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南昌航空大學科技學院學士學位論文
Optimal design of hydraulic support
m. oblak. Harl and b. butinar
Abstract :This paper describes a procedure for optimal determination of two groups of parameters of a hydraulic support employed in the mining industry. The procedure is based on mathematical programming methods . In the first step, the optimal values of some parameters of the leading four-bar mechanism are found in order to ensure the desired motion of the support with minimal transversal displacements. In the second step, maximal tolerances of the optimal values of the response of hydraulic support wil be satisfying.
Keywords: four-bar mechanism, optimal design, mathematical programming \, approximation method, tolerance
1 Introduction
The designer aims to find the best design for the mechanical system considered. Part of thie effort is the optimal choice of some selected parameters of a system. Methods of mathematical programming can be used, Of course, it depends on the type of the systemWith this foemulation, good computer support is assured to look for optimal parameters of the system.
The hydraulic support (Fig.1) described by Harl (1998) is a part of the mining industry equipmenr port in the mine Velenje-Slovenia, used for protection of working environment in the gallery. It consists of four-bar mechanisms FEDG and AEDB as shown in Fig.2. The mechanism AEDB defines the path of coupler point C and the mechanism FEDG is used to drive the support by a hydraulic actuator。
Fig. 1 Hydraulic support
It is required that the motion of the support,more precisely, the motion of the point C in Fig.2, is vertical with minimal transversal displacements. If this is not the case, the hydraulic support will not work properly because it is stranded on removal of the earth machine.
A prototype of the hydraulic support was tested in a laboratory (Grm 1992). The support exhibited large transversal displacements, which reduce its employability. Thetefore, a redesign was necessary. The project should be improved with minimal cost if possible. It was decided to find the best values for the most problematic
Fig.2 Two four-bar mechanisms
Parameters of the leading four-bar me AEDB with methods of mathematical programming. Otherwise it chanisms would be necessary to change the project, at least mechanism AEDB.
The solution of above problem will give us the response of hydranlic support for the ideal system. Real response will be different because of tolerances of various parmeters of the system, which is why the maximal allowed tolerances of paramentsa1,a2,a3,a4 will be calculated support. ,with help of mathematical programming.
2 The deterministic model of the hydraulic support
At fist it is necessary to develop an appropriate metical model of the hydraulic support.It could be based on the following assumptions:
- the links are rigid bodies,
- the motion of individual is relatively slow.
The hydraulic support is a mechanism with one degree of freedom. Its kinematics can be model consists of four-bar mechanisms FEDG and AEDB (Oblak et al. 1998).The leading four-bar mechanisms AEDB with methods of mathematical programming. Otherwise it would be necessary to change the project, at least mechanism AEDB. It is required that the motion of the support,more precisely, the motion of the poit C. Therefore, the path of coupler point C is as near as possible to the desired trajectory k.
The synthesis of the four-bar mechanism 1 has been performed with help of motion given by Rao Dukkipati(1989). The general situation is depicted in Fig,3.
Fig.3 Trajectory L of the point C
Equations of trajectory L of the point C will be written in the coordinate frame considered. Coordinates x and y of the point C will be written with the typical parameters of a four-bar mechanism a1,a2,….a6.The coordinates of points B and D are
xBcos (1)
yB=sin (2)
xD=cos() (3)
yD=sin() (4)
The parameters …are related to each other by
xB2+ (5)
α1)2+ yD2= (6)
By substituting (1) － (4) into （5）－（6）the response equations of the support are obtained as
(xcos)2+ (y- sin)2- =0 (7)
[x- cos()-]2+[ y- sin()]2- =0 (8)
This equation represents the mathematical model for calculating the optimal values of paramerters a1,a2,a4.
2.1 Mathematical model
The mathemtial model of the system will be formulated in the from proposed by Haug and Arora (1979):
gi(u,v)0, i=1,2,…,l, (10)
and response equations
hi(u,v)=0, j=1,2,…,m. (11)
The vector u=[u1,u2,…,un]T is called the vector of design variables, v=[v1,v2,…,vm]Tis the vector of response variables and f in(9)is the objective function.
