Quality and uniformity control of fiber web by roller card system. Part I: dynamic modeling formulation and experimental identification

Abstract

This paper focuses on the uniformity of fiber web output by roller cards, because control of uniform carded fiber webs becomes very important when we want faster and more accurate textile spinning. It will theoretically build the model of the dynamic system, and experimentally verifies the model with an implemented sensor. The system analysis, according to working properties and the active region, is divided into three parts: feed input, main carding, and doffer output. The system input is the mass flow rate of the fiber web per unit area. Firstly, the mathematical model of fibers at the main carding will be obtained, and then the mathematical models of the mass flow rate of fiber web at the input and output will be deduced. The mathematical model of the entire roller card system is finally derived by integrating the aforementioned three parts. The computer simulation confirms that the roller card is a stable system, as was shown in the real system. In the experimental verification, this paper measures the output of fiber web at the doffer with an infrared ray sensor, finds the relationship between the sense signals and mass flow rate of the fiber web, and compares the simulated output of the mathematical model with actual output, in order to verify the accuracy of the theoretical modeling of this system. This theoretic model of the roller card system will be used to control the quality and uniformity of the fiber web, as shown in the subsequent parts of this series of papers.

Keywords

roller card system uniformity of fiber web delay time model mass flow rate of fibers infrared sensor

In the textile industry, the carding process is the core process, from raw materials to finished products. It has a significant effect on the subsequent processes, including spinning speed, quality, and uniformity, which are the most important factors related to the quality of the finished products of spinning. Under higher demand on the quality of textile fabrics, it is necessary to adopt automation design of the roller card to improve the quality of carding. This study focuses on the control of the uniformity of the fiber web by the roller card. In past studies, the delay response in the dynamic display of the mathematical model for fibers in the carding process of the roller card is difficult to accurately express, thereby resulting in the difference between the mathematical model and the actual system, and affecting the design of the controller. Higuchi and Takahashi¹ established a first-order mathematical model of a roller card system, which employed the final value theorem and power spectral density analysis to obtain the delay time and time constant. They analyzed the impacts of four controllers on delay time and external interference, and constructed a mathematical model of the work roller module. Rust and Gutierrez² established a mathematical model of a carding unit in a roller card, and derived the whole roller card system. Their experiments verified and analyzed the variations of output of fiber web density, in order to model the work roller-controlled carding process, improve output response, and enhance productivity. Kuo and Wang³ built a mathematical model of a carding unit, which eliminated interference and noise with state feedback designing the probability observer. They further established a mathematical model of the whole roller card, and utilized a filter to improve the noise interference of the sensor. Hlava⁴ constructed a non-linear mathematical model of a roller card system with delay time, which assumed the control variable to be a constant, rationalized the delay time function, and outputted it as fiber web density. The model was for flat top card. With the focus on the roller card system, this paper examines the transfer function of the relationship between feed roller speed and fiber web output, with the control target as the mass flow rate of fiber web, to deduce the mathematical model of the roller card.

The roller card deduced by this paper is presented in Figure 1. The carding procedure includes feed input, main carding, and doffer output. Through the feed plate, fibers enter the feed roller, which brings fibers into the first take-in for opening and dust removal, and then into the cylinder by the second take-in. After carding by the work-stripper roller and cylinder, fibers are cleaned by the clothing-stripper roller. By the centrifugal effect of the cylinder revolving and the relative action of the flat and doffer at the top of cylinder, fiber web fibers after carding are transferred to the doffer, which outputs fiber web and completes the carding process of the whole roller card. Before establishing the mathematical model of the roller card, this paper defines the physical parameters and symbols of the roller card as follows.
Figure 1.
Schematic diagram of the planar mechanism of the roller card.

