Study on the mechanical properties of guide shoe on the transverse vibration of high-speed elevator

Abstract

The vibration of car systems in high-speed elevators is analyzed especially the mechanical properties of the rolling guide shoe. As a guiding mechanism, the rolling guide shoe plays a crucial role in car vibration. The mechanical properties of rolling guide shoes mainly include the dynamics of the guide shoe mechanism, emphasizing the complex multi-degree-of-freedom system vibration of the car-frame, as well as the nonlinear mechanical properties of the guide shoe lining rubber and guide rail contact. Taking the parameters such as guide shoe lining material and contact thickness into account a mechanical model for the guide shoe is established using the finite element method. To analyze the transverse vibration of the car-frame, an8-degree-of-freedom dynamic model is established using Lagrange equations. The Newmark numerical solution algorithm is used to study the impact of different mechanical parameters of guide shoes on the transverse vibration of cars. The results indicate that using a guide shoe with small stiffness and large damping can effectively reduce the transverse vibration of the elevator. The methodology and findings can provide valuable reference for the design of guide shoes for high-speed elevators.

Keywords

High-speed elevator rolling guide shoe finite element analysis dynamic modeling transverse vibration

Introduction

With the continuous development of high-rise buildings, the demand for high-speed elevators is growing. High-speed elevators have a high natural frequency density and are prone to resonance and vibration. In addition, the demand for comfort and stricter standards of elevators is increasing.¹ Elevator vibration is the most important factor affecting comfort and safety.² In operation, the vibration response is longitudinal and transverse vibrations,^3,4 with passengers being particularly sensitive to the transverse vibrations of elevator cars.⁵

Studies have shown that the transverse vibration of elevators is significantly influenced by many factors. Two important factors are: the high-speed elevator guiding system,⁶ and the excitation caused by guide rail unevenness. The influence of air resistance on the vibration characteristics of high-speed elevators should not be underestimated.⁷ The guiding system mainly includes guide rail, guide wheel, and guide shoe, with the guide shoe as a guide device and a core vibration isolation component to reduce transverse vibration.⁸ For high-speed elevators greater than 6 m/s, a rolling guide shoe is generally used to reduce the friction between the guide shoe and guide rail. The rolling guide shoe also provides elastic support to compensate for any defects in the guide rail.⁹ Therefore, it is important to evaluate the vibration isolation of the rolling guide shoe to improve the comfort of elevators and improve their competitiveness in the market. Song et al.⁸ developed a dynamic model of a guide shoe-car system and validated its accuracy through theoretical and simulation methods. The effects of elevator velocity, guide shoe stiffness and damping on the elevator vibration were investigated. Based on Bernoulli Euler theory, Cao et al.¹⁰ established differential equations of the forced vibration of elevator guide rails, and derived vibration equations for elevator guide shoes and cars using the d'Alembert principle. Considering the time-varying characteristics of contact stiffness, Qin et al.¹¹ established a 17 degrees-of-freedom dynamic model to analyze the time-varying characteristics of contact stiffness under aerodynamic loads. Zhang et al.¹² studied the dynamic characteristics of the guideway, guide shoe, and car of elevators. A coupled system model of guide rail, guide shoe and car was established, and the impact of three parameters: length, weight per unit length and bending stiffness was analyzed. The results showed that the bending stiffness of the guideway mainly affected the vibration displacement, while the length of the guideway had a significant effect on vibration displacement and guideway chatter. Combined the Hertzian contact theory with the Bouc-Wen hysteresis model, Zhang et al.¹³ developed a nonlinear vibration model of an elevator car system to analyze the vibration of the elevator car under different variations of random parameters and excitations. The results indicated that the randomness of geometric parameters had the most significant influence on the transverse acceleration. Yang et al.¹⁴ established a four degree-of-freedom model for the transverse vibration of the elevator, aiming to using computational fluid dynamics to investigate the effects of various parameters under different working conditions on the aerodynamic forces on the transverse surface of the elevator. Liu et al.¹⁵ established a transverse vibration model for high-speed elevators based on gas-solid coupling to analyze the transverse vibration characteristics of high-speed elevators with or without gas-solid coupling.

Previous studies on elevator transverse vibration mainly focused on the dynamic modeling of the four guide shoes and the car system as a whole.^4,6 There has been limited research specifically focused on the vibration isolation of guide shoes. Most studies treat the guide shoes as rigid bodies, neglecting the elastic deformation of the guide shoe lining and the damping effect of the rubber on the outer edge of the rollers. Furthermore, research on how the design parameters of the guide shoe affect its mechanical properties is limited. The dynamic models used for the car-frame were relatively simple, with limited degrees of freedom, and do not fully account for the rotational effect of the car frame.

An 8-degree of freedom (DOF) dynamic model for the transverse vibration of high-speed elevators is proposed. This model takes into account the rotational degrees of freedom of the car and frame, as well as the translational degrees of freedom of each guide shoe. The mechanical properties of the guide shoe and their influencing factors are analyzed using the finite element method. Based on the dynamic model established in the section 3, the influence of the stiffness and damping of the guide shoe on the acceleration is analysed using the Newmark’s numerical integration method. Section 4 provides certain theoretical support for the vibration damping design of guide shoe.

