Vitaly Volpert
email: volpert@math.univ-lyon1.fr

EXISTENCE OF STATIONARY SOLUTIONS FOR SOME SYSTEMS OF INTEGRO-DIFFERENTIAL EQUATIONS WITH SUPERDIFFUSION

Keywords: 2010 AMS Mathematics subject classification. Primary 35J05, 35P30, 47F05 Integro-differential equations, non Fredholm operators, Sobolev spaces

In this article, we establish the existence of solutions of a system of integro-differential equations arising in population dynamics in the case of anomalous diffusion. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without the Fredholm property in unbounded domains are used.

Introduction.

In the present work, we address the existence of stationary solutions of the system of integro-differential equations

∂u s ∂t = -D s √ -∆u s + ∫ R d K s (x -y)g s (u(y, t)) dy + f s (x), (1.1) 1 ≤ s ≤ N,
appearing in cell population dynamics. We believe that such a model is relatively new. The single equation analogous to (1.1) with the standard Laplacian in the diffusion term was studied in [START_REF]Existence of stationary solutions for some nonlocal reactiondiffusion equations[END_REF]. Herein, the space variable x corresponds to the cell genotype, u s (x, t) are densities for different groups of cells as functions of their genotype and time, and u(x, t) = 3D(u 1 (x, t), u 2 (x, t), . . . , u N (x, t)) T . The right side of system (1.1) describes the evolution of cell densities by means of cell proliferation, mutations and cell influx. The anomalous diffusion terms here correspond to the change of genotype via small random mutations, and the nonlocal terms describe large mutations. In general, the anomalous diffusion comprises the case of the subdiffusion when the problem involves a fractional derivative with respect to time, and the superdiffusion when the negative Laplacian raised to a fractional power as in (1.1) is involved. Functions g s (u) are the rates of cell 2 VITALI VOUGALTER AND VITALY VOLPERT birth dependent upon u (density dependent proliferation), and the functions K s (x -y) show the proportion of newly born cells changing their genotype from y to x. Let us assume here that they depend on the distance between the genotypes. The last term on the right side of (1.1) describes cell influxes for different genotypes.

The square root of minus Laplacian in system ( Let us set all D s = 1 and study the existence of solutions of the system of equations

(1.2) - √ -∆u s + ∫ R d K s (x -y)g s (u(y)) dy + f s (x) = 0, 1 ≤ s ≤ N.
We consider the case where the linear part of operator (1.2) does not satisfy the Fredholm property such that conventional methods of nonlinear analysis may not be applicable. Solvability conditions for non Fredholm operators, which is the novelty of our approach, along with the method of contraction mappings, will be used. Possible applications of our results are studies of nonlinear elliptic problems in unbounded domains involving operators with the essential spectrum containing the origin.

Let us consider the equation . In particular, in the case of a = 0, the operator A is Fredholm in some properly chosen weighted spaces, see [2, 3, 4, 5, 6]. However, the situation where a ̸ = 0 is significantly different, and the approach developed in these articles cannot be applied. Front propagation problems with superdiffusion were extensively treated in recent years, see e.g., [START_REF] Volpert | Exact solutions in front propagation problems with superdiffusion[END_REF][START_REF]Fronts in anomalous diffusion-reaction systems[END_REF].

(1.3) -∆u + V (x)u -au = f, with u ∈ E = H 2 (R d ) and f ∈ F = L 2 (R d ), d ∈ N,
We set K s (x) = ε s H s (x) with ε s ≥ 0, ε := max 1≤s≤N ε s ,
and we suppose that the next assumption holds.

Assumption 1.1. Let 1 ≤ s ≤ N be such that f s (x) : R 3 → R, f s (x) ∈ L 1 (R 3 ) and ∇f s (x) ∈ L 2 (R 3 ). Furthermore, f s (x) is nonzero for a certain s. Assume also that H s (x) : R 3 → R is such that H s (x) ∈ L 1 (R 3 ) and ∇H s (x) ∈ L 2 (R 3 ). Moreover, H 2 : = N ∑ s=1 ∥H s ∥ 2 L 1 (R 3 ) > 0
and

Q 2 : = N ∑ s=1 ∥∇H s ∥ 2 L 2 (R 3 ) > 0.
We choose space dimension d = 3, which is related to the solvability conditions for the linear Poisson equation (3.1) stated in Lemma 3.1. The results obtained below can be generalized to d > 3. From the point of view of applications the space dimension is not limited to d = 3 because the space variable is correspondent to cell genotype but not to the usual physical space.

