Issue 
Mechanics & Industry
Volume 20, Number 3, 2019



Article Number  305  
Number of page(s)  11  
DOI  https://doi.org/10.1051/meca/2019013  
Published online  29 May 2019 
Regular Article
Study on horizontal vibration characteristics of highspeed elevator with airflow pressure disturbance and guiding system excitation
College of Mechanical and Electronic Engineering, Shandong Jianzhu University, 250101 Ji'nan, PR China
^{*} email: zhangruijun@sdjzu.edu.cn
Received:
14
January
2018
Accepted:
17
September
2018
This paper aims to study the horizontal vibration dynamic characteristics of highspeed elevator by considering the combined effect of airflow and guiding system. The relationships of lateral force and overturning moment with horizontal displacement, deflection angle displacement and rated speed of the car are mathematically solved, and the horizontal vibration dynamic model of the car under the two excitations is established. For case model, the natural frequency and horizontal vibration response of the car are studied, and the guiderail excitation frequency and car natural frequency are compared and analyzed. The results indicate that the higher the rated speed is, the more obvious the resonance phenomenon between the guiderail and car will be in a certain range; the effect of airflow on horizontal vibration acceleration of the car with a speed lower than 6 m/s is small, but when the speed is over 6 m/s, the airflow will greatly affect the singlepeak value of horizontal vibration acceleration, which is approximately a quadratic relationship; the deflection angle displacement has an increasing influence on horizontal vibration dynamic response with the increasing speed. The conclusions provide a theoretical guidance for the research and control on the horizontal vibration of highspeed elevator.
Key words: Horizontal vibration / highspeed elevator / airflow pressure disturbance / guiding system
© AFM, EDP Sciences 2019
1 Introduction
The highrise and super highrise buildings are constantly emerging, making the highspeed and largetravel become the inevitable trend of the development of the elevator. With the increase of elevator speed, the resulting horizontal vibration is becoming more and more intense, which has become an important factor affecting the ride comfort of the elevator [1]. The related literatures indicate that the response value of the horizontal vibration acceleration of the elevator is directly proportional to its running speed [2]. In addition, when the highspeed elevator is running at a high speed, the rapid compression of the airflow in the hoistway will cause strong transient pressure shock and tunnel inverse aerodynamic effect, which will further aggravate the horizontal vibration response. Strong horizontal vibration will not only reduce the lifespan of elevator, but also have a serious impact on its operation safety [3], which restrict the rapid development of the highspeed elevator. Therefore, it is very important to study the horizontal vibration characteristics of the elevator at high speed.
As a complex electromechanical system, the highspeed elevator has many causes for its horizontal vibration. Under normal conditions, the main causes of the horizontal vibration can be attributed to two types: the airflow pressure disturbance and the guiding system excitation [4]. The guiding system is mainly composed of the guiderails and the guideshoes, and the excitation generated by the guiding system mainly depends on the timevarying stiffness of the guiderail (the periodic variation of the bending stiffness of the guiderail in its height direction), the profile deviation and the installation error. Taking the rail unevenness as horizontal displacement excitation, Xia et al. [5] established a horizontal vibration model of the elevator car by considering the variation of realtime interfacial stiffness between the guiderail and the guidewheel, and studied the horizontal vibration characteristics of the elevator car. Transforming the factors such as rail roughness, rail bending, guideshoe defect and so on to the contact force of the guiderail to the guideshoe and considering the excitation at rail joints, Yin et al. [6] studied the horizontal vibration dynamic characteristics of the multidegree of freedom of the highspeed elevator. Mei et al. [7] fitted the irregularity excitation between the guideroller and the guiderail by considering the rail profile deviation and the roller roundness tolerance, and solved the horizontal vibration response of the highspeed traction elevator by using the numerical analysis method finally. Utsunomiya et al. [8] analyzed the vibration spectrum of the guiderail, which indicated that the disturbance of the rail mainly concentrated in the lowfrequency areas, and there was a main disturbance frequency which was directly proportional to the running speed of the elevator and inversely proportional to the guiderail length. Colón et al. [9] used Karhunen LoèVE and polynomial chaos method to study the dynamic characteristics and control of the horizontal vibration of elevator car caused by rail irregularity through the establishment of the irregular stochastic model of the rail profile.
