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Optimization of dynamic parameters for a traction-type passenger elevator using a dynamic byte coding genetic algorithm D Mei∗ , X Du, and Z Chen Institute of Advanced Manufacturing Engineering, Zhejiang University, Hangzhou, People’s Republic of China The manuscript was received on 21 April 2008 and was accepted after revision for publication on 28 July 2008. DOI: 10.1243/09544062JMES1149
Abstract: To decrease the vibration and improve the dynamic performance of traction-type passenger elevators, an accurate vertical dynamic model with nine degrees of freedom for a gearless 2 : 1 traction-type passenger elevator system is presented. Then, an optimization model of dynamic parameters for this type of passenger elevator is proposed based on the dynamic model. The optimization model takes the amplitude of vibration acceleration response of the elevator cage as the objective function. Various exciting forces and various working conditions can be considered by using corresponding weight coefficients in the objective function. To get higher efficiency and stronger robustness of optimized value, a dynamic byte coding genetic algorithm in solving the optimization model by combining the advantages of binary coding method with dynamic parameter encoding method is proposed, and the optimal solution is verified by sensitivity analysis. A practical engineering optimization for a 2 : 1 traction-type passenger elevator system shows that the optimization model and method of dynamic parameters proposed in this article are effective. Keywords: traction-type elevator, ride quality, dynamic parameter optimization, dynamic byte coding genetic algorithm, sensitivity verification
1
INTRODUCTION
310027, People’s Republic of China. email:
[email protected]
performances of high-speed elevators because of the additional dynamic load on its driving system. Additionally, the vibration and sound interfere with the comfort and health of the human passengers [2]. Thus, suppressing the vibration and improving the dynamic behaviour of elevator is a very important task. A traction-type elevator is a complex multi-degree system, which is shown in Fig. 1, comprised of cage, frame, driving machine, counterweight, cable, tension system, and so on [3]. According to the ratio of the guiding sheave’s tangential velocity to the velocity of the cage, the traction-type elevators can be grouped into two general types: the first one is a 1 : 1 traction elevator that has one sheave and one pulley and the second one is a 2 : 1 traction elevator that has one sheave and three pulleys. The vertical vibration of the 2 : 1 traction elevator is much more easily induced than that of the 1 : 1 traction elevator, because the former one has more pulleys that cause the cage vibration than the latter one. Thus, the 2 : 1 traction-type elevator was chosen for the examination.
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Elevators are the most commonly used mode of vertical transportation in modern buildings with more than three storeys. With the development of urbanization, elevators, especially traction-type passenger elevators, have played an important role in human life. More than a decade ago, it was estimated that there were more than half a million passenger elevators in the United States transporting people day and night throughout the year [1]. The advent of highrise buildings in modern cities requires high-speed elevator systems to provide quick access between floors in buildings, but with the increase of running speed and travelling height, the vibration of elevators become more and more serious. The concomitant excessive vibration and impact worsen the ∗ Corresponding
author: Department of Mechanical Engineering,
Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang Province
596
D Mei, X Du, and Z Chen
Fig. 2 Fig. 1
Main components of a typical traction-type passenger elevator [3]
To suppress the vibration and improve the dynamic performance of traction-type passenger elevators, an accurate vertical dynamic model with nine degrees of freedom (nine-DOF) for a 2 : 1 traction-type passenger elevator system is presented. Then, a new optimization method of dynamic parameters for this traction-type passenger elevator is proposed based on the dynamic model. The optimization model is solved by a dynamic byte coding genetic algorithm (GA), and the optimal solution is also verified by sensitivity analysis. A practical engineering optimization for a 2 : 1 traction elevator system shows that the optimization model and method of dynamic parameters proposed in this article is effective.