By virtue of the standard Sobolev inequality, see e.g., [11, page 183], under Assumption 1.1, we have

f s (x) ∈ L 2 (R 3 ), 1 ≤ s ≤ N.
We use the Sobolev space of vector functions

H 2 (R 3 , R N ) := {u(x) : R 3 -→ R N | u s (x) ∈ L 2 (R 3 ), ∆u s ∈ L 2 (R 3 ), 1 ≤ s ≤ N }, equipped with the norm (1.4) ∥u∥ 2 H 2 (R 3 ,R N ) := N ∑ s=1 ∥u s ∥ 2 H 2 (R 3 ) = N ∑ s=1 {∥u s ∥ 2 L 2 (R 3 ) + ∥∆u s ∥ 2 L 2 (R 3 ) }.
Also,

∥u∥ 2 L 2 (R 3 ,R N ) := N ∑ s=1 ∥u s ∥ 2 L 2 (R 3 ) .
The Sobolev embedding implies

(1.5) ∥ϕ∥ L ∞ (R 3 ) ≤ c e ∥ϕ∥ H 2 (R 3 ) ,
where c e > 0 is the constant of the embedding. The hat symbol will denote the standard Fourier transform such that

(1.6) ϕ(p) = 1 (2π) 3/2 ∫ R 3 ϕ(x)e -ipx dx.
When all nonnegative parameters ε s = 0, we obtain the linear Poisson equations

(1.7) √ -∆u s = f s (x), 1 ≤ s ≤ N.
By virtue of Lemma 3.1, along with Assumption 1.

∥∆u s ∥ 2 L 2 (R 3 ) = ∥∇f s (x)∥ 2 L 2 (R 3 )
we derive for the unique solution u 0,s (x) of (1.7

) that u 0,s (x) ∈ H 2 (R 3 ) such that u 0 (x) = (u 0,1 (x), u 0,2 (x), . . . , u 0,N (x)) T ∈ H 2 (R 3 , R N ).
We look for the resulting solution of the nonlinear system of equations (1.2) as

(1.8) u(x) = u 0 (x) + u p (x), with u p (x) = (u p,1 (x), u p,2 (x), . . . , u p,N (x)) T .
In a straightforward manner, we derive the perturbative system of equations

(1.9) √ -∆u p,s = ε s ∫ R 3 H s (x -y)g s (u 0 (y) + u p (y)) dy, 1 ≤ s ≤ N.
We introduce a closed ball in our Sobolev space as

(1.10) B ρ := {u(x) ∈ H 2 (R 3 , R N ) | ∥u∥ H 2 (R 3 ,R N ) ≤ ρ}, 0 < ρ ≤ 1.
We now look for the solution of (1.9) as the fixed point of the auxiliary nonlinear system

(1.11) √ -∆u s = ε s ∫ R 3 H s (x -y)g s (u 0 (y) + v(y)) dy, 1 ≤ s ≤ N,
in ball (1.10). For a given vector function v(y), this is a system of equations with respect to u(x). The left side of (1.11) contains the non Fredholm operator

√ -∆ : H 1 (R 3 ) → L 2 (R 3
). Since its essential spectrum fills the nonnegative semi-axis [0, +∞), this operator has no bounded inverse. The analogous situation appeared in articles [25, 28], but as distinct from the present work; the problems treated there required orthogonality relations. The fixed point technique was applied in [START_REF] Vougalter | On threshold eigenvalues and resonances for the linearized NLS equation[END_REF] to estimate the perturbation to the standing solitary wave of the nonlinear Schrödinger (NLS) equation where either the external potential or the nonlinear term in the NLS were perturbed but the Schrödinger operator involved in the nonlinear problem there had the Fredholm property, see [START_REF] Cuccagna | Spectra of positive and negative energies in the linearized NLS problem[END_REF], [20, Assumption 1]. We define a closed ball in the space of N dimensions (1.12)

I := {z ∈ R N | |z| ≤ c e ∥u 0 ∥ H 2 (R 3 ,R N ) + c e }.
We now introduce the following quantities with 1 ≤ s, j ≤ N ,

a 2,s,j := sup z∈I ∇ ∂g s ∂z j , a 2,s := N ∑ j=1 a 2 2,s,j , a 2 := max 1≤s≤N a 2,s . Also, a 1,s := sup z∈I |∇g s (z)|, a 1 := max 1≤s≤N a 1,s .
The next assumption follows from the nonlinear part of problem (1.2).