Modeling of guide shoe mechanics

This section first establishes a two-dimensional model of the contact between the guide shoe and the guide rail, and a mathematical formula for the stiffness of the guide shoe is derived. Then a three-dimensional finite element model of the guide shoe contact is developed. The results from the stiffness calculation using the mathematical formula are compared with the simulation results from the three-dimensional finite element model to validate the feasibility of the three-dimensional finite element model.

Mechanical modeling of the guide shoe mechanism

Rolling guide shoe is installed at the four upper corners of the elevator car frame for functionality. The guide shoe consists of three rollers that contact the three surfaces of the “T” rail in high-speed elevators. The interaction mechanics between the guide shoe and guide rail determines the transverse vibration of the elevator car system. A schematic of the interaction relationship is shown in Figure 1, and the corresponding force diagram is shown in Figure 2.

Figure 1.

Simplified diagram of the guide shoe-guide rail fit for high-speed elevators.

Figure 2.

Static force analysis diagram of the guide shoe.

Neglecting the inertia of the individual of the guide shoe, the hydrostatic equilibrium relationship is obtained as follows:

{\begin{cases} F_{A} = k x_{1} \\ F_{C X} + F_{A} = F_{D} \\ F_{D} L_{B C} = F_{A} L_{A C} \end{cases}

(1)

{\begin{cases} \begin{array}{l} \begin{array}{l} F_{A} = k x_{1} \\ F_{D} = k_{3} x_{2} \end{array} \\ F_{C X} = F_{D} - F_{A} \end{array} \\ k_{3} x_{2} L_{B C} = k x_{1} L_{A C} \end{cases}

(2)

where x₁ represents the deformation of the spring; x₂ represents the deformation of the guide shoe lining; k₃ denotes the stiffness of the contact between the wheel and rail; L_AC refers to the length of the lever; L_BC represents the distance from the center of the guide wheel axis to the end of the lever; k represents the elasticity coefficient of the guide shoe spring; F_A represents the elastic force on the lever of the spring; F_CX represents the constraint force on the lever, and F_D represents the reaction force exerted by the guide rail on the guide wheel.

The deformation in the horizontal direction of the guide rail is set as x. The triangular similarity theorem is used to determine the elasticity generated by the guide wheel hub and the extrusion deformation of the guide rail on the guide shoe lining. Based on the Newton’s second law, the elasticity exerted by the guide shoe lining on the guide wheel and the reaction force exerted by the guide rail on the guide shoe lining are the same. The stiffness of the guide lining is incorporated into the mechanics model of the guide shoe, leading to an improvement of the original static equations of the guide shoe. Equation (1) can be obtained by correcting equation (2).

High-speed elevators mainly use rolling guide shoe, and the interaction between the rolling guide shoe and the contact points of the guide rail belongs to rolling contact. Therefore, the contact area between rolling elements is the Hertz contact force.¹² According to the Hertz elastic contact theory,¹⁶ as shown in Figure 3, the contact between the guide shoe and roller under the normal load F can be treated as a problem of two rolling cylinders with parallel axes. This results in a rectangular contact area, where the axial length L of the cylinder is equal. The force per unit length of an elastic cylinder p = F/L, where F is the total load on the cylinder, and L is the axial length of the elastic cylinder. The contact is formed on a rectangular strip with a width of 2a.

E^{*} = \frac{E_{1} E_{2}}{(1 - {v_{1}}^{2}) E_{2} + (1 - {v_{2}}^{2}) E_{1}}

(3)

where R₁ and R₂ are the radius of the two cylinders; E₁andE₂arethe modulus of elasticity of the guide shoe and the guide rail, respectively; v₁ and v₂ is the Poisson ratio of the guide shoe and the guide rail, respectively. Therefore, the contact stiffness between the guide shoe and the guide rail is as follows:

k = π L E^{*} / 2

(4)

Figure 3.

Schematic diagram of the guide shoe-guide contact model.

There is a displacement excitation y_ri between the guide shoe and the corresponding guide rail, and y_di is the translational degree of freedom of each guide shoe. The normal force between the guide shoe and the guide rail is as follow:

F_{i} = - k (y_{d i} - y_{r i}) - c ({\dot{y}}_{d i} - {\dot{y}}_{r i}); i = 1, 2, 3, 4

(5)

Contact mechanics modeling of guide shoe-guide rail

To comprehensively analyze the variations in the stiffness of the guide shoe, a three-dimensional model of the guide shoe-guide rail contact is established. Figure 4 shows a standard model of the guide shoe-guide rail contact, characterized by a trapezoidal rubber cross-section. The guide wheel axle model is simplified as a cylinder to intercept the track of the wheel-rail contact area, which is simplified to facilitate the subsequent delineation and calculation of solid meshes.

Figure 4.

Three-dimensional modeling of the guide shoe-guide rail contact.

The mesh technology used for finite element analysis of three-dimensional models directly affects the accuracy and efficiency. Here a multi-area hexahedral mesh is used. The hexahedral mesh has fewer elements, faster calculation than the tetrahedral mesh, meeting the basic requirements of accuracy, detailed as follows:

Material: using rubber material for the structure of guide shoe lining, and using structural steel processing to test the structure;

Mesh: a hexahedral mesh, the global mesh is3mm, which is encrypted on the guide shoe lining with a basic size of 1 mm, and the total number of mesh is 178806;

Boundary conditions: the position of the guide wheel is fixed to prevent offset; X and Y displacements of the guide rail is also fixed, and a Z-axis upward force is applied to the bottom of the guide rail (the Z-axis is vertically upward when viewing in the forward direction, and X and Y-axes are left-right and forward-backward directions). The guide shoe deforms significantly and is simulated using a hyperelastic material, while the rest of the structural components are modeled using the default structural steel material. To improve the accuracy of the calculation results, local encryption processing is applied for guide shoe lining. The diagram of the three-dimensional mesh is shown in Figure 5.