Collecting power

Collecting power is defined as the proportion of fibers transferring from the surface of one roller in two neighboring running rollers to another roller due to different relative speeds.³ In this paper, collecting power is calculated as shown in Figure 2, where point A is the action point of the first work roller and cylinder; point B is the action point of the stripper roller and cylinder; point C is the action point of the work roller and cylinder; point D is the action point of the stripper roller and work roller. $Q (t)$ denotes fibers on the cylinder leaving point A for point C. When fibers flow through point B, a part of the fibers is stripped by stripper roller, denoted as m_B, and other fibers passing through point C on the cylinder are carded by the work roller, denoted as m, so that fibers through point B to point C are $m_{B} + Q (t)$ . Collecting power $η_{w}$ denotes the ratio of fibers carded by the work roller and through point C of the cylinder to total fibers (the sum of fibers stripped by the stripper roller and fibers from point A and point B):
$η_{w} = \frac{m}{m_{B} + Q (t)}$
(1)

Figure 2.
Action of the work-stripper cylinder roller.

Attenuation factor

Since each roller has different rotary speeds, when fibers undergo carding through the cylinder and work roller, some fibers remain at the surface of the work roller, while other fibers on the surface of the cylinder are carded by the next group of work rollers.

If k_a is expressed as the attenuation factor, denoting the proportion of fibers on the surface of roller i attenuating to the surface of another roller j due to different surface linear speeds of two neighboring rollers, then the attenuation factor can be calculated as:²
$k_{a} = \frac{v_{i}}{v_{j}}$
(2)
where v_i is the surface linear speed of roller i and v_j is the surface linear speed of roller j.

Delay time

As presented in Figure 3, when fibers are moving in the roller card, rotary frictions of all rollers act with each other, so the conveying time of fibers is delayed. This study deduces the delay times of each roller of the roller card. In rollers of the work-stripper cylinder, action angles $θ,_{} α,_{} β$ of carding by each roller are calculated by the cosine formula, where the rotary speeds are $ω_{w},_{} ω_{s},_{} ω_{c}$ , the radiuses are $r_{w},_{} r_{s},_{} r_{c}$ , and the delay times are $τ_{θ},_{} τ_{α},_{} τ_{β}$ . The gaps between the stripper-work roller, stripper-cylinder roller, and work-cylinder roller are $r_{ws},_{} r_{sc},_{} r_{wc}$ , respectively.
Figure 3.
Carding unit of the work-stripper cylinder.

Dynamic analysis of the roller card

This paper divides a roller card into three parts – input, main carding, and output doffer – to analyze the action flow of fibers at each part of roller card system step by step, and to deduce the mathematical model of the whole roller card.

Feed input

The feed input is as presented in Figure 4, which comprises the feed roller, take-in and work roller.
Figure 4.
Feeding and opening of the roller card.

The system input is defined as the mass flow rate of fibers from the feed plate to the feed roller:
$P_{f} (t) = V_{f} (t) S_{f} \begin{matrix} (g / \sec) \end{matrix}$
(3)
where $S_{f} = \frac{x}{L} (g / m)$ is the fiber count, $V_{f} (t) = r_{f} ω_{f} (t)$ is the speed of the feed roller, $ω_{f} (t)$ is the angular speed of the feed roller, and r_f is the radius of the feed roller.

Fiber web output is drafted by the draft roller of the roller card and the weight of its fiber web is defined as x and the length is defined as L. The mass flow rate of fiber output at the feed roller is $Q_{f} (t)$ :
$Q_{f} (t) = \frac{n π r_{t 1} V_{f} (t) S_{f}}{v_{t 1}}$
(4)
where $v_{t 1}$ is the speed of the first take-in, $r_{t 1}$ is the radius of the first take-in, and n is the revolutions per minute of the first take-in (rpm).

The mass flow rate of fibers flowing through two take-ins is $Q_{t 2 - c} (t)$ ; the attenuation factors are k₁ and k₂. k₁ is the attenuation factor between the first take-in and the second take-in; k₂ is the attenuation factor between the second take-in and the cylinder:
$Q_{t 2 - c} (t) = k_{1} k_{2} Q_{f} (t - τ_{t 1} - τ_{t 2})$
(5)

The delay times passing through two take-ins are $τ_{t 1} = θ_{t 1} / ω_{t 1}$ , $τ_{t 2} = θ_{t 2} / ω_{t 2}$ , respectively, and thus $ω_{t 1}$ is the angular speed of the first take-in and $ω_{t 2}$ is the angular speed of the second take-in.

