(14)
where z+ and z- correspond to the maximum value max(z) and minimum value min(z) separately. Then the fluctuating amplitude zA and mean indicating value zM of the sensor can be written as
zA =
r x− − C x− 2 z+ − z− r x+ − C x+ 2 = − 2 2 1 − x+ 1 − x− 2
(15)
zM =
z+ + z− r x+ − C x+ 2 r x− − C x− 2 = + . 2 1 − x+ 2 1 − x− 2
(16)
Eqs. (15) and (16) can be rearranged as
C−
zM ± zA zM ± zA x± 2 − r x± + = 0. 2 2
(17)
Thus, Eq. (17) can be solved as
x± =
r±
r 2 − (zM ± zA )[2C − (zM ± zA )] 2C − (zM ± zA )
(18)
Substituting Eq. (18) into Eq. (12) yields
R+E±e=
B ( 1 − x± 2 ) + 2Ax± . 1 + x± 2
(19)
Then, the accurate analytic expression of the eccentricity e and the section radius R can be got as followed
e=
B ( 1 − x− 2 ) + 2Ax− B ( 1 − x+ 2 ) + 2Ax+ − 2 2 ( 1 + x+ ) 2 ( 1 + x− 2 )
(20)
R=
B ( 1 − x+ 2 ) + 2Ax+ B ( 1 − x− 2 ) + 2Ax− + − E, 2 ( 1 + x+ 2 ) 2 ( 1 + x− 2 )
(21)
where x ± can be calculated from Eq. (18) by using the fluctuating amplitude zA and mean indicating value zM . The above mathematical derivation process has indicated that the analytic relationship between the sensor indicating value z and the section eccentricity e is accurate absolutely. Actually, the mathematical relationships of all the variables are absolutely accurate and this algorithm, which is called as closed analytic algorithm in the following statement, doesn’t have any theoretical principle error. Meanwhile, the radius R of the measured workpiece section can be obtained from this algorithm as well. 3. Sensitivity auditing In order to ensure the measurement accuracy, it is necessary to understand how variation of different parameters, including the basic structural parameters of the lever mechanism and the indicating value of the sensor, affect on the eccentricity e. Therefore, a sensitivity auditing of the parameters must be carried out. Without loss of generality, the sensitivity can be discussed by using following parameter: C = 15 mm, r = 65 mm, E = 20 mm, A = 130 mm, B = 32 mm, R = 25 mm, then one has zA = 0.2543 mm and zM = 6.5389 mm at e = 0.5 mm. Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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3.1. Sensitivity auditing for the proportional algorithm (1) The sensitivity of the fluctuating amplitude zA can be expressed as
∂e = k0 . ∂ zA
(22)
According to the above parameters, one has əe/əzA = 2. (2) The sensitivity of the lever structural size A can be expressed as
∂ e zA = ∂A r
(23)
One has əe/əA= 0.0039. (3) The sensitivity of the lever structural size r can be expressed as
AzA ∂e =− 2 ∂r r
(24)
One has əe/ər=−0.0078. 3.2. Sensitivity auditing for the closed analytic algorithm (1) The sensitivity of the fluctuating amplitude zA can be solved as
∂e ∂ e ∂ x+ ∂ e ∂ x− = + ∂ zA ∂ x+ ∂ zA ∂ x− ∂ zA
(25)
One has əe/əzA = 1.9659 at the above parameters. (2) The sensitivity of the mean indicating value zM can be solved as
∂e ∂ e ∂ x+ ∂ e ∂ x− = + ∂ zM ∂ x+ ∂ zM ∂ x− ∂ zM
(26)
One has əe/əzM =−0.0025. (3) The sensitivity of the lever structural size A is
x− ∂e x+ = − ∂ A 1 + x2+ 1 + x2−
(27)
One has əe/əA= 0.0039. (4) The sensitivity of lever structural size B is
1 − x2− 1 − x2+ ∂e = − 2 ∂ B 2(1 + x+ ) 2(1 + x2− )
(28)
One has əe/əB=−0.0 0 040. (5) The sensitivity of lever structural size C can be solved as
∂e ∂ e ∂ x+ ∂ e ∂ x− = + ∂ C ∂ x+ ∂ C ∂ x− ∂ C
(29)
One has əe/əC= 0.0 0 078. (6) The sensitivity of lever structural size r can be solved as
∂e ∂ e ∂ x+ ∂ e ∂ x− = + ∂ r ∂ x+ ∂ r ∂ x− ∂ r
(30)
One has əe/ər=−0.0076. It can be seen clearly that the sensitivity of the eccentricity e to the same parameter is almost equal in two algorithms, and the eccentricity e is the most sensitive to parameter zA because əe/əzA has the maximum value which is close to 2. Therefore, in order to ensure the precision of the eccentricity e, the measurement accuracy of the sensor must be strictly controlled. The sensitivity of the eccentricity e to the structural sizes of the measuring device are all less than 0.008, which indicates the measurement accuracy is less affected by the precise of machining and assembling of measuring mechanism, and the measuring method has high adaptability. 4. Experimental validation The effectiveness of current closed analytic algorithm in the way of improving the measurement accuracy needs to be verified by experiment. Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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Fig. 5. A stepped shaft for experimental investigation.