Assumption 1.2. Let 1 ≤ s ≤ N be such that g s (z) : R N → R with g s (z) ∈ C 2 (R N ).
We also assume that g s (0) = 0, ∇g s (0) = 0 and a 2 > 0.

Here, C 2 (R N ) denotes the space of twice continuously differentiable functions on R N . It follows that a 1 , as defined above, is also positive; otherwise, all functions g s (z) will be constants in the ball I, and then, a 2 = 0. For example, g s (z) = z 2 , z ∈ R N , obviously satisfies Assumption 1.2.

We introduce operator T g such that u = T g v, where u is a solution of the system of equations (1.11).

Our main result is as follows.

Theorem 1.3. Let Assumptions 1.1 and 1.2 hold. Then, system (1.11) defines the map T g : B ρ → B ρ , which is a strict contraction for all 0 < ε < ε * for some ε * > 0. The unique fixed point u p (x) of the map T g is the only solution of the system of equations (1.9) in B ρ .

It would be natural to conjecture here that this fixed point continuously depends on parameters of the contracting operator, namely, on functions g s for the operator T g . We leave this as an open question.

Obviously, the resulting solution of system (1.2) given by (1.8) will be nontrivial due to the fact that the source terms f s (x) are nontrivial for a certain s = 1, . . . , N , and all g s (0) = 0, as assumed.

The next, trivial, lemma will be needed in our applications.

Lemma 1.4. Consider the function φ(R)

:= αR + β/R 2 for R ∈ (0, +∞) with the constants α, β > 0. It achieves the minimal value at R * = (2β/α) 1/3 , which is given by φ(R * ) = (3/2 2/3 )α 2/3 β 1/3 .
We now proceed to the proof of our main statement.

The existence of the perturbed solution.

Proof of Theorem 1.3. We arbitrarily choose v(x) ∈ B ρ and denote the terms involved in the integral expressions on the right side of system (1.11) as

G s (x) := g s (u 0 (x) + v(x)), 1 ≤ s ≤ N.
Applying standard Fourier transform (1.6) to both sides of system (1.11) yields

u s (p) = ε s (2π) 3/2 H s (p) G s (p) |p| , 1 ≤ s ≤ N.
Thus, for the norm, we obtain

(2.1) ∥u s ∥ 2 L 2 (R 3 ) = (2π) 3 ε 2 s ∫ R 3 | H s (p)| 2 | G s (p)| 2 p 2 dp.
Obviously, for any ϕ(x)

∈ L 1 (R 3 ), (2.2) ∥ ϕ(p)∥ L ∞ (R 3 ) ≤ 1 (2π) 3/2 ∥ϕ(x)∥ L 1 (R 3 ) .
In departure from articles [25, 28] involving the standard Laplacian operator in the diffusion term, here we do not try to control the norms

H s (p) |p| L ∞ (R 3 ) .
We estimate the right side of (2.1), applying (2.2) with R > 0 as

(2.3) (2π) 3 ε 2 s ∫ |p|≤R | H s (p)| 2 | G s (p)| 2 p 2 dp + (2π) 3 ε 2 s ∫ |p|>R | H s (p)| 2 | G s (p)| 2 p 2 dp ≤ ε 2 s ∥H s ∥ 2 L 1 (R 3 ) { 1 2π 2 ∥G s (x)∥ 2 L 1 (R 3 ) R + 1 R 2 ∥G s (x)∥ 2 L 2 (R 3 )
} .

Since v(x) ∈ B ρ , we obtain

∥u 0 + v∥ L 2 (R 3 ,R N ) ≤ ∥u 0 ∥ H 2 (R 3 ,R N ) + 1,
and, by means of Sobolev embedding (1.5),

|u 0 + v| ≤ c e ∥u 0 ∥ H 2 (R 3 ,R N ) + c e .
We use the identity formula

G s (x) = ∫ 1 0 ∇g s (t(u 0 (x) + v(x))) • (u 0 (x) + v(x)) dt, 1 ≤ s ≤ N.
Throughout this paper, • stands for the scalar product of two vectors in R N . With ball I defined in (1.12) we derive

|G s (x)| ≤ sup z∈I |∇g s (z)||u 0 (x) + v(x)| ≤ a 1 |u 0 (x) + v(x)|.
Hence,

∥G s (x)∥ L 2 (R 3 ) ≤ a 1 ∥u 0 + v∥ L 2 (R 3 ,R N ) ≤ a 1 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1).
In addition, for t ∈ [0, 1] and 1 ≤ j ≤ N ,

∂g s ∂z j (t(u 0 (x) + v(x))) = ∫ t 0 ∇ ∂g s ∂z j (τ (u 0 (x) + v(x))) • (u 0 (x) + v(x)) dτ.