The mass flow rate of fibers flowing through point A of the cylinder is $Q_{c 1} (t)$ and the delay time by the action angle $θ_{A' A}$ of fibers on the cylinder is $τ_{θ}$ , then
$Q_{c 1} (t) = (1 - η_{wl}) Q_{t 2 - c} (t - τ_{θ})$
(6)
where $η_{w 1}$ is the collecting power of fibers through point A in Figure 4.

By integrating Equations (4)–(6), the relation of output and input of the mass flow rate of feeding fibers at input is
$Q_{c 1} (t) = (1 - η_{w 1}) k_{1} k_{2} \frac{n π r_{t 1}}{v_{t 1}} S_{f} V_{f} (t - τ_{t 1} - τ_{t 2} - τ_{θ} - τ_{sensor})$
(7)
where $τ_{sensor}$ is the time of the fiber web output flowing to the sensor.

Main carding

As presented in Figure 5, structure of main carding includes a cylinder, two worker-stripper rollers, a clothing stripper roller and a doffer.
Figure 5.
Structure of main carding.

Assume the mass flow rate of fibers at the main carding is:⁵
$Q_{c 2} (t) = Q_{c 1} (t) + Q_{w} (t) - Q_{s} (t)$
(8)
where $Q_{c 2} (t)$ is the mass flow rate of fibers passing through point B of the cylinder. After the cylinder operates for a period of time, the mass flow rate $Q_{s} (t)$ of fibers through the clothing-stripper roller results in impurities, whose mass is far less than the sum of $Q_{c 1} (t)$ of the input cylinder and recycling mass flow rate of fiber web $Q_{w} (t)$ :
$Q_{s} (t) << Q_{c 1} (t) + Q_{w} (t)$
(9)

Thus, we can obtain
$Q_{c 2} (t) = Q_{c 1} (t) + Q_{w} (t)$
(10)

By analyzing the flow process of raw fiber fibers between the cylinder, work-stripper roller, clothing-stripper roller, and stripper roller, this paper proposes the following assumptions.⁵
The proportionality constant between $Q_{d} (t)$ and $Q_{c 2} (t)$ is

$K = \frac{Q_{d} (t)}{Q_{c 2} (t)}$
(11)

After the cylinder runs for n rounds, $Q_{w} (t)$ and $Q_{s} (t)$ can be neglected. Deduce with Equations (8) and (11) and the cylinder load, doffer transference, and the mass flow rate of the fiber web at cylinder feedback are as shown in Table 1.

Substituting the mass flow rates of fibers in the last column of Table 1 into Equation (10), we obtain⁵
$\frac{1}{K} Q_{c 1} (t) = Q_{c 1} (t) + \frac{Q_{c 1} (t) [1 - K]}{K}$
(12)

The assumption of Equation (10) is met and the assumption of this paper is proved to be valid and correct.

In the case of a simple system with take-in, cylinder, and doffer-take interaction, after n revolutions within time t, the increase rate ${\overset{\cdot}{Q}}_{d} (t)$ of the fiber web output at the doffer is proportional to the difference of the mass flow rate $Q_{c 1} (t)$ of the fiber web fibers flowing through point A of the cylinder minus the mass flow rate $Q_{d} (t)$ of the fiber web output at the doffer:⁵
${\overset{\cdot}{Q}}_{d} (t) = K [Q_{c 1} (t) - Q_{d} (t)]$
(13)

Thus, the solution can be obtained:
$Q_{d} (t) = Q_{c 1} (t) [1 - e^{- Kt}]$
(14)

Next, we consider a complex system, such as that shown in Figure 1, with take-in, cylinder, doffer, and three stripper-worker rollers.