2
OPTIMIZATION MODEL OF DYNAMIC PARAMETERS FOR A TRACTION-TYPE PASSENGER ELEVATOR
2.1 Vertical dynamic model of a 2 : 1 traction-type passenger elevator A traction-type passenger elevator is a complex system containing both mechanical and electrical elements. The dynamics of a 1 : 1 traction-type passenger elevator has been analysed deeply in the past, and its dynamical model was built by considering its major DOF [4, 5]. However, so far, there have been a few Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science
Schematic diagram of a 2 : 1 traction-type passenger elevator
studies on the dynamics of the 2 : 1 traction-type passenger elevator. As a 2 : 1 traction-type passenger elevator shown in Fig. 2, the mechanical part of the system is made up of six major inertial elements: the traction machine, counterweight sheave, car sheave, frame, cage, and counterweight (m1 − m6 ). There are nine major DOF in this system: rotation of the traction machine, counterweight sheave, car sheave (θ1 − θ3 ), translation of the traction machine, counterweight sheave, car sheave, frame, cage, and counterweight (x1 − x6 ). Based on the above configuration, a nine-DOF vertical dynamic model for a 2 : 1 traction-type passenger elevator can be built as shown in Fig. 3. I1 − I3 and r1 − r3 , respectively, denote the rotary inertia and the radius of the driving sheave, counterweight sheave, and car sheave; k0 and c0 denote the stiffness and damping of the rubber under the traction machine; k1 and c1 the stiffness and damping of the driving rope segments connected with counterweight sheave; k2 and c2 the stiffness and damping of the driving rope segments connected with car sheave; k3 and c3 the stiffness and damping of the thimble rod spring; k4 and c4 the stiffness and damping of the car spring; k5 and c5 the stiffness and damping of the rubber under the cage; and k6 and c6 the stiffness and damping of counterweight spring. According to Lagrange’s equation, a nine-DOF vertical dynamic model for a 2 : 1 traction-type passenger elevator can be established as M · x¨ + C · x˙ + K · x = q
(1) JMES1149 © IMechE 2009
Dynamic parameters for a traction-type passenger elevator
where x is the system displacement vector, x = [x1 , x2 , x3 , x4 , x5 , x6 , θ1 , θ2 , θ3 ]T , q the system loading vector, and
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M, C, and K are the system mass, damping, and stiffness matrices, respectively, given by
M = diag[m1 , m2 , m3 , m4 , m5 , m6 , I1 , I2 , I3 ] ⎡
c0 + c 1 + c 2
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C=⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎡ k 0 + k1 + k2 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ K=⎢ ⎢ ⎢ ⎢ ⎢ ⎣
−c1 c1 + ce1 + c6
−c2 0 c 2 + ce2 + c4
0 0 −c4 c4 + c5
Symmetry
−k1 k1 + ke1 + k6
−k2 0 k2 + ke2 + k4
0 0 −k4 k4 + k5
0 −c6 0 0 0 c6
0 0 0 −k5 k5
Symmetry
where ke1 = (k1 k3 )/(k1 + k3 ), ce1 = (c1 c3 )/(c1 + c3 ), ke2 = (k2 k3 )/(k2 + k3 ), ce2 = (c2 c3 )/(c2 + c3 ), and kt and ct , respectively, denote the rotational stiffness and rotation damping of the driving sheave.