This implies

∂g s ∂z j (t(u 0 (x) + v(x))) ≤ sup z∈I ∇ ∂g s ∂z j |u 0 (x) + v(x)| = a 2,s,j |u 0 (x) + v(x)|. EQUATIONS WITH SUPERDIFFUSION 9
Hence, by virtue of the Schwarz inequality,

|G s (x)| ≤ |u 0 (x) + v(x)| N ∑ j=1 a 2,s,j |u 0,j (x) + v j (x)| ≤ a 2 |u 0 (x) + v(x)| 2 , such that ∥G s (x)∥ L 1 (R 3 ) ≤ a 2 ∥u 0 (x) + v(x)∥ 2 L 2 (R 3 ,R N ) ≤ a 2 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1)
2 . Thus, we arrive at the upper bound for the right side of (2.3), given by (2.4)

ε 2 s ∥H s ∥ 2 L 1 (R 3 ) (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 2 { a 2 2 2π 2 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 2 R + a 2 1 R 2 } ,
with R ∈ (0, +∞). Lemma 1.4 gives us the minimal value of (2.4). Therefore,

∥u s ∥ 2 L 2 (R 3 ) ≤ 3 2 4/3 π 4/3 ε 2 ∥H s ∥ 2 L 1 (R 3 ) (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 3(1/3) a 2/3 1 a 4/3 2 , such that (2.5) ∥u∥ 2 L 2 (R 3 ,R N ) ≤ 3 2 4/3 π 4/3 ε 2 H 2 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 3(1/3) a 2/3 1 a 4/3 2 .
Obviously, by means of (1.11),

-∆u s = ε s √ -∆ ∫ R 3 H s (x -y)G s (y) dy, 1 ≤ s ≤ N.
Using (2.2), we easily obtain

∥∆u s ∥ 2 L 2 (R 3 ) ≤ ε 2 a 2 2 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 4 ∥∇H s ∥ 2 L 2 (R 3 ) . Hence, (2.6) N ∑ s=1 ∥∆u s ∥ 2 L 2 (R 3 ) ≤ ε 2 a 2 2 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 4 Q 2 .
The definition of norm (1.4), along with inequalities (2.5) and (2.6), give

∥u∥ H 2 (R 3 ,R N ) ≤ ε(∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 2 a 2/3 2 × √ 3 2 4/3 π 4/3 H 2 a 2/3 1 + a 2/3 2 Q 2 ≤ ρ,

VITALI VOUGALTER AND VITALY VOLPERT

for all ε > 0 sufficiently small; therefore, u(x) ∈ B ρ as well. If, for a certain v(x) ∈ B ρ , there are two solutions u 1,2 (x) ∈ B ρ of system (1.11), their difference w(x

) := u 1 (x) -u 2 (x) ∈ L 2 (R 3 , R N ) solves √ -∆w = 0.
Due to the fact that the operator √ -∆ does not possess nontrivial square integrable zero modes, w(x) = 0 almost everywhere in R 3 . Hence, system (1.11) defines a map T g : B ρ → B ρ for all ε > 0 small enough.

Our goal is to show that this map is a strict contraction. We arbitrarily choose v 1,2 (x) ∈ B ρ . By means of the argument above we have u 1,2 = T g v 1,2 ∈ B ρ as well. By virtue of system (1.11),

√ -∆u 1,s = ε s ∫ R 3 H s (x -y)g s (u 0 (y) + v 1 (y)) dy, 1 ≤ s ≤ N, (2.7) √ -∆u 2,s = ε s ∫ R 3 H s (x -y)g s (u 0 (y) + v 2 (y)) dy, 1 ≤ s ≤ N. (2.8) We define G 1,s (x) := g s (u 0 (x) + v 1 (x)), G 2,s (x) := g s (u 0 (x) + v 2 (x)), 1 ≤ s ≤ N,
and apply the standard Fourier transform (1.6) to both sides of systems (2.7) and (2.8). We obtain

u 1,s (p) = ε s (2π) 3/2 H s (p) G 1,s (p) |p| , u 2,s (p) = ε s (2π) 3/2 H s (p) G 2,s (p) |p| .
In addition,