From Equation (1), we assume that all fiber webs on the work roller are completely stripped by the stripper roller, then
$m_{B} (t) = m (t) = \frac{η_{w} Q (t)}{1 - η_{w}}$
(15)

Since the mass flow rate is proportional to mass:
$\overset{\cdot}{m} \propto m$
(16)

The mass flow rate of fibers at the first group of work-stripper rollers is
$\frac{Q_{c 1} (t)}{K} [\frac{η_{w}}{1 - η_{w}}]$
(17)

The mass flow rate of fibers at the third group of work-stripper rollers is
$\frac{Q_{c 1} (t)}{K} [\frac{3 η_{w}}{1 - η_{w}}]$
(18)

The total mass flow rate of fibers transferring to the doffer with three group work-stripper rollers is
$\frac{Q_{c 1} (t)}{k} = \frac{Q_{c 1} (t)}{K} [1 + (\frac{3 η_{w}}{1 - η_{w}})]$
(19)

$\frac{1}{k} = \frac{1}{K} [1 + (\frac{3 η_{w}}{1 - η_{w}})]$
(20)
where k is the corresponding delay factor.

From Equations (19) and (20), Equation (14) can be rewritten as
$Q_{d} (t) = Q_{c 1} (t) [1 - e^{- Kt}]$
(21)

Substituting Equation (7) into Equation (21), the equation of the mass flow rate between the input and fiber web output of the cylinder is
$\begin{matrix} Q_{d} (t) = Q_{c 1} (t) [1 - e^{- kt}] \\ = (1 - η_{w 1}) k_{1} k_{2} \frac{n π r_{t 1}}{v_{t 1}} S_{f} V_{f} (t - τ_{t 1} - τ_{t 2} - τ_{θ} - τ_{sensor}) \\ \times [1 - e^{- kt}] \end{matrix}$
(22)

Doffer output

The doffer output is as presented in Figure 6. This part discusses the relationship between doffer input and fiber web output. The mass flow rate of the fiber web at the doffer is $Q_{d} (t)$ and the collection power of the doffer is $η_{d}$ . The mass flow rate of the fiber web output by the roller card is $Q_{o} (t)$ ; between this and $Q_{d} (t)$ , the attenuation factor is k_d, the delay time passing through the doffer is $τ_{d}$ , l₁ is the length of the path of the fiber web moving off the first take-in under running, l₂ is the length of the path of the cylinder running, $V_{c} (t)$ is the speed of cylinder. The mass flow rate of the doffer output can be calculated as:⁴
$Q_{o} (t) = k_{d} η_{d} Q_{d} (t - τ_{d})$
(23)

$\frac{Q_{o} (S)}{Q_{d} (S)} = k_{d} η_{d} e^{- τ_{d} s}$
(24)

$k_{d} = \frac{r_{c} ω_{c}}{r_{d} ω_{d}}$
(25)
where $r_{d},_{} ω_{d}$ are the radius and rotary speeds of the doffer, respectively, and
$η_{d} = \frac{l_{1} r_{d} ω_{d}}{l_{1} r_{d} ω_{d} + l_{2} r_{c} ω_{c}},$
(26)

$τ_{d} = \frac{θ_{d}}{ω_{d}}$
(27)

$l_{1} = v_{t 1} t$
(28)

$l_{2} = V_{c} t$
(29)

Figure 6.
Structure of the output doffer.

By integrating Equations (20) and (21), the dynamic equation between the feed roller output of the roller card and the fiber web output of the doffer can be obtained:
$\begin{matrix} Q_{o} (t) = k_{d} η_{d} Q_{d} (t - τ_{d}) = (1 - η_{w 1}) k_{1} k_{2} k_{d} η_{d} \frac{n π r_{t 1}}{v_{t 1}} \\ \times S_{f} V_{f} (t - τ_{t 1} - τ_{t 2} - τ_{θ} - τ_{sensor} - τ_{d}) [1 - e^{- kt}] \end{matrix}$
(30)

The delay time of fibers flowing through the whole roller card is
$T = τ_{t 1} + τ_{t 2} + τ_{θ} + τ_{sensor} + τ_{d}$
(31)

Taking the Laplace transform $V_{f} (t - T) [1 - e^{- kt}]$ :
$L {V_{f} (t - T)} = \frac{e^{- Ts}}{s} V_{f} (s)$
(32)
where $L {f (t)}$ denotes the Laplace transform of $f (t)$ .

