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Abstract

A quasi-static method to analyze equilibrium path and trajectory deflections for the load of polar crane in nuclear power plant is presented based on three-dimensional analyses. The governing equations of equilibrium path and the compatibility conditions of the rope–pulley system are provided in the method; properties and formulations of the variables involved in these equations are studied in detail. A method to describe general rope–pulley system is given, and a numerical method is derived to solve the common tangent between spatial pulleys, which was conforming to the rope reeving. Moreover, the rope lengths wound by the drum and left in the system are computed, and the transitive relation for rope velocities between two sides of the pulley is derived. Two numerical examples prove that the proposed method is reasonable and valid, and to a large extent, the method is universal and can be applied to general rope–pulley systems.

Keywords

Rope–pulley system equilibrium path trajectory deflections common tangent between spatial pulleys polar crane

Introduction

Polar cranes in nuclear power plants (NPP) are type of special cranes for replacing the reactor fuel or maintaining the heavy equipment. Due to the special working circumstance, the cranes should locate the load in a high degree of accuracy. For example, the lateral displacement of the load for some type of polar cranes in service should not be even more than 5 mm while lifting 10 m height. To meet the accurate and reliable requirements, one should know how the load to motion when the crane works.

At present, there are many researches about cranes. Some studies^1–4 focused on the control of the load deflection considering the trolley moving while the crane hoisting or lowering, while some studies^5–8 discussed the dynamic responses of the crane structure. These studies solved many problems about the operation schemes and structural designs of the cranes. However, most of these studies simplified the rope–sheave system as a pendulum or double pendulum system or considered the plurality of ropes as a flexible cable,^9–12 which is obviously not adapted to the cranes requiring high accuracy. In fact, the load’s movement is determined by the interactions between ropes and pulleys in the hoisting system. Although many achievements on the interactions have been accumulated, much work about combining the achievements with the large hoisting system of specific characteristics remains to be finished.

McDonald and Peyrot^13,14 presented a pulley element for exact analysis of cables rolling on pulley, in which the internal forces and the tangent stiffness matrix corresponding to an equilibrium configuration are provided. Based on the study, the further work aiming at the possible application to practical cable systems is advanced,¹⁵ and the finite element analysis with three-node element is conducted.¹⁶ However, the rope–pulley system was treated in a plane while deriving the internal force equilibrium in these papers. Aufaure^17,18 presented a finite element formulation for a length of cable passing through a pulley to study the deformation and dynamic behavior of structures and treated the pulley as a particle to analyze the movement, which neglected many details. Zhou et al.¹⁹ developed sliding cable elements and pulley models using the massless spring approach for the analysis of parachute systems. By previously creating markers in the MSC.ADAMS software package, Wang et al.²⁰ analyzed the rope wound along helix based on the concentrated mass theory with multi-degree of freedom and established a parametric model which is applicable to the case of less pulleys. Ju and Choo²¹ proposed a parameterized super element formulation for modeling the multiple-pulley cables; however, there was no further study about the movement of the pulley. Hong and Cipra²² presented a program method to analyze the motion of complex cable–pulley mechanism, which could obtain the relationships of each variable; however, the method could not be applied to spatial rope–pulley system.

The trajectory deflections of the load are crucial parameters for the polar cranes requiring precise location. Meanwhile, the trajectory deflections are systematic, which are determined by the factors such as the operating modes, the models, and layouts of pulleys and the reeving ways of ropes. Due to the position changes of the movable pulleys in hoisting, the tension of each rope changes constantly, which makes it difficult to solve the load trajectory. Based on three-dimensional (3D) analyses, this article presents a quasi-static method for analyzing the load trajectory deflections of polar crane in NPP taking into account the detailed structure of the hoisting mechanism. The method considers the reeving way of each rope that connects spatial pulleys and then can accurately calculate the forces on the hook block. Compatibility conditions of the rope lengths are provided to make the solving conditions complete. In the method, the load trajectory under uniform lifting or lowering speed is discussed through analyzing the attitude and displacement of the hook block.

Equilibrium path equations

Generally, the hoisting equipment of the polar crane in NPP consists of drum, fixed pulleys, movable pulleys mounted on the hook block, and ropes reeving through the pulleys. Figure 1 shows the schematic diagram of the main hoisting mechanism for some type of polar crane in NPP. We can see that two ropes start from the drum, go through the pulleys in a certain order, and anchor to fixed points. The movable pulleys are identified by numbers 1–12 and the fixed pulleys by numbers 13–24, and number 22 identifies a guide pulley which can be treated as a special fixed pulley. The load ascends or descends in accordance with the drum winding or releasing the ropes.

Figure 1.

Schematic view of the main rope–pulley system for polar crane in NPP.

Like most of the cranes, polar crane in NPP cannot ensure the load hoisting in the vertical direction. The lateral displacement of the load is the trajectory deflection. The bigger accelerator of the hook block causes greater deflection in general. Therefore, while the system is working, the operators strive to keep the motion smooth for the hook block. Under this condition, the external forces on the hook block are in equilibrium, and the trajectory of the hoisting point is the equilibrium path of the polar crane. As the directions of the rope tensions probably change while hoisting, the trajectory of the load would naturally be out of vertical. As a result, the trajectory deflections are irrelevant to the operation process but mainly determined by the mechanic designs, the relative positions of the drum and pulleys, and the reeving ways of the ropes, which should necessarily be taken into account in designing the polar crane in NPP.

