Measurement of axial motion yarn tension based on transverse vibration frequency

Abstract

In the process of textile production, the detection of yarn tension is very important to ensure product quality. In order to detect yarn tension efficiently and conveniently, a non-contact detection method based on transverse frequency was established. Firstly, the governing equation of yarn motion was derived by using the Hamilton principle and solved by the Galerkin truncation method. Then, the natural frequencies of yarns with different linear density were calculated according to the yarn characteristic equation. Finally, the fitting formulas of tension, speed, and frequency of yarn axial motion were derived. By measuring the natural frequency of yarn, the yarn tension could be calculated conveniently and quickly. A high-speed camera was used to measure the transverse vibration displacement of yarn and detect the free vibration frequency of yarn. Through experiments, the results of tension sensor and yarn tension detection based on vibration frequency were compared, and the results show that the error was within 5%, which verifies the effectiveness and accuracy of this method.

Keywords

Yarn tension detect non-contact transverse vibration natural frequency high-speed camera

The research on the transverse vibration and stability of moving structures has important application value in practical engineering practice, and its related research has a history of more than 100 years. This problem has always been a challenging subject of concern for the industry, and now there are still a large number of scholars paying close attention to, and investing in, the research on this problem.^1,2 At present, the yarn axial motion system appears in various fields, such as conveyor belt, cable, cableway, tape, paper tape winding, etc.³ Due to its wide application and importance, many scholars have studied the modeling of yarn axial motion system.^4,5 Therefore, using yarn vibration theory as the theoretical basis of yarn non-contact measurement is an effective method. Its principal research is relatively mature and the test equipment is built quickly.

Although the research on the transverse vibration of moving structures now has achieved a series of research results, most of the research focuses on theory and simulation, and there are relatively few experimental research results on the transverse vibration of moving structures. The detection and analysis of vibration signal of moving yarn is an important step in the experimental study of yarn transverse vibration. The detection system is mainly composed of three parts: measurement part, signal acquisition part, and signal analysis and processing part. How to obtain yarn vibration information quickly and accurately is a major challenge for vibration experimental research.

Liang⁶ used the pxi-4472b data acquisition card to install an acceleration sensor at the end of the beam to obtain the acceleration of transverse vibration at the free end of the beam. Jun⁷ selected inductive sensors to measure vibration. In order to avoid interference in the measurement process, the vibration signals were processed and analyzed in the time domain and the frequency domain. Vedrines et al.⁸ measured the transverse vibration of the beam with the Baumer laser displacement sensor. Xia et al.⁹ installed multiple laser displacement sensors at different positions of the yarn to measure the transverse and longitudinal displacement of the axial motion yarn. Doignon et al.¹⁰ proposed a method to measure the vibration amplitude and frequency of the moving plate in real time based on the fast vision system. All of these experiments proved that the vibration frequency of the structure was closely related to the tension and motion speed.

The vibration signal is not useless in engineering. It can be used to calculate structural tension. The method of calculating tension based on vibration measurement has been applied to many occasions. Nam and Nghia¹¹ estimated the tension of the cable by using the measured cable vibration frequency. Dan et al.¹² first established the relationship between bridge stay cable tension and vibration frequency, geometric parameters, and material parameters. The stay cable tension could be calculated by obtaining the stay cable vibration frequency through experiments. Li et al.¹³ proposed a method using five sensors to measure the dynamic behavior of the beam to estimate the tension of the Euler beam with unknown boundary conditions.