Figure 5.

Mesh diagram of the guide shoe-guide rail contact model.

The main focus is on the stiffness of the guide shoe- rail contact. In accordance with Newton’s second law, the force between the two is mutual. A force of 50N is applied to the guide rail, and the displacements in the X- and Y-directions are constrained on both sides of the guideway. The maximum time-varying stiffness between the guide shoe and rail is determined based on the deformation of the guide shoe lining. The deformation cloud is shown in Figure 6.

Figure 6.

Deformation of the guide shoe lining (The maximum deformation at a maximum load of 0.94218 mm).

As shown in Figure 6, the contact area experienced the highest stress and presented a semi-elliptical configuration. The maximum deformation occurs at the central of the ellipse, where it shows a radial reduction in all directions, forming an elliptical shape. As the magnitude of the external force increases, the maximum deformation of the lining increases. The stiffness of the guide shoe lining made of neoprene material can be theoretically calculated using equations (3) and (5), as shown in Table1.

Table 1.

Stiffness verification.

Methods	Stiffness/(N/m)
Theoretical calculation	45960
Finite element calculation	30908-50333

As shown in Table1, it is evident that the stiffness values obtained through the theory fall within the stiffness range calculated using finite element analysis. This observation indicates that the 3D finite element model fully complies with the calculation standards.

Modeling of elevator transverse vibration

Considering the y-direction translational and rotational degrees of freedom (DoF) of the car and the frame, as well as the y-direction translational DoF of the four guide shoes, an eight-degree-of-freedom dynamics model of the car system is established through the Lagrange principle. The accuracy of the dynamics model is verified by using the Adams software. Section 3.2 provides the excitation model for the unevenness excitation of the guide rail.

Establishment of differential equations of motion

In an actual elevator system, dampers are used to connect the car to the car frame, while the guide shoe is installed on the car frame to facilitate the correct movement of the car. However, due to uneven excitation and installation errors of the guide rail, the elevator car will generate transverse random excitations, which would affect the stability of the elevator. The main excitation sources for elevator transverse vibration are as follows: (1) The centers of mass of the car and frame coincide with their geometrical centers; (2) The two sets of damping rubbers located at the bottom and side of the car are replaced by two spring damping systems that have different stiffness and damping coefficients. The design of the rolling guide shoe is modified to consist of a solid mass block connected to the spring damping system. It is found that the stiffness and damping coefficients of the four rolling guide shoes are equal. The simplified schematic diagram of the transverse vibration is shown in Figure 7.

Figure 7.

Simplified structure diagram of transverse vibration.

y₁ is the translational degrees of freedom, and $θ_{1}$ is the rotational degrees of freedom of the car. y₂is the translational degrees of freedom and $θ_{2}$ is the rotational degrees of freedom of the car frame. The transverse translational degrees of freedom y_d1, y_d2, y_d3 and y_d4 of the four guide shoes are taken into account. y_r1, y_r2, y_r3, and y_r4 are the guiderail unevenness excitations. An eight-degree-of-freedom dynamics model of transverse vibration is established.

As shown in Figure 7, O₁ and O₂ are the center-of-mass origin positions of the car and the frame; m₁, J₁, m₂ and J₂ are the mass and moment of inertia of the car and the frame, respectively; $y_{1}, θ_{1}, y_{2}, θ_{2}$ are the transverse displacement and rotation angle of the car around the y-axis, as well as the transverse displacement and rotation angle of the frame around the y-axis, respectively; $y_{d 1}, y_{d 2}, y_{d 3}, y_{d 4}$ are the transverse displacement of the four guide shoes, respectively. k₁ and c₁ are the stiffness and damping coefficient of the damping rubber on both sides of the car;k₂ and c₂ are the stiffness and damping coefficient of the vibration-damping rubber at the bottom of the car, respectively; k₃ and c₃ are the equivalent stiffness and damping coefficient of the rolling guide shoe-roller, respectively; F₁, F₂, F₃ and F₄ are the equivalent contact forces stimulated by the unevenness of the guide rail, respectively; h₁ and h₂ are the perpendicular distance of the guide shoes to the center of mass of the car frame, respectively; h₃ and h₄ are the vertical distances from the guide shoes to the center of mass of the car, respectively;h₅ and h₆are the vertical distances from the damping rubber on both sides of the car to the center of mass of the car frame, respectively; l is the transverse distance from the damping rubber at the bottom of the car to the center of mass of the car.

Based on the above assumptions, it is obtained according to the Lagrange equation:

\frac{d}{d t} (\frac{\partial T}{\partial {\dot{q}}_{i}}) - \frac{\partial T}{\partial q_{i}} = Q

(6)

where, q_i is the generalized coordinate degree of freedom of the system; Q is the generalized force of the system, and T is the kinetic energy of the system.

For the elevator system, there is a damping link that causes energy dissipation in the system, making it a non-conservative system. Q is composed of the generalized potential force, the generalized linear damping force, and the generalized excitation force. V is the potential energy, and W is the dissipation energy.