The external force on the hook block contains load weight, own gravity, and the contact forces acting on the movable pulleys, which can be equivalent to the rope tensions at both sides of the pulleys.

To describe the load trajectory, we create a reference coordinate system $({e'}_{1}, {e'}_{2}, {e'}_{3})$ consolidated on the hook block at the hanging point as shown in Figure 2. $(e_{1}, e_{2}, e_{3})$ is the initial coordinate system and parallel with the global coordinate system. $e_{1}$ , $e_{2}$ , and $e_{3}$ are the unit vectors toward the front, the upward, and the left, respectively. Regarding the hook block as a rigid body, its position can be described by the origin displacement $u$ and the rotation vector $θ$ at any given time. $u$ and $θ$ can be expressed in component forms as follows

u = u_{1} e_{1} + h e_{2} + u_{3} e_{3}

(1)

θ = θ_{1} e_{1} + θ_{2} e_{2} + θ_{3} e_{3}

(2)

where $u_{1}$ denotes the fore-and-aft trajectory, $u_{3}$ denotes the right-and-left trajectory, and h is the hoisting height. h is a known parameter to describe the equilibrium path of the load.

Figure 2.

Motion and stress of the hook block.

The variable $θ$ describes the entire rotation of the hook block, and it is so small that the relevant transforming matrix, which transforms a vector from the current coordinate system to the global coordinate system, can be approximated to²³

R \approx [\begin{matrix} 1 & - θ_{3} & θ_{2} \\ θ_{3} & 1 & - θ_{1} \\ - θ_{2} & θ_{1} & 1 \end{matrix}]

(3)

The external forces on the hook block include the weight $G$ of the load, the weight $m_{b} g$ of the hook block, the resultant force $F_{i}$ , and resultant moment $M_{i}$ on each movable pulley’s shaft as shown in Figure 2. While the hook block is in balance, all the external forces form an equilibrium system

\sum_{i} F_{i} + m_{b} g + G = 0

(4)

\sum_{i} (q_{i} \times F_{i} + M_{i}) + m_{b} q_{c} \times g = 0

(5)

where $q_{i}$ is the center of pulley i in the current coordinate system and $q_{c}$ is the centroid vector of the hook block. Initially, they are constant and denoted by ${\bar{q}}_{i}$ and ${\bar{q}}_{c}$ , respectively. Then, we have the relations between the two sets of vectors as follows

q_{i} = R {\bar{q}}_{i}; q_{c} = R {\bar{q}}_{c}

(6)

It can be seen that the directions of the forces in equation (5) is coupled with angles $θ_{1}$ , $θ_{2}$ , and $θ_{3}$ . In fact, the forces in equations (4) and (5) are determined by the position of hook block and updated with the pace of load hoisting. The resultant force $F_{i}$ and resultant moment $M_{i}$ are functions of the displacements $u_{1}$ , h, and $u_{3}$ and the rotation angles $θ_{1}$ , $θ_{2}$ , and $θ_{3}$ . Equations (4) and (5) describe the conditions that the displacement of the hanging point and the rotation angles of the hook block should be satisfied in implicit form, which are so-called the equilibrium path equations.

Tangent between spatial pulleys

While hoisting, the rope starts to contact the pulley at the rope inlet point and to leave the pulley at the rope outlet point. The positions of the inlet and outlet points, which determine the application points and directions of rope tensions, are essential parameters to describe the rope–pulley system. Comparing to the load, the rope connecting two pulleys has a very small mass and can be considered as a straight line. Besides, the width of the pulley groove is quite small in contrast with the hoisting height. Therefore, we can treat the pulleys as spatial circles and the rope as common tangent between the circles to solve the tangent points, which are the inlet point and outlet point.

As shown in Figure 3, suppose two spatial circles with radii $r_{1}$ and $r_{2}$ denote their centers by vectors $o_{1}$ and $o_{2}$ , respectively, in the global coordinate and establish the reference coordinates $(e_{1}, e_{2})$ and $({\bar{e}}_{1}, {\bar{e}}_{2})$ in each circle’s surface. $e_{3} = e_{1} \times e_{2}$ and ${\bar{e}}_{3} = {\bar{e}}_{1} \times {\bar{e}}_{2}$ are the normal vectors of these circles. $r_{1}$ and $r_{2}$ are the radius vectors from the circle centers to the corresponding tangent points. Generally, the distance between the two centers is bigger than the sum of the two radii. To describe the tangent point, the orientation angle is introduced as follows: taking circle $o_{1}$ , for example, a line coinciding with $e_{1}$ rotates from $e_{1}$ to the tangent point in the circle and sweeps out an angle $β_{1}$ , where $β_{1}$ is the orientation angle of the tangent point, whose range is $[0, 2 π]$ . If we order $x_{1} = \cos β_{1}$ , $y_{1} = \sin β_{1}$ , $x_{2} = \cos β_{2}$ , and $y_{2} = \sin β_{2}$ , the radius vectors will be

r_{1} = r_{1} (y_{1} e_{2} + x_{1} e_{1})

(7)

r_{2} = r_{2} (y_{2} {\bar{e}}_{2} + x_{2} {\bar{e}}_{1})

(8)

Figure 3.