4.1. Experimental calibration method The experimental calibration is performed by using a standard smooth shaft with a known radius R and cross-section eccentricity e. If the actual measurement value of the sensor is zc , the fluctuating amplitude and mean value are respectively
zAe = e zM =
max(zc ) − min(zc ) 2
(31)
max(zc ) + min(zc ) . 2
(32)
The theoretical fluctuating amplitude and mean value are separately
zAL = L zM =
r y1 − C y1 2 r y2 − C y2 2 − 1 − y2 2 1 − y1 2
(33)
r y2 − C y2 2 r y1 − C y1 2 + , 2 1 − y2 1 − y1 2
(34)
where y1 and y2 can be calculated by
y1 =
y2 =
A−
A−
A2 + B2 − ( R + E − e ) B + (R + E − e )
(35)
A2 + B2 − ( R + E + e ) . B + (R + E + e )
(36)
2
2
The conversion coefficients of fluctuating amplitude and mean value can be defined as
KA = zAL /zAe
(37)
L e KM = zM − zM .
(38)
As a result, the calibrated fluctuating amplitude and mean value can be determined by
zA = KA
max(zc ) − min(zc ) 2
zM = KM +
max(zc ) + min(zc ) . 2
(39) (40)
4.2. Experimental results and analysis The following instrument and equipment were used in experiment: a lever mechanism for measuring shaft bending deflection, a sensor and its software and hardware supporting system, vernier caliper, steel ruler, micrometer and its magnetic meter base, a stepped shaft specimen with preliminary bending deformation. The structure of the stepped shaft need to satisfy that each section diameter is shown in Fig. 5, the length of each shaft segment is almost equal, the total length is determined by clamping condition, it should be as long as possible. After the shaft being machined, a preliminary bending processing is required, which makes the runout 2e be greater than 6 mm at least at the diameter Ф80mm shaft segment. Eleven sets of data were got by measuring all segment sections of the stepped shaft, and the results are listed in Table 1, where 2e is the practical eccentricity measured by micrometer; 2e0 is the eccentricity calculated by proportional algorithm; 2ep is the eccentricity calculated by closed analytic algorithm; and ∗ represents the data used for experimental calibration. In the experimental system, the fundamental structure parameters are r = 65 mm, A = 130 mm, B = 32 mm, C = 12.9 mm, E = 21.8 mm. By comparing the error values in Table 1, it can be clearly seen that closed analytic algorithm can remarkably reduce the measurement error and the measurement accuracy is improved greatly. Thus, the correctness and validity of current closed analytic algorithm can been fully testified. In addition, the measurement error caused by the closed analytic algorithm is much smaller than the one caused by the proportional algorithm, which shows that closed analytic algorithm has high precision and is effective in shaft deflection measurement. Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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Table 1 Experimental measuring results of the eccentricity. Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Conversion: KA = 0.52354,KM = 9.53839,k0 = 1.00424 ∗
∗
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Conversion: KA = 0.52478, KM = 9.40507, k0 = 1.00659 ∗
1.183 1.985 2.758 2.935 2.548 1.919 1.225 2.348 3.725 5.175 6.275
0.035 0.064 0.077 0.111 0.065 0.069 0.044 0.082 0.172 0.208 0.228
0.0 0 0 0.007 −0.003 0.028 −0.008 0.015 0.010 0.016 0.069 0.062 0.050
1.183 1.985∗ 2.758 2.935 2.548 1.919 1.225 2.348 3.725 5.175 6.275
0.033 0.060∗ 0.071 0.104 0.059 0.065 0.042 0.077 0.164 0.196 0.214
−0.004 0.0 0 0∗ −0.013 0.018 −0.017 0.009 0.006 0.008 0.056 0.044 0.029
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Conversion: KA = 0.52115,KM = 9.35226,k0 = 0.99961
Conversion: KA = 0.52683, KM = 9.32650, k0 = 1.01053
1.183 1.985 2.758 2.935 2.548∗ 1.919 1.225 2.348 3.725 5.175 6.275
0.041 0.073 0.090 0.124 0.076∗ 0.078 0.050 0.093 0.189 0.231 0.256
0.004 0.013 0.005 0.037 0.0 0 0∗ 0.021 0.014 0.024 0.080 0.078 0.069
1.183 1.985 2.758 2.935 2.548 1.919∗ 1.225 2.348 3.725 5.175 6.275
0.028 0.052 0.060 0.093 0.049 0.058∗ 0.037 0.068 0.150 0.177 0.190
−0.009 −0.009 −0.025 0.005 −0.028 0.0 0 0∗ 0.0 0 0 −0.002 0.039 0.021 0.0 0 0
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Conversion: KA = 0.53254,KM = 9.34726,kB = 1.02139 1.183 1.985 2.758 2.935 2.548 1.919 1.225 2.348 3.725∗ 5.175 6.275 ∗
0.016 0.031 0.031 0.063 0.023 0.038 0.024 0.044 0.112∗ 0.123 0.124
Conversion: KA = 0.52923, KM = 9.38275, kB = 1.01490 −0.022 −0.030 −0.055 −0.027 −0.056 −0.020 −0.013 −0.027 0.0 0 0∗ −0.034 −0.067
1.183 1.985 2.758 2.935 2.548 1.919 1.225 2.348 3.725 5.175∗ 6.275
0.023 0.044 0.049 0.081 0.039 0.050 0.032 0.058 0.135 0.155∗ 0.163
-0.014 -0.017 -0.037 -0.007 -0.039 -0.008 -0.005 -0.012 0.024 -0.0 0 0∗ -0.026
The experimental data used for calibration.