From the second shifting theorem:
$L {V_{f} (t - T) e^{- kt}} = \frac{e^{- Ts} e^{- kT}}{(s + k)} V_{f} (s)$
(33)

Taking the Laplace transform Equation (30), based on Equations (31)–(33):
$Q_{o} (s) = (1 - η_{w 1}) k_{1} k_{2} k_{d} η_{d} \frac{n π r_{t 1}}{v_{t 1}} S_{f} V_{f} (s) [\frac{e^{- Ts}}{s} - \frac{e^{- Ts}}{s + k} \frac{1}{e^{kT}}]$
(34)

Approximate the delay time function:
$[\frac{e^{- Ts}}{s} - \frac{e^{- Ts}}{s + k} \frac{1}{e^{kT}}] = \frac{(1 - \frac{1}{e^{kT}}) s + k}{s (s + k) (Ts + 1)}$
(35)

Replacing Equation (35) with Equation (34), we have:
$Q_{o} (s) = (1 - η_{w 1}) k_{1} k_{2} k_{d} η_{d} \frac{n π r_{t 1} S_{f}}{v_{t 1}} V_{f} (s) \frac{[(1 - \frac{1}{e^{kT}}) s + k]}{{Ts}^{2} + (Tk + 1) s + k}$
(36)

The open-loop transfer function of the whole roller card system is
$\frac{Q_{o} (s)}{V_{f} (s)} = (1 - η_{w 1}) k_{1} k_{2} k_{d} η_{d} \frac{n π r_{t 1} S_{f}}{v_{t 1}} \frac{[(1 - \frac{1}{e^{kT}}) s + k]}{{Ts}^{2} + (Tk + 1) s + k}$
(37)

Result and discussion

In this paper, the relationship between feed roller input of the roller card and fiber web output of the doffer is mathematically derived. The evaluation of fiber web uniformity is the critical issue in this study. Huang and Lin⁶ have proposed an image process technique for inspecting defects on nonwovens. However, the inspected defect indirectly relates to uniformity. In this paper, the uniformity of the output fiber web can be denoted by the coefficient of variation (CV). The larger the CV, the more non-uniform the fiber web is:
$CV = \frac{σ}{x} \cdot 100 %$
(38)
where σ is the standard deviation and $\bar{x}$ is the mean:
$where σ = \sqrt{\frac{\sum_{i = 1}^{n} (x_{i} - \bar{x})^{2}}{n_{s} - 1}}$
(39)
where n_s is the number of samples in the experiment.

The design of a cotton web density sensing method

The parameters of the roller of the roller-type carding machine are as shown in Table 2: the cotton web fiber material is polypropylene (76 mm, 3 d), and the cotton web density measurement hardware design is as shown in Figure 7. As can be seen, the infrared light and photo-sensors are fixed on the same plane in fixed distance d (3.5 mm in this paper) to prevent the sensors from the interference of the cotton web and avoid the impact on the sensor signal measurement by the cotton web residuals at the transmitting and the receiving ends. Adjusting d₁ (1.3 mm in this paper) can realize the effective measurement of the sensors and allow the sensors to read the optimal original data to project the infrared light to the carding roller. The sensors then collect the signals reflected by the carding roller to completely collect the maximum energy released by the reflected light.
Figure 7.
The cotton web density measurement hardware design.

Table 1.
Mass flow rate of fiber web load of each part of the roller card

Rotation Cylinder output Doffer Web feedback

$Q_{c 1} (t)$ $Q_{c 2} (t)$ $Q_{d} (t)$ $Q_{w} (t)$

1 $Q_{c 1} (t)$ $Q_{c 1} (t)$ ${kQ}_{c 1} (t)$ $[1 - k] Q_{c 1} (t)$

2 $Q_{c 1} (t)$ $Q_{c 1} (t) {1 + [1 - k]}$ ${kQ}_{c 1} (t) {^{} 1 + [1 - k]}$ $[1 - k] {1 + [1 - k]} Q_{c 1} (t)$

3 $Q_{c 1} (t)$ $Q_{c 1} (t) {1 + [1 - k] + [1 - k]^{2}}$ $Q_{c 1} (t) {^{} 1 + [1 - k] + [1 - k]^{2}}$ $[1 - k] {^{} 1 + [1 - k] + [1 - k]^{2}} Q_{c 1} (t)$