Common tangent of two spatial pulleys.

Since the common tangent is perpendicular to the radius vectors, we have

r_{1} \cdot (o_{1} - o_{2} + r_{1} - r_{2}) = 0

(9)

r_{2} \cdot (o_{1} - o_{2} + r_{1} - r_{2}) = 0

(10)

Thus, the equation for common tangent is

{\begin{matrix} p_{12} y_{1} + p_{11} x_{1} - r_{2} w = - r_{1} \\ p_{22} y_{2} + p_{21} x_{2} + r_{1} w = r_{2} \\ x_{1}^{2} + y_{1}^{2} = 1 \\ x_{2}^{2} + y_{2}^{2} = 1 \\ w = x_{1} (a_{11} x_{2} + a_{12} y_{2}) + y_{1} (a_{21} x_{2} + a_{22} y_{2}) \end{matrix}

(11)

where the coefficients $a_{ij} = e_{i} \cdot {\bar{e}}_{j}$ , $p_{1 k} = (o_{1} - o_{2}) \cdot e_{k}$ , and $p_{2 k} = (o_{1} - o_{2}) \cdot {\bar{e}}_{k}$ . This is a nonlinear equation group hardly to get the analytical solution; meanwhile it is difficult to solve the numerical solution.

There are four types of common tangent between two spatial pulleys, which are separately called the common tangent of right external, left external, right internal, and left internal corresponding to k = 1, k = 2, k = 3, and k = 4 as shown in Figure 4. It is meaningless for the solution inconsistent with the type of the common tangent. The nonlinear equation (11) has four solutions, and we had to consider the initial value approximate to the specified solution as much as possible to obtain the specified solution.

Figure 4.

Types of the common tangent between pulleys.

While the two spatial pulleys are coplanar or paralleled to each other, the equation for common tangent has analytical solutions. We take the corresponding analytical solutions as the initial value to solve equation (11) for the specified numerical solution, which is proved to be feasible and effective. How to solve the initial value is expounded in Appendix 1, and a number matrix to describe the rope–pulley system is introduced in detail, which is convenient for programming solution.

Transmission of tensions at two sides of pulley

The rope in the branch can be divided into two parts: ropes wrapped on pulleys and ropes hung in space. As for pulley i, the dividing points of the rope are the inlet point and outlet point denoted by $p_{i}^{s}$ and $p_{i}^{t}$ , respectively, as shown in Figure 5. Generally, the pulley rotates slowly, and the frictional force between the pulley and rope is big enough to prevent relative slip. The rope’s influence on the pulley’s movement is transmitting the rope tension $f_{i}^{s}$ and $f_{i}^{t}$ to the pulley.

Figure 5.

Inlet point and outlet point of pulley.

The rope changes from straight to curving at inlet point of the pulley, and at outlet point changes from curving to straight. Because of the rope’s some resistance to deformation, there is a small transition curve between the straight rope and the curving rope. The transition curve diminishes the distance between the straight rope and the pulley’s center at the outlet point, otherwise extends the distance at the inlet point. The diminished distance is denoted by $δ r_{i}$ , and the extended distance is approximated to $δ r_{i}$ , where $δ$ is a ratio coefficient and $r_{i}$ is the pulley’s radius as shown in Figure 6.

Figure 6.

Stiffness resistance of rope.

Regarding the pulley as a rigid body, the resultant force $F_{i}$ and resultant moment $M_{i}$ on the pulley’s shaft can be written as follows

F_{i} = f_{i}^{s} + f_{i}^{t} + m_{i} g + m_{i} a_{i}

(12)

M_{i} = (1 + δ) r_{i}^{s} \times f_{i}^{s} + (1 - δ) r_{i}^{t} \times f_{i}^{t} - {\bar{J}}_{i} ω_{i} (ω \times h_{i})

(13)

where $m_{i} g$ is the pulley’s weight; $a_{i}$ is the pulley’s accelerator; and $r_{i}^{s}$ and $r_{i}^{t}$ are the vectors from the pulley’s center to the inlet point and outlet point, respectively. The last terms of equations (12) and (13) denote the inertia force and the gyro effect of the pulley, respectively. As the hoisting speed is slow, the pulley’s inertia force and the gyro effect can be neglected.