The continuous improvement of elevator running speed will cause many complicated aerodynamic problems when the highspeed elevator car is running in the long hoistway. Therefore, the aerodynamic characteristics of highspeed and ultrahighspeed elevator have attracted wide attention from scholars at home and abroad in recent years, and they mainly focused on the test bench simulation experiments and numerical simulations. Based on the dimensionless analysis of parameters affecting the tunnel effect of highspeed elevator, Duan et al. [10] designed and constructed a simplified experimental equipment for aerodynamic characteristics of highspeed elevator, and then carried out the experimental simulations under different elevator conditions such as different blockage ratios and reserved ratios. Bai et al. [11] built a physical experiment model in a certain proportion to study the aerodynamic characteristics of super highspeed elevator, measured the instantaneous speed of a falling elevator car at five distinct positions as well as the average pressure in front of and behind the car along the hoistway, and finally introduced a new concept of “tunnel effected shape drag”. Wang et al. [12] simulated the threedimensional turbulent flow of highspeed elevator using the static uncompressible NavierStokes equations, studied the aerodynamic characteristics of elevators with different blockage ratios, and obtained the conclusion that pressure and drag forces of the highspeed elevator are positively related to the blocking ratio. In order to optimize the aerodynamic characteristics of highspeed elevator and find a reasonable flowguiding cover for the elevator car, Li et al. [13] simulated and optimized a highspeed elevator with loading capacity of 1000 kg and speed of 6 m/s by CFD. Shi et al. [14] conducted dynamic simulations of highspeed elevator using twodimensional model, and studied the transient changes of the aerodynamic forces acting on the car before and after the interlacing of counterweight and elevator car. Chen et al. [15] compared and analyzed the models of 2Dhoistway, 3Dhoistway and 3Dhoistway with landing sill through ANSYS, and studied the aerodynamic characteristics of the hoistway flow field finally.
Although the above studies of the effects of airflow pressure disturbance and guiding system excitation on horizontal vibration characteristics of highspeed elevator have taken substantive results, the aerodynamic characteristics of hoistway flow field and mechanical characteristics of the car under this effect are mainly concentrated when studying the influence of the airflow pressure, few scholars applied the airflow influence to study the dynamic characteristics of elevator horizontal vibration, and they took little consideration of the combined effect of airflow pressure disturbance and guiding system excitation. In addition, due to the limitation of computational conditions, the twodimensional physical model was taken as the research object in the analysis of the airflow pressure within the hoistway usually, and the influence of the horizontal displacement of the elevator car was not taken into consideration, which ignored the essential features of the highspeed elevator.
In view of the above problems, this paper considers the combined effect of airflow pressure disturbance and guiding system excitation, establishes a 3D calculation model of highspeed elevator system, and simulate the model to solve the numerical relationships between the aerodynamic forces/moments and the horizontal displacement and deflection angle displacement of the car using FLUENT by analyzing the flow field characteristics, which is used as the input of the airflow pressure disturbance, and uses the guiderail bending deformation as the input of guiding system excitation, establishes a horizontal vibration dynamics model of highspeed elevator based on the airflow pressure disturbance and guiding system excitation, then solves the dynamic model using the Newmarkbeta method, and finally compares and analyses the natural frequency and horizontal vibration dynamic response of the highspeed elevator car with or without considering the airflow pressure disturbance with an case model, respectively.
2 Analysis of the airflow pressure disturbance within the hoistway
2.1 Mathematical model of 3D flow field within the hoistway
2.1.1 Governing equations of flow field within the hoistway
During the normal operation of highspeed elevator, the airflow in the hoistway presents a flow state of low Mach number (Ma < 0.3) and high Reynolds number (Re > 2320), which can be regarded as the turbulent flow of incompressible gas [13]. Regardless of energy exchange, in the Cartesian coordinate system (x, y, z), the governing equations that describe the flow filed are the incompressible NavierStokes equations, i.e., the mass conservation equation and the momentum conservation equations.
The mass conservation equation:(1)
The momentum conservation equations:(2) (3) (4)where U is air velocity vector, u_{a} , v_{a} and w_{a} are the velocity components of U in x, y and z coordinate directions, t is time, ρ is air density, v is air kinematic viscosity, p is pressure, div is divergence, grad is gradient.