Fig. 3
0 0 0 −c5 c5
A nine-DOF vertical dynamic model for a 2 : 1 traction-type passenger elevator
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2.2
0 −k6 0 0 0 k6
(c1 − c2 )r1 −c1 r1 c1 r1 0 0 0 (c1 + c2 )r12 + ct
c 1 r2 (−c1 + ce1 )r2 0 0 0 0 c1 r 1 r 2 (c1 + ce1 )r22
(k1 − k2 )r1 −k1 r1 k2 r1 0 0 0 (k1 + k2 )r12 + kt
⎤ −c2 r3 0 ⎥ (c2 − ce2 )r3 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ c 2 r1 r3 ⎥ ⎥ ⎦ 0 (c2 + ce2 )r32
k1 r2 (−k1 + ke1 )r2 0 0 0 0 k1 r1 r2 (k1 + ke1 )r22
⎤ −k2 r3 0 ⎥ (k2 − ke2 )r3⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ k 2 r1 r3 ⎥ ⎥ ⎦ 0 (k2 + ke2 )r32
Optimization model of dynamic parameters for a 2 : 1 traction-type passenger elevator with multiple exciting forces
Several significant factors that contribute to the ride quality of a traction-type passenger elevator can be sorted mainly into functional parameters, noise parameters, and flexible dynamic parameters [6]. Compared with other parameters, flexible dynamic parameters contribute much more to its dynamic performance and ride quality. They can be changed in a large range to modify the dynamic characteristics of an elevator system. These parameters include the stiffness and damping of the rubber under traction machine and cage, the thimble rod spring, the car spring, and so on. They may be continuous or discrete according to practical design requirements. Thus, the flexible dynamic parameters can be chosen as the design variables for the optimization of dynamic performance for a 2 : 1 traction-type passenger elevator. Due to the harmonic property of exciting forces of vertical vibration mostly generated by the asymmetry of the driving sheave, guide sheave, counterweight sheave, and so on, the vibration response of the cage is strongly harmonic, which is obvious in practical measurements. Otherwise, vibration acceleration is generally adopted as the criterion for evaluating the dynamic behaviour of elevator products. Therefore, the amplitude of the vibration acceleration response is appropriate to be selected as the objective function of the optimization of dynamic parameters for a 2 : 1 traction-type passenger elevator. Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science
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Based on equation (1), the optimization model of dynamic parameters for a 2 : 1 traction-type passenger elevator with single exciting force can be established as min s.t.
RSI (u, q)
u = {u1 , u2 , . . . , un }T
ui ∈ {ui,1 , ui,2 , . . . , ui,ki } uj,min uj uj,max
i = 1, 2, . . . , n1
(2)
u = {u1 , u2 , . . . , un }T ui ∈ {ui,1 , ui,2 , . . . , ui,ki } uj,min uj uj,max
3
j = n1 + 1, . . . , n
where RSI (u, q) is the optimization object, and the subscript ‘SI’ the amplitude of vibration acceleration response of the elevator cage excited with a single exciting force; q is the exciting forces vector, in which only one element is non-zero; u = {u1 , u2 , . . . , un }T the design variables (flexible dynamic parameters) vector, the first n1 components of which are discrete variables, and the others are continuous variables; {ui,1 , ui,2 , . . . , ui,ki } is a set composed of ki optional values of the discrete variable ui ; and uj,min and uj,max the lower and upper limits of the continuous variable uj . In practice, several exciting forces usually have significant effect on the vibration of elevator cage, and the frequencies of them are distinct from each other. In these cases, equation 2 cannot precisely describe the dynamic behaviour of traction-type elevators. Thus, the weighted root-mean-square value of all the amplitudes of vibration acceleration response of each single force is taken as the optimization object, and the optimization model of dynamic parameters with multiple exciting forces for a 2 : 1 traction-type passenger elevator can be established as min RMU = [wfi Rf2i ,SI (u, q fi )],
s.t.
±10 per cent range to evaluate the deviation of the objective function. The final optimal solution should have small response amplitude, be insensitive to every design variable’s small change, and can satisfy any practical requirements.
