Issue 
Mechanics & Industry
Volume 22, 2021



Article Number  28  
Number of page(s)  11  
DOI  https://doi.org/10.1051/meca/2021027  
Published online  12 April 2021 
Regular Article
Analysis of longitudinal vibration acceleration based on continuous timevarying model of highspeed elevator lifting system with random parameters
School of Mechanical and Electrical Engineering, Shandong Jianzhu University, Jinan 250101, Shandong Province, PR China
^{*} email: zhangqing@sdjzu.edu.cn
Received:
4
June
2019
Accepted:
1
March
2021
In this paper, for studying the influence of the randomness of structural parameters of highspeed elevator lifting system (HELS) caused by manufacturing error and installation error, a continuous timevarying model of HELS was constructed, considering the compensation rope mass and the tension of the tensioning system. The Galerkin weighted residual method is employed to transform the partial differential equation with infinite degrees of freedom (DOF) into the ordinary differential equation. The fiveorder polynomial is used to fit the actual operation state curve of elevator, and input as operation parameters. The precise integration method of timevarying model of HELS is proposed. The determination part and the random part response expression of the longitudinal dynamic response of HELS are derived by the random perturbation method. Using the precise integration method, the sensitivity of random parameters is determined by solving the random part response expression of timevarying model of HELS, and the digital characteristics of the acceleration response are analyzed. It is found that the line density of the hoisting wire rope has the maximum sensitivity on longitudinal vibration velocity response, displacement response and acceleration response, and the sensitivity of the elastic modulus of the wire rope is smallest.
Key words: Highspeed elevator lifting system / timevarying / random parameters / longitudinal vibration / acceleration response
© Q. Zhang et al., Published by EDP Sciences 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
As a “vertically moving car”, the elevator has been widely used in highrise buildings and super highrise buildings. In developed countries, the number of people taking elevators per day is more than that of other means of transportation, and elevators have become one of the symbols to measure the degree of modernization of the country. Withthe development of elevator toward high speed and large strokes, various vibration phenomena inevitably appear in elevators, and a large part of themis related to the elevator's lifting system.
Manufacturing error and installation error in the highspeed elevator lifting system are objective. The random parameters such as wire rope density, and elastic modulus existing in the lifting system cause the vibration of the HELS to be random vibration. The random vibration system not only affects the eigenvalues and eigenvectors of the various modes of the system, but also affects the statistical characteristics of the response [1]. In addition, studies have shown that when the initial conditions are consistent, the longitudinal vibration of the lifting system has a much greater impact on the system than the lateral vibration. Therefore, it is of great significance to study the dynamic response of the longitudinal vibration random parameters of the HELS on the elevator car vibration reduction, random parameter sensitivity analysis, and safety assessment.
At present, the research on HELS mainly focuses on the dynamic characteristics of deterministic parameters [2–7]. It is rare to consider the random parameters of HELS [1,8–10]. The research on the random parameters of longitudinal vibration of HELS is rarer. Lin et al. [11] established elevator virtual prototype model through Solid Works, and analyzed the dynamics of the highspeed elevator car with the ADAMS, then the dynamic model of iDOF of the elevator system in the vertical direction was established, and the sensitivity analysis is used to optimize the elevator dynamic parameters. Feng et al. [12] considered the timevarying characteristics of the elevator traction rope stiffness, established an elevator dynamics model with 8 DOF coupled vibrationand performed modal analysis on the system, according to the relationship between the natural frequency of the dynamic structure system and the excitation frequency difference, the failure mode of the system resonance is defined, and the reliability sensitivity analysis was performed on the random variables of the system. Wu et al. [4] used virtual prototyping technology to analyze and simulate the elevator operation dynamically, the 11 DOF vertical vibration model of the elevator system was established, through the sensitivity analysis of the highspeed elevator vibration signal, the influence of the main dynamic parameters on the highspeed elevator vibration was obtained. Although the above literatures consider random parameters for the study of longitudinal vibration of elevators, its research is based on the discrete model of elevator concentration parameters, but its research is based on the elevator lumped parameter model. The ordinary differential control equations established by this type of model are simple, easy to understand, andsolve. However, because such models ignore the continuous characteristics of the wire rope, they cannot better reflect the dynamic characteristics of the elevator lifting system.
The establishment of the distributed parameter model of the HELS draws on the research theory of the axially moving string, which is simplified into a section of axial motion string with concentrated mass, which can better describe the flexible timevarying characteristics of the traction wire rope, so it is gradually being applied. Zhang et al. [13] simplified the elevator hoisting rope to a variable length axial motion string with a certain mass attached to one end, the differential equations and energy equations for the vertical vibration of the HELS were established by the energy method and the Hamilton principle. Bao et al. [2,14] used the Hamliton principle to construct a lateral vibration control equation for flexible wire rope without external excitation and external excitation, and evaluated the theoretical model through experiments, the experimental results well agree with the theoretical predictions. In addition [15], considering the interaction between the rigid motion and the deformation motion of the steel wire rope, the differential equation of the wire rope motion of the lifting system is constructed, and the model is analyzed. However, the above literature does not consider the effect of the compensation rope mass and the tension of the tensioning system on the vibration of the HELS.