Q = - \frac{\partial V}{\partial q_{i}} - \frac{\partial W}{\partial {\dot{q}}_{i}} + F_{i}

(7)

Correlating equations (6) and (7), we have:

\frac{d}{d t} (\frac{\partial T}{\partial {\dot{q}}_{i}}) - \frac{\partial T}{\partial q_{i}} + \frac{\partial V}{\partial q_{i}} + \frac{\partial W}{\partial {\dot{q}}_{i}} = F_{i}

(8)

The 8-DOF dynamics model for transverse vibration is obtained as follows:

[M] {\ddot{q}} + [C] {\dot{q}} + [K] {q} = F_{g u i d e}

(9)

{\begin{cases} m_{1} {\ddot{y}}_{1} = 2 k_{1} [(2 y_{2} + h_{5} θ_{2} - h_{6} θ_{2}) - (2 y_{1} + h_{3} θ_{1} - h_{6} θ_{1})] + 2 c_{1} [(2 {\dot{y}}_{2} + h_{5} {\dot{θ}}_{2} - h_{6} {\dot{θ}}_{2}) - (2 {\dot{y}}_{1} + h_{3} {\dot{θ}}_{1} - h_{4} {\dot{θ}}_{1})] \\ J_{1} {\ddot{θ}}_{1} = - k_{1} [(2 y_{2} - 2 h_{5} θ_{2}) - (2 y_{1} - 2 h_{3} θ_{1})] h_{3} - c_{1} [(2 {\dot{y}}_{2} - 2 h_{5} {\dot{θ}}_{2}) - (2 {\dot{y}}_{1} - 2 h_{3} {\dot{θ}}_{1})] h_{3} + k_{1} [(2 y_{2} + 2 h_{6} θ_{2}) \\ - (2 y_{1} - 2 h_{4} θ_{1})] h_{4} + c_{1} [(2 {\dot{y}}_{2} + 2 h_{6} {\dot{θ}}_{2}) - (2 {\dot{y}}_{1} - 2 h_{4} {\dot{θ}}_{1})] h_{4} + k_{2} * 2 L (θ_{2} - θ_{1}) L + c_{2} * 2 L ({\dot{θ}}_{2} - {\dot{θ}}_{1}) L \\ m_{2} {\ddot{y}}_{2} = k_{3} [(\sum_{i = 1}^{4} y_{d i} - 4 y_{2} - 2 h_{1} θ_{2} + 2 h_{2} θ_{2})] + c_{3} [(\sum_{i = 1}^{4} {\dot{y}}_{d i} - 4 {\dot{y}}_{2} - 2 h_{1} {\dot{θ}}_{2} + 2 h_{2} {\dot{θ}}_{2})] \\ - 2 k_{3} (2 y_{2} + h_{5} θ_{2} - h_{6} θ_{2} - 2 y_{1} - h_{3} θ_{1} + h_{4} θ_{1}) - 2 c_{3} (2 {\dot{y}}_{2} + h_{5} {\dot{θ}}_{2} - h_{6} {\dot{θ}}_{2} - 2 {\dot{y}}_{1} - h_{3} {\dot{θ}}_{1} + h_{4} {\dot{θ}}_{1}) \\ J_{2} {\ddot{θ}}_{2} = [k_{3} (y_{d 1} - y_{2} - h_{1} θ_{2}) + c_{3} ({\dot{y}}_{d 1} - {\dot{y}}_{2} - h_{1} {\dot{θ}}_{2}) + F_{1}] h_{1} \\ + [k_{3} (y_{d 2} - y_{2} - h_{1} θ_{2}) + c_{3} ({\dot{y}}_{d 2} - {\dot{y}}_{2} - h_{1} {\dot{θ}}_{2}) + F_{2}] h_{1} \\ - [k_{3} (y_{d 3} - y_{2} + h_{2} θ_{2}) + c_{3} ({\dot{y}}_{d 3} - {\dot{y}}_{2} - h_{2} {\dot{θ}}_{2}) + F_{3}] h_{2} \\ - [k_{3} (y_{d 4} - y_{2} + h_{2} θ_{2}) + c_{3} ({\dot{y}}_{d 4} - {\dot{y}}_{2} + h_{2} {\dot{θ}}_{2}) + F_{4}] h_{2} \\ m_{d 1} {\ddot{y}}_{d 1} = F_{1} - [k_{3} (y_{d 1} - y_{2} - h_{1} θ_{2}) + c_{3} ({\dot{y}}_{d 1} - {\dot{y}}_{2} - h_{1} {\dot{θ}}_{2})] \\ m_{d 2} {\ddot{y}}_{d 2} = F_{2} - [k_{3} (y_{d 2} - y_{2} - h_{1} θ_{2}) + c_{3} ({\dot{y}}_{d 2} - {\dot{y}}_{2} - h_{1} {\dot{θ}}_{2})] \\ m_{d 3} {\ddot{y}}_{d 3} = F_{3} - [k_{3} (y_{d 3} - y_{2} + h_{2} θ_{2}) + c_{3} ({\dot{y}}_{d 3} - {\dot{y}}_{2} - h_{2} {\dot{θ}}_{2})] \\ m_{d 4} {\ddot{y}}_{d 4} = F_{4} - [k_{3} (y_{d 4} - y_{2} + h_{2} θ_{2}) + c_{3} ({\dot{y}}_{d 4} - {\dot{y}}_{2} + h_{2} {\dot{θ}}_{2})] \\ F_{1} = k_{3} y_{r 1} + c_{3} {\dot{y}}_{r 1}; F_{2} = k_{3} y_{r 2} + c_{3} {\dot{y}}_{r 2}; F_{3} = k_{3} y_{r 3} + c_{3} {\dot{y}}_{r 3}; F_{4} = k_{3} y_{r 4} + c_{3} {\dot{y}}_{r 4} \end{cases}

(10)

where

[M]

is the mass matrix;

[C]

is the damping matrix;

[K]

is the stiffness matrix, and

F_{g u i d e}

is the uneven excitation of the guide rail.