2.1.2 Turbulence model
A turbulence model is required for numerical simulation of the turbulent flow field. At present, the most widely used turbulent numerical simulation method in engineering is Reynolds Averaged NavierStokes (RANS). Its turbulent models mainly include mixed long model, oneequation model and standard twoequation model, and the application of standard twoequation model is the most common. Therefore, RANS and the standard twoequation model are used in this paper to simulate the flow field within the hoistway.
The turbulent kinetic energy k equation:(5)The turbulent dissipation rate ε equation:(6)where μ is air dynamic viscosity, μ_{t} is turbulent viscosity coefficient and
(7) C_{μ} , C _{1}, C _{2}, σ_{k} , σ_{ε} are empirical constants, and C_{μ} = 0.09, C _{1} = 1.44, C _{2} = 1.92, σ_{k} = 1.0, σ_{k} = 1.3.
If φ is introduced, then equations (1)–(6) can be written as the following general governing equation:(8)where φ is generic variable (when φ are 1, [u_{a} , v_{a} , w_{a} ], k and ε, Eq. (8) represents the mass conservation equation, the momentum conservation equations, the turbulence kinetic energy k equation and the turbulent dissipation rate ε equation respectively), Γ_{ φ } is generalized diffusion coefficient, S_{φ} is generalized source term.
2.2 Numerical simulation and analysis of flow field within the hoistway
The horizontal vibration of the elevator car includes translation in its horizontal direction and rotation about the centroid. When the elevator car deviates from the symmetrical position in the horizontal direction, the lateral force and overturning moment will be generated in the car centroid due to the asymmetric distribution of the airflow on both sides [16]. In order to meet the essential characteristics of highspeed elevator to a greater extent, the paper focus on the highspeed elevator with a car size of 1600 mm × 1600 mm × 2300 mm, a hoistway crosssection of 2000 mm × 2100 mm, and neglects the influence of counterweight and the overall flow of airflow in the hoistway. The hoistway crosssection is assumed to be a rectangle, only the car is considered in the elevator model and the influence of other structures are neglected, then the simplified calculation model of the elevator is set up as shown in Figure 1. The calculation area of the whole flow field is established in the relative coordinate system. The status of the car is considered to be stationary, the speeds of the hoistway wall and airflow are equal to that of the simulated car, and the direction is opposite. The upper and lower surfaces of the hoistway are the air outlet and inlet boundaries respectively. The tetrahedral structured grid is used to mesh the calculated area. The size of the lateral grid around the car is 0.01 m and the rest is 0.04 m, the size of the longitudinal grid is 0.08 m, and the number of grids is about 520 000.
In this paper, the finite volume method is used to discretize the calculation area of the flow field, and equation (8) is integrated on the control volume. The discrete algebraic equations are generated on the control body nodes by the secondorder upwind scheme, and the equations are solved by SIMPLE algorithm of the Fluent solver. The convergence standards for calculating the physical quantities of the flow field are 0.1%. The paper takes the uniform speed stage of highspeed elevator as the research object, simulates the lateral force and overturning moment of the car centroid at different horizontal displacements, deflection angle displacements and rated speeds of the elevator car, and finally studies the relationship of the lateral force and overturning moment with the horizontal displacement, deflection angle displacement and rated speed.
Fig. 1 Calculation model of highspeed elevator system. 
2.2.1 The influence of car horizontal displacement on the lateral force and overturning moment
Letting the car horizontal displacement of elevator be x (right is forward direction), the rated speed of elevator car be v. In order to ensure the accuracy of the simulation results, in this paper, the car horizontal displacements (x) are taken as 0.002 m, 0.004 m, 0.006 m, 0.008 m and 0.01 m, which are combined with the rated speeds (v) of 4 m/s, 6 m/s, 8 m/s and 10 m/s respectively, and then the effects of the horizontal displacement and rated speed on the lateral force and overturning moment of the car centroid are studied. It is calculated that at the uniform speed stage, the aerodynamic force/moment of the car centroid in a state are certain values, as shown in Figure 2.