Aiming at the transverse vibration of yarn in the process of textile production, through the combination of theoretical modeling, numerical analysis and experimental verification, this article reports our studies of the transverse vibration characteristics of moving yarn according to the complex working conditions in the actual production process. For the yarn motion equation, the coupling of transverse displacement and axial displacement is considered, and the Galerkin method is used to discretize the motion equation to obtain the characteristic equation of the problem, so as to calculate the transverse vibration characteristics of the moving yarn in the frequency domain; the time accumulation integration method is introduced to calculate and analyze the displacement response characteristics of yarn transverse vibration in time domain. For the demand of fast non-contact measurement of moving yarn tension, a yarn tension estimation method based on transverse vibration frequency measurement is proposed, and an open structure yarn transmission experimental platform is designed and built for experimental verification. Through experiments, the non-contact yarn tension measurement and tension sensor proposed in this paper measure and compare the tension of the yarn under the conditions of 40 cN, 60 cN, and 80 cN respectively, and the moving speeds of 2 m/s, 4 m/s, and 6 m/s respectively. The results show that this method can quickly realize the non-contact measurement of yarn tension in textile equipment.

Yarn dynamics model and solution

Model building

The system model during yarn movement is shown in Figure 1. Set the yarn length $L (m)$ , the linear density $ρ (tex)$ , the elastic modulus $E (Gpa)$ , the Poisson's ratio $μ$ , the movement speed $V (m / s)$ , and the initial tension $T (c N)$ . For the convenience of research, this study made the following agreement: the direction of yarn movement was x, and the direction perpendicular to the yarn was y. The variables u and w represented the displacement in the x and y directions respectively, and suppose the moving yarn is Euler-Bernoulli beam.

Figure 1.

Yarn motion system model.

As shown in Figure 1, due to the deformed of the yarn during the movement, when a point $D$ on the yarn moved to $D'$ , u and w respectively indicated the displacement in x and y directions. Since u and w were functions of coordinates x and y, and time t, they could be expressed as:¹⁴

u = u (x, y, t), w = w (x, y, t)

(1)

The displacement vector of point D after yarn deformation was expressed as:

r = (x + u) i + (y + w) j

(2)

The velocity vector of point D could be obtained from the derivative of displacement vector to time:

v = (V + V \frac{\partial u}{\partial x}) i + (\frac{\partial w}{\partial t} + V \frac{\partial w}{\partial x}) j

(3)

where i and j represented unit vectors in the x and y directions.

Considering the geometric nonlinearity caused by yarn deformation, the relationship between strain and displacement of yarn was given according to Karman strain theory:¹⁴

\begin{array}{l} ε_{x} = \frac{\partial u}{\partial x} - y \frac{\partial^{2} w}{\partial x^{2}} + \frac{1}{2} [{(\frac{\partial u}{\partial x} - y \frac{\partial^{2} w}{\partial x^{2}})}^{2} + {(\frac{\partial^{2} w}{\partial x^{2}})}^{2}] \\ ε_{y} = - \frac{1}{2} (\frac{\partial u}{\partial x} - y \frac{\partial^{2} w}{\partial x^{2}}) \frac{\partial w}{\partial x}, ε_{x y} = - \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2} \end{array}

(4)

The yarn stress is:

\begin{matrix} σ_{x} = \frac{E}{1 - μ^{2}} (\frac{\partial u}{\partial x} + μ \frac{\partial w}{\partial y}), σ_{y} = \frac{E}{1 - μ^{2}} (\frac{\partial w}{\partial y} + μ \frac{\partial u}{\partial x}), \\ σ_{x y} = \frac{E}{2 (1 + μ)} (\frac{\partial u}{\partial y} + \frac{\partial w}{\partial x}) \end{matrix}

(5)

According to the Hamilton variation principle:

\int_{t_{1}}^{t_{2}} (δ K - δ U + δ W - δ M) = 0

(6)

where

t_{1}

and

t_{2}

was random time,

δ

was the variation sign,

K

was the yarn kinetic energy,

U

was the potential energy,

W

was the work done by the non-conservative force,

M

was the momentum transferred by the boundary.