M = d i a g ([m_{1}, J_{1}, m_{2}, J_{2}, {m_{d}}_{1}, {m_{d}}_{2}, {m_{d}}_{3}, {m_{d}}_{4}])

(11)

q = {[y_{1}, θ_{1}, y_{2}, θ_{2}, {y_{d}}_{1}, {y_{d}}_{2}, {y_{d}}_{3}, {y_{d}}_{4}]}^{T}

(12)

K 1 = (\begin{array}{c} 4 k_{1} & 2 k_{1} (h_{3} - h_{4}) & - 4 k_{1} & 2 k_{1} (h_{6} - h_{5}) \\ 2 k_{1} (h_{3} - h_{4}) & 2 k_{1} ({h_{3}}^{2} - {h_{4}}^{2}) + 2 k_{2} l^{2} & 2 k_{1} (h_{3} - h_{4}) & - 2 k_{1} (h_{3} h_{5} + h_{4} h_{6}) - 2 k_{2} l^{2} \\ - 4 k_{1} & 2 k_{1} (h_{3} - h_{4}) & 4 (k_{1} + k_{3}) & 2 k_{3} (h_{1} - h_{2}) + 2 k_{1} (h_{5} - h_{6}) \\ 2 k_{1} (h_{6} - h_{5}) & - 2 k_{1} (h_{3} h_{5} + h_{4} h_{6}) - 2 k_{2} l^{2} & 2 k_{3} (h_{1} - h_{2}) + 2 k_{1} (h_{5} - h_{6}) & 2 k_{3} ({h_{1}}^{2} + {h_{2}}^{2}) + 2 k_{1} ({h_{5}}^{2} + {h_{6}}^{2}) + 2 k_{2} l^{2} \end{array})

(13)

K 2 = (\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - k_{3} & - k_{3} & - k_{3} & - k_{3} \\ - k_{3} h_{1} & - k_{3} h_{1} & k_{3} h_{2} & k_{3} h_{2} \end{array}) K 3 = (\begin{array}{l} 0 & 0 & - k_{3} & - k_{3} h_{1} \\ 0 & 0 & - k_{3} & - k_{3} h_{1} \\ 0 & 0 & - k_{3} & k_{3} h_{2} \\ 0 & 0 & - k_{3} & k_{3} h_{2} \end{array}) K 4 = (\begin{array}{l} k_{3} & 0 & 0 & 0 \\ 0 & k_{3} & 0 & 0 \\ 0 & 0 & k_{3} & 0 \\ 0 & 0 & 0 & k_{3} \end{array})

(14)

K = (\begin{array}{l} K 1 & K 2 \\ K 3 & K 4 \end{array})

(15)

C 1 = (\begin{array}{c} 4 c_{1} & 2 c_{1} (h_{3} - h_{4}) & - 4 c_{1} & 2 c_{1} (h_{6} - h_{5}) \\ 2 c_{1} (h_{3} - h_{4}) & 2 c_{1} ({h_{3}}^{2} - {h_{4}}^{2}) + 2 c_{2} l^{2} & 2 c_{1} (h_{3} - h_{4}) & - 2 c_{1} (h_{3} h_{5} + h_{4} h_{6}) - 2 c_{2} l^{2} \\ - 4 c_{1} & 2 c_{1} (h_{3} - h_{4}) & 4 (c_{1} + c_{3}) & 2 c_{3} (h_{1} - h_{2}) + 2 c_{1} (h_{5} - h_{6}) \\ 2 c_{1} (h_{6} - h_{5}) & - 2 c_{1} (h_{3} h_{5} + h_{4} h_{6}) - 2 c_{2} l^{2} & 2 c_{3} (h_{1} - h_{2}) + 2 c_{1} (h_{5} - h_{6}) & 2 c_{3} ({h_{1}}^{2} + {h_{2}}^{2}) + 2 c_{1} ({h_{5}}^{2} + {h_{6}}^{2}) + 2 c_{2} l^{2} \end{array})

(16)

C 2 = (\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - c_{3} & - c_{3} & - c_{3} & - c_{3} \\ - c_{3} h_{1} & - c_{3} h_{1} & c_{3} h_{2} & c_{3} h_{2} \end{array}) C 3 = (\begin{array}{l} 0 & 0 & - c_{3} & - c_{3} h_{1} \\ 0 & 0 & - c_{3} & - c_{3} h_{1} \\ 0 & 0 & - c_{3} & c_{3} h_{2} \\ 0 & 0 & - c_{3} & c_{3} h_{2} \end{array}) C 4 = (\begin{array}{l} c_{3} & 0 & 0 & 0 \\ 0 & c_{3} & 0 & 0 \\ 0 & 0 & c_{3} & 0 \\ 0 & 0 & 0 & c_{3} \end{array})