Therefore, under the premise of comprehensively considering the influence of compensation rope mass and tension of the tensioning system, the timevarying continuous model of HELS is constructed by combining energy method and Hamilton principle. Using random perturbation method to derive the dynamic equation of system response under random parameters. Applying the precise integration method of HELS to analyze the sensitivity and standard deviation of structural random parameters of elevator operation process. Then the influences of each structural parameters on the dynamic characteristics of the lifting system are analyzed.
2 Establishment of longitudinal timevarying model for highspeed traction elevator lifting system
To study the timevarying characteristics of the longitudinal vibration of traction rope conveniently, the modeling and solution of this paper are based on the following three assumptions:

Hoisting ropes are continuous and uniform, with constant crosssectional area A and elastic modulus E during movement;

The influence of lateral vibration from the hoisting ropes is ignored, and elastic deformation caused by the vertical vibration of the hoisting ropes is smaller than the length of the ropes;

The influences of bending rigidity on hoisting ropes, friction force, and airflow are ignored.
Figure 1 shows the timevarying model of longitudinal vibration in a highspeed traction elevator lifting system. The hoisting rope of the highspeed traction elevator is simplified as a variablelength string along the axial force and movement. The specified structure of the car is ignored and the structure is simplified into a rigid weight block of mass m connected to the lower end of the cord. ρ_{1} is the density of the hoisting rope, A is the crosssectional area, E is the elastic modulus, and ρ_{2} is the density of compensation rope. The origin of the coordinates is the tangent point of the traction sheave and the hoisting rope, and the direction vertically downward is the positive direction of the X axis. The length of hoisting rope at the top of the car from the origin of the coordinate is l (t). The vibration displacement at the string x (t) is y (x (t) , t), and v (t) is the operating speed of the highspeed traction elevator. l_{0} is the maximum lift height (highspeed traction elevators are generally used in super highrise buildings. The height of the car is very small compared to the super highrise buildings, so the height of the car is negligible).
By using the finite deformation theory of continuum, the displacement vector and velocity vector of x (t) in the Xaxis are as follows:$$r=[x(t)+y(x(t),t)]j\text{,}$$(1) $$V=[v(t)+{y}_{t}(x(t),t)]j\text{,}$$(2)where j is the unit vector in the Xaxis direction, y (x (t) , t) and y_{t} (x (t) , t) are the partial derivatives of t, and y, y_{t} represent y (x (t) , t) and y_{t} (x (t) , t), respectively.
Similarly, the displacement vector and velocity vector of the car in the direction of Xaxis are as follows:$${r}_{c}=\left[l\left(t\right)+y\right]j\text{,}$$(3) $${V}_{c}=\left[v\left(t\right)+{y}_{t}\right]j\text{,}$$(4)
The kinetic energy of the system can be expressed as follows:$${E}_{\text{k}}=\frac{1}{2}m{V}^{2}x=l(t)+\frac{1}{2}{\rho}_{1}{\int}_{0}^{l(t)}{V}^{2}ds\text{,}$$(5)
The elastic potential of the system is:$${E}_{\text{s}}={\int}_{0}^{l(t)}(P{y}_{x}+\frac{1}{2}EA{{y}_{x}}^{2})ds\text{,}$$(6)where y_{x} (x (t) , t) is the partial derivative of x, and y_{x} represents y_{x} (x (t) , t).
P is the tension of the rope during a static balance of tension. While the hoisting rope is subjected to its own gravity and the gravity of the car, it is also subjected to the gravity of the tensioning rope and the pretensioning force f of the tensioning device. Thus, tension P in the static balance can be expressed as:$$P=\left[m+{\rho}_{1}\left(l\left(t\right)x\right)+{\rho}_{2}\left({l}_{0}l\left(t\right)\right)\right]g+f\text{,}$$(7)
The gravitational potential energy of the system is expressed as:$${E}_{\text{g}}={\int}_{0}^{l(t)}{\rho}_{1}gdtmgyx=l(t)\text{,}$$(8)
According to the Hamilton principle:$$I={\int}_{{t}_{1}}^{{t}_{2}}\left[\delta {E}_{k}\delta {E}_{s}\delta {E}_{g}\right]dt=0\text{,}$$(9)
The longitudinal vibration dynamics equation of the highspeed traction elevator lifting system arederived as follows:$${\rho}_{1}\left({y}_{tt}+a\right){P}_{x}{\rho}_{1}gEA{y}_{xx}=0,\left(0<x<l\left(t\right)\right)\text{,}$$(10) $$m\left(a+{y}_{tt}\right)+{\rho}_{1}v\left(v+{y}_{t}\right)+EA{y}_{x}+Pmg=0,\left(x=l\left(t\right)\right)\text{,}$$(11)
Equation (11) is the boundary condition where the string is at x = l (t).