It can be seen from Figure 2 that with the increasing of horizontal displacement and rated speed of the car, both the lateral force and overturning moment suffered by the car centroid show an upward trend, and at the same rated speed, the lateral force and overturning moment are direct proportion to the horizontal displacements of the car respectively. Now, we take the ratios of lateral force and overturning moment to horizontal displacement at different rated speeds as the influence coefficients of the horizontal displacement on the lateral force and overturning moment respectively, as shown in Figure 3.
Fig. 2 Lateral force and overturning moment generated by different rated speeds and horizontal displacements. 
Fig. 3 Influence coefficients of the horizontal displacement on lateral force and overturning moment at different rated speeds. 
2.2.2 The influence of car deflection angle displacement on the lateral force and overturning moment
Letting the car deflection angle displacement be θ (anticlockwise is the forward direction). In this paper, the car deflection angle displacements (θ) are 0.5 π/180 rad, π/180 rad, 1.5 π/180 rad, 2 π/180 rad and 2.5 π/180 rad, which are combined with the rated speeds of 4 m/s, 6 m/s, 8 m/s, 10 m/s and 12 m/s respectively, and then we study the effects of the deflection angle displacement and rated speed on the lateral force and overturning moment of the car centroid. It is calculated that at the uniform speed stage, the aerodynamic force/moment of the car centroid in a state are certain values, as shown in Figure 4.
As can be seen from Figure 4, both the lateral force and overturning moment suffered by the car centroid also increase linearly with the increasing of deflection angle displacement and rated speed of the car, and at the same speed, the lateral force and overturning moment are direct proportion to the deflection angle displacements respectively. Now, we take the ratios of lateral force and overturning moment to deflection angle displacement at different rated speeds as the influence coefficients of the deflection angle displacement on the lateral force and overturning moment respectively, as shown in Figure 5.
Fig. 4 Lateral force and overturning moment generated by different rated speeds and deflection angle displacements. 
Fig. 5 Influence coefficients of the deflection angle displacement on lateral force and overturning moment at different rated speeds. 
2.2.3 Solution of lateral force and overturning moment of the car centroid
From the analysis of Figure 3 and Figure 5, it can be seen that both the horizontal displacement and deflection angle displacement of the car will produce a lateral force and an overturning moment on the car centroid and affect the lateral force and overturning moment to a certain extent. In addition, compared with the car horizontal displacement, the influence of deflection angle displacement on the lateral force and overturning moment is greater. Therefore, according to Figure 3 and Figure 5, combining with the classical theory of aerodynamics, this paper fits the curves of the influence coefficients in Figure 3 and Figure 5 by quadratic curve respectively, and the corresponding expressions that are related to the uniform speed are C_{fx} = 157.9 v ^{2}, C_{mx} = 86.79v ^{2}, C_{fθ} = 1575 v ^{2} and C_{mθ} = 761.6 v ^{2}. Because each force/moment has a linear relationship with x and θ through the corresponding influence coefficients, and the direction of the lateral forces generated by the horizontal displacement and the deflection angle displacement of the car are opposite to the positive direction of the xaxis, so the lateral force (F_{fx} ) and overturning moment (M_{fx} ) acting on the car centroid caused by the horizontal displacement and the lateral force (F_{fθ} ) and overturning moment (M_{fθ} ) acting on the car centroid caused by the deflection angle displacement at different uniform speeds can be written as follows:(9) (10) (11) (12)
3 Analysis of the guiding system excitation
The final expression form of guiderail horizontal displacement excitation is the overall straightness deviation, and the rail bending deformation is the most important and common cause of the straightness deviation [17]. Because the main object of this paper is the car horizontal vibration, so we only consider the bending deformation in the onedimensional direction (horizontal direction) when analyzing the guiderail horizontal displacement excitation. In this paper, the sine wave mode signal is used as the bending deformation deviation, as shown in Figure 6, where A is the amplitude of guiderail bending deformation, ΔL is vertical distance between two guiderail supports.