5. Conclusion and discussion A contact-detection method with low cost and high accuracy was investigated for the bending deflection measurement of circle-section parts in this paper. The following conclusions are obtained: (1) For the measurement of shaft bending deflection, current closed analytic algorithm is based on accurate mathematical relationships, it doesn’t have any theoretical principle error and the shaft section radius can be measured as well. (2) The measurement accuracy of eccentricity obtained by current measuring method is less affected by the size precision of the measurement mechanism, but it is more sensitive to the accuracy of displacement sensor. (3) It is validated by experiment investigation that the measuring method proposed is efficient and accurate. The present work can not only serve the bending deformation monitoring of circular-section parts, in order to control quality of machining and straightening process, but also on this basis, through calculating the coordinates of the section Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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center and further algorithm development, can implement the error measurements of straightness, coaxiality, roundness and cylindricity, which are the important precision measurement items in mechanical engineering. Acknowledgment This work was supported by P.R. China Jilin Province 2012 Postdoctorate Research Starting Funding Project [Grant No. 2012-133-26], and Changchun Research Institute for Mechanical Science Co. Ltd. References [1] H. Martins Gomes, F.J. Flores de Almeida, An analytical dynamic model for single-cracked beams including bending, axial stiffness, rotational inertia, shear deformation and coupling effects, J. Appl. Math. Model. 38 (2014) 938–948. [2] Z.Q. Zhang, Y.H. Yan, H.L. Yang, A simplified model of maximum cross-section flattening in continuousrotary straightening process of thin-walled circular steel tubes, J. Mater. Process. Technol. 238 (2016) 305–314. [3] H.J. Pahk, J.S. Park, I. Ye, Development of straightness measurement technique using the profile matching method, Int. J. Mach. Tools Manuf. 37 (1997) 135–147. [4] E.H.K Fung, Y.K Wong, H.F Ho, Marc P Mignolet, Modelling and prediction of machining errors using ARMAX and NARMAX structures, Appl. Math. Model. 27 (2003) 611–627. [5] M. Sasaki, H. Takebe, K. Hane, Transmission-type position sensor for the straightness measurement of a large structure, J. Micromech. Microeng 9 (1999) 429–433. [6] N.K. Rana, R.R. Sawant, A.H. Moon, A non-contact method for rod straightness measurement based on quadrant laser sensor, in: Proceedings of the IEEE International Conference on Industrial Technology, Mumbai, India, 2006, pp. 2292–2297. [7] Q. Feng, B. Zhang, C. Kuang, A straightness measurement system using a single-mode fiber-coupled laser module, Optics Laser Technol. 36 (2004) 279–283. [8] P. Schalk, R. Ofner, P. O’Leary, Pipe eccentricity measurement using laser triangulation, Image Vis. Comput. 25 (2007) 1194–1203. [9] R.S. Lu, Y.F. Li, Q. Yu, On-line measurement of the straightness of seamless steel pipes using machine vision techique, Sens. Actuators A 94 (2001) 95–101. [10] J. Derganc, B. Likar, F. Pernuš, A machine vision system for measuring the eccentricity of bearings, Comput. Ind. 50 (2003) 103–111.
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An improved high precision measuring method for shaft bending deflection Cong-Hui Wang, Yong-Chen Pei∗, Qing-Chang Tan, Jia-Wei Wang School of Mechanical Science and Engineering, Jilin University, Nanling Campus, Changchun 130025, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 6 September 2016 Revised 4 January 2017 Accepted 22 February 2017 Available online xxx Keywords: Shaft bending deflection Eccentricity measuring High precision measuring Measuring algorithm Analytic model Experimental verification
a b s t r a c t The circular-section columnar parts, such as shaft, rod and pipe etc., are widely used in all kinds of machines. The bending deformation of such parts will seriously affect the assembly precision and working performance of the machines. A high efficiency and precision measuring method for bending deflection is needed. With a classic contact-detection mechanism at low cost composed of a displacement sensor, a lever and a shaft supporting device, a high precision measuring algorithm is proposed and deduced after strict mathematical analysis in this paper, and which is a closed analytic algorithm and will not produce any principle error. The parameters sensitivity auditing is carried out and the result shows the measuring method has high adaptability. Furthermore, the corresponding experimental calibration method of the current measurement technology is provided, and an experimental validation on the bending deformation of a particular shaft is carried out. The result shows in depth the proposed method has high accuracy. © 2017 Elsevier Inc. All rights reserved.