⋮ ⋮ ⋮ ⋮ ⋮

$n \to \infty$ $Q_{c 1} (t)$ $Q_{c 1} (t) \frac{{^{} 1 - [1 - k]^{n}}}{k}$ $Q_{c 1} (t) {^{} 1 - [1 - k]^{n}}$ $[1 - k] \frac{{^{} 1 - [1 - k]^{n}}}{k} Q_{c 1} (t)$

$[1 - k]^{n} \to 0$ $Q_{c 1} (t)$ $\frac{1}{k} Q_{c 1} (t)$ $Q_{c 1} (t)$ $\frac{[1 - k]}{k} Q_{c 1} (t)$

Table 2.
The parameters of the roller of the roller-type carding machine parts of the roller gauge (unit: 1/1000 inch)

Freed roller and take-in 12”

Take-in and cylinder 9”

Stripper roller and cylinder 10”

Work roller and cylinder 12”

Freed and work roller 4”

Cylinder and doffer 6”

Clothing take-in roller and cylinder 7”

Infrared light optic sensor

The infrared light optical sensor used in this experiment outputs analogous transmission signals. After voltage conversion by using the internal amplifier, it outputs the analogous voltage. Then, the voltage measured by the sensor, the relationship curve of the voltage measurement, and cotton web density can be displayed by a PC in real time after the analog-to-digital conversion card. Due to the sensor sensitivity and interference of the motor input, it results in apparent noise. Hence, the output signals must be filtered by the filter. For this experiment we selected the Butterworth filter after obtaining the corner frequency in accordance with the transfer function of the system mathematical model.

Direct current servo motor and driver

The direct current (DC) servo motor used in this experiment has speed feedback, and thus the output torque within the torque range is linear. The servo motor includes the speed reduction gear and encoder. The motor speed can be controlled by a DC servo motor controller receiving the signals of the servo motor encode. The output voltage of the DC servo motor is in the range of 0–24 V and the output voltage of the A/D-D/A card is in the range of 0–10 V. When the output voltage of the A/D-D/A card is 10 V, the servo motor will output the voltage of 24 V. The control input voltage is in the range of 0–10 V.

Motion control interface card

In this experiment, a motion control interface card was combined with a real simulation software VIsSim to establish a hardware-in-the-loop (HIL) real time graphical control system where the design control parameters can be dynamically changed online to observe the system response behaviors in real time. The specifications of the motion control interface card are as shown in Table 3.
Table 3.
The specifications of the motion control interface card

Pulse Width Modulation (PWM) Output 4CH 12 Bit resolution

Incremental encoder input 4CH 32 Bit counter

Analog output 4CH 12 Bit

Analog input 16CH 12 Bit

Digital output / input 32CH two-way

Counter 2CH 16 Bit

Cotton web input/output and voltage data

The signals acquired by the infrared optical sensor may differ under different testing environments. Therefore, prior to the experimental signal measurement, the basic value measurement of the carding machine without load should be conducted to create the data accuracy in the experimental process. The basic voltage measured after correction was the benchmark value for signal measurement. The experimental steps are as follows: the data read when the empty carding machine operates at 3.5 ms is the acquired signal; when the measurement time is 100 s, since the speed of the carding machine can only be stabilized in a period of time after the start of the empty carding machine, the average value of the operating data of the empty carding machine is the corrected value; the cotton web data read when measuring the cotton web density should subtract the corrected value to obtain the real cotton web relative output voltage.

The relationship between cotton web density output and voltage

In this experiment, the sensors were installed in the draft roller. Firstly, the basic voltage values were measured with the readings of the sensors as the target values. When the signal reached the target value, the weight sensor was used to measure the output cotton web weight. The measurement area was cotton web length at 9.734 m, and cotton web width at the distance between the sensor light source and transmitter of 1.3 mm. The mass flow rate of each unit area of $0.002 m^{2}$ was the cotton web density. According to the relationship between cotton web density and the voltage data corresponding to the sensors, a linear regression equation is obtained as $y = 0.012 x - 0.011$ , where y is the voltage reading and x is the cotton web density, as shown in Figure 8.
Figure 8.
Average voltage corresponding to fiber web density.