i = 1, 2, . . . , n1
(3)
j = n1 + 1, . . . , n
where RMU is the optimization object, the subscript ‘MU’ the equivalent amplitude of vibration acceleration response of elevator cage excited by multiple forces, wfi the weight coefficient of the exciting force fi , Rfi ,SI the amplitude of vibration acceleration response of the exciting force fi , and in exciting forces vector q fi , only the exciting force fi is non-zero. Taking advantage of some numerical methods, the optimization model of dynamic parameters in equations 2 or 3 can be solved. However, it is impossible to adjust the practical dynamic parameters that are precisely equal to the optimal results. For a feasible solution, the objective function should be insensitive to each design variable’s deviation in a small scope, so that it can be verified by sensitivity analysis. In this research, after a series of optimal solutions were searched, their sensitivity analyses were also conducted by letting every design variable change in a Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science
3.1
OPTIMIZATION METHOD USING DYNAMIC BYTE CODING GA Dynamic byte coding GA
GA, first proposed by Holland [7], is a stochastic searching algorithm based on Darwinian models of natural selection and evolution. The algorithm encodes a potential solution to a specific problem on a simple genome-like data structure and applies recombination operators to these structures so as to preserve critical information. The operators mainly include selection, crossover, and mutation, which are called as genetic operators. By simulating the processes of natural selection and mutation in the biological evolution, GA provides a new thought for optimizing large-scale and complex real-world problems that are difficult for traditional mathematical method to solve [8]. Encoding is a primary issue in GA. The choice of encoding method determines not only the genome form of individuals, but also the transformation of individuals from genotype in the search space to representation code in the solution space. It also determines the genetic operations of the GA population and the efficiency of GA to a great extent. At present, there are mainly four types of encoding mechanisms: binary coding, Gray coding, floating-point encoding, and dynamic parameter encoding [9, 10]. However, these encoding methods have their own merits and demerits, such as Hamming cliffs in binary coding and low global search ability in floating-point encoding. Therefore, each of them has limited scope of application. To get higher efficiency and stronger robustness of optimized value, a new encoding GA optimization method (dynamic byte coding GA) was proposed in this research, which combines the advantages of binary coding method and dynamic parameter encoding method. Binary coding method has strong global search ability, high efficiency, and strong robustness of optimized value, whereas dynamic parameter encoding method has good local constringency. Dynamic byte coding GA is based on the combination of these two encoding methods and its basic mechanism is introduced as follows. 1. Each design variable is represented by an unsigned byte integer (eight bits), whether it is a discrete variable or a continuous one. This representation is another kind of binary coding method, which essentially adopts an eight-bit integer to represent JMES1149 © IMechE 2009
Dynamic parameters for a traction-type passenger elevator
the gene since the gene with such coding length has high convergent precision compared to that in dynamic parameter encoding [11]. 2. With unsigned byte genetic representations, the crossover operator and mutation operator are implemented by the rule of logic operation bit by bit. Therefore, the efficiency is higher than that of normal binary coding method that uses single integer to represent the gene. 3. Adjustment strategy of search space: the continuous evolution of GA population will drive the population close to maturation. To improve the convergent precision, the search space of parameters should be adjusted. As for unsigned byte integer encoding, the decoding method of single gene is given by y =o+
g ·h 255
(4)
where y denotes the parameter in the search space, g the gene parameter corresponding to the parameter y, o the minimum value of the search space of the parameter, and h the range of the search space. The search space of each parameter may resize based on the solution, which has the maximum fitness in the asymptotically mature population and is set as the centre of the new search space. When the offset distance of one parameter relative to the current range centre is estimated, its search region can be adjusted dynamically according to the magnitude of the distance. The offset distance is defined as d = |g − 127.5|
(5)
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adjustment of the search space are critical in dynamic parameter encoding. Whitley et al. [12] have proposed a method to judge the population maturity according to the average number of different genes in all individuals, which is obtained by comparing the genome between two individuals and counting the amount of different genes they have. If the average number is smaller than a certain threshold value, the population should be considered mature and the search space of the design variable should be adjusted. However, it will take much computing time to judge the population maturity. Suppose there are n individuals in the population, each individual has b genes (design variables) and each gene is represented by a p-bit string, then the total logic operation time for the judgment of the population maturity is n(n − 1)bp/2. For unsigned byte integer encoding, there are 256 × 256 = 65 536 kinds of possible combinations. It is applicable to buildup a look-up table before the iterative evolution of the population for the comparison of genes. In this way, the computing time will decrease remarkably. 3.2
Procedures to solve the optimization model of dynamic parameters
Using dynamic byte encoding GA, the optimization model of the elevator’s dynamic parameters for traction-type elevator in equations 2 or 3 can be solved; its flow chart is shown in Fig. 4 and its detailed solving procedures are as follows.
where the function rand(0.9, 1.1) will return a random number between 0.9 and 1.1. 4. The judgment of the population maturity: the judgment of the population maturity and the dynamical
Step 1: preparation. Initialize the search space of each parameter. Establish a look-up table for gene comparison and the globally optimum fitness is set as a negative with sufficiently large absolute value. Step 2: initialization and evaluation. N individuals of initial population are generated at random. Each individual’s fitness is calculated according to the selected cost function. Step 3: resize. If the population is close to mature, resize the search space and generate new population at random. Then, the evaluation is executed again. Step 4: selection. The individual with high fitness has more chance to be selected. Step 5: crossover. Each selected individual does crossover operation. Step 6: mutation. Some digits of certain individuals in population mutate by adding a Gaussian stochastic variable. Step 7 : evaluation. Each individual’s fitness is calculated and compared to convergence criteria of the optimization. If the convergence criteria are satisfied, then stop the iterative process. Otherwise, go to Step3. Step 8: end of optimization process.