When the elevator car moves in negative direction of zaxis and with a rated speed of v, the excitation at the lower guideshoe is the same as that at the upper guideshoe, but there is an advanced time that t _{0} = l/v, and l is the vertical distance between the upper and lower guideshoes. If the horizontal displacements of the left and right guiderail at the upper guideshoe are respectively x_{dlu} = A _{1}sin(2πf_{r}t) and x_{dru} = B _{1}sin(2πf_{r}t), then the horizontal displacements of the left and right guiderail at the lower guideshoe are x_{dld} = A _{1}sin(2πf_{r} (t+t _{0})) and x_{drd} = B _{1}sin(2πf_{r} (t+t _{0})), respectively. Where A _{1}, B _{1} are horizontal displacement amplitudes of the left and right guiderail, f_{r} is guiderail excitation frequency and f_{r} = v/λ, λ is wavelength.
In order to avoid the errors caused by the different amplitude and phase of the guiderails on both sides, the amplitude of guiderail bending deformation is usually processed by integrating the bending deformation to one side. That is, the bending deformation of one side (left) guiderail is regarded as a sinusoidal excitation signal, and the other side (right) guiderail is considered as the ideal rail. That is, x_{dlu} = A _{1}sin(2πf_{r}t), x_{dld} = A _{1}sin(2πf_{r} (t+t _{0})), x_{dru} = 0 and x_{drd} = 0.
Fig. 6 Bending deformation deviation of guiderail. 
4 The dynamic model of the car horizontal vibration under the airflow pressure disturbance and guiding system excitation
This paper studies the horizontal vibration of highspeed elevator car system, and mainly considers the movement of the car in the horizontal direction and the rotation about the centroid in the plane, depends on the airflow pressure disturbance and guiding system excitation as the external force, simplifies the four guidewheel and guideshoe systems as parallel springdamping systems [18], and finally establishes the horizontal vibration dynamic model of the highspeed elevator car system as shown in Figure 7.
The movement of the elevator car is decomposed into the translation in the horizontal direction and the rotation about the car centroid. Letting car mass be m_{c} , car moment of inertia be I_{c} , the equivalent stiffness and equivalent damping of the rolling guideshoe be k_{e} and c_{e} , the vertical distances between car centroid and the upper and lower rolling guideshoe be h _{1} and h _{2}, the horizontal force of the guiding system on the car centroid be F_{f} and the moment of the guiding system on the car centroid be M_{f} . According to Lagrange equation, the differential equations of highspeed elevator horizontal vibration can be written as follows:(13) (14)
Now equations (13) and (14) are converted into the universal form of dynamic equations of the lumped mass system with multipledegreeoffreedom as follows:(15) with where M is mass matrix, C is damping matrix, K is stiffness matrix, F is total excitation matrix acting on the car, F _{1} is guiderail excitation matrix and,
F _{2} is airflow excitation matrix caused by horizontal displacement and,
F _{3} is airflow excitation matrix caused by deflection angle displacement and,
Fig. 7 Dynamic model of horizontal vibration of the system. 
5 Case study
5.1 Selection of input parameters
This paper selects the input parameters referring a type of the highspeed elevators of Shandong Fuji Elevator Co., Ltd., as shown in Table 1, where Δm is car rated load, L is single guiderail length.
Parameter values of highspeed elevator system.
5.2 Modal analysis of horizontal vibration
The vibration behavior of a highspeed elevator car belongs to a small damping vibration whose natural frequency can be approximated to the natural frequency of an undamped system. Therefore, the influence of damping and external excitation can be neglected [17]. The characteristic equation and natural frequency of the car system are equations (16) and (17), respectively.(16) (17)where ω is natural angular frequency, f_{n} is natural frequency.
Based on equations (16) and (17), we get the first two order natural frequencies of the elevator car under the conditions of noload, mediumload and fullload, as shown in Table 2. The guiderail excitation frequency at different rated speeds can be obtained from f_{r} = v / λ = v / 2ΔL, as shown in Table 3.
Natural frequency of highspeed elevator car/Hz.
Guiderail excitation frequency at different rated speeds.
5.3 Dynamic response analysis
Taking A _{1} = 0.8 × 10^{−3} m and mediumload condition, for equation (15), we simulate the dynamic response of the car under only the action of the guiding system with different rated speeds and the combined effect of the airflow pressure disturbance and guiding system excitation at different rated speeds based on the Newmarkbeta method, respectively. The simulation results are shown in Figure 8.