∥u 1,s (x) -u 2,s (x)∥ 2 L 2 (R 3 ) = ε 2 s (2π) 3 ∫ R 3 | H s (p)| 2 | G 1,s (p) -G 2,s (p)| 2 |p| 2 dp.
Furthermore, it can be estimated by using (2.2) from above that

ε 2 s ∥H s ∥ 2 L 1 (R 3 ) { 1 2π 2 ∥G 1,s (x) -G 2,s (x)∥ 2 L 1 (R 3 ) R + ∥G 1,s (x) -G 2,s (x)∥ 2 L 2 (R 3 ) 1 R 2 } ,
where R ∈ (0, +∞). For 1 ≤ s ≤ N , we use the identity

G 1,s (x) -G 2,s (x) = ∫ 1 0 ∇g s (u 0 (x) + tv 1 (x) + (1 -t)v 2 (x)) • (v 1 (x) -v 2 (x)) dt.
Clearly, for v 1,2 (x) ∈ B ρ and t ∈ [0, 1], we easily obtain the upper bound for

∥v 2 (x) + t(v 1 (x) -v 2 (x))∥ H 2 (R 3 ,R N ) as t∥v 1 (x)∥ H 2 (R 3 ,R N ) + (1 -t)∥v 2 (x)∥ H 2 (R 3 ,R N ) ≤ ρ.
Thus, v 2 (x) + t(v 1 (x) -v 2 (x)) ∈ B ρ as well. We derive

|G 1,s (x) -G 2,s (x)| ≤ sup z∈I |∇g s (z)||v 1 (x) -v 2 (x)| = a 1,s |v 1 (x) -v 2 (x)|.
Therefore,

∥G 1,s (x) -G 2,s (x)∥ L 2 (R 3 ) ≤ a 1,s ∥v 1 (x) -v 2 (x)∥ H 2 (R 3 ,R N ) .
In addition, for 1 ≤ j ≤ N ,

∂g s ∂z j (u 0 (x) + tv 1 (x) + (1 -t)v 2 (x)) = ∫ 1 0 ∇ ∂g s ∂z j (τ [u 0 (x) + tv 1 (x) + (1 -t)v 2 (x)]) • (u 0 (x) + tv 1 (x) + (1 -t)v 2 (x)) dτ.
Hence,

∂g s ∂z j (u 0 (x) + tv 1 (x) + (1 -t)v 2 (x)) ≤ sup z∈I ∇ ∂g s ∂z j {|u 0 (x)| + t|v 1 (x)| + (1 -t)|v 2 (x)
{ |u 0 (x)| + 1 2 |v 1 (x)| + 1 2 |v 2 (x)| } |v 1,j (x) -v 2,j (x)| ≤ a 2,s { |u 0 (x)| + 1 2 |v 1 (x)| + 1 2 |v 2 (x)| } |v 1 (x) -v 2 (x)|.
The Schwarz inequality gives the upper bound for

∥G 1,s (x) -G 2,s (x)∥ L 1 (R 3 ) as a 2,s { ∥u 0 (x)∥ L 2 (R 3 ,R N ) + 1 2 ∥v 1 (x)∥ L 2 (R 3 ,R N ) + 1 2 ∥v 2 (x)∥ L 2 (R 3 ,R N ) } × ∥v 1 (x) -v 2 (x)∥ L 2 (R 3 ,R N ) ≤ a 2 {∥u 0 (x)∥ H 2 (R 3 ,R N ) + 1}∥v 1 (x) -v 2 (x)∥ H 2 (R 3 ,R N ) .
This allows us to estimate the norm ∥u

1 (x) -u 2 (x)∥ 2 L 2 (R 3 ,R N ) by (2.9) ε 2 H 2 ∥v 1 -v 2 ∥ 2 H 2 (R 3 ,R N ) { a 2 2 2π 2 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 2 R + a 1 2 R 2 } .
By virtue of Lemma 1.4, we minimize (2.9) over R > 0 in order to obtain that ∥u