When there are significant changes in the surface structure of the fiber web, voltages of the sensor would also change greatly; thus, the CV becomes larger. The uniformity obtained by the CV of voltages of the sensor is called fiber web uniformity.

This paper verifies the mathematical model by experiments, in which the physical parameters of each part of the roller card are as presented in Table 4.
Table 4.
Parameters of each part of the roller card

Components Radius (mm) Velocity (rpm)

Feed roller 20 10

Take-in roller 65 98

Cylinder 190 182

Carding roller 55 10

Stripper roller 33 370

Doffer 160 10

Draft roller 310 12

Substituting the physical parameters of the roller card from Tables 5 and 6 into (31), we can obtain the second-order open-loop transfer function $Q_{o} (s) / V_{f} (s)$ of the whole roller card system as follows:
$\frac{Q_{o} (s)}{V_{f} (s)} = \frac{0.0328 s + 0.01774}{7.091 s^{2} + 4.758 s + 0.53}$
(40)

Table 5.
Angles of the active region of roller card fibers

Symbol Angle (degree)

$θ_{t 1}$ 90

$θ_{t 2}$ 120

$θ_{A' A}$ 22.3

α 74.2

β 83.5

$θ_{d}$ 120

Table 6.
Parameter table of the whole roller card

Delay time (s) $τ_{c}$ $τ_{w}$ $τ_{s}$ $τ_{t 1}$ $τ_{t 2}$ $τ_{θ}$ $τ_{sensor}$ $τ_{d}$

0.02 4.76 0.125 0.176 0.22 0.025 0.0178 6.67

Attenuation factor $\begin{matrix} k_{1} = 1 & k_{2} = 0.311 & k_{d} = 21.61 \end{matrix}$

Collecting power $η_{w 1} = 0.54, η_{w} = 0.54,_{} η_{d} = 0.0857$

Transfer factor $k = 0.53$

Delay factor $D = 2.02$

Feeding speed $V_{f} = 0.0137$

Sensor gain $k_{s} = 6$

Fiber count $S_{f} = 0.3355$

Cylinder speed $n = 25.97 rpm$

Its unit step response is as presented in Figure 9.
Figure 9.
The simulation response of the roller card.

Figure 10 presents the comparison of the simulation response of the mathematical model of the fiber web of roller card with the fiber web output of the actual dynamic roller card. The output is the voltages of the infrared ray sensor. It proves that the output of the proposed mathematical model agrees with the output of the real system.
Figure 10.
Comparison of actually measured output response and mathematical model output.

Conclusion

This paper deduces the mathematical model by the variations in the mass flow rate of raw fibers flowing through each part of the roller part, and uses the changes in voltages of the infrared ray sensor to determine the fiber web output. The unit step response confirms the accuracy of the mathematical model of the deduced second-order system. In addition, the output response indicates that this system is a stable system with delay time. The mathematical model of this roller card can be used to design a controller based on the variation rate of the fiber web in order to improve the output fiber web uniformity.

Rotation	Cylinder output	Doffer	Web feedback
$Q_{c 1} (t)$	$Q_{c 2} (t)$	$Q_{d} (t)$	$Q_{w} (t)$
1	$Q_{c 1} (t)$	$Q_{c 1} (t)$	${kQ}_{c 1} (t)$	$[1 - k] Q_{c 1} (t)$
2	$Q_{c 1} (t)$	$Q_{c 1} (t) {1 + [1 - k]}$	${kQ}_{c 1} (t) {^{} 1 + [1 - k]}$	$[1 - k] {1 + [1 - k]} Q_{c 1} (t)$
3	$Q_{c 1} (t)$	$Q_{c 1} (t) {1 + [1 - k] + [1 - k]^{2}}$	$Q_{c 1} (t) {^{} 1 + [1 - k] + [1 - k]^{2}}$	$[1 - k] {^{} 1 + [1 - k] + [1 - k]^{2}} Q_{c 1} (t)$
⋮	⋮	⋮	⋮	⋮
$n \to \infty$	$Q_{c 1} (t)$	$Q_{c 1} (t) \frac{{^{} 1 - [1 - k]^{n}}}{k}$	$Q_{c 1} (t) {^{} 1 - [1 - k]^{n}}$	$[1 - k] \frac{{^{} 1 - [1 - k]^{n}}}{k} Q_{c 1} (t)$
$[1 - k]^{n} \to 0$	$Q_{c 1} (t)$	$\frac{1}{k} Q_{c 1} (t)$	$Q_{c 1} (t)$	$\frac{[1 - k]}{k} Q_{c 1} (t)$