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and the calculation of the new search space is described as
αhc o = max ymin , y − 2
αhc h = min y + , ymax − o 2
(6)
where ymin and ymax are assumed to be the minimum and maximum value of the original search space of the parameter, respectively, hc the range of the current search space, and α the adjustment coefficient that is defined as
α =1+
1.1d − 0.9 · rand(0.9, 1.1) 127.5
(7)
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Fig. 4
Flow chart of dynamic byte coding GA
Besides the operations of dynamic byte coding method discussed above, there are some other important concerns when solving the optimization model of dynamic parameters for traction-type passenger elevator as follows. 1. Decoding of the parameter: use equation (4) to decode the continuous parameters. However, the minimum value of the initial search space is zero for the discrete parameter and the range of the search space is 1.0. During iterative evolution the search space of discrete parameters holds the line. Let there be z optional values in the definition domain of a certain discrete parameter, so that its decoding is specified by
g z d yd = P (int) 255.0
(8)
3. Evaluation of the population: the evaluation of the population mainly includes the following: (a) the design parameters obtained by decoding each individual are used for harmonic response analysis of the elevator’s vertical vibration. The individual fitness is calculated according to the magnitude of the harmonic response; (b) make a descending index of the individual fitness of the population; (c) if the fitness of the optimum individual in the current population exceeds the fitness of the globally optimum individual, the latter will be substituted by this individual; (d) do the judgment of the population maturity. 4. Genetic operators: selection operator may follow the sequencing selection method presented by Back [13], which can effectively avoid the prematurity of the population. Combination of intergene crossover with innergene crossover is introduced as crossover operator. In mutation operator, a certain bit in one gene of one individual is selected randomly to switch between 0 and 1. 5. Diagnosis of over regulating the search space: overregulation is inevitable for dynamic parameter encoding. If the population has been asymptotically matured for nover times and the maximum individual fitness is lower than the maximum one in the last asymptotic maturation time, overregulation of the search space is confirmed. Then, the original search space is set as the current search space, the globally optimum individual is set as the current optimum individual, and the search space is resized again according to equations (6) and (7). New population is generated at random to begin the next iteration of evolution. 6. Convergence criteria: if the population has been asymptotically matured for nconvergent times and the maximum individual fitness is lower than the globally optimum individual fitness, the iteration is halted and the optimized solution is obtained. The same result can be achieved when the overall iteration times reach N .
4
AN APPLICATION EXAMPLE
where yd denotes the discrete parameter, gd the gene integer representation corresponding to the discrete parameter yd , and P an array representing the definition domain of the discrete parameter, which has z elements. 2. Design of the fitness function: directly take the inverse value of the objective function as the fitness function.
The optimization model of dynamic parameters for traction-type passenger elevator and optimization method by using dynamic byte coding GA proposed in this article has been applied successfully in some practical engineering problems. To demonstrate the effectiveness, an application example that was finished in 2004 can be introduced as follows.
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Dynamic parameters for a traction-type passenger elevator
4.1
Problem description
A gearless 2 : 1 traction-type passenger elevator serving in a grand building had a serious vibration problem. Figure 5 shows the practical measurements of vibration acceleration of the elevator car finished with PMT [14]. In working condition of no load, the peak– peak value of vertical (Z -direction) vibration acceleration of the elevator car was up to 4.45 × 10−1 m/s2 (44.5 mg), and the A95 [15] value was up to 3.96 × 10−1 m/s2 (39.6 mg), as shown in Fig. 5(a). Through fast Fourier transform (FFT) analysis of the serious-vibration section A of vibration acceleration in Z -direction, the maximum peak value is up to 1.007 × 10−1 m/s2 (10.07 mg) at 2.5 Hz, as shown in Fig. 5(b). To decrease the vibration and improve the dynamic performance of this elevator that was in severe vibration conditions, the optimization of dynamic parameters for it was conducted by using a dynamic byte coding GA.