Tobperform the optimal design of the leading four-bar mechanism AEDB,the vector of design variables is defined as
u=[ ]T, (12)
and the vector of response variables as
v=[xy]T. (13)
The dimensions α3，α5，α6 of the corresponding links are kept fixed.
The
f(u,v) =max[g0(y)-f0(y)]2, (14)
where x= g0(y) is the equation of the curve K and x= f0(y) is the equation of the curve L.
Suitable limitations for our system will be chosen.The system must satisfy the well-known Grasshoff conditions
(15)
(16)
Inequalities (15) and (16) express the property of a four-bar mechanism, where the links may only oscillate.
The condition：
(17)
Prescribes the lower upper bounds of the design variables.
The problem (9)–(11)is not dirrctly solvable with the usual gradient-based optimization methods. This could be cirumvented by int express the property of the objective function be written with the typical parameters be written as
minun+1 (18)
sobject to
gi(u,v) 0, i=1,2,…,l, (19)
f(u,v)- un+10, (20)
and response equations
hj(u,v)=0, j=1,2,…,m, (21)
where：
u=[u1 … un un+1]T
v=[v1 … vn vn+1]T
A nonlinear programming problem of the leading four-bar mechanism AEDB can therefore be difined as
mina7, (22)
sobject to constraints
(23)
(24)
,
(25)
(26)
And respose equationt
(27)
(28)
3.The stochastic model of the hydraulic support
The mathematical model can be used to calculate the parameters of L and K to ensure that the track such as to maintain the distance between the minimum. However the endpoint C calculation L may deviate from the track, because of the movements in the presence of some interference factor. Look at these deviations from what should or not lies in the deviation is in the parametric tolerance tolerance range.Response function (27) - (28) allows us to consider the response variable V vector, the vector of dependent variable V vector design. This means that v = H ( V, H ) function is a mathematical model (22) - (28) foundation, because it describes a response variable V vector and V vector as well as design variables and the mathematical model of the relationship between v. Similarly, the function H used to consider the parameter errors in the value of the maximum permissible value.In the stochastic model, design variable vector u=[u1, ... , un]T can be viewed in U=[U1, ... , Un]T random vector, which means the response variable vector v=[v1, ... , vn]T is a random vector V=[V1, V2, ... , Vn]TV=h ( U ) (29)Suppose design variable U1, ... , Un from probability theory and the classification of normal function of Uk ～( k=1,2, ... , n ) of independence. The main parameters and ( k=1,2, ... , n ) can be associated with such as the measurement of this kind of scientific concepts and tolerance to link, such as a =,. So as long as the choice of suitable existence probability, k=1,2, ... , n (30)Type (30) is calculated the results of.Random vector V probability distribution function is search for dependent random vector U probability distribution function and its actual computability. Therefore, random vector V is described by mathematical properties, and the properties were identified using Taylor on u=[u1, ... , un]T h approximation function description, or with the aid of Oblak and Harl in the Monte Carlo method.
3.1 The mathematical model
Used to calculate the allowable error of hydraulic support optimization mathematical model will be nonlinear problem of independent variable
w=[ ] (31)
and objective function
(32)
With conditions
(33)
,
(34)
In（33），E is the maximal allowed standard deviation of coordinate x of the point C and
A={1,2,4} (35)
The nonlinear programming problem for calculating the optmal tolerances could be therefore defined as ：
(36)
Subject to constraints
(37)
,
(38)
4.Numerical examply
The carrying of the hydraulic support is 1600kN. Both four-bar AEDB andFEDG must fulfill the following demand：
－they must allow minimal transversal displacements of the point C, and，
－they must provide sufficient side stability.
The parameters of the hydraulic support (Fig.2) are given in Table 1.
The drive mechanism FEDG is specified by the vector
(mm) (39)
And the mechanism AEDB by
(mm)
In(39),the parameter d is a walk of the support with maximal value of 925 mm. Parameters for the shaft of the mechanism AEDB are given in Table 2.