The kinetic energy of the moving yarn could be expressed as:

K = \frac{1}{2} ρ \int_{A} v \cdot v d A

(7)

The potential energy of the yarn was:

U = ρ \int_{A} (σ_{x} ε_{x} + σ_{x} ε_{x} + 2 σ_{x y} ε_{x y}) d A

(8)

The work done by non-conservative forces was:¹⁵

W = ρ \int_{A} w \cdot (\frac{\partial w}{\partial t} + V \frac{\partial w}{\partial x}) d A

(9)

where

A = ρ L

The boundary transfer momentum was:

M = ρ L V \int_{0}^{L} v \cdot r d x

(10)

In this study, only the yarn transverse vibration was concerned, the bending stiffness of yarn was ignored, which was a function of coordinates x and y. From Equation (6)–Equation (9) and Equation (10), the yarn motion equation was deduced as follows:

\begin{matrix} ρ L (\frac{\partial^{2} w}{\partial t^{2}} + 2 V \frac{\partial^{2} w}{\partial x \partial t} + V^{2} \frac{\partial^{2} w}{\partial x^{2}}) - ρ \frac{\partial}{\partial x} (σ_{x} w_{x} + σ_{x y} w_{y}) \\ - ρ \frac{\partial}{\partial x} (σ_{y} w_{y} + σ_{x y} w_{x}) = 0 \end{matrix}

(11)

During the actual transmission process of yarn, the yarn moving in the rail was considered as the research object, driving the yarn movement along the x axis at the left and right boundaries, so the transverse displacement of the $x = 0$ and $x = L$ yarn was a fixed value, meanwhile, the yarn stress was T in the direction along the x axis, so the boundary conditions was at x = 0 and x = L:

N_{x x} = T, w = 0, at x = 0, L

(12)

Model solution

In the paper, the Galerkin method was used to solve the equation for the control of yarn motion by multiplying Equation (11) by a weight function and then integrating over the entire yarn area to obtain the equivalent integral weak term form of the equation for yarn transverse motion:

\begin{matrix} ρ L \bar{w} (\frac{\partial^{2} w}{\partial t^{2}} + 2 V \frac{\partial^{2} w}{\partial x \partial t} + V^{2} \frac{\partial^{2} w}{\partial x^{2}}) \\ - ρ \bar{w} (σ_{x} \frac{\partial w}{\partial x} + σ_{x y} \frac{\partial w}{\partial y}) \frac{d w}{d x} \\ - ρ \bar{w} (σ_{y} \frac{\partial w}{\partial y} + σ_{x y} \frac{\partial w}{\partial x}) \frac{d w}{d x} = 0 \end{matrix}

(13)

where

\bar{w}

was the weight function of the transverse displacement.

Since the boundary conditions of the system were considered when deriving the weak form of the equivalent integral of the equation of motion, and the trial function satisfying the boundary conditions could be used as the basis function of the displacement, the yarn transverse movement displacement function could be defined as:

w (x, y, t) = \sum_{i = 0}^{N_{x}} \sum_{j = 0}^{M_{y}} G_{i j}^{w} (t) \sin \frac{(i + 1) π x}{L} Y_{j} (y)

(14)

where

N_{x}

and

M_{y}

were the number of transverse displacement basis functions;

G_{i j}^{w}

was generalized coordinates related to time

t

;

Y_{j} (y)

was trial functions satisfying boundary conditions, and the Legendre polynomial was used:

\begin{matrix} Y_{j} (y) = \sum_{r = 0}^{R} {(- 1)}^{r} \frac{(2 j - 2 r)!}{2^{q} r! (j - r)! (j - 2 r)!} {(y - 1)}^{i - 2 r}, \\ j = 0, 1 \dots M_{y}, \end{matrix}

(15)

R = {\begin{cases} j / 2 if j is even, \\ (j - 1) / 2 if j is odd, \end{cases}

(16)

The corresponding weight function $\bar{w}$ of the transverse displacement $w$ was expressed as:

\bar{w} = \sum_{i = 0}^{N_{x}} \sum_{j = 0}^{M_{y}} {\bar{G}}_{i j}^{w} (t) \sin \frac{(r + 1) π x}{L} Y_{j}^{w} (y)

(17)

where

{\bar{G}}_{i j}^{w} (t)

was an arbitrary time function.