Fig. 1 Timevarying model of the hoisting rope in an elevator lifting system. 
3 Galerkin discretization of timevarying partial differential equations for the HELS
The algebraic equation coefficient matrix obtained by Galerkin discrete method is symmetric, and approximation accuracy is higher than those of the other methods. Therefore, the Galerkin method is used to discretize the partial differential control equation.
For facilitating the discrete method, a dimensionless parameter ξ is introduced, and normalize the original variables, that is, $\xi =x/l\left(t\right)$. The time domain of x becomes the fixed domain [0,1] of ξ. Assuming that the solution of equation (10) can be represented by infinite DOF distribution function y:$$y\left(x,t\right)={\displaystyle \sum _{i=1}^{n}}{\varphi}_{i}\left(\xi \right){q}_{i}\left(t\right)={\displaystyle \sum _{i=1}^{n}}{\varphi}_{i}\left(\frac{x}{l\left(t\right)}\right){q}_{i}\left(t\right)\text{,}$$(12)
ϕ_{i (ξ)} is the trial function, and q_{i} (t) is the timedependent generalized coordinates$${\varphi}_{i}\left(\xi \right)=\sqrt{2}\mathrm{sin}\left(\frac{2i1}{2}\pi \xi \right)\left(i=\mathrm{1,2},\cdots ,n\right)\text{,}$$(13)
Then,$$\begin{array}{l}{y}_{x}=\frac{1}{l(t)}{\displaystyle \sum _{i=1}^{n}}{{\varphi}^{\prime}}_{i}(\xi ){q}_{i}(t),\text{\hspace{0.17em}}{y}_{xx}=\frac{1}{l(t)}{\displaystyle \sum _{i=1}^{n}}{{\varphi}^{\u2033}}_{i}(\xi ){q}_{i}(t)\\ {y}_{t}={\displaystyle \sum _{i=1}^{n}}{\varphi}_{i}(\xi ){{\displaystyle \dot{q}}}_{i}(t)\frac{\xi v}{l(t)}{\displaystyle \sum _{i=1}^{n}}{{\varphi}^{\prime}}_{i}(\xi ){q}_{i}(t),\\ {y}_{tt}={\displaystyle \sum _{i=1}^{n}}{\varphi}_{i}(\xi ){{\displaystyle \stackrel{\mathrm{}}{q}}}_{i}(t)\frac{2\xi v}{l(t)}{\displaystyle \sum _{i=1}^{n}}{{\varphi}^{\prime}}_{i}(\xi ){{\displaystyle \dot{q}}}_{i}(t)+\frac{2\xi {v}^{2}}{{l}^{2}(t)}{\displaystyle \sum _{i=1}^{n}}{{\varphi}^{\prime}}_{i}(\xi ){q}_{i}(t)\\ \frac{a\xi}{l(t)}{\displaystyle \sum _{i=1}^{n}}{{\varphi}^{\prime}}_{i}(\xi ){q}_{i}(t)+\frac{{\xi}^{2}{v}^{2}}{{l}^{2}(t)}{\displaystyle \sum _{i=1}^{n}}{{\varphi}^{\u2033}}_{i}(\xi ){q}_{i}(t)\end{array}$$(14)
Substitute equation (14) into the kinetic equation (11), and multiply both sides by ϕ_{j} (ξ), and integrate ξ in the range [0,1]. Substitute equation (14) into the boundary condition (12), and multiply both sides by ϕ_{j} (1) after transformation. The original partial differential equations are discretized into the following equation by using the weighted residual method:$$M{{\displaystyle \stackrel{\mathrm{}}{q}}}_{j}+C{{\displaystyle \stackrel{}{q}}}_{j}+K{q}_{j}=F\text{,}$$(15)
where, q_{j} = [q_{1} (t) , q_{2} (t) , ⋯ , q_{n} (t)], is the generalized coordinate vector, M, C, K, and F are the mass, damping, stiffness, and generalized force matrices, respectively. And,$$M={\rho}_{1}{\delta}_{IJ}+\frac{m}{l}{\varphi}_{i}\left(1\right){\varphi}_{j}\left(1\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}C=\frac{2{\rho}_{1}v}{l}{\int}_{0}^{1}\xi {{\varphi}^{\prime}}_{i}{\varphi}_{j}d\xi +\frac{{\rho}_{1}v}{l}{\varphi}_{i}\left(1\right){\varphi}_{j}\left(1\right)\text{,}$$ $$K=\frac{m{v}^{2}}{{l}^{3}}{{\varphi}_{i}}^{\u2033}(1){\varphi}_{j}(1)\frac{{\rho}_{1}a}{l}{\int}_{0}^{1}\xi {{\varphi}^{\prime}}_{i}{\varphi}_{j}d\xi \frac{{\rho}_{1}{v}^{2}}{{l}^{2}}{\int}_{0}^{1}{\xi}^{2}{{\varphi}^{\prime}}_{i}{{\varphi}^{\prime}}_{j}d\xi \frac{EA}{{l}^{2}}{\int}_{0}^{1}{{\varphi}_{i}}^{\u2033}{\varphi}_{j}d\xi \text{,}$$ $$F={\rho}_{1}a{\int}_{0}^{1}{\varphi}_{j}d\xi \frac{ma}{l}{\varphi}_{j}(1)\frac{{\rho}_{1}{v}^{2}}{l}{\varphi}_{j}(1)\frac{{\rho}_{2}g({l}_{0}l)+f}{l}{\varphi}_{j}(1)\text{.}$$