As shown in Figure 5, the tangent directions at the inlet point and the outlet point in the pulley’s surface can be expressed as follows

t_{i}^{s} = \frac{1}{r_{i}} (h_{i} \times r_{i}^{s})

(14)

t_{i}^{t} = \frac{1}{r_{i}} (h_{i} \times r_{i}^{t})

(15)

The corresponding tangent components of the rope tensions are

T_{i}^{s} = - f_{i}^{s} \cdot t_{i}^{s}

(16)

T_{i}^{t} = f_{i}^{t} \cdot t_{i}^{t}

(17)

Dot-multiplying both sides of equation (13) by the unit vector $h_{i}$ leads to the equation for moment equilibrium in the direction of the pulley’s shaft

M_{f} = [T_{i}^{t} (1 - δ) - T_{i}^{s} (1 + δ)] r_{i}

(18)

In fact, the left term of the equation is resisting moment, which is caused by friction between the pulley and the shaft and can be expressed by the frictional resistance coefficient $η_{f}$

M_{f} = η_{f} T_{i}^{s} r_{i}

(19)

In practical engineering, $η_{f}$ is usually between 0.014 and 0.0145.²⁴ $M_{f}$ is the moment that is transmitted to the hook block. By ordering $M_{f} = 0$ in equation (18), the increment of the tension at outlet point can be obtained

T_{i}^{t} - T_{i}^{s} = \frac{2 δ}{1 - δ} T_{i}^{s} = η_{e} T_{i}^{s}

(20)

where the tension increment is caused merely by stiffness effect of the rope, and $η_{e}$ is the stiffness resisting coefficient which is usually between 0.0055 and 0.06 in practical engineering.²⁴ And the relation between $η_{e}$ and $δ$ is

δ = \frac{η_{e}}{2 + η_{e}}

(21)

Taking account of all the resistances, the relationship of the tensions at both sides of the pulley will be derived by equations (18)–(21) as follows

T_{i}^{s} = \frac{2}{(2 + η_{e}) η_{f} + 2 (1 + η_{e})} T_{i}^{t} = η T_{i}^{t}

(22)

where $η$ is the efficiency coefficient.

According to equations (16) and (17) and the previous analysis, the rope tensions $f_{i}^{s}$ and $f_{i}^{t}$ at both sides of pulley i satisfy the following expression

f_{i}^{s} = - \frac{n_{i}^{t} \cdot t_{i}^{t}}{n_{i}^{s} \cdot t_{i}^{s}} η f_{i}^{t} = {\bar{η}}_{i} f_{i}^{t}

(23)

where $n_{i}^{s}$ and $n_{i}^{t}$ are the corresponding directions of the rope tensions.

The gravity of the hook block is sufficient to tighten the rope hung in space. Therefore, the tangent direction between pulleys can be considered as the tension direction of the rope, and the tangent point is the inlet point or the outlet point.

Without considering rope mass, the tensions of the rope hung in space are equal at every point, and the rope tensions at both sides of pulleys satisfy equation (23). Therefore, giving the rope tension T that connected the drum, we can obtain the tension of each rope hung in space in sequence as shown in Figure 7.

Figure 7.

Directions of rope tensions and transitive relationships.

Compatible condition of rope length

As shown in Figure 8, a duplex drum is usually used in practical engineering, which has a groove of helical line and is difficult to confirm the trajectory of the rope tangent point on the drum. In this section, compatible conditions of rope length are proposed to solve the positions of the rope tangent points on the drum.

Figure 8.

Rope length wound by the drum.

The rope usually has very high tensile strength, and the elongation can be neglected. If the longitudinal displacement of the tangent point is $z_{k}$ (k = 1 stands for the rope on the left side of drum and k = 2 stands for the rope on the right side of drum), the length of the rope wound by the drum is

{\bar{L}}_{k} = \frac{z_{k}}{d} \sqrt{4 π^{2} r_{d}^{2} + d^{2}}

(24)

where $r_{d}$ is the drum radius and d is the pitch of rope groove on the drum.

As the starting points of the two ropes are connecting to the same drum, the winding loops of the two rope branches are the same at one moment. However, the instantaneous orientation angles $β_{1}$ and $β_{2}$ of the ropes on the drum are not the same. Neither are the initial orientation angles $β_{1}^{0}$ and $β_{2}^{0}$ . Thus, the longitudinal displacements of the tangent points are not the same, but they satisfy the following relationships

{\begin{matrix} z_{1} = \frac{d}{2 π} (ϕ + β_{1} - β_{1}^{0}) \\ z_{2} = \frac{d}{2 π} (ϕ + β_{2} - β_{2}^{0}) \end{matrix}

(25)

where $ϕ$ is the rotational angle of the drum. The cross section of the drum can be treated as spatial circles to solve $β_{1}^{0}$ , $β_{2}^{0}$ , $β_{1}$ , and $β_{2}$ , where $β_{1}$ and $β_{2}$ are coupled with the parameters $z_{1}$ and $z_{2}$ .

By eliminating $ϕ$ in equation (25), the relation between the two longitudinal displacements is derived as follows

z_{1} = z_{2} + \frac{d}{2 π} (β_{1} - β_{2} - β_{1}^{0} + β_{2}^{0})

(26)

The total rope length in the branch is

L_{k} = \sum_{i = 1}^{n} l_{i} + \sum_{i = 1}^{m} s_{i}

(27)

where n is the number of the ropes in space; $l_{i}$ is the rope length which can be written out by the coordinates of the tangent points; m is the number of the pulleys; and $s_{i}$ is the length of the rope wrapped on pulley i, which can be derived by multiplying the pulley radius $r_{i}$ by the wrap angle $α_{i}$

s_{i} = α_{i} r_{i}

(28)

In general, the rope bypasses movable pulley from the bottom and fixed pulley from the top as shown in Figure 9. The vectors from the pulley’s center to the two tangent points are denoted by $b_{1}$ and $b_{2}$ . The upward unit vector of coordinate system fixed on pulley i is denoted by $e_{y}$ . Then, the wrap angle on movable pulley is

α_{i} = acos (- b_{1} \cdot e_{y}) + acos (- b_{2} \cdot e_{y})

(29)

Figure 9.