(17)

C = (\begin{array}{l} C 1 & C 2 \\ C 3 & C 4 \end{array})

(18)

\begin{array}{l} F_{g u i d e} = (\begin{array}{c} 0 \\ 0 \\ k_{3} y_{r 1} + c_{3} {\dot{y}}_{r 1} + k_{3} y_{r 2} + c_{3} {\dot{y}}_{r 2} + k_{3} y_{r 3} + c_{3} {\dot{y}}_{r 3} + k_{3} y_{r 4} + c_{3} {\dot{y}}_{r 4} \\ k_{3} [(y_{r 1} + y_{r 2}) h_{1} - (y_{r 3} + y_{r 4}) h_{2}] + c_{3} [({\dot{y}}_{r 1} + {\dot{y}}_{r 2}) h_{1} - ({\dot{y}}_{r 3} + {\dot{y}}_{r 4}) h_{2}] \\ k_{3} y_{r 1} + c_{3} {\dot{y}}_{r 1} \\ k_{3} y_{r 2} + c_{3} {\dot{y}}_{r 2} \\ k_{3} y_{r 3} + c_{3} {\dot{y}}_{r 3} \\ k_{3} y_{r 4} + c_{3} {\dot{y}}_{r 4} \end{array}) \end{array}

(19)

A solid model of the car and frame is established using the ADAMS software. Subsequently, relevant parameters are input to generate the intrinsic frequency for the car and frame, as shown in Table 2.

Table 2.

Comparison of natural frequency.

Methods	First order natural frequency/Hz	Second order natural frequency/Hz	Third order natural frequency/Hz	Fourth order natural frequency/Hz
Numerical calculation	1.3851	1.7315	14.5236	14.8954
ADAMS simulation	1.4028	1.7519	14.2354	15.1021

As shown in Table 2, the error between the numerical calculation results and the ADAMS simulation results is within 10%. The errors of the first, second, third, and fourth order natural frequencies are 1.3%, 1.2%, 1.98% and 1.4% respectively. This indicates that the proposed model is reasonable and feasible.

Establishment of the random guide rail excitation model

When the elevator runs in the elevator shaft, the unevenness of guide rails will significantly affect the transverse vibration of the elevator. Unevenness of guide rails was found to approximately follow a normal distribution.^17–19 The Gaussian white noise (with a mean value of 0, a standard deviation of 0.06 mm², as shown in Figure 8) is used to simulate the uneven excitation of guide rails.

Figure 8.

Uneven excitation (a) yr1 and yr2, (b) yr3 and yr4 of guide rails.

Influence of guide shoe parameters on guide shoe stiffness

Based on the 3D finite element model in Section 2.2, the deformation of the guide shoe lining is obtained by changing some parameters of the structure. The stiffness is obtained from the deformation and force, which is basic for analyzing the acceleration response in Section 5. All the data in Section 4 are derived from the simulation of the finite element 3D model established in Section 2. The data are obtained by assigning different material properties to the structure, meshing, setting boundary conditions, and different load steps. The results can provide some theoretical support for the design of vibration damping guide shoe.

Different guide shoe lining materials

The guide shoe lining is the main object of deformation caused by the contact force between the guide shoe and the guide wheel. The super elasticity of the rubber material of the guide shoe lining has a great influence on the stiffness of the guide shoe lining. The determination of the elastic modulus of a material depends on its mechanical properties, and the elastic modulus directly affects the stiffness coefficient of the material. To study the effect of different guide shoe lining materials on the stiffness of the guide shoe, four different guide shoe lining materials are selected for finite element analysis while keeping other factors constant.

As shown in Figure 9, different guide shoe lining materials will directly affect the stiffness coefficient of the material. The neoprene material has the largest deformation, while the natural rubber material has the largest time-varying stiffness. To obtain the maximum time-varying stiffness, the time step is set through the finite element to obtain the discrete points. The MATLAB software is used to obtain the slopes between the points, and then the largest slopes are the maximum time-varying stiffness.

Figure 9.

Deformation force diagram of guide shoe lining with different guide shoe lining materials.

As shown in Table 3, the properties of the guide shoe lining materials have a large effect on the stiffness of guide shoes.

Table 3.

Deformation forces on the guide shoe lining with different guide shoe lining materials.

Materials	Maximum deformation displacement (mm)	Maximum time-varying stiffness (N/m)
Rubber	1.0484	74629.7
Natural rubber	0.91088	85353.4
Neoprene rubber	1.3932	53333.3
Butyl rubber	0.94218	56944.3

Differences in contact thickness of guide shoe lining

The stiffness coefficient of an object is influenced by the geometric shape of its cross-section, and objects made of the same material with different geometric shapes deforms differently. This subsection mainly studies the impact of different contact thicknesses between the guide shoe lining and the guide rail on the stiffness of the guide shoe lining.

From Figure 10, for different thicknesses of guide shoe lining, the slope of the tangent at each point on the relationship curve is different. It indicates that the thickness of the guide shoe lining has a certain influence on the variation of nonlinear contact stiffness. An increase in the thickness of guide shoe lining will increase the variation of nonlinear contact stiffness.

Figure 10.