Notation A, B A0 , B0 C, r, E e, e0 k0 KA , KM R zc zA zM zAe e zM
x and y coordinates of the rotating center x and y coordinates of the geometric center structural size of the lever mechanism shaft section eccentricity lever proportional coefficient conversion coefficients for calibration shaft section radius actual measured value of the sensor fluctuating amplitude of the sensor indicating value sensor mean indicating value fluctuating amplitude in experiment mean value in experiment
zAL
theoretical fluctuating amplitude
L zM
theoretical mean value eccentricity deviation tilt angle between the lever and x axis
ε ϑ
∗
Corresponding author. E-mail address: [email protected] (Y.-C. Pei).
http://dx.doi.org/10.1016/j.apm.2017.02.050 0307-904X/© 2017 Elsevier Inc. All rights reserved.
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ϕ
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tilt angle for the equilibrium position eccentric angle of shaft section
1. Introduction As widely-used important components in machines, shaft, rod, pipe and other circular-section columnar parts have many functions and applications, such as transferring movement and power, supporting rotating parts, bearing loads, and so on. The bending deflection of such parts will severely affect the assembly precision and working performances of the machine in the way of centrifugal vibration, operational accuracy, even reliability and security [1]. In order to test, control or straighten the bending deformation of shaft parts [2] induced in the process of manufacturing or using, the bending deflection measuring is very important and necessary [3,4]. And in order to meet the requirements for automatic mass production, a high efficiency, high precision and automatic measuring method for bending deflection is needed. Manual measuring bending deflection of a shaft with micrometer and V type block is clearly unable to meet requirement of modern high efficiency production. So Minoru et al. [5] has developed a transmission-type position sensor for the straightness measurement, but it could only been applied for a large structure. Rana et al. [6] has proposed a non-contact method for rod straightness measurement based on transmitting the laser light from a laser source and detecting on the other end with quadrant laser sensor, but the ranges of the length and the bent of the measured rod were limited. Feng et al. [7] has used a single-mode fiber-coupled laser module for straightness measurement to enlarge the measuring range. In fact, bending deflection of shaft, pipe and rod parts can be evaluated by the cross-section eccentricity. Schalk et al. [8] has presented a pipe eccentricity measurement system based on laser triangulation. However, the laser collimator technique has been difficult to guarantee the measurement precision because of the influence produced by laser beam drift, reflection of light and air turbulence so on; and its larger size and more expensive price have prevented it from being widely used as a testing system. Lu et al. [9] and Derganc et al. [10] have tried to use machine vision technique to measure bending deflection of rotating parts, but the measuring capability is determined by the development of image processing ability. In present work, basing upon a classic measuring system at low cost composed of a displacement sensor, a lever and a shaft supporting device, a contact measurement technique for shaft bending deflection was made, which can avoid the concentric alignment error in some optical measurement techniques. And a high precision algorithm to obtain the deflection was presented, which was investigated by analyzing the accurate analytic relationship between the shaft cross-section eccentricity and the sensor indicating value. Sensitivity of the parameters in the algorithm was audited and a verification experiment was done on the effectiveness and high accuracy of the measuring method proposed. 2. Mechanism and model analysis 2.1. A contact-detection mechanism of bending deflection A contact-detection mechanism of bending deflection is simplified as shown in Fig. 1. It is composed of a displacement sensor, a lever and a shaft supporting device. Since the bending deformation of the measured shaft can be described by the eccentricities at several cross-sections, this mechanism focuses on the measurement of section eccentricity. When the shaft rotates, the circumference of a measured shaft section will always contact with the measuring lever tightly. The eccentricity of the measured section can be transformed into the fluctuation of the indication value of the displacement sensor by the swing of the lever. If the mathematical relationship between the value of the sensor and the eccentricity is established, the eccentricity of the shaft section can be obtained by tracking the sensor value. Comparing with some no-contact methods using laser or machine vision technique, the current contact detection with the lever mechanism cannot been interfered by light and air and can better control the influence factors of measurement accuracy. Moreover, a small range displacement sensor can be selected because of the proportional reduction of the displacement signal, thereby the measuring mechanism cost is low. As shown in Fig. 1, the rotating center of the lever mechanism is defined as the origin of Cartesian coordinates, recording as O(0,0). And C, r, E are the basic structural dimensions of the lever mechanism. The radius of a measured cross-section can be noted as R. The rotating center of the tip can be noted as O (A,B), the geometric center of the section is noted as O (A0 ,B0 ). Obviously, A0 =A +ecos(ϕ ), B0 =B +esin(ϕ ), where e and ϕ are the eccentricity and phase angle respectively. θ is the title angle between the lever and x axis, z is the sensor indicating value. If the measured cross-section is not eccentric and the radius R =B−E, it is easy to deduce that the lever mechanism is always in horizontal position, i.e. θ is always equal to 0, and the indicating value z remains unchanged as the workpiece rotates, as shown in Fig. 1(a). If the measured section is eccentric, the indicating value z will fluctuate with the lever mechanism swinging during the workpiece rotating. But when the section radius R =B−E, the equilibrium position of the lever mechanism is in horizontal line; when R=B-E, the equilibrium position of the lever mechanism will deviate from horizontal line, as shown in Fig. 1(b) and (c). According to the principle of plane geometry and trigonometric function, line L0 , L1 and L2 presented in the Fig. 1(b) can respectively be expressed as
L0 : x = −r
(1)
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Fig. 1. A lever mechanism for shaft bending deflection measuring. (a): R =B-E; (b): R >B-E; (c): R
L1 : y = −xtan(θ ) − C/cos(θ )
(2)
L2 : y = −xtan(θ ).