The yarn force balance equation was obtained from Equation (14) – Equation (17):

\sum_{i = 0}^{N_{x}} \sum_{j = 0}^{M_{y}} [\begin{array}{l} m_{i j}^{u} 0 \\ 0 m_{i j}^{w} \end{array}] [\begin{array}{l} {\bar{G}}_{i j}^{u} \\ {\bar{G}}_{i j}^{w} \end{array}] + [\begin{array}{l} c_{i j}^{u} 0 \\ 0 c_{i j}^{u} \end{array}] [\begin{array}{l} {\hat{G}}_{i j}^{u} \\ {\hat{G}}_{i j}^{w} \end{array}] = [\begin{array}{l} f_{i j}^{u} \\ 0 \end{array}]

(18)

Equation (18) could be written as a matrix vector form:

M^{u w} {\bar{G}}^{u w} (t) {+ K}^{u w} {\hat{G}}^{u w} (t) {= F}^{u w}

(19)

where “

-

” in the upper part of

G^{u w}

represented the differential of time, and

G^{u w}

was the vector quantity to be solved for transverse displacement, which was defined as:

G^{u w} = {[\begin{array}{l} G_{00}^{u}, G_{01}^{u}, \dots, G_{N_{y}}^{u} G_{10}^{u}, G_{1 M_{y}}^{u}, \dots, G_{N_{x} M_{y}}^{u} \\ G_{00}^{w}, G_{01}^{w}, \dots, G_{M_{y}}^{w} G_{10}^{w}, G_{1 M_{y}}^{w}, \dots, G_{N_{x} M_{y}}^{u} \end{array}]}^{T}

(20)

$M^{u w}$ and $K^{u w}$ were the mass matrix and the damping matrix of the yarn, respectively, defined as:

M^{u w} = [\begin{matrix} m_{i j}^{u} & 0 \\ 0 & m_{i j}^{w} \end{matrix}] K^{u w} = [\begin{matrix} c_{i j}^{u} & 0 \\ 0 & c_{i j}^{w} \end{matrix}] {,F}^{u w} = [\begin{matrix} \begin{matrix} \begin{matrix} f_{i j}^{u} \\ 0 \end{matrix} \end{matrix} \end{matrix}]

(21)

The elements in Equation (21) were defined as:

\begin{matrix} m_{i j}^{u} = \int_{0}^{L} \int_{t_{1}}^{t_{2}} (ρ L X_{i}^{u} Y_{j}^{u}) d x d y, \\ m_{i j}^{w} = \int_{0}^{L} \int_{t_{1}}^{t_{2}} (ρ L X_{i}^{w} Y_{j}^{w}) d x d y \end{matrix}

(22)

\begin{matrix} c_{i j}^{u} = \int_{0}^{L} \int_{t_{1}}^{t_{2}} (ρ L X_{i}^{u} Y_{j}^{u}) d x d y, \\ c_{i j}^{w} = \int_{0}^{L} \int_{t_{1}}^{t_{2}} (ρ L X_{i}^{w} Y_{j}^{w}) d x d y \end{matrix}

(23)

where

X_{i}^{u}

Y_{j}^{u}

X_{i}^{w}

Y_{i}^{w}

was the basis function to satisfy the heuristic function.

The discrete equation of the transverse vibration of the yarn can be expressed as:

\sum_{i = 0}^{N_{x}} \sum_{j = 0}^{M_{y}} (m_{i j}^{w} {\bar{G}}_{i j}^{w} + 2 V c_{i j}^{w} {\hat{G}}_{i j}^{w} + V^{2} G_{i j}^{w}) = 0

(24)

Equation (24) was written as a matrix vector form:

M^{w} {\bar{G}}^{w} (t) + 2 V K^{w} {\hat{G}}^{w} (t) + V^{2} G^{w} (t) =0

(25)

where

G^{w}

was the vector quantity related to the transverse vibration displacement, defined as:

G^{w} = {[G_{00}^{w}, G_{01}^{w}, \dots, G_{0 M_{y}}^{w}, G_{10}^{w}, G_{11}^{w}, \dots, G_{1 M_{y}}^{w}, \dots, G_{N_{x} M_{y}}^{w}]}^{T}

(26)

Calculation of vibration frequency and modal shape

To obtain the frequency and modal shape of the yarn transverse vibration, the eigenvalues of the vibration discrete Equation (25) needed to be considered, assuming that the solution of Equation (25) was:

G^{w} (t) = G_{0} e^{λ_{n} t}

(27)

where

λ_{n}

was the eigenvalue and

G_{0}

was the eigenvector.