Rope wrapped on pulleys: (a) movable pulley and (b) fixed pulley.

The wrap angle on fixed pulley is

α_{i} = acos (b_{1} \cdot e_{y}) + acos (b_{2} \cdot e_{y})

(30)

Then, the length of rope wound by the drum should equal the length that reduced in the branch. Then, we have

{\begin{matrix} {\bar{L}}_{1} + L_{1} - L_{1}^{0} = 0 \\ {\bar{L}}_{2} + L_{2} - L_{2}^{0} = 0 \end{matrix}

(31)

where $L_{k}^{0}$ is the initial rope length in branch k.

In the above analysis, we know that there are nine unknown parameters to resolve: seven degrees of freedoms to determine the movement of the mechanism ( $u_{1}$ , $u_{3}$ , $θ_{1}$ , $θ_{2}$ , $θ_{3}$ , $z_{1}$ , and $z_{2}$ ) and two rope tensions ( $T_{1}$ and $T_{2}$ ). The equilibrium provides six equations in equations (4) and (5); meanwhile, the compatible conditions provide three equations in equations (26) and (31); therefore, the solving conditions are complete.

Moment equilibrium for guide pulley

In order to avoid rope interferences and reduce deviation components of the rope tensions, a guide pulley is usually used to connect two movable pulleys with large distance span. Different from ordinary pulleys, the guide pulley is connected on a horizontal shaft $e_{3}$ with the hanging point C as shown in Figure 10. The entire guide pulley can be rotated around the horizontal shaft.

Figure 10.

Guide pulley.

While the load is hoisting, the rotation angle of the guide pulley around the horizontal shaft is changing and then the angle is the 10th unknown parameter, which means that a supplementary equation should be provided to confirm the equilibrium path.

If the guide pulley rotates around the horizontal shaft by an angle of $α$ , the normal vector of the guide pulley is

n = \cos α e_{1} + \sin α e_{2}

(32)

where the unit vector $e_{1}$ is toward front and $e_{2}$ toward upward.

The vector from the pulley’s center to the point C is

r_{c} = - b \cos α e_{2} + b \sin α e_{3}

(33)

where b is the distance between the pulley’s center and the hanging point.

Neglecting the friction of the horizontal shaft, the reaction moment on the shaft should be zero and we have

[m_{0} r_{c} \times g + (r_{c} + r_{t}) \times f_{t} + (r_{c} + r_{s}) \times f_{s}] \cdot e_{3} = 0

(34)

where $m_{0} g$ is the weight of the guide pulley; $r_{t}$ and $r_{s}$ are the vectors from the pulley’s center to the outlet point and the inlet point, respectively; and $f_{t}$ and $f_{s}$ are the rope tensions at two sides of the pulley.

$α$ can be a degree of freedom to describe global rotation angle of the guide pulley. The moment equilibrium equation (34), which implies $α$ , can be as a supplementary equation to solve $α$ .

Angular speed of pulley

Rope speed is one of the important designing parameters in many aspects, such as the selection of wire rope, the designing of pulleys, the power system of drum, and so on. The rope speed cannot be estimated by multiplying factor for the hoisting system that requires precise control. Based on the velocities of tangent points obtained in previous sections, a method to resolve the rope speeds and the angular speeds of pulleys along the rope branch is proposed in this section.

Figure 11 shows a pulley with radius r and center $o$ . $p_{s}$ and $p_{t}$ are the tangent points at the inlet point and outlet point, respectively. $p_{1}$ is a point on the rope hung in space and $p_{2}$ is a point on the adjacent rope wrapped on pulley. Taking the rope segment between the two points to investigate, the rope length between the two points can be denoted as follows

l = l_{1} + l_{2}

(35)

where $l_{1}$ is the rope length between $p_{s}$ and $p_{1}$ and can be expressed as follows

l_{1} = (p_{s} - p_{1}) \cdot n_{s}

(36)

where $n_{s}$ is the unit direction vector of the rope hung in space and can be expressed by the tangent points.

Figure 11.

Rope speed and angular speed of pulley.

The change rate of $l_{1}$ is

{\overset{\cdot}{l}}_{1} = {\overset{\cdot}{p}}_{s} \cdot n_{s} - v_{s}

(37)

where $v_{s} = {\overset{\cdot}{p}}_{1} \cdot n_{s}$ is the rope speed at inlet point of the pulley and ${\overset{\cdot}{p}}_{s}$ is the velocity of the tangent point.