(3)
Due to the section being always tangent to the lever, the distance between O (A0 ,B0 ) and L2 is equal to R +E, which can also be obtained from the distance equation from dot to straight line. So, the following equation can be got
A0 sin(θ ) + B0 cos(θ ) = R + E.
(4)
From Eq. (4), θ can be solved as Eq. (5)
θ = sin−1
R+E A0 2 + B0 2
− tan−1
B 0
A0
.
(5)
Substituting Eq. (1) into Eq. (2), and removing the sensor initial position value -C, the sensor absolute indicating value z can be expressed as
z = r tan(θ ) − C/cos(θ ) + C.
(6)
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a
b
c
d
Fig. 2. Fluctuations of lever tilt angle and sensor indicating value. (a): R = 0.5(B-E), e = 0.5 mm; (b): R =B-E, e = 0.5 mm; (c): R = 2(B-E), e = 0.5 mm; (d): R = 2(B-E), e = 2 mm.
According to Eq. (6), when θ = 0 and z = 0, as shown in Fig. 1(a); when θ > 0 and z > 0, as shown in Fig. 1(b); when θ < 0 and z < 0, as shown in Fig. 1(c). For example, the basic structural dimensions of the lever mechanism are followed as C = 15 mm, r = 65 mm, E = 20 mm, A = 130 mm and B = 32 mm, the tilt angle θ of the detection mechanism and the sensor indicating value z can be separately calculated based on the above equations. And the fluctuations of both with the phase angle ϕ are illustrated as Fig. 2 theoretically. It can be seen that both the tilt angle θ and the indicating value z vary periodically when the workpiece rotates, that the larger difference between R and B −E, the larger tilt angle θ 0 for the equilibrium position of the lever mechanism, and that the larger eccentricity e of the workpiece section, the larger swinging range of the lever mechanism. A mathematical model must be set up to find the relationship between the cross-section eccentricity and the sensor indicating value. 2.2. A direct proportional algorithm Intuitionally, if a direct proportional algorithm is used to calculate the section eccentricity, the eccentricity e is the production of zA and k0 , which can be expressed as
e = k0 zA ,
(7)
where zA =[max(z)−min(z)]/2, which is the fluctuating amplitude of the sensor indicating value z; and k0 =A/r, which is the lever proportional coefficient. Practice shows that when the section radius R is close to the value of (B-E) and the eccentricity is small, e calculated by Eq. (7) is less deviation from the actual eccentricity. Whereas, when the difference between the radius R and the value of (B-E) is larger, or when the section eccentricity is larger, e obtained from Eq. (7) will show a larger deviation from the actual value, and the deviation is even much larger than the mechanical system precision and the measurement accuracy of the sensor. By assuming that the actual eccentricity of the shaft cross section is e0 , the eccentricity obtained from the Eq. (7) is e, the deviation ε between e and e0 is expressed as ε =e −e0. When the following parameters are used: C = 15 mm, r = 65 mm, E = 20 mm, A = 130 mm, B = 32 mm, the variation of the deviation ε with the eccentricity e0 and the section radius R is shown as Figs. 3 and 4. It is evident that the deviation caused by the proportional algorithm is sharply increasing with the eccentricity e0 increasing or the difference between R and (B−E) being larger. Although it decreases to some extent when R =B−E, the deviation ε still increases rapidly with the eccentricity e0 increasing. Thus, the system detection accuracy is covered by the deviation ε caused by proportional algorithm. In addition, ε > 0 (i.e. e >e0 ) shows that proportion algorithm Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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Fig. 3. Deviation ε varying with the eccentricity e0 and the radius R of shaft section.
Fig. 4. Deviation contour map with the change of eccentricity e0 and the radius R of shaft section.
is overestimating the section eccentricity in fact. The analysis results are consistent with the practice situation perfectly. Therefore, it is particularly significant to develop a high precision algorithm for calculating the eccentricity of shaft section. 2.3. An improved closed analytic algorithm In order to solve the problem of large error produced in the direct proportional algorithm, an investigation on the accurate analytic relationship between the section eccentricity e and the sensor indicating value z is carried out. Since the measured indicating value z has maximum value max(z) and minimum value min(z) at əz/əϕ = 0, the partial derivation of z with respect to ϕ can be written as
∂z ∂ z ∂θ r − C sin(θ ) ∂θ = = = 0, ∂ϕ ∂θ ∂ϕ cos2 (θ ) ∂ϕ
(8)
where [r-Csin(θ )]/cos(θ )=0 cannot be always true, thus there is
∂θ −e cos(θ + ϕ ) = = 0. ∂ϕ A cos(θ ) − B sin(θ ) + e cos(θ + ϕ )
(9)
And Eq. (9) can be established by
θ + ϕ = nπ + π / 2 .