4.1Four connecting rod AEDA optimization
Four link model AEDA related data in equation (22) - (28) are expressed. ( Fig 3). C lemniscate of maximum horizontal offset for65mm. That is why type (26) for the
(41)
Rod and bar between AE AA angle in the range of 76.8o and 94.8o, will be a number… successively introduced formula (41) obtained results are listed in table 3.
These points corresponding to the angle of…in the range of [76.8o,94.8o] and they each angle difference of1
The design variables of the minimum and maximum range is
(mm) (42)
(mm) (43)
Nonlinear design problems in equation (22) and (28) in the form of. This problem by
Kegl et al (1991) based on the approximation of the optimal approximation solution. By using the direct method of distinguishing to calculate design derived data.
Design variables for the initial value
(mm) (44)
Optimization of design parameters through calculation is repeated 25 times
Table3 node C corresponding to the X and Y values
角度
x初值(mm)
y初值(mm)
x終值(mm)
y終值(mm)
76.8
66.78
1784.87
69.47
1787.50
77.8
65.91
1817.67
68.74
1820.40
78.8
64.95
1850.09
67.93
1852.92
79.8
63.92
1882.15
67.04
1885.07
80.8
62.84
1913.85
66.12
1916.87
81.8
61.75
1945.20
65.20
1948.32
82.8
60.67
1976.22
64.29
1979.44
83.8
59.65
2006.91
63.46
2010.43
84.8
58.72
2037.28
62.72
2040.70
85.8
57.92
2067.35
62.13
2070.87
86.8
57.30
2097.11
61.73
2100.74
87.8
56.91
2126.59
61.57
2130.32
88.8
56.81
2155.80
61.72
2159.63
89.8
57.06
2184.74
62.24
2188.67
90.8
57.73
2213.42
63.21
2217.46
91.8
58.91
2241.87
64.71
2246.01
92.8
60.71
2270.08
66.85
2274.33
93.8
63.21
2298.09
69.73
2302.44
94.8
66.56
2325.89
70.50
2330.36
(mm)
In Table 3 of C x values and Y values respectively corresponding to the design variables and the variables of optimization design line )。
Fig 4diagram represents the endpoint C started the lemniscate locus L ( dotted line) and perpendicular to the ideal trajectory K ( solid line).
Lateral displacement and the trajectory of Figure 4graph represents the endpoint C started the lemniscate locus L ( dotted line) and perpendicular to the ideal trajectory K ( solid line).
Fig.4 Trajectories of the point C
4.2 Four connecting rod mechanism AEDA optimal error
In the nonlinear problem (36) - (38), selection of independent variables of the minimum and maximum value
(mm) (46)
(mm) (47)
Independent variable initial value
(mm) (48)
Trajectory deviation to chose two cases E=0.01and E=0.05. In the first case, the design variables of the ideal of tolerance after 9times of repeated calculations, early results. Second situations are7 times repeated calculation to obtain the ideal value. These results are listed in Table 4and table 5.
Figure 5 and Figure6of the standard deviation from Monte Carlo method is calculated and expressed in the diagram (shown in double dots line below) while Taylor approximation curve ( solid line).
Fig.5 Standard deviations for E=0.01
Optimal tolerances for the design variables a1,a2,a4 were calculated after 9 iterations. For E=0.05 the optimum was obtained after 7 iterations.The results are given in Table 4 and 5.
In Fig.5 and 6 the staylor ndard deviations are calculated by the Monte Carlo method and with Taylor approximation (full line represented Taylor approximation),respectively.
Fig.6 Standard deviations for E=0.05
5.Conclusins
With a suitable mathematical model of the system and by employing mathematical programming,the design of the hydraulic support was improved, and better performance was achieved.However, due to the results of optimal tolerances,it might be reasonable to take into consideration a new construction. This is especially true for the mechanism AEDB, since very small tolerances raise the costs of production.
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