$l_{2}$ is the length of the rope wrapped on pulley between $p_{s}$ and $p_{2}$ , which can be expressed as

l_{2} = α r = (β_{2} - β_{s}) r

(38)

where $α$ is the wrapped angle and $β_{2}$ and $β_{s}$ are the orientation angles of $p_{2}$ and $p_{s}$ , respectively. As $p_{2}$ is consolidated on the pulley, the change rate of $β_{2}$ is equal to the angular speed $ω$ of the pulley. Then, the change rate of $l_{2}$ can be written as follows

{\overset{\cdot}{l}}_{2} = (ω - {\overset{\cdot}{β}}_{s}) r

(39)

Neglecting the rope elongation, we have that ${\overset{\cdot}{l}}_{1} + {\overset{\cdot}{l}}_{2} = 0$ . Then, we can obtain

ω = \frac{1}{r} (v_{s} - {\overset{\cdot}{p}}_{s} \cdot n_{s}) + {\overset{\cdot}{β}}_{s}

(40)

The equation is the relationship of the angular speed $ω$ and the rope speed $v_{s}$ at inlet point of pulley. Similarly, the rope speed $v_{t}$ at outlet point of pulley can be expressed as follows

v_{t} = (ω - {\overset{\cdot}{β}}_{t}) r + {\overset{\cdot}{p}}_{t} \cdot n_{t}

(41)

and the difference between $v_{s}$ and $v_{t}$ is

v_{t} - v_{s} = ({\overset{\cdot}{β}}_{s} - {\overset{\cdot}{β}}_{t}) r + {\overset{\cdot}{p}}_{t} \cdot n_{t} - {\overset{\cdot}{p}}_{s} \cdot n_{s}

(42)

After solving the change rates of the tangent points, the transitive relationship of the rope speeds at sides of pulley can be obtained by equation (42). In the rope branch, the speed of the rope with the end anchored to a fixed point is equal to zero; therefore, starting from the end of the branch, one can obtain the rope speeds and the angular speeds of the pulleys successively by equations (40)–(42).

Numerical examples

Auxiliary rope–pulley system of the polar crane for NPP

To verify the methodology presented in this article, we developed a general MATLAB program, by which parameters input, model calculation, and processing can be carried out, and an auxiliary rope–pulley system with less pulleys is analyzed first as shown in Figures 12 and 13. The parameters of the system are shown in Tables 1 and 2. The positions of the drum and pulleys are given in the global coordinate (X, Y, Z) and X toward front, Y toward up, and Z toward left. The weight of the hook block is 1000 kg. The centroid of the hook block is (0, −20, 0) m and the hanging point is (0, −20.24, 0) m in the global coordinate. The working condition that hoisting a load of 20 tons to arise 15 m was studied. The efficiency coefficient of pulley is 0.98, and the frictional resistance coefficient is 0.01455.

Figure 12.

Schematic diagram of auxiliary rope–pulley system.

Figure 13.

Simulation model of auxiliary rope–pulley system in MATLAB.

Table 1.

Initial parameters of the drum in auxiliary rope-pulley system.

Drum	Diameter (mm)	Pitch	Center position (mm)	Rotation shaft
1	570	22	(0, 0, 1796)	(0, 0, 1)
2	570	22	(0, 0, −1796)	(0, 0, 1)

Table 2.

Initial parameters of the pulleys.

Number	Diameter (mm)	Center position (mm)	Rotation direction
1	180	(530, 25, 250)	(1, 0, 0)
2	355	(750, −295, 0)	(1, 0, 0)
3	315	(850, −295, 0)	(1, 0, 0)
4	180	(530, 25, −317)	(1, 0, 0)
5	355	(542, −19,728, 227)	(0, 0, 1)
6	450	(542, −19,728, 160)	(0, 0, 1)
7	450	(542, −19,728, −160)	(0, 0, 1)
8	355	(542, −19,728, −227)	(0, 0, 1)

The trajectories of the mount point as well as the rotational angles of the hook block versus the height are depicted in Figure 14 in blue solid curve, and the red dotted curves in this figure denote vertical lines crossing the initial mount point. The distances between the two kinds of curves signify the trajectory deflections.

Figure 14.

Trajectory deflections and rotational angles of the mount point.

It can be seen that (1) the trajectory is not exactly vertical and the deflections get greater and greater in general. (2) Because the rotation directions of the movable pulleys are along the Z direction, which is the direction of the frictional resistance moments, the rotational angle around Z-axis is greater than those around X-axis and Y-axis.

Because of the roughly right-and-left symmetrical measurements of the rope–pulley system, the positions of the pulleys, and the reeving way of the rope, the rope speeds almost keep unchanging relative to the hoisting speed, and the curves shown in Figure 15 are nearly horizontal lines whose values approach to the corresponding multiplying factors. The curves in Figure 16 are the rope speeds at the tangent point of the drum, which become smaller slightly as the space ropes diverge from the vertical.

Figure 15.

Rope speeds at outlet points of pulleys.

Figure 16.

Rope speeds on the winch.

Figure 17 shows that the orientation angles on the drum become larger as the hook block gets closer to the drum. From Figure 18, it can be seen that the rope tensions on two sides of the drum are not equal as the two rope branches are connected to fixed points directly. However, the sum of the two tensions equals approximately to 1/4 of the weight of the load and the hook block.

Figure 17.

Orientation angles on the winch.

Figure 18.

Tensions at two sides of drum.