(10)
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Substituting Eq. (10) into Eq. (4), Eq. (11) will be yielded.
A sin(θ ) + B cos(θ ) = R + E ± e,
(11)
when tan(θ /2) is used to express all the trigonometric functions, Eq. (11) can be resolved as shown in Eq. (12). The maximum value and minimum value for tan(θ /2) are denoted as x+ and x− respectively as followed
tan
θ 2
=
A−
A2 + B2 − ( R + E ± e ) = x± , B + (R + E ± e ) 2
(12)
where A2 +B2 -(R +E +e)2 must be greater than 0, the measurable condition of the lever mechanism is
R+e<
A2 + B2 − E.
(13)
Substituting Eq. (12) into Eq. (6), the following Eq. (14) can be obtained.
r tan(θ ) − C/ cos(θ ) + C = 2
r x± − C x± 2 = z± , 1 − x± 2
(14)
where z+ and z- correspond to the maximum value max(z) and minimum value min(z) separately. Then the fluctuating amplitude zA and mean indicating value zM of the sensor can be written as
zA =
r x− − C x− 2 z+ − z− r x+ − C x+ 2 = − 2 2 1 − x+ 1 − x− 2
(15)
zM =
z+ + z− r x+ − C x+ 2 r x− − C x− 2 = + . 2 1 − x+ 2 1 − x− 2
(16)
Eqs. (15) and (16) can be rearranged as
C−
zM ± zA zM ± zA x± 2 − r x± + = 0. 2 2
(17)
Thus, Eq. (17) can be solved as
x± =
r±
r 2 − (zM ± zA )[2C − (zM ± zA )] 2C − (zM ± zA )
(18)
Substituting Eq. (18) into Eq. (12) yields
R+E±e=
B ( 1 − x± 2 ) + 2Ax± . 1 + x± 2
(19)
Then, the accurate analytic expression of the eccentricity e and the section radius R can be got as followed
e=
B ( 1 − x− 2 ) + 2Ax− B ( 1 − x+ 2 ) + 2Ax+ − 2 2 ( 1 + x+ ) 2 ( 1 + x− 2 )
(20)
R=
B ( 1 − x+ 2 ) + 2Ax+ B ( 1 − x− 2 ) + 2Ax− + − E, 2 ( 1 + x+ 2 ) 2 ( 1 + x− 2 )
(21)
where x ± can be calculated from Eq. (18) by using the fluctuating amplitude zA and mean indicating value zM . The above mathematical derivation process has indicated that the analytic relationship between the sensor indicating value z and the section eccentricity e is accurate absolutely. Actually, the mathematical relationships of all the variables are absolutely accurate and this algorithm, which is called as closed analytic algorithm in the following statement, doesn’t have any theoretical principle error. Meanwhile, the radius R of the measured workpiece section can be obtained from this algorithm as well. 3. Sensitivity auditing In order to ensure the measurement accuracy, it is necessary to understand how variation of different parameters, including the basic structural parameters of the lever mechanism and the indicating value of the sensor, affect on the eccentricity e. Therefore, a sensitivity auditing of the parameters must be carried out. Without loss of generality, the sensitivity can be discussed by using following parameter: C = 15 mm, r = 65 mm, E = 20 mm, A = 130 mm, B = 32 mm, R = 25 mm, then one has zA = 0.2543 mm and zM = 6.5389 mm at e = 0.5 mm. Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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3.1. Sensitivity auditing for the proportional algorithm (1) The sensitivity of the fluctuating amplitude zA can be expressed as
∂e = k0 . ∂ zA
(22)
According to the above parameters, one has əe/əzA = 2. (2) The sensitivity of the lever structural size A can be expressed as
∂ e zA = ∂A r
(23)
One has əe/əA= 0.0039. (3) The sensitivity of the lever structural size r can be expressed as
AzA ∂e =− 2 ∂r r
(24)
One has əe/ər=−0.0078. 3.2. Sensitivity auditing for the closed analytic algorithm (1) The sensitivity of the fluctuating amplitude zA can be solved as
∂e ∂ e ∂ x+ ∂ e ∂ x− = + ∂ zA ∂ x+ ∂ zA ∂ x− ∂ zA
(25)
One has əe/əzA = 1.9659 at the above parameters. (2) The sensitivity of the mean indicating value zM can be solved as
∂e ∂ e ∂ x+ ∂ e ∂ x− = + ∂ zM ∂ x+ ∂ zM ∂ x− ∂ zM
(26)
One has əe/əzM =−0.0025. (3) The sensitivity of the lever structural size A is
x− ∂e x+ = − ∂ A 1 + x2+ 1 + x2−
(27)
One has əe/əA= 0.0039. (4) The sensitivity of lever structural size B is
1 − x2− 1 − x2+ ∂e = − 2 ∂ B 2(1 + x+ ) 2(1 + x2− )
(28)
One has əe/əB=−0.0 0 040. (5) The sensitivity of lever structural size C can be solved as
∂e ∂ e ∂ x+ ∂ e ∂ x− = + ∂ C ∂ x+ ∂ C ∂ x− ∂ C
(29)
One has əe/əC= 0.0 0 078. (6) The sensitivity of lever structural size r can be solved as
∂e ∂ e ∂ x+ ∂ e ∂ x− = + ∂ r ∂ x+ ∂ r ∂ x− ∂ r
(30)
One has əe/ər=−0.0076. It can be seen clearly that the sensitivity of the eccentricity e to the same parameter is almost equal in two algorithms, and the eccentricity e is the most sensitive to parameter zA because əe/əzA has the maximum value which is close to 2. Therefore, in order to ensure the precision of the eccentricity e, the measurement accuracy of the sensor must be strictly controlled. The sensitivity of the eccentricity e to the structural sizes of the measuring device are all less than 0.008, which indicates the measurement accuracy is less affected by the precise of machining and assembling of measuring mechanism, and the measuring method has high adaptability. 4. Experimental validation The effectiveness of current closed analytic algorithm in the way of improving the measurement accuracy needs to be verified by experiment. Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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Fig. 5. A stepped shaft for experimental investigation.