4 Precise integration method for vertical vibration model of the timevarying system in the highspeed traction elevator
For the timevarying dynamic model of the highspeed elevator traction hoisting system established above, because of its strong timevarying characteristics, the mass, damping and stiffness of the system are changing every moment. It is difficult for the general numerical method to achieve high accuracy for this kind of problem. Precise integration method, due to its explicit stability and high accuracy [16], has been widely used in solving dynamics of nonlinear timevarying systems, and achieved good results [17,18]. Therefore, for the highspeed elevator timevarying model established in this paper, the precise integration method of the longitudinal timevarying model of HELS is proposed to analyze the model, so as to make the result more accurate.
First, follow the introduction of the dual variable of Hamiltonian system [19],$$p=M{\displaystyle \stackrel{}{x}}+C(t)x/2\text{\hspace{0.17em}}\text{or}{\displaystyle \stackrel{}{}}={M}^{1}p{M}^{1}C(t)x/2$$(16)
By substituting the equation (16) into the dynamic equation, the following equation can be obtained:$$\stackrel{}{p}}=\left(C(t){M}^{1}C(t)/4K(t)\right)xC(t){M}^{1}p/2+f(t)$$(17)
The above equations are written in the general form of a linear system$$\begin{array}{c}\hfill {\displaystyle \stackrel{}{x}}=Ax+Cp+{r}_{x}\hfill \\ \hfill {\displaystyle \stackrel{}{\mathit{\text{p}}}}=Bx+Dp+{r}_{p}\hfill \end{array}$$(18) where, A = − M ^{−1}C(t)/2, B = C(t)M ^{−1}C(t)/4 − K(t), C = − C(t)M ^{−1}/2, D = M ^{−1}, r_{p} = 0, r_{x} = f (t).
Therefore,$$\stackrel{}{\mathit{\text{z}}}}=Hz+\varphi (t)$$(19)where, $z=\left[\begin{array}{c}\hfill x\hfill \\ \hfill p\hfill \end{array}\right]$, $H=\left[\begin{array}{cc}\hfill A\hfill & \hfill D\hfill \\ \hfill B\hfill & \hfill \text{C}\hfill \end{array}\right]$, $\varphi (t)=\left[\begin{array}{c}\hfill {\text{r}}_{x}\hfill \\ \hfill {r}_{p}\hfill \end{array}\right]$.
Assume that the nonhomogeneous term is linear in time step (t_{k}, t_{k} + 1), the equation is$$\stackrel{}{z}}=Hz+{\varphi}_{k}+{\displaystyle \stackrel{}{{\varphi}_{k}}}(t{t}_{k})$$(20)
Then, the solution at t_{k}_{+1} moment can be written as$${\text{z}}_{k+1}={T}_{\text{k}}\left[{z}_{k}+{{H}_{k}}^{1}({\varphi}_{k}+{{H}_{k}}^{1}{\displaystyle \stackrel{}{{\varphi}_{k}}})\right]{{H}_{k}}^{1}[{\varphi}_{k}+{{H}_{k}}^{1}{\displaystyle \stackrel{}{{\varphi}_{k}}}+{\displaystyle \stackrel{}{{\varphi}_{k}}}({t}_{k+1}{t}_{k})]$$(21)
where, T_{k} = e ^{Hk(tk+1−tk)}_{.}
Then the solution of the equation is transformed into the solution matrix T_{k}, and the accuracy of the matrix T_{k} becomes the key to solving the equation. Zhong [19] proposed a 2^{N} algorithm by using the additive theorem. For the above formula, there is,$${T}_{\text{k}}={e}^{{H}_{k}\mathrm{\Delta}t}={({e}^{{H}_{k}\frac{\mathrm{\Delta}t}{m}})}^{m}={({e}^{{H}_{k}\tau})}^{m}$$(22)
Among them, choose m = 2^{N}, usually, Δt is a small time interval, so τ = Δt/m is a very small time interval, for τ, there is,$${e}^{{H}_{k}\tau}\approx I+{H}_{k}\cdot \tau +\frac{{({H}_{k}\cdot \tau )}^{2}}{2}=I+{T}_{a}$$(23)
where I is the identity matrix,T_{a} = (H_{k} ⋅ τ) ⋅ (I+H_{k} ⋅ τ/2).