It can be seen from Figure 8 that the horizontal vibration acceleration response of the car is similar to the horizontal vibration acceleration curve of the car caused by the sinusoidal guiderail displacement excitation in reference [19]. And the former part of each vibration acceleration curve is the coexistence stage of damped vibration and forced vibration, and the latter part is the stage where only the forced vibration exists. Besides, the damped vibration frequency is equal to the first natural frequency of the system theoretically. From Figure 8, we can see that the damped vibration frequency is about 3.5 Hz (the cyclic number of damped vibration per second is about 3.5), which is very close to 3.4656 Hz in Table 2 (a certain amount of error is caused by the system model assumptions and so on), which verifies the correctness of the model and numerical calculation method used in this paper. From the analysis of Figure 8, we can obtain the singlepeak values of the car horizontal vibration acceleration with different rated speeds with or without considering the influence of airflow pressure disturbance, as shown in Table 4 and Figure 9.
Fig. 8 Horizontal vibration acceleration of elevator car at different rated speeds. 
Singlepeak value of horizontal vibration acceleration/(m/s^{2}).
Fig. 9 Singlepeak value of horizontal vibration acceleration at different rated speeds. 
5.4 Results and discussion
From Table 2, it can be seen that the natural frequencies of the car horizontal vibration are between 3 and 4.2 Hz. When the car is noloaded, mediumloaded and fullloaded, the first natural frequency shows a slight decrease trend; the second natural frequency is invariant. In addition, through the contrast analysis between Tables 2 and 3, we can know that with the increase of the elevator car rated speed, the guiderail excitation frequency is getting closer to the car natural frequency within a certain range, that is, the higher the speed is, the more obvious the resonance phenomenon between the guiderail and the car system will be.
As can be seen from Table 4 and Figure 9, the singlepeak value of the car horizontal acceleration is getting larger and larger with the increasing of rated speed. And when the speed is less than 6 m/s, the singlepeak value considering the airflow pressure disturbance shows an increasing trend compared with noconsideration on the airflow pressure disturbance, but the variation is very small; while when the speed is greater than 6 m/s, compared to the noconsideration on the airflow pressure disturbance, the singlepeak value is increased at a rate of quadratic when considering the flow pressure disturbance. And according to the analysis of Section 2.2, it can be seen that the faster the elevator car rated speed, the more obvious the effect of deflection angle displacement on the horizontal vibration acceleration of elevator car.
6 Concluding remarks
Aiming at the two main excitation sources of horizontal vibration for highspeed elevator − the airflow pressure disturbance and guiding system excitation, through the numerical analysis of 3D calculation model of highspeed elevator system, this paper studies the relationship of the lateral force and the overturning moment acting on the car centroid with the horizontal displacement, deflection angle displacement and rated speed respectively. The effect of horizontal displacement and deflection angle displacement cannot be neglected, and finally establishes a horizontal vibration dynamics model under the combined action of above two main excitation sources.
In this paper, by analyzing the influence coefficients of horizontal displacement and deflection angle displacement on the lateral force and overturning moment of the car respectively, we conclude that the effect of the deflection angle displacement on the lateral force and the overturning moment is more obvious under the constraint of the guiderails on both sides. Therefore, in the study of horizontal vibration control of highspeed elevator, the control of deflection angle displacement should be taken as the main point.
Through the study of natural frequency and dynamic response of horizontal vibration with and without the consideration of the airflow pressure disturbance under the same guiding system excitation, we conclude that in a certain range, the higher the rated speed is, the more obvious the resonance phenomenon between the guiderail and car will be; when the elevator car under noload, mediumload and fullload condition, the first natural frequency shows a trend of slight decrease, while the second natural frequency is invariant; for the elevator with rated speed less than 6 m/s, the influence of airflow pressure disturbance on the singlepeak value of horizontal vibration acceleration is negligible; however, when the rated speed is over 6 m/s, the influence of airflow pressure disturbance on the singlepeak value is approximately a rate of quadratic ratio relationship, and with the increasing rated speed, the influence of deflection angle displacement on the horizontal vibration acceleration is more and more obvious. Therefore, the influence of airflow should not be ignored in the study of horizontal vibration. The above conclusions have some theoretical significance in the research on the horizontal vibration characteristics and control of highspeed elevator.