4.1. Experimental calibration method The experimental calibration is performed by using a standard smooth shaft with a known radius R and cross-section eccentricity e. If the actual measurement value of the sensor is zc , the fluctuating amplitude and mean value are respectively
zAe = e zM =
max(zc ) − min(zc ) 2
(31)
max(zc ) + min(zc ) . 2
(32)
The theoretical fluctuating amplitude and mean value are separately
zAL = L zM =
r y1 − C y1 2 r y2 − C y2 2 − 1 − y2 2 1 − y1 2
(33)
r y2 − C y2 2 r y1 − C y1 2 + , 2 1 − y2 1 − y1 2
(34)
where y1 and y2 can be calculated by
y1 =
y2 =
A−
A−
A2 + B2 − ( R + E − e ) B + (R + E − e )
(35)
A2 + B2 − ( R + E + e ) . B + (R + E + e )
(36)
2
2
The conversion coefficients of fluctuating amplitude and mean value can be defined as
KA = zAL /zAe
(37)
L e KM = zM − zM .
(38)
As a result, the calibrated fluctuating amplitude and mean value can be determined by
zA = KA
max(zc ) − min(zc ) 2
zM = KM +
max(zc ) + min(zc ) . 2
(39) (40)
4.2. Experimental results and analysis The following instrument and equipment were used in experiment: a lever mechanism for measuring shaft bending deflection, a sensor and its software and hardware supporting system, vernier caliper, steel ruler, micrometer and its magnetic meter base, a stepped shaft specimen with preliminary bending deformation. The structure of the stepped shaft need to satisfy that each section diameter is shown in Fig. 5, the length of each shaft segment is almost equal, the total length is determined by clamping condition, it should be as long as possible. After the shaft being machined, a preliminary bending processing is required, which makes the runout 2e be greater than 6 mm at least at the diameter Ф80mm shaft segment. Eleven sets of data were got by measuring all segment sections of the stepped shaft, and the results are listed in Table 1, where 2e is the practical eccentricity measured by micrometer; 2e0 is the eccentricity calculated by proportional algorithm; 2ep is the eccentricity calculated by closed analytic algorithm; and ∗ represents the data used for experimental calibration. In the experimental system, the fundamental structure parameters are r = 65 mm, A = 130 mm, B = 32 mm, C = 12.9 mm, E = 21.8 mm. By comparing the error values in Table 1, it can be clearly seen that closed analytic algorithm can remarkably reduce the measurement error and the measurement accuracy is improved greatly. Thus, the correctness and validity of current closed analytic algorithm can been fully testified. In addition, the measurement error caused by the closed analytic algorithm is much smaller than the one caused by the proportional algorithm, which shows that closed analytic algorithm has high precision and is effective in shaft deflection measurement. Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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Table 1 Experimental measuring results of the eccentricity. Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Conversion: KA = 0.52354,KM = 9.53839,k0 = 1.00424 ∗
∗
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Conversion: KA = 0.52478, KM = 9.40507, k0 = 1.00659 ∗
1.183 1.985 2.758 2.935 2.548 1.919 1.225 2.348 3.725 5.175 6.275
0.035 0.064 0.077 0.111 0.065 0.069 0.044 0.082 0.172 0.208 0.228
0.0 0 0 0.007 −0.003 0.028 −0.008 0.015 0.010 0.016 0.069 0.062 0.050
1.183 1.985∗ 2.758 2.935 2.548 1.919 1.225 2.348 3.725 5.175 6.275
0.033 0.060∗ 0.071 0.104 0.059 0.065 0.042 0.077 0.164 0.196 0.214
−0.004 0.0 0 0∗ −0.013 0.018 −0.017 0.009 0.006 0.008 0.056 0.044 0.029
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Conversion: KA = 0.52115,KM = 9.35226,k0 = 0.99961
Conversion: KA = 0.52683, KM = 9.32650, k0 = 1.01053
1.183 1.985 2.758 2.935 2.548∗ 1.919 1.225 2.348 3.725 5.175 6.275
0.041 0.073 0.090 0.124 0.076∗ 0.078 0.050 0.093 0.189 0.231 0.256
0.004 0.013 0.005 0.037 0.0 0 0∗ 0.021 0.014 0.024 0.080 0.078 0.069
1.183 1.985 2.758 2.935 2.548 1.919∗ 1.225 2.348 3.725 5.175 6.275
0.028 0.052 0.060 0.093 0.049 0.058∗ 0.037 0.068 0.150 0.177 0.190
−0.009 −0.009 −0.025 0.005 −0.028 0.0 0 0∗ 0.0 0 0 −0.002 0.039 0.021 0.0 0 0
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Practical eccentricity 2e (mm)