Therefore, the matrix T_{k} can be decomposed as follows,$${T}_{k}={(I+{T}_{a})}^{{2}^{N}}={[{T}_{a}+I]}^{{2}^{N1}}\times {[{T}_{a}+I]}^{{2}^{N1}}$$(24)
According to the flow chart, as shown in Figure 2, T_{k} = I + T_{a}. Then, according to equation (21), given initial condition z_{0}, the steps are gradually performed to obtain z_{1}, z_{2},⋯ , z_{k},⋯, which is a typical “selfstarting” algorithm.
Fig. 2 Operation flow chart. 
5 The quintic polynomial fitting for the running state curve of the elevator
According to the actual operation state of the elevator, the description of the operation state stage of the elevator when it goes up is shown in Table 1 below. The fifthorder polynomial (25) is used to fit the actual operation state of the elevator, and the operation curves of the elevator in each stage can be obtained, as shown in Figure 3 below.$${l}_{i}(t)={C}_{0}^{i}+{C}_{1}^{i}t+{C}_{2}^{i}{t}^{2}+{C}_{3}^{i}{t}^{3}+{C}_{4}^{i}{t}^{4}+{C}_{5}^{i}{t}^{5}$$(25)
Stage division of elevator operation state curve.
Fig. 3 Elevator running state curve. 
6 Sensitivity analysis of random parameters of lifting system based on random perturbation method
The structural parameters (such as the line density of traction rope ρ_{1}, crosssectional area of traction rope A, elastic modulus of traction rope E) of HELS have certain randomness. Therefore, M, C, K, F in the differential equation of system dynamics has stochastic property. The following transformations are required.$$\begin{array}{l}M={M}_{d}+\u03f5{M}_{r}\\ C={C}_{d}+\u03f5{C}_{r}\\ K={K}_{d}+\u03f5{K}_{r}\\ Y={Y}_{d}+\u03f5{Y}_{r}\\ F(t)={F}_{d}(t)+\u03f5{F}_{r}(t)\end{array}$$(26)where ϵ is a small parameter [20,21]. The subscripts d and r respectively represent the determined part and the random part of the random variable.
Substituting equation (26) into (16) and expand to compare ε with the same power coefficient. Omitting higherorder terms above O (ϵ^{2}) the following equations are obtained.$${\u03f5}^{0}:{M}_{d}{{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}+{C}_{d}{{\displaystyle \stackrel{}{q}}}_{d}+{C}_{d}{q}_{d}={F}_{d}(t)$$(27) $${\u03f5}^{1}:{M}_{d}{{\displaystyle \stackrel{\mathrm{}}{q}}}_{r}+{C}_{d}{{\displaystyle \stackrel{}{q}}}_{r}+{K}_{d}{q}_{r}={F}_{r}\left(t\right)\left({M}_{r}{{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}+{C}_{r}{{\displaystyle \stackrel{}{q}}}_{d}+{K}_{r}{q}_{d}\right)$$(28)
Equations (27) and (28) represent the deterministic part and the random part of the response, respectively. For convenience, the random response {q_{r}} is divided into two parts:$${q}_{r}={q}_{r1}+{q}_{r2}$$(29)where {q_{r1}} and {q_{r2}} respectively satisfy the following equation.$${M}_{d}{{\displaystyle \stackrel{\mathrm{}}{q}}}_{r1}+{C}_{d}{{\displaystyle \stackrel{}{q}}}_{r1}+{K}_{d}{q}_{r1}={F}_{r}(t)$$(30) $${M}_{d}{{\displaystyle \stackrel{\mathrm{}}{q}}}_{r2}+{C}_{d}{{\displaystyle \stackrel{}{q}}}_{r2}+{K}_{d}{q}_{r2}=({M}_{r}{{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}+{C}_{r}{{\displaystyle \stackrel{}{q}}}_{d}+{K}_{r}{q}_{d})$$(31)
Equations (30) and (31) represent random responses due to randomness of excitation and randomness of parameters, respectively. For equation (31), the random variable can be Taylorexpanded near the determined portion b_{dj} (j = 1,2,…m) of the random parameter [22,23].$${{\displaystyle \stackrel{\mathrm{}}{q}}}_{r2}={\displaystyle \sum _{j=1}^{m}}\frac{\partial {{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}}{\partial {b}_{j}}\cdot {b}_{rj}$$(32) $${{\displaystyle \stackrel{}{q}}}_{r2}={\displaystyle \sum _{j=1}^{m}}\frac{\partial {{\displaystyle \stackrel{}{q}}}_{d}}{\partial {b}_{j}}\cdot {b}_{rj}$$(33) $${q}_{r2}={\displaystyle \sum _{j=1}^{m}}\frac{\partial {q}_{d}}{\partial {b}_{j}}\cdot {b}_{rj}$$(34) $${M}_{r}={\displaystyle \sum _{j=1}^{m}}\frac{\partial {M}_{d}}{\partial {b}_{j}}\cdot {b}_{rj}$$(35) $${C}_{r}={\displaystyle \sum _{j=1}^{m}}\frac{\partial {C}_{d}}{\partial {b}_{j}}\cdot {b}_{rj}$$(36) $${K}_{r}={\displaystyle \sum _{j=1}^{m}}\frac{\partial {K}_{d}}{\partial {b}_{j}}\cdot {b}_{rj}$$(37)