By Equation (25) and Equation (27):

(λ_{n}^{2} M^{w} + 2 V K^{w} λ_{n} + V^{2}) G_{0} =0

(28)

The eigenvalue was generally complex and the vibration frequency was the imaginary part of the eigenvalue. For the motion state of the yarn system, the vibration mode was generally dynamic. In order to facilitate calculation, the vibration discrete Equation (25) was transformed into the spatial state form:

[\begin{array}{l} K^{w} M^{w} \\ M^{w} 0 \end{array}] {\begin{cases} {\hat{G}}^{w} \\ {\bar{G}}^{w} \end{cases}} + [\begin{array}{l} K^{w} 0 \\ 0 - M^{w} \end{array}] - {\begin{cases} G^{w} \\ {\hat{G}}^{w} \end{cases}} =0

(29)

Abbreviated as:

A \hat{S} (t) + B S (t) = 0

(30)

Eigenvalues and eigenvectors could be calculated from Equation (28), therefore, the natural frequencies of the yarn under different velocities and tensions could be obtained. On this basis, only the fitting formulas between vibration frequency, motion speed, and tension were derived, and the approximate tension of yarn could be calculated by using the natural frequencies measured in practice.

Experimental process

In order to verify the accuracy and effectiveness of the non-contact yarn tension detecting method based on vibration frequency proposed in this article, an open yarn tension test platform was established, and the physical figure of the platform is shown in Figure 2. It mainly consists of yarn feeder, guide rail, yarn receiving shaft, load type tension sensor, and high-speed camera. The load type tension sensor was used to measure the actual tension of the yarn with an accuracy of ±0.5%. In the experiment, the transverse displacement of the yarn was measured by pco.dimax high-speed camera, and the sampling frame number was 100 fps. Data acquisition, A/D conversion and data processing were all carried out by UMAC and MATLAB software.

Figure 2.

Test platform for yarn tension.

The calculation method of yarn tension is shown in Figure 3. Firstly, the geometric parameters of the moving yarn were determined $(L, ρ, E, μ)$ , and these parameters were brought into the dynamic equation of the yarn transverse vibration. Through numerical calculation, the free vibration frequency of the yarn under different tension and different moving speed was obtained. According to the numerical calculation results, the fitting formula among the yarn vibration frequency, tension and moving speed was established. The moving speed and transverse displacement position of the yarn were obtained through experiments. Then, the free vibration frequency and forced vibration frequency of the yarn were obtained by processing the vibration signal with the help of the fast Fourier transform (FFT), and then the free vibration frequency of the yarn was obtained by filtering out the forced vibration frequency. Finally, the yarn tension could be obtained by substituting the measured vibration frequency of the yarn into the fitting formula.

Figure 3.

Calculation of yarn tension by vibration method.

The physical parameters of cotton yarn used in the experiment were length $L = 1.2$ m, linear density $ρ = 32$ tex, elastic modulus $E = 22.48$ Gpa,¹⁶ Poisson's ratio $μ = 0.3$ .¹⁶ The yarn moving speed was changed from 0 to 14 m/s, and the yarn tension was changed from 20 cN to 120 cN, so the first order vibration frequency of the yarn under different tension and different speed was obtained. According to the numerical calculation results, the curve between the first order vibration frequency and the tension and speed of the yarn was obtained, as shown in Figure 4. From Figure 4, we can see that the yarn tension is positively related to the system vibration frequency, and the yarn moving speed is negatively correlated with the system vibration frequency.