Error of proportional algorithm 2e-2e0 (mm)
Error of closed analytic algorithm 2e-2ep (mm)
Conversion: KA = 0.53254,KM = 9.34726,kB = 1.02139 1.183 1.985 2.758 2.935 2.548 1.919 1.225 2.348 3.725∗ 5.175 6.275 ∗
0.016 0.031 0.031 0.063 0.023 0.038 0.024 0.044 0.112∗ 0.123 0.124
Conversion: KA = 0.52923, KM = 9.38275, kB = 1.01490 −0.022 −0.030 −0.055 −0.027 −0.056 −0.020 −0.013 −0.027 0.0 0 0∗ −0.034 −0.067
1.183 1.985 2.758 2.935 2.548 1.919 1.225 2.348 3.725 5.175∗ 6.275
0.023 0.044 0.049 0.081 0.039 0.050 0.032 0.058 0.135 0.155∗ 0.163
-0.014 -0.017 -0.037 -0.007 -0.039 -0.008 -0.005 -0.012 0.024 -0.0 0 0∗ -0.026
The experimental data used for calibration.
5. Conclusion and discussion A contact-detection method with low cost and high accuracy was investigated for the bending deflection measurement of circle-section parts in this paper. The following conclusions are obtained: (1) For the measurement of shaft bending deflection, current closed analytic algorithm is based on accurate mathematical relationships, it doesn’t have any theoretical principle error and the shaft section radius can be measured as well. (2) The measurement accuracy of eccentricity obtained by current measuring method is less affected by the size precision of the measurement mechanism, but it is more sensitive to the accuracy of displacement sensor. (3) It is validated by experiment investigation that the measuring method proposed is efficient and accurate. The present work can not only serve the bending deformation monitoring of circular-section parts, in order to control quality of machining and straightening process, but also on this basis, through calculating the coordinates of the section Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050
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center and further algorithm development, can implement the error measurements of straightness, coaxiality, roundness and cylindricity, which are the important precision measurement items in mechanical engineering. Acknowledgment This work was supported by P.R. China Jilin Province 2012 Postdoctorate Research Starting Funding Project [Grant No. 2012-133-26], and Changchun Research Institute for Mechanical Science Co. Ltd. References [1] H. Martins Gomes, F.J. Flores de Almeida, An analytical dynamic model for single-cracked beams including bending, axial stiffness, rotational inertia, shear deformation and coupling effects, J. Appl. Math. Model. 38 (2014) 938–948. [2] Z.Q. Zhang, Y.H. Yan, H.L. Yang, A simplified model of maximum cross-section flattening in continuousrotary straightening process of thin-walled circular steel tubes, J. Mater. Process. Technol. 238 (2016) 305–314. [3] H.J. Pahk, J.S. Park, I. Ye, Development of straightness measurement technique using the profile matching method, Int. J. Mach. Tools Manuf. 37 (1997) 135–147. [4] E.H.K Fung, Y.K Wong, H.F Ho, Marc P Mignolet, Modelling and prediction of machining errors using ARMAX and NARMAX structures, Appl. Math. Model. 27 (2003) 611–627. [5] M. Sasaki, H. Takebe, K. Hane, Transmission-type position sensor for the straightness measurement of a large structure, J. Micromech. Microeng 9 (1999) 429–433. [6] N.K. Rana, R.R. Sawant, A.H. Moon, A non-contact method for rod straightness measurement based on quadrant laser sensor, in: Proceedings of the IEEE International Conference on Industrial Technology, Mumbai, India, 2006, pp. 2292–2297. [7] Q. Feng, B. Zhang, C. Kuang, A straightness measurement system using a single-mode fiber-coupled laser module, Optics Laser Technol. 36 (2004) 279–283. [8] P. Schalk, R. Ofner, P. O’Leary, Pipe eccentricity measurement using laser triangulation, Image Vis. Comput. 25 (2007) 1194–1203. [9] R.S. Lu, Y.F. Li, Q. Yu, On-line measurement of the straightness of seamless steel pipes using machine vision techique, Sens. Actuators A 94 (2001) 95–101. [10] J. Derganc, B. Likar, F. Pernuš, A machine vision system for measuring the eccentricity of bearings, Comput. Ind. 50 (2003) 103–111.
Please cite this article as: C.-H. Wang et al., An improved high precision measuring method for shaft bending deflection, Applied Mathematical Modelling (2017), http://dx.doi.org/10.1016/j.apm.2017.02.050