Substituting equations (31)–(37) into equation (31) and comparing the coefficients of b_{rj} $${M}_{d}\frac{\partial {{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}}{\partial {b}_{j}}+{C}_{d}\frac{\partial {{\displaystyle \stackrel{}{q}}}_{d}}{\partial {b}_{j}}+{K}_{d}\frac{\partial {q}_{d}}{\partial {b}_{j}}=\left(\frac{\partial {M}_{d}}{\partial {b}_{j}}{{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}+\frac{\partial {C}_{d}}{\partial {b}_{j}}{{\displaystyle \stackrel{}{q}}}_{d}+\frac{\partial {K}_{d}}{\partial {b}_{j}}{q}_{d}\right)\left(j=\mathrm{1,2,\dots},m\right)$$(38)
Using the precise integration method for vertical vibration model of the timevarying system in the highspeed traction elevator to solve the equation (38), the sensitivity of the system response, $\frac{{{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}}{\partial {b}_{j}}$, $\frac{{{\displaystyle \stackrel{}{q}}}_{d}}{\partial {b}_{j}}$, and $\frac{\partial {q}_{d}}{\partial {b}_{j}}$ can be get.
7 Analysis of mean and standard deviation of longitudinal vibration acceleration of highspeed elevators with random parameters
Define the covariance matrix of the displacement response of the continuous timevarying model as N_{q}, the random parameter covariance matrix as N_{b}, and the displacement response sensitivity matrix as $\left[\frac{\partial {q}_{d}}{{\partial}_{b}}\right]$.$${N}_{q}=\left[\begin{array}{cccc}\hfill Var\left({q}^{\left(1\right)}\right)\hfill & \hfill Cov\left({q}^{\left(2\right)},{q}^{\left(1\right)}\right)\hfill & \hfill \cdots \hfill & \hfill Cov\left({q}^{\left(k\right)},{q}^{\left(1\right)}\right)\hfill \\ \hfill Cov\left({q}^{\left(2\right)},{q}^{\left(1\right)}\right)\hfill & \hfill Var\left({q}^{\left(1\right)}\right)\hfill & \hfill \cdots \hfill & \hfill Cov\left({q}^{\left(k\right)},{q}^{\left(2\right)}\right)\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill Cov\left({q}^{\left(k\right)},{q}^{\left(1\right)}\right)\hfill & \hfill Cov\left({q}^{\left(k\right)},{q}^{\left(2\right)}\right)\hfill & \hfill \cdots \hfill & \hfill Var\left({q}^{\left(k\right)}\right)\hfill \end{array}\right]$$(39) $${N}_{b}=\left[\begin{array}{cccc}\hfill Var\left({b}_{1}\right)\hfill & \hfill Cov\left({b}_{2},{b}_{1}\right)\hfill & \hfill \cdots \hfill & \hfill Cov\left({b}_{m},{b}_{1}\right)\hfill \\ \hfill Cov\left({b}_{2},{b}_{1}\right)\hfill & \hfill Var\left({b}_{2}\right)\hfill & \hfill \cdots \hfill & \hfill Cov\left({b}_{m},{b}_{2}\right)\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill Cov\left({b}_{m},{b}_{1}\right)\hfill & \hfill Cov\left({b}_{m},{b}_{2}\right)\hfill & \hfill \cdots \hfill & \hfill Var\left({b}_{m}\right)\hfill \end{array}\right]$$(40) $$\left[\frac{\partial {q}_{d}}{\partial b}\right]=\left[\begin{array}{cccc}\hfill \frac{\partial {q}_{d}}{\partial {b}_{1}}\hfill & \hfill \frac{\partial {q}_{d}}{\partial {b}_{2}}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial {q}_{d}}{\partial {b}_{m}}\hfill \end{array}\right]$$(41)
where Var(q ^{(k)}) represents the variance of the k_{th} element in the vector q, and Cov represents the covariance.$${N}_{q}=\left[\frac{\partial {q}_{d}}{\partial b}\right]{N}_{b}{\left[\frac{\partial {q}_{d}}{\partial b}\right]}^{T}$$(42)
The standard deviation of displacement response can be obtained by solving equation (36).$${\sigma}_{q}^{i}={\left({\displaystyle \sum _{j=1}^{m}}{\displaystyle \sum _{k=1}^{m}}\frac{\partial {q}_{d}^{i}}{\partial {b}_{j}}\frac{\partial {q}_{d}^{i}}{\partial {b}_{k}}{\sigma}_{bj}{\sigma}_{bk}{\rho}_{jk}\right)}^{1/2}$$(43)
where ${\sigma}_{q}^{i}$ is the standard deviation ${\left[Var\left({q}^{\left(i\right)}\right)\right]}^{1/2}$ of the ith element in vector X, ρ_{jk} is the correlation coefficient between b_{j} and b_{k}.