Fig. 5
4.2
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Optimization model
According to equation (3), by taking the working conditions of different location in the hoistway and the different load of the cage (total weight of people in the cage) into consideration, an optimization model of dynamic parameters for the 2 : 1 traction-type passenger elevator can be established as
min
RMU =
3 k=1
⎡ ⎣w1k
3
⎞⎤ 5 ⎝waj wfi Rf2i ,SI (u, q fi ) ⎠⎦ ⎛
j=1
i=1
u = {u1 , u2 , . . . , u5 }T s.t.
(9a)
u1 ∈ {7.105 × 106 , 2.82 × 107 } u2 ∈ {1.1869 × 105 , 2.12 × 105 , 6.2196 × 105 } 1.0 × 106 u3 2.0 × 107
Practical measurements of the elevator before optimization: (a) measurements of Z -direction vibration acceleration before optimization, (b) FFT of Z -direction vibration acceleration (section A) before optimization
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Table 1 Weights of different working conditions Item
Working condition No load Half load Full load Bottom Centre Top
Wl Wa
Table 3 No.
u1
u2
u3 (107 N/m)
u4 (106 N/m)
u5 (106 N/m)
RMU (mm/s2 )
1 2 3 4 5 6 7 8 9 10
2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 2 3 3 2
2.00 1.94 2.00 1.84 2.00 2.00 2.00 2.00 2.00 2.00
1.17 1.48 1.16 1.10 1.16 1.16 1.60 1.16 1.16 1.61
1.22 1.18 1.23 1.18 1.23 1.23 1.21 1.23 1.23 1.20
12.40 12.80 12.40 12.50 12.40 12.40 13.05 12.40 12.40 13.05
Weight 0.40 0.40 0.20 0.35 0.45 0.20
Table 2 Vertical harmonic exciting forces Exciting source
wf
Amplitude
Frequency (Hz)
Driving sheave asymmetry Guide sheave asymmetry Car sheave asymmetry Counterweight sheave asymmetry Torque ripple
0.35 0.1 0.2 0.05
2350 N 500 N 2470 N 1940 N
2.50 3.21 1.01 1.01
0.30
62 N m
2.50
Optimal solution of dynamic parameters
4.0 × 105 u4 2.0 × 106 4.0 × 106 u5 1.0 × 105 (9b) where the five design variables u1 , u2 , u3 , u4 , and u5 are the stiffness of the rubber under traction machine, a single thimble rod spring, the rubber under cage, the car spring, and the counterweight spring, respectively. The units of them are uniformly N/m. wl , wa , and wf are the weight of load, location, and exciting force of the elevator, respectively. Values of wl and wa are listed in Table 1. Values of wf and the properties of harmonic exciting forces of this elevator are listed in Table 2.
4.3
Optimal solution and its verification by sensitivity analysis
Fig. 6
4.4
Sensitivity analysis of the optimal solution
Modification with optimal parameters
After the modification of this elevator with the optimal parameters, the measurements of Z -direction vibration acceleration of this elevator show the fine effect of optimization, as shown in Fig. 7. The peak–peak value of Z -direction vibration acceleration of elevator car was decreased from 4.45 × 10−1 m/s2 (44.5 mg) to 1.55 × 10−1 m/s2 (15.5 mg), and the A95 value was decreased from 3.96 × 10−1 m/s2 (39.6 mg) to 8.2 × 10−2 m/s2 (8.2 mg), as shown in Fig. 7(a). Through FFT analysis of the serious-vibration section B of vibration acceleration in Z -direction, the maximum peak value at 2.5 Hz has disappeared, as shown in Fig. 7(b). Compared with the result before optimization, the ride quality of this elevator has been improved remarkably after optimization and modification.