Force diagram of the guide shoe lining deformation with different thickness of guide shoe lining.

As shown in Table 4, when subjected to the same force, the guide shoes with different thicknesses have different maximum deformation displacements and produce different maximum time-varying stiffness.

Table 4.

Deformation force of guide shoe lining with different thicknesses.

Different thicknesses of guide shoe lining (mm)	Maximum deformation displacement (mm)	Maximum time-varying stiffness (N/m)
12	1.0484	74629.7
14	0.94717	81850.2
16	0.8768	89585.2
18	0.82555	92919.5

Different coefficients of friction between guide shoe and guide rail

The friction coefficient between two objects in contact directly affects their relationship. In the finite element analysis, the guide shoe lining and guide rail are considered to have frictionless contact. However, in actual production, the factors such as the surface roughness of the guide rail and the adhesive strength of the rubber of the guide shoe lining may cause friction. Several representative friction coefficients are selected for analysis, especially 0.4 and 0.6, and other factors remain constant.

From Figure 11, it can be seen that the relationship between different friction coefficients and the force deformation of the guide shoe lining is not significant. The linear contact stiffness increases with the friction coefficient increases, while the influence is small. As the friction coefficient increases, the adhesion of the guide wheel on the guide rail increases, and the maximum tangential force in the wheel-rail contact area also increases. The normal force in the contact area is not significantly affected, with a little impact on the transverse contact stiffness.

Figure 11.

Deformation force diagram of the guide shoe lining under different friction coefficients.

From Table 5, the maximum deformation displacements are close to each other and the maximum time-varying stiffness is almost equal under different friction coefficients. This indicates that the friction coefficients have little effect on the contact stiffness of the guide shoe.

Table 5.

Deformation Force of the Guide Shoe lining under Different Friction Coefficients.

Friction coefficients	Maximum deformation displacement (mm)	Maximum time-varying stiffness (N/m)
0	1.0484	74629.7
0.4	1.0331	77390.3
0.6	1.0326	77609.2

Different contact cross section between guide shoe and guide rail

The unevenness of the guide rail has a significant influence on the contact model of the guide shoe. Various models have been proposed to represent excitations caused by guide rail unevenness, including sinusoidal excitation, impulse excitation, and triangular excitation models. In this section, three types of guide rail shapes, namely normal guide rail, inclined guide rail, and concave guide rail, are selected for finite element analysis, aiming to investigate the impact of different guide rail shapes on the contact model of the guide shoe.

From Figure 12, there is not much difference in stiffness between normal and angled rails, while the influence of the concave track on stiffness is more significant.

Figure 12.

Deformation force diagram of the guide shoe lining under different rail shapes.

From Table 6, it is evident that there is minimal variation in the maximum time-varying stiffness between the normal guide rail shape and the inclined guide rail shape. However, there is a significant difference in the maximum deformation stiffness of the concave guide rail compared to the above two. The results indicate that changing the configuration of the guide rail has a significant impact on the stiffness of the contact model of the guide shoe.

Table 6.

Deformation force on the guide shoe lining with different rail shapes.

Different guide rail shapes	Maximum deformation displacement (mm)	Maximum time-varying stiffness (N/m)
Normal guide rail	1.0544	76657.4
Inclined guide rail	1.0484	74629.7
Concave guide rail	0.88915	67349.1

Effects of stiffness and damping of guide shoe spring on the transverse vibration

The Newmark numerical integration method is used to simulate the dynamics model established in Section 3 to solve and compare the influence of different stiffness and damping of the guide shoe on the transverse vibration of the car system. For equation (9), the Newmark numerical integration method is an implicit integration method, calculated as follows:

\begin{array}{l} q_{i + 1} = {[\frac{1}{α {(h)}^{2}} m + \frac{β}{α h} c + k]}^{- 1} \\ \times {F_{i + 1} + m [\frac{1}{α {(h)}^{2}} {\dot{q}}_{i} + \frac{1}{α h} q_{i} + (\frac{1}{2 α} - 1) {\ddot{q}}_{i}] + c [\frac{β}{α h} q_{i} + (\frac{β}{α} - 1) {\dot{q}}_{i} + (\frac{β}{α} - 2) \frac{h}{2} {\ddot{q}}_{i}]} \end{array}

(20)

{\ddot{q}}_{i + 1} = \frac{1}{α h^{2}} (q_{i + 1} - q_{i}) - \frac{1}{α h} {\dot{q}}_{i} - (\frac{1}{2 α} - 1) {\ddot{q}}_{i}

(21)

{\dot{q}}_{i + 1} = {\dot{q}}_{i} + (1 - β) h {\ddot{q}}_{i} + β h {\ddot{q}}_{i + 1}

(22)

The Newmark integration method is based on the assumption that the acceleration varies linearly between two instantaneous times.

The main implementation steps are:

(1) From the known initial conditions $q_{0}$ and ${\dot{q}}_{0}$ , ${\ddot{q}}_{0}$ is obtained through Equation (9);

(2) Choose the appropriate values for h, $α$ and $β$ , where h is the time step;

(3) Start from i = 0, and use Equation (20) to find the displacement vector q_i+1;

(4) Based on Equations (21) and (22), obtain the acceleration vector ${\ddot{q}}_{i + 1}$ and velocity vector ${\dot{q}}_{i + 1}$ at the moment t_i+1.

(5) Obtain the acceleration response for the set time by iterating through Steps (3) and (4).