Similarly, the standard deviations of velocity and acceleration responses can be obtained:$${\sigma}_{{\displaystyle \dot{q}}}^{i}={({\displaystyle \sum _{j=1}^{m}}{\displaystyle \sum _{k=1}^{m}}\frac{{{\displaystyle \stackrel{}{q}}}_{d}^{i}}{\partial {b}_{j}}\frac{{{\displaystyle \stackrel{}{q}}}_{d}^{i}}{\partial {b}_{k}}{\sigma}_{bj}{\sigma}_{bk}{\rho}_{jk}}^{1/2}$$(44) $${\sigma}_{{\displaystyle \stackrel{}{q}}}^{i}={({\displaystyle \sum _{j=1}^{m}}{\displaystyle \sum _{k=1}^{m}}\frac{{{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}^{i}}{\partial {b}_{j}}\frac{{{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}^{i}}{\partial {b}_{k}}{\sigma}_{bj}{\sigma}_{bk}{\rho}_{jk}}^{1/2}$$(45)
8 Case analysis
Taking a HELS as an example. Its maximum operating speed is v=5 m/s, the elevator operating parameters are as shown in Section 4. the elastic modulus of the wire rope is E = 8 × 10^{10} N/m^{2}, lifting wire rope crosssectional area A = 89.344 cm^{2}, compensation rope line density ρ_{2} = 0.343 kg/m, tension F = 300 N. The longitudinal timevarying model of hoisting system has independent random parameters (the lift mass, the elastic modulus of the wire rope, and the wire rope density) and obeys normal distribution. The coefficient of variation takes CV = 0.02, and each random parameter is shown in Table 2 below.
HELS random parameter value.
8.1 Random parameter sensitivity analysis
The precise integration method is employed to solve the response sensitivity equation of the random parameter system, equation (38). The vibration displacement, vibration velocity and acceleration response sensitivity of each random parameter are obtained. After taking the absolute value and calculating the mean values $E\left\frac{\partial {{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}}{\partial {b}_{j}}\right$, $E\left\frac{\partial {{\displaystyle \stackrel{}{q}}}_{d}}{\partial {b}_{j}}\right$, $E\left\frac{\partial {q}_{d}}{\partial {b}_{j}}\right$, the calculation results are shown in Table 3.
It can be seen from Table 3 that, among the three random parameters, the linear density of the hoisting wire rope is the most sensitive to the vibration velocity response, the vibration displacement response, and the vibration acceleration response. And the three random parameters of the lift mass, the elastic modulus of the lifting steel wire rope and the wire rope density are the most sensitive to the acceleration response, followed by the vibration velocity response, and the sensitivity to the vibration displacement response is the smallest.
Random parameter sensitivity mean.
8.2 Random parameter acceleration and jerk response analysis
Using the response expression constructed by the perturbation theory, the precise integral method is used to solve the equation (38) to obtain the response acceleration ${{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}$, the longitudinal vibration acceleration response of the highspeed elevator is determined as shown in Figure 4a. The longitudinal vibration jerk response is determined as shown in Figure 4b.
Take the random part of the random parameter b_{rj} = ±σ_{bj}, and substitute it with the obtained ${{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}$ into equation (38), Solve the random part of the acceleration response, and then superimpose with ${{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}$ to get the total acceleration response, as shown in Figure 5a. In addition, the total jerk response as shown in Figure 5b.
Analysis the acceleration and jerk response determination part of the HELS, it can be seen that the absolute value of the longitudinal maximum acceleration and jerk increases after considering the influence of the random parameter, which is increased by about 50%. The corresponding values of the longitudinal acceleration and jerk total response have different degrees of dispersion at each moment, indicating that the dispersion of the longitudinal acceleration and jerk response is increased after considering the randomness of the parameters.