The optimization model shown in equations (9a) and (9b) has been solved ten times by using the dynamic byte coding GA, and its optimal solutions are listed in Table 3. The figures in u1 and u2 column denote the index of discrete variable u1 and u2 in the set of {ui,1 , ui,2 , . . . , ui,ki }. After the general observation of ten optimal solutions, the conclusion can be observed in which all the optimal solutions are closely equal to u = {2.82 × 107 , 6.2196 × 107 , 2.00 × 107 , 1.16 × 106 , 1.23 × 106 }T . The optimal object is 12.40 mm/s2 , which is much less than 72.61 mm/s2 – the result calculated with the original parameters. This solution can be verified by sensitivity analysis, as shown in Fig. 6, and the result shows that the objective function is insensitive to each design variable’s deviation in a ±10 per cent range at optimal solution. Thus, the optimal solution gained is proper and practical.
In the theoretical calculation process, some parameters are very difficult to be measured precisely, such as stiffness, damping, exciting forces, and so on. Therefore, these dynamic parameters can be evaluated approximately, and sometimes there is a big
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4.5
Discussion of the difference between the theoretical result and testing result
Dynamic parameters for a traction-type passenger elevator
Fig. 7
603
Practical measurements of the elevator after optimization: (a) measurements of Z -direction vibration acceleration after optimization, (b) FFT of Z -direction vibration acceleration (section B) after optimization
difference between theoretical value and practical value. Otherwise, in the modelling process, although most components and elements of the traction-type passenger elevator have been considered, some components and elements are still simplified or neglected since a traction-type passenger elevator is a very complex system containing both mechanical and electrical elements. Another very important reason is the installation of the elevator system; in some cases, the dynamic behaviour of some same type elevators are very different, but the installation errors of the elevator system are also difficult to be described precisely in the modelling process. Due to the above causes, the theoretical optimal result of the vibration response, 12.40 mm/s2 , is substantially different from the testing result, 1.55 × 10−1 m/s2 . On the contrary, in this research, actually the difference between theoretical and testing results is not very pivotal because both can, respectively,
To suppress the vibration and improve the dynamic performance of traction-type passenger elevators, an accurate vertical dynamic model with nine-DOF
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validate the optimization model and method of dynamic parameters proposed in this article. As for the theoretical calculation result, the theoretical optimal result of the vibration response is 12.40 mm/s2 , which is much less than 72.61 mm/s2 – the result calculated with the original parameters. As for the testing result, the peak–peak value of vertical vibration acceleration of elevator car was decreased from 4.45 × 10−1 to 1.55 × 10−1 m/s2 after optimization. Therefore, both the theoretical result and testing result show that the optimization model and method of dynamic parameters proposed in this article are effective.
5
CONCLUSIONS
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D Mei, X Du, and Z Chen
for a 2 : 1 traction-type passenger elevator system is presented. Then, a new optimization model of dynamic parameters for this type of passenger elevator is proposed based on the dynamic model. The optimization model takes the amplitude of vibration acceleration response of the elevator cage as an objective function.Various exciting forces and working conditions can be considered by using corresponding weight coefficients in objective function. To get higher efficiency and stronger robustness of optimized value, a dynamic byte coding GA was proposed in solving the optimization model by combining the advantages of binary coding method with dynamic parameter encoding method, and the optimal solution is also verified by sensitivity analysis. The optimization model and method was used in a gearless 2:1 traction-type passenger elevator system that was in severe vibration condition. After optimization and modification, the peak–peak value of vertical vibration acceleration of the elevator car was decreased from 4.45 × 10−1 to 1.55 × 10−1 m/s2 , and the A95 value was decreased from 3.96 × 10−1 to 8.2 × 10−2 m/s2 . The application example shows that the optimization model and method of dynamic parameters proposed in this article is effective. Although the optimization model of dynamic parameters for a 2 : 1 traction-type passenger elevator and optimization method by using dynamic byte coding GA proposed in this article have been applied successfully in some practical engineering problems, some relevant research work still needs to be investigated in the future. First, considering more components and elements, a more accurate dynamic model for a 2 : 1 traction-type passenger elevator system needs to be investigated. Second, other than considering the vibration in the vertical direction, the dynamic modelling in the horizontal direction also should be considered. Finally, the measuring technologies for some dynamics parameters, which are very difficult to be measured precisely need to be investigated in the future such as the stiffness, damping, exciting force, and so on, since the measuring errors of these parameters will directly determine the result of the simulation of the dynamic behaviour for the elevator system.