Based on the above theory, the dynamics of a high-speed elevator prototype with a rated speed of 6 m/s in an elevator test tower is analyzed. The Newmark numerical integration method is used to solve the dynamic equations, and key specifications of the high-speed elevator are shown in Table 7.

Table 7.

Main parameters of the high-speed elevator.

Main parameters	Values	Main parameters	Values
Mass of the carm₁ [kg]	1805	Stiffness of damping springs on both sides of the car k₁ [N/m]	2e6
Moment of inertia of the car J₁ [kg•m²]	3689	Damping of damping springs on both sides of the car c₁ [N•s/m]	500
Mass of the frame m₂ [kg]	2282	Stiffness of damping spring at the bottom of the car k₂ [N/m]	9.8e5
Moment of inertia of the frame J₂ [kg•m²]	2406	Damping of damping spring at the bottom of the car c₂ [N•s/m]	500
Mass of the guide shoe m_di [kg]	13	Guide shoe spring stiffness k₃ [N/m]	8e4
		Guide shoe spring stiffness c₃ [N•s/m]	500

Effect of the stiffness of guide shoe spring on transverse vibration

To study the effect of guide shoe spring stiffness on the transverse vibration of the elevator car, all other parameters remain unchanged. The range of guide shoe stiffness is determined based on the data obtained from an actual elevator. The stiffness of the guide shoe spring varies at the values of 60000 N/m, 80000 N/m, 100000 N/m, and 120000 N/m. The transverse vibration acceleration response of the car is shown in Figure 13.

Figure 13.

Acceleration response of transverse vibration of a car with different guide shoe stiffness.

The maximum peak to peak acceleration values of guide shoe springs with different stiffness are shown in Table 8.

Table 8.

Peak-to-peak values of transverse vibration acceleration of the car with different guide shoe stiffness.

Stiffness (N/m)	60000	80000	100000	120000
Maximum peak-to-peak acceleration (m/s²)	0.0710	0.1094	0.1471	0.1735

(1) The guide shoe stiffness significantly influences the acceleration of transverse vibration in the elevator car. As the guide shoe stiffness increases, the lateral vibration continues to escalate, resulting in a continuous increase in the maximum peak to peak acceleration response.

(2) The use of guide shoe springs with lower stiffness can more effectively reduce the transverse vibration of high-speed elevators.

Effect of guide shoe spring damping on transverse vibration

To study the effect of guide shoe spring damping on transverse vibration of elevator cars, all other parameters are unchanged. The range of guide shoe stiffness is determined based on the data obtained from an actual elevator. The damping coefficient for the guide shoe spring varies at 300 N•s/m, 500 N•s/m, 700 N•s/m, and 900 N•s/m. The transverse vibration acceleration response of the car is shown in Figure 14.

Figure 14.

Acceleration response of transverse vibration of car with different guide shoe damping.

The maximum peak to peak values of damping acceleration for different guide shoe springs are shown in Table 9.

Table 9.

Peak-to-peak values of transverse vibration acceleration of the car with different guide shoe damping.

Damping (N•s/m)	300	500	700	900
Maximum peak-to-peak acceleration (m/s²)	0.1157	0.1094	0.1065	0.1053

(1) The guide shoe damping has the least impact on the transverse vibration acceleration of the elevator car. As the guide shoe damping increases, the transverse vibration gradually decreases, resulting in a decrease in the maximum peak to peak value of the acceleration response.

(2) The use of guide shoe springs with large damping can more effectively reduce the transverse vibration of high-speed elevators.

Conclusions

The transverse vibration dynamics model of a high-speed elevator car system is analyzed, and the accuracy of the model is verified by simulation and calculation. In addition, the finite element analysis method is used to analyze the factors affecting the stiffness variation of the guide shoe. This study proposes an effective model and numerical algorithm, aiming to design and optimize the mechanical parameters of rolling guide shoes. This work also provides a theoretical foundation for enhancing the comfort of elevator rides. The main conclusions can be drawn as follows:

Model rationality

The mechanical model of the rolling guide shoe-rail is introduced into the transverse vibration dynamics model. The Newmark numerical integration method is used to solve the response. The simulation results have a strong correlation with the numerical calculation results, indicating that the proposed transverse vibration dynamics model for high-speed elevators is reasonable and feasible. This model can effectively predict the transverse vibration response of the high-speed elevator car system.

Influence of mechanical characteristics on the stiffness of guide shoes

Through the finite element model of guide shoe-rail contact, the impact of different factors on the guide shoe stiffness can be analyzed, including considering the nonlinear properties of guide shoe stiffness. This is crucial for accurately predicting the lateral vibration of automotive systems.

Influence of stiffness damping on transverse vibration of car system

The analysis results show that the guide shoe stiffness has a significant influence on the transverse vibration, while the guide shoe damping has a small one. Within a certain range, increasing the guide shoe stiffness will intensify the transverse vibration of the car and reduce ride comfort. On the contrary, increasing the guide shoe damping within a certain range will make the lateral vibration of the car smooth. Therefore, using a guide shoe with higher damping and lower stiffness is beneficial for reducing the transverse vibration of elevators. The results indicate that the guide shoe parameters have a significant impact on the vibration of elevators. Based on the current vibration situation of elevators, implementing active vibration control is expected to achieve good results.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Zhongxu Tian

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