Figure 4.

First order yarn vibration frequency under different speed and tension.

When the yarn was in a condition of rest, the fitting formula between the first order vibration frequency of the yarn transverse vibration and the tension is as follows:

f_{1} = k_{1} T_{1}^{a}

(31)

When the yarn was in the state of motion, the fitting formula of the first order vibration frequency, moving speed and tension of the yarn transverse vibration is as follows:

f_{2} = k_{2} T_{1}^{b} + k_{3} \frac{V^{2}}{T_{1}^{b}}

(32)

The sum of the coefficient depended on the physical parameters of the yarn. Therefore, once the moving speed and the first vibration frequency of the yarn were obtained, the tension of the yarn can be calculated according to Equation (31) and Equation (32).

Experimental measurement and comparison of yarn tension

Static yarn tension measurement

The detection and analysis of yarn vibration signal in the process of yarn transmission is an important link in the research of yarn transverse vibration. The yarn transverse vibration detection system mainly includes measurement part, signal acquisition part and signal processing part. In the process of the experiment, the high-speed camera was used to measure the transverse vibration displacement of the yarn. The speed measurement was directly obtained from the motor coder driven by the yarn receiving shaft. At the same time, the tension sensor on the experimental platform was used to measure the tension of the yarn directly. The measurement results of the tension sensor were compared with the measurement results of the tension calculation method provided in this article. During the experiment, in order to ensure the reliability of the data, the vibration data of the moving yarn was collected after the vibration was stable, and multiple tests were carried out to ensure its scientific and repeatable.

For the static yarn, the tension change range was set to 20 cN∼120 cN. In order to cause yarn vibration, a certain feature point on the yarn was instantaneously touched within each tension level, and then the yarn displacement was measured with a high-speed camera. According to the time displacement curve of yarn vibration, it was easy to obtain the amplitude-frequency response curve. When the yarn tension was 80 cN, the time displacement curve and the amplitude-frequency response curve of the yarn transverse vibration are shown in Figure 5. The first order vibration frequency of the yarn was detected to be 28.54 Hz. From the fitting Equation (32), the yarn tension could be obtained to be 82.21 cN, with an error of 3.13%. In order to verify the reliability of the tension fitting formula, the yarn density was replaced for testing. Figure 5 shows the displacement response and amplitude-frequency response curve when the linear density was 18 tex and the tension was 80 cN. Figure 6 shows the displacement response and amplitude-frequency response curve when the linear density was 32 tex and the tension was 80 cN. It was found that when the tension was the same, the higher the yarn linear density, then the lower the transverse vibration frequency. Taking these three types of yarn as an example, we changed the yarn tension, and compared the tension of the yarn measured based on vibration in this article with the result directly measured by the tension sensor, as shown in Table 1. The yarn tension measured in this study was consistent with the result measured by the tension sensor, and the error between the two was within 5%, which showed that this method could achieve accurate measurement of yarn tension with different linear density.

Figure 5.

Time curve and amplitude-frequency response curve of yarn vibration displacement when the linear density is 18 tex and the tension is 50 cN. (a) Time displacement curve and (b) amplitude-frequency response curve.

Figure 6.

Time curve and amplitude-frequency response curve of yarn vibration displacement when the linear density is 32 tex and the tension is 50 cN. (a) Time displacement curve and (b) amplitude-frequency response curve.

Table 1.

Comparison between calculation results of static yarn tension and measurement results of tension sensor under different linear density

Yarn linear density (tex)	Tension sensor measurement results $T (C N)$	Vibration frequency $f_{1} (H z)$	Vibration measurement tension $T_{v} (c N)$	Error (%) $\| T_{v} - T \| / T$
18	20	23.17	20.26	1.30
	40	24.86	40.85	2.12
	60	25.42	62.37	3.95
	80	28.15	83.24	4.05
	100	33.62	104.21	4.21
32	20	13.95	20.47	2.35
	40	15.42	40.64	1.60
	60	16.74	62.32	3.87
	80	18.21	81.79	2.24
	100	21.85	104.61	4.61
45	20	8.11	19.74	1.30
	40	9.25	41.45	363
	60	10.47	61.73	2.88
	80	11.62	82.52	3.15
	100	12.79	98.82	1.18