2 Gang, X. Y., Mei, D. Q., and Chen, Z. C. A horizontal vibration wave model for high speed elevators. Key Eng. Mater., 2005, 297–300, 1585–1591. 3 Rildova. Seismic performance of rail-counterweight system of elevator in buildings. PhD Thesis, Virginia Polytechnic Institute and State University, 2004. 4 Nai, K., Forsythe, W., and Goodall, R. M. Improving ride quality in high-speed elevators. Elevator World, 1997, 45, 88–93. 5 Nai, K., Forsythe, W., and Goodall, R. M. Modeling and simulation of a lift system. In Proceedings of the International Conference on Control, Modeling, Computation and Information, Institute of Maths and its applications, Manchester, September 1992, pp. 6–11. 6 Lorsbach, G. P. Analysis of elevator ride quality, vibration. Elevator world, 2003, 51, 108–111. 7 Holland, J. H. Adaptation in nature and artificial system, 1992 (MIT Press, Cambridge, MA). 8 Chen, G. L. Genetic algorithm and its application, 1996 (People’s Posts & Telecom Press, Beijing). 9 David, B. F. An introduction to simulated evolutionary optimization. IEEE Trans., Neural Netw., 1994, 5, 3–14. 10 Schraudolph, N. N. and Belew, R. K. Dynamic parameter encoding for genetic algorithms. Mach. Learn., 1992, 9, 9–21. 11 Robert, J. S. and Robert, J. M. Dynamic fuzzy control of genetic algorithm parameter coding. IEEE Trans.Syst. Man Cybern. B, 1999, 29, 426–433. 12 Whitley, D., Mathias, K., and Fitzhorn, P. Delta coding: an iterative search strategy for genetic algorithms. In Proceedings of the 4th International Conference on Genetic Algorithms, San Mateo, CA, 1991, pp. 77–84. 13 Back, T. The interaction of mutation rate, selection and self-adaptation within a genetic algorithm. In Proceedings of the 2nd Parallel Problem Solving from Nature, North Holland, 1992, pp. 85–94. 14 PMT, EVA-625 system operations manual, version 6 for Windows™, 2001. 15 International Standard ISO 18738. Lifts (elevators) – measurement of lift ride quality, 2003.
ACKNOWLEDGEMENTS
C ct C D G
This research work was supported by National Natural Science Foundation of China (Grant No. 50775203) and Science and Technology Key Special Program of Zhejiang Province (Grant No. 2006C11251). REFERENCES 1 Swerrie, D. A. The San Francisco earthquake of 1989 and an inspector’s reflections. Elevator World, 1990, XXXVIII, 14–22. Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science
APPENDIX Notation B
hc H I K kt K m
the number of genes in each individuals damping (N s/m) rotation damping (N s m) system damping matrix offset distance the gene parameter corresponding to the parameter y the range of the current search space the range of the search space rotary inertia (kg m2 ) stiffness (N/m) rotation stiffness (N m) system stiffness matrix mass (kg) JMES1149 © IMechE 2009
Dynamic parameters for a traction-type passenger elevator
M n n1 N o p P q r R u u w x X y
system mass matrix the number of individuals in the population the number of discrete design variables individuals of initial population the minimum value of the search space the number of bit for each gene an array representing the definition domain of the discrete parameter yd system exciting force vector radius (m) optimization object function design variable design variable vector weight coefficient linear displacement (m) system displacement vector the parameter in the search space
JMES1149 © IMechE 2009
z
the number of the elements in the definition domain of yd
α θ
adjustment coefficient angular displacement (rad)
605
Subscripts a convergent d f ki l max min MU over SI
location of the elevator the iteration process is convergent discrete variable exciting force of the elevator the number of optional values of the discrete variable load of the elevator upper limit lower limit multiple exciting forces the search space is over regulated single exciting force
Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science