Main rope–pulley system of the polar crane for NPP

In this section, numerical examples for the main rope–pulley system as shown in Figure 1 are given to verify the application of the methodology. The global coordinate is established as that in section “Auxiliary rope–pulley system of the polar crane for NPP.” The hook-block-fixed coordinate is established at the centroid of the hook block, whose initial origin is (0, −45, 0) m in global coordinate. The initial parameters about the drum and pulleys are displayed in Tables 3 –5. Among the parameters, the positions of the movable pulleys are given in the hook-block-fixed coordinate, while the others are given in the global coordinate. The weight of the hook block is 5000 kg as well as the hanging point is (0, −1.222, 0) m in the hook-block-fixed coordinate.

Table 3.

Initial parameters of the drum in main rope-pulley system.

Drum	Diameter (mm)	Pitch (mm)	Center (mm)	Rotation direction
1	2100	38	(−1848, 755, 3366)	(0, 0, −1)
2	2100	38	(−1848, 755, −3175)	(0, 0, 1)

Table 4.

Initial parameters of fixed pulleys.

Pulley	Diameter (mm)	Center (mm)	Rotation direction
13	900	(5.02, 0, 615.74)	(0.10, 0, 0.99)
14	900	(−5.02, 0, 520.26)	(0.10, 0, 0.99)
15	900	(0, 0, 234)	(0, 0, 1)
16	900	(0, 0, 137)	(0, 0, 1)
17	900	(0, 0, 0)	(0, 0, 1)
18	900	(0, 0, −137)	(0, 0, 1)
19	900	(0, 0, −234)	(0, 0, 1)
20	900	(5.02, 0, −520.26)	(0.10, 0, 0.99)
21	900	(−5.02, 0, −615.73)	(0.10, 0, 0.99)
22	1100	(914, −525, 95)	(1, 0, 0)
23	457	(634.4, −714, 48)	(1, 0, 0)
24	457	(−457.4, −714, −892)	(1, 0, 0)

Table 5.

Initial parameters of movable pulleys.

Pulley	Diameter (mm)	Center (mm)	Rotation direction
1	900	(0, 556, 542.5)	(0, 0, 1)
2	900	(0, 556, 446.5)	(0, 0, 1)
3	1016	(0, 556, 350.5)	(0, 0, 1)
4	900	(0, 556, 236.5)	(0, 0, 1)
5	900	(0, 556, 140.5)	(0, 0, 1)
6	900	(0, 556, 44.5)	(0, 0, 1)
7	900	(0, 556, −44.5)	(0, 0, 1)
8	900	(0, 556, −140.5)	(0, 0, 1)
9	1016	(0, 556, −236.5)	(0, 0, 1)
10	900	(0, 556, −350.5)	(0, 0, 1)
11	900	(0, 556, −446.5)	(0, 0, 1)
12	900	(0, 556, −542.5)	(0, 0, 1)

In order to compare the influence of different load weights on the trajectory deflections, we analyzed the equilibrium paths of the polar crane in NPP, which lifted 30 m height with weights of empty, 5, 20, 200 tons, respectively.

The trajectory deflections of the hanging point in the left-and-right direction and fore-and-aft direction versus the lifting height are shown in Figures 19 and 20. It can be seen that (1) the deflections affected by lifting weight are small, and (2) the deflection under empty load is a little greater than that under some lifting weight.

Figure 19.

Deflections in the left-and-right direction.

Figure 20.

Deflections in the fore-and-aft direction.

To compare the effects of lifting and lowering on the trajectory deflections, we analyzed the equilibrium path of the polar crane, which first lifted 20 m height with 200 tons weight and then lowered 20 m to the initial height. The trajectory deflections in the left-and-right direction and fore-and-aft direction are shown in Figures 21 and 22. It can be seen that the deflection in lifting is greater than that in lowering.

Figure 21.

Contrast of lifting and lowering for deflection in the left-and-right direction.

Figure 22.

Contrast of lifting and lowering for deflection in the fore-and-aft direction.

Discussion

Through the previous analysis, we know that the measurements of the pulleys and drums as well as the layouts and the reeving ways of the rope–pulley system determine the directions and magnitude relations of the rope tensions and thus affect the trajectory deflections of the load. The trajectory deflections are systematic and have several key properties as follows:

The deflections and their increment speeds become greater as the load gets nearer to the top. This is because the distance between fixed pulleys and movable pulleys becomes closer, and some rope directions diverge from the vertical, which increase the horizontal components of the rope tensions, thus the greater deflections are caused.

Because the deflection is systematic, different lifting weights have little influence on the deflection.

The deflections in lifting are greater than that in lowering. The reason is that the outlet side and inlet side of the pulleys in lowering is contrary to that in lifting, and the ropes approached to vertical while lowering which reduced the horizontal components of the rope tensions.

Conclusion

There are systematic trajectory deflections for the load of the polar crane in NPP. The model selections of the pulleys and drums, the layouts of the system components, and the reeving way of ropes would affect the load trajectory of the polar cranes. To reduce the trajectory deflections and locate the load precisely, a reasonable mechanical design for rope–pulley system is necessary. This article provides a quasi-static method to solve equilibrium path of the hook block, which is a systematic method and takes into account the detailed structure of the rope–pulley system. The method and related program that provide a simulation test platform can be a reference for the mechanic design, especially the model selections of the pulleys and drums, the layouts of the system components, and the reeving way of ropes.