Fig. 4 Lifting system longitudinal acceleration and jerk response determination section. (a) Acceleration response (b) Jerk response. 
Fig. 5 Overall response of the longitudinal acceleration and jerk of the system. (a) Acceleration response (b) Jerk response. 
8.3 Longitudinal acceleration and passenger comfort analysis of HELS
Select 33–33.5 s with the largest vibration acceleration as the research object of acceleration response, determine the response ${{\displaystyle \stackrel{\mathrm{}}{q}}}_{d}$ as the acceleration response mean $\stackrel{\u203e}{{\displaystyle \stackrel{\mathrm{}}{q}}}$. Calculate the standard deviation ${\sigma}_{{\displaystyle \ddot{x}}}$ due to the randomness of the parameters by combining (30) and Precise integration method for vertical vibration model of the timevarying system in the highspeed traction elevator, and calculate the coefficient of variation CV, the results are shown in Table 4.
As can be seen from the Table 4, in the case where the coefficient of variation of the random parameter is 0.02, the coefficient of variation of the longitudinal vibration acceleration response of highspeed elevators varies greatly. Compare the longitudinal vibration acceleration determination part and the total acceleration image obtained in the previous section, the actual response is more discrete, and the randomness of the system parameters is more obvious to the longitudinal vibration acceleration of highspeed elevators.
The whole process of car operation is selected as the research object, and the vibration does VDV is used to detect the passenger comfort [24]. VDV is defined as:$$VDV={\left[{\int}_{0}^{T}{{\displaystyle a}}_{w}^{4}(t)dt\right]}^{1/4}$$(46)
where T is the duration of the vibration signal and ${{\displaystyle a}}_{w}^{4}$ is the acceleration the vibration signal after the weighting of the frequency meter. The VDV values of acceleration determination part and total acceleration of highspeed elevator lifting system are calculated respectively, as shown in Table 5.
As can be seen from the Table 5, the VDV values of both are more than 0.5. Compared with the acceleration determination part of highspeed elevator lifting system, the VDV value increases by 4.71% after considering the influence of random parameters, which indicates that the influence of random parameters reduce the comfort of passengers.
Acceleration response mean, standard deviation and coefficient of variation.
The VDV value of the determination part and total acceleration.
9 Conclusion

In this paper, considering the mass of the compensation rope and the tension provided by the tensioning system, based on the axial string theory, combined with the energy method and the Hamilton principle, the longitudinal vibration timevarying continuous model of the HELS is constructed. The Galerkin method is used to transform the infinite dimensional partial differential equation into the ordinary differential equation with finite DOF. The fifthorder polynomial is used to fit the actual operating state parameters of the highspeed elevator and input as parameters of the dynamic equation. The precise integration method for longitudinal vibration model of the timevarying system in the highspeed traction elevatoris proposed, and the whole running process of elevator random dynamics is calculated.

The determination and the random part response expression of the longitudinal dynamic response of the HELS are established by random perturbation theory. The displacement, velocity and acceleration sensitivity expressions of the random parameters are determined by solving the random response expression. The response sensitivity expression is used to solve the sensitivity values of each random parameter. It is found that the hoisting wire rope density has the highest sensitivity to longitudinal vibration velocity response, displacement response and acceleration response, the second is the lift mass, and the elastic modulus of the hoisting wire rope is the least sensitive. In the elevator manufacturing and installation process, parameters with strict sensitivity should be strictly controlled to improve the longitudinal dynamic performance of the highspeed elevator.

Through the analysis of the digital characteristics of the acceleration response, the acceleration response and VDV values generated by the random parameters are calculated, which accurately reflects the dispersion degree of the longitudinal acceleration response and passenger comfort of the highspeed elevator under the influence of random parameters.
Acknowledgments
This research was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2017MEE049), the Introduce urgently needed talents project for the western economic uplift belt and the key areas of poverty alleviation and development in Shandong Province.
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Cite this article as: Q. Zhang, T. Hou, H. Jing, R. Zhang, Analysis of longitudinal vibration acceleration based on continuous timevarying model of highspeed elevator lifting system with random parameters, Mechanics & Industry 22, 28 (2021)
All Tables
All Figures
Fig. 1 Timevarying model of the hoisting rope in an elevator lifting system. 

In the text 
Fig. 2 Operation flow chart. 

In the text 
Fig. 3 Elevator running state curve. 

In the text 
Fig. 4 Lifting system longitudinal acceleration and jerk response determination section. (a) Acceleration response (b) Jerk response. 

In the text 
Fig. 5 Overall response of the longitudinal acceleration and jerk of the system. (a) Acceleration response (b) Jerk response. 

In the text 
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