Components	Radius (mm)	Velocity (rpm)
Feed roller	20	10
Take-in roller	65	98
Cylinder	190	182
Carding roller	55	10
Stripper roller	33	370
Doffer	160	10
Draft roller	310	12

Symbol	Angle (degree)
$θ_{t 1}$	90
$θ_{t 2}$	120
$θ_{A' A}$	22.3
α	74.2
β	83.5
$θ_{d}$	120

Delay time (s)	$τ_{c}$	$τ_{w}$	$τ_{s}$	$τ_{t 1}$	$τ_{t 2}$	$τ_{θ}$	$τ_{sensor}$	$τ_{d}$
Attenuation factor	$\begin{matrix} k_{1} = 1 & k_{2} = 0.311 & k_{d} = 21.61 \end{matrix}$
Collecting power	$η_{w 1} = 0.54, η_{w} = 0.54,_{} η_{d} = 0.0857$
Transfer factor	$k = 0.53$
Delay factor	$D = 2.02$
Feeding speed	$V_{f} = 0.0137$
Sensor gain	$k_{s} = 6$
Fiber count	$S_{f} = 0.3355$
Cylinder speed	$n = 25.97 rpm$

Footnotes

Funding

This work was supported by the National Science Council of the Republic of China (Grant No. NSC 97-2221-E-011 -030 -MY3).

References

Higuchi

Takahashi

. Automatic control of roller cards. J Textil Mach Soc Jpn 1964; 10: 50–58.

Rust

Gutierrez

. Mathematical modeling and simulation for carding. Textil Res J 1994; 64: 573–578.

Kuo

CFJ

Wang

. Stochastic control of card output density. Textil Res J 2001; 71: 195–200.

Hlava

. Anisochronic modeling of a flat-top carding machine. the 12th international DAAAM symposium, intelligent manufacturing & automation: Focus on precision engineering 24–27th October 2001, pp. 24–27.

Carl

. Fundamentals of spun yarn technology. London, CRC Press, 2003.

Huang

Lin

. Image inspection of nonwoven defects using wavelet transforms and neural networks. Fibers Polym 2008; 9: 633–638.

Freed roller and take-in	12”
Take-in and cylinder	9”
Stripper roller and cylinder	10”
Work roller and cylinder	12”
Freed and work roller	4”
Cylinder and doffer	6”
Clothing take-in roller and cylinder	7”

Pulse Width Modulation (PWM) Output	4CH 12 Bit resolution
Incremental encoder input	4CH 32 Bit counter
Analog output	4CH 12 Bit
Analog input	16CH 12 Bit
Digital output / input	32CH two-way
Counter	2CH 16 Bit

Delay time (s)	$τ_{c}$	$τ_{w}$	$τ_{s}$	$τ_{t 1}$	$τ_{t 2}$	$τ_{θ}$	$τ_{sensor}$	$τ_{d}$
Delay time (s)	0.02	4.76	0.125	0.176	0.22	0.025	0.0178	6.67
Attenuation factor	$\begin{matrix} k_{1} = 1 & k_{2} = 0.311 & k_{d} = 21.61 \end{matrix}$
Collecting power	$η_{w 1} = 0.54, η_{w} = 0.54,_{} η_{d} = 0.0857$
Transfer factor	$k = 0.53$
Delay factor	$D = 2.02$
Feeding speed	$V_{f} = 0.0137$
Sensor gain	$k_{s} = 6$
Fiber count	$S_{f} = 0.3355$
Cylinder speed	$n = 25.97 rpm$