Because the tension of each rope changes constantly as the load hoisting, to solve the equilibrium path of the hook block is rather difficult and complicated. This article made some attempts on the problem. However, the mass and the deformation of the rope are without considering, and the dynamics and control of the load are uninvestigated, which can be the target of further research and development.

Novelty indication

At present, there are many researches about the load trajectories of cranes. However, some of these researches considered the load and the plurality of ropes as a flexible pendulum, and some of them made many simplifications for the rope–pulley system, and the results cannot meet the high accuracy requirements in locating the loads for the polar cranes in NPPs. This article presents a quasi-static method to analyze equilibrium path and trajectory deflections are systematic for the load of polar crane in NPP based on 3D analyses. A numerical method is derived to solve the common tangent between spatial pulleys, which was conforming to the rope reeving. As the method considers the reeving way of each rope that connects spatial pulleys and the tension transmission at two ends of each pulley, then it can accurately calculate the forces on the hook block exactly. The systematic method is theoretical, and the simulation results verify the rationality of the method, which can be a reference for the mechanic design such as the model selections of the pulleys and drums, the layouts of the system components, and the reeving way of ropes. To a large extent, the method is universal and can be applied to general rope–pulley systems.

Footnotes

Appendix 1

Four analytical values for the tangent points of two paralleled pulleys are written in the following. As for the external tangent, we have $β_{2} = β_{1}$ , as shown in Figure 23, then

{\begin{matrix} x_{1} = x_{2} = \frac{1}{p} (p_{11} ρ_{r} + p_{12} ρ_{p}) \\ y_{1} = y_{2} = \frac{1}{p} (p_{12} ρ_{r} - p_{11} ρ_{p}) \end{matrix}

{\begin{matrix} x_{1} = x_{2} = \frac{1}{p} (p_{11} ρ_{r} - p_{12} ρ_{p}) \\ y_{1} = y_{2} = \frac{1}{p} (p_{12} ρ_{r} + p_{11} ρ_{p}) \end{matrix}

where $ρ_{r} = r_{2} - r_{1}$ , $ρ_{p} = \sqrt{p - {ρ_{r}}^{2}}$ , and $p = p_{11}^{2} + p_{12}^{2} + p_{13}^{2}$ .

As for the internal tangent, $| β_{2} - β_{1} | = π$ , as shown in Figure 24, then

{\begin{matrix} x_{1} = - x_{2} = - \frac{1}{p} (p_{11} ρ_{r} + p_{12} ρ_{p}) \\ y_{1} = - y_{2} = - \frac{1}{p} (p_{12} ρ_{r} - p_{11} ρ_{p}) \end{matrix}

{\begin{matrix} x_{1} = - x_{2} = - \frac{1}{p} (p_{11} ρ_{r} - p_{12} ρ_{p}) \\ y_{1} = - y_{2} = - \frac{1}{p} (p_{12} ρ_{r} + p_{11} ρ_{p}) \end{matrix}

where $ρ_{r} = r_{2} + r_{1}$ and $ρ_{p} = \sqrt{p - ρ_{r}^{2}}$ .

Taking these analytical solutions as the initial value, one can make the numerical solution of equation (11) in accordance with the reeving way between the rope and pulleys.

With the pulley numbers and the types of common tangents between the spatial pulleys, we can describe the rope–pulley system by a number matrix. For the main rope–pulley system as shown in Figure 1, one could take the following two matrixes to describe the reeving ways of the two rope branches, respectively

K_{1} = [\begin{matrix} 3 & 14 & 2 & 13 & 1 & 22 & 10 & 20 & 11 & 21 & 12 & 24 \\ 4 & 1 & 2 & 1 & 2 & 3 & 1 & 2 & 1 & 2 & 1 & 2 \end{matrix}]

and

K_{2} = [\begin{matrix} 9 & 19 & 8 & 18 & 7 & 17 & 6 & 16 & 5 & 15 & 4 & 23 \\ 4 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 1 \end{matrix}]

The first row stores the numbers of the pulleys through which the rope reeves from the drum in sequence, while the second row stores the numbers of the corresponding tangent types as shown in Figure 4. For example, the first number 3 in $K_{1}$ represents the number of the pulley connects to the left side of the drum, while the first (or the corresponding) number in the second row means the reeving way of the pulley connecting to the system.

As for the auxiliary rope–pulley system as shown in Figure 16, the matrix can be written as follows

K_{1} = [\begin{matrix} 5 & 2 & 8 & 4 \\ 4 & 3 & 1 & 2 \end{matrix}], K_{2} = [\begin{matrix} 7 & 3 & 6 & 1 \\ 4 & 1 & 3 & 4 \end{matrix}]

Handling Editor: Jining Sun

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 receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China, P. R. China (no. 11372057).

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Equilibrium path and load trajectory deflections for polar cranes in nuclear power plant

Abstract

Keywords

Introduction

Equilibrium path equations

Tangent between spatial pulleys

Transmission of tensions at two sides of pulley

Compatible condition of rope length

Moment equilibrium for guide pulley

Angular speed of pulley

Numerical examples

Auxiliary rope–pulley system of the polar crane for NPP

Main rope–pulley system of the polar crane for NPP

Discussion

Conclusion

Novelty indication

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References