Tension measurement of moving yarn

When the yarn was in the state of motion, the influence of the yarn speed on the vibration frequency should be considered, especially in the case of high-speed motion. Setting the yarn tension as 40 cN, 60 cN, and 80 cN, and the movement speed as 2 m/s, 4 m/s, and 6 m/s, respectively. When the yarn tension was 40 cN and its moving speed was 6 m/s, the vibration displacement curve and the amplitude-frequency response curve of the yarn can be seen in Figure 7. It can be clearly seen from the amplitude-frequency response curve that the free vibration frequency and forced vibration frequency of the yarn were detected. In the experiment, the forced vibration of the yarn mainly came from the rotation of the motor. Therefore, the forced vibration frequency could be calculated according to the motor speed. The short-time Fourier transform (STFT) was used to filter out the forced vibration frequency and retain the free vibration frequency, as shown in Figure 8. Finally, by substituting the measured natural frequency into Equation (32), the change of the tension of the moving yarn with time could be obtained, as shown in Figure 9.

Figure 7.

Time curve and amplitude-frequency response curve of yarn vibration displacement under tension of 40 cN and speed of 6 m/s. (a) Time displacement curve and (b) amplitude-frequency response curve.

Figure 8.

Frequency spectrum at tension of 40 cN and speed of 6 m/s.

Figure 9.

Comparison between tension measurement results based on frequency detection and sensor detection at tension 40 cN and speed 6 m/s.

STFT was carried out in the discrete frequency range when calculating the amplitude-frequency characteristics, so the calculated free vibration frequency was also discrete, changing from one discrete value to another. As the STFT was used to calculate the yarn vibration frequency, it could be observed that the yarn tension curve calculated from the vibration frequency in Figure 9 was not smooth and continuous, but there was a step change. According to the experimental steps and signal data processing steps, the tension calculation results could be obtained when the yarn tension was 60 cN and the speed was 2 m/s, as shown in Figure 10.

Figure 10.

Comparison between tension measurement results based on frequency detection and sensor detection at tension 60 cN and speed 2 m/s.

Due to the uncertainty of the yarn geometry parameters, the error of the fitting formula and the accuracy of the measurement of the yarn transverse displacement by the high-speed camera and the vibration frequency obtained by the data processing, there were some errors in the measurement of the yarn tension based on the vibration. However, the measurement accuracy of this method could also be further improved, such as using more accurate signal processing technology to process the vibration displacement signal. At the same time, although this method can effectively measure the yarn tension, it still has a certain range of applications. With the increase of the speed, the first order frequency of the system vibration decreased gradually, and the first order vibration frequency disappeared at the critical speed. Therefore, in order to detect the first vibration frequency of the system vibration, the yarn moving speed was lower than the instability critical speed.

Conclusion

This article presents a non-contact method to measure the yarn tension based on the vibration frequency. According to the characteristic equation, the natural frequency of the yarn was calculated and fitting formula between the first-order vibration frequency, yarn tension, and motion speed was derived. Based on the fitting formula, the yarn tension can be directly calculated by using the measured yarn vibration frequency and motion speed. Compared with the detection results of contact tension sensor, the error of the method was controlled within 5%, which proved the accuracy and effectiveness of the method proposed in this article. Meanwhile, the method takes into account the effect of material diameter and is applicable to thin materials with different diameter; this method is a non-contact tension detection method, which can quickly detect the tension of the yarn without any impact on the dynamic behavior of the yarn and is convenient for layout. With the improvement of the accuracy of this method, more and more different industrial fields will find it applicable.

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 author(s) disclosed the receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by Zhejiang Provincial Postdoctoral Science Foundation.

ORCID iDs

Yang Li

Qiuyang Zheng

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