1 (x) -u 2 (x)∥ 2 L 2 (R 3 ,R N ) is bounded from above by (2.10) ε 2 H 2 ∥v 1 -v 2 ∥ 2 H 2 (R 3 ,R N ) 3 2 4/3 a 4/3 2 π 4/3 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 4/3 a 2/3 1 .
By means of (2.7) and (2.8) we have

-∆(u 1,s -u 2,s ) = ε s √ -∆ ∫ R 3 H s (x-y)[G 1,s (y)-G 2,s (y)] dy, 1 ≤ s ≤ N.
Hence, via (2.2), we obtain

∥∆(u 1,s -u 2,s )∥ 2 L 2 (R 3 ) ≤ ε 2 ∥∇H s ∥ 2 L 2 (R 3 ) ∥G 1,s (x) -G 2,s (x)∥ 2 L 1 (R 3 ) ≤ ε 2 ∥∇H s ∥ 2 L 2 (R 3 ) a 2 2 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 2 ∥v 1 -v 2 ∥ 2 H 2 (R 3 ,R N ) , such that N ∑ s=1 ∥∆(u 1,s -u 2,s )∥ 2 L 2 (R 3 ) is estimated from above by (2.11) ε 2 Q 2 a 2 2 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1) 2 ∥v 1 -v 2 ∥ 2 H 2 (R 3 ,R N )
. By virtue of inequalities (2.10) and (2.11)

the norm ∥u 1 -u 2 ∥ H 2 (R 3 ,R N )
π 4/3 + Q 2 a 2/3 2 ] 1/2 ∥v 1 -v 2 ∥ H 2 (R 3 ,R N ) .
EQUATIONS WITH SUPERDIFFUSION

13

Hence, the map T g : B ρ → B ρ , defined by system (1.11), is a strict contraction for all values of ε > 0 small enough. Its unique, fixed point u p (x) is the only solution of system (1.9) in B ρ . The resulting u(x) ∈ H 2 (R 3 , R N ), given by (1.8), is a solution of the system of equations (1.2).

Auxiliary results.

We recall the solvability conditions for the linear Poisson type equation

is bounded from above by εa 2 / 3 2

 23 (∥u 0 ∥ H 2 (R 3 ,R N ) + 1)

Lemma 3 . 1 .

 31 (3.1) √ -∆ϕ = f (x), x ∈ R d , d ∈ N,established in[START_REF]Existence of stationary solutions for some integro-differential equations with superdiffusion[END_REF]. We denote the inner product as(3.2) (f (x), g(x)) L 2 (R d ) := ∫ R d f (x)g(x) dx,with a slight abuse of notation, when the functions involved in (3.2) are not square integrable, for example, the functions present in orthogonality conditions (3.3) and(3.4).Indeed, if f (x) ∈ L 1 (R d ) and g(x) ∈ L ∞ (R d ),then the integral on the right side of (3.2) makes sense. The technical result below was easily proved [30] by applying the standard Fourier transform to problem (3.1). Let f (x) ∈ L 2 (R d ), d ∈ N. (i) When d = 1 and |x|f (x) ∈ L 1 (R), equation (3.1) admits a unique solution ϕ(x) ∈ H 1 (R) if and only if the orthogonality condition(3.3) (f (x), 1) L 2 (R) = 0 holds. (ii) When d = 2 and |x|f (x) ∈ L 1 (R 2 ), problem (3.1) possesses a unique solution ϕ(x) ∈ H 1 (R 2 ) if and only if the orthogonality relation (3.4) (f (x), 1) L 2 (R 2 ) = 0 holds. (iii) When d ≥ 3 and f (x) ∈ L 1 (R d ), equation (3.1) has a unique solution ϕ(x) ∈ H 1 (R d ).

7, 15], surface diffusion [12, 13], semiconductors [14], and

  1.1) represents a particular example of superdiffusion actively studied in relation to different applications in plasma physics and turbulence [so on. The physical meaning of superdiffusion is that the random process occurs with longer jumps in comparison with normal diffusion. The moments of jump length distribution are finite in the case of normal diffusion, but this is not the case for superdiffusion. The operator √ -∆ is defined via spectral calculus. A similar problem in the presence of the standard Laplacian in the diffusion term was recently treated [

29, 30, 31]

. A single equation analogous to system (1.1) is addressed in [30].