Nomenclature
A : Amplitude of guiderail bending deformation (m)
A _{1} : Horizontal displacement amplitude of left guiderail (m)
B _{1} : Horizontal displacement amplitude of right guiderail (m)
C_{fx,} C_{mx,}C_{fθ,}C_{mθ} : Influence coefficient
C_{μ} , C _{1}, C _{2},σ_{k} ,σ_{ε} : Empirical constant
c_{e} : Equivalent damping of rolling guideshoe (N.s.m^{−1})
F : Total excitation matrix acting on the car
F _{1} : Guiderail excitation matrix
F _{2} : Airflow excitation matrix caused by horizontal displacement
F _{3} : Airflow excitation matrix caused by deflection angle displacement
F_{f} : Horizontal force of guiding system on car centroid (N)
F_{fx} : Lateral force caused by horizontal displacement (N)
F_{fθ} : Lateral force caused by deflection angle displacement (N)
f_{n} : Car natural frequency (Hz)
f_{r} : Guiderail excitation frequency (Hz)
h _{1} : Vertical distance between car centroid and upper guideshoe (m)
h _{2} : Vertical distance between car centroid and lower guideshoe (m)
I_{c} : Car moment of inertia (kg.m^{−2})
k : Turbulent kinetic energy (J)
k_{e} : Equivalent stiffness of rolling guideshoe (N.m^{−1})
L : Single guiderail length (m)
l : Vertical distance between upper and lower guideshoes (m)
M_{f} : Moment of guiding system on car centroid (N.m)
M_{fx} : Overturning moment caused by horizontal displacement (N.m)
M_{fθ} : Overturning moment caused by deflection angle displacement (N.m)
_{ Sφ } : Generalized source term
t _{0} : Advanced time between upper and lower guideshoes (s)
u_{a} , v_{a} , w_{a} : Velocity component of U in x, y, z coordinate directions (m.s^{−1})
v : Carrated speed (m.s^{−1})
x : Car horizontal displacement (m)
x_{dlu} : Horizontal displacement of left guiderail at upper guideshoe (m)
x_{dru} : Horizontal displacement of right guiderail at upper guideshoe (m)
x_{dld} : Horizontal displacement of left guiderail at lower guideshoe (m)
x_{drd} : Horizontal displacement of right guiderail at lower guideshoe (m)
θ : Car deflection angle displacement (rad)
v : Air kinematic viscosity (m^{−2}.s^{−1})
ε : Turbulent dissipation rate
_{ μ } : Air dynamic viscosity (N.s.m^{−2})
μt : Turbulent viscosity coefficient (N.s.m^{−2})
_{Γφ } : Generalized diffusion coefficient
_{ΔL } : Vertical distance between two guiderail supports (m)
_{Δm } : Elevatorrated load (kg)
ω : Natural angular frequency (Hz)
Acknowledgments
The authors are grateful for the financial support by Shandong Provincial Natural Science Foundation, China (ZR2017MEE049) and the Introduction of Urgently Needed Talents for Economic Rising Zone in Western China and the Key Areas for Poverty Alleviation and Development in Shandong Province in 2017.
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Cite this article as: J. Liu, R. Zhang, Q. He, Q. Zhang, Study on horizontal vibration characteristics of highspeed elevator with airflow pressure disturbance and guiding system excitation, Mechanics & Industry 20, 305 (2019)
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All Figures
Fig. 1 Calculation model of highspeed elevator system. 

In the text 
Fig. 2 Lateral force and overturning moment generated by different rated speeds and horizontal displacements. 

In the text 
Fig. 3 Influence coefficients of the horizontal displacement on lateral force and overturning moment at different rated speeds. 

In the text 
Fig. 4 Lateral force and overturning moment generated by different rated speeds and deflection angle displacements. 

In the text 
Fig. 5 Influence coefficients of the deflection angle displacement on lateral force and overturning moment at different rated speeds. 

In the text 
Fig. 6 Bending deformation deviation of guiderail. 

In the text 
Fig. 7 Dynamic model of horizontal vibration of the system. 

In the text 
Fig. 8 Horizontal vibration acceleration of elevator car at different rated speeds. 

In the text 
Fig. 9 Singlepeak value of horizontal vibration acceleration at different rated speeds. 

In the text 
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