2, 3, 4, 5, 6]. Schrödinger

  -type operators without the Fredholm property were studied via the methods of spectral and scattering theory [

	the Fredholm property. Its image is not closed, for d > 1, the dimension
	of its kernel and the codimension of its image, are not finite.
	In this paper, we study some properties of operators of this kind.
	Recall that elliptic equations with non Fredholm operators were ex-
	tensively treated in recent years. Approaches in weighted Sobolev and
	Hölder spaces were developed [
	of problem (1.3)
	contains the origin. As a consequence, this operator does not satisfy

a is a constant and the scalar potential function V (x) is either vanishing in the whole space or converging to 0 at ∞. For a ≥ 0, the essential spectrum of the operator A : E → F corresponding to the left side

16, 17, 21, 22, 23]. The

  Laplacian operator with drift, from the point of view of non Fredholm operators, was treated in [

26] and linearized Cahn-Hilliard problems in [24, 27]. Ar- ticles [25, 28] were

  devoted to the studies of nonlinear, non Fredholm, elliptic problems. Significant applications to the theory of reactiondiffusion equations were developed [

9, 10]. Non

	Fredholm operators
	also appear when studying wave systems with an infinite number of
	localized traveling waves, see [1]


Jihui Zhou

Qingsong Cao

Faxiong Sun

Lan Bi

Feasibility of Coordinate Measuring System Based on Wire Driven Robot

Keywords: Wire-driven parallel kinematic manipulators (PKMs), Coordinate measuring system, Feasibility

The abstract should summarize the contents of the paper and should contain at least 70 and at most 150 words. It should be set in 9-point font size and should be inset 1.0 cm from the right and left margins. The coordinate measuring machine systems (CMMS) are widely used for part inspections in manufacturing plants, which mostly include bridge CMMS and horizontal CMMS driven by motor and portable CMMS operated manually. Wire-driven parallel kinematic manipulators (PKMs) have advantages of flexibility, accuracy and large workspace. This paper investigates the feasibility of the coordinate measuring system based on wire-driven PKM. The forward and inverse position solutions are obtained by MATLAB simulation through the kinematic analysis of the manipulator. The controllable workspace with tension conditions and stiffness conditions are obtained by means of the static analysis. Analysis results show that the manipulator have high precision, high speed and large workspace, these advantages just meet the measure requirements of coordinate measuring systems. Finally, a primary experiment is presented to show that the measuring platform of this manipulator can move smoothly and its measurement error is relatively low. The feasibility of taking the wire-driven PKM as coordinate measuring system has been validated by theoretical and experimental study. This research maybe lays a foundation for the further application of wire-driven PKM in the field of coordinate measuring.

Introduction

As one of the most indispensable metrological instruments for inspections and quality control in manufacturing plants, coordinate measuring machine systems (CMMS) are widely used in many fields such as mechanical manufacturing, automotive industry, electronic industry, aerospace industry and national defense industry. The first CMM was developed by the Ferranti Company of Scotland in the 1950s, although this machine only had 2 axes. The first 3-axis models began appearing in the 1960s and computer control debuted in the early 1970s. After the 1970s, CMMS have made rapid development [START_REF] Liu | Present state and developing trend of three-coordinate measuring machines[END_REF]. With the development of modern industry, fast and accurate measurement is an important research subject [START_REF] Zhang | The development tendency of coordinate measuring machines[END_REF]. At the same time, novel type mechanisms need to be designed. Wire-driven parallel kinematic manipulators (PKMs) possess a number of promising advantages over the conventional rigid-link manipulators, such as simple and lightweight mechanical structure, high-loading capacity, large workspace, low moment inertia and high speed motion. The early study on wire-driven PKMs is the application in cargo handling. Over the years, a number of different wire-driven PKMs designed for a wide variety of applications in large-scale manufacturing such as welding, cutting, grinding, assembly, fixturing, paint stripping, and machining. An example of wire driven robots that are currently in use is the NIST Robot Crane [START_REF] Edward | Summary of Modeling and Simulation for NIST RoboCrane Applications[END_REF]. The NIST Robot Crane is a large-workspace robot for painting and maintaining aircraft which is also suitable for material handling in warehouses and storage facilities [START_REF] Goodwin | RoboCrane Construction of Bridges[END_REF]. Recently, six degrees of freedom wire-driven PKMs in the application of a large radio telescope is presented [START_REF] Zheng | Research survey and development tendency of wiredriven parallel manipulators[END_REF]. The potential for cable robots are given to be used in a variety of applications, the studies on the wire-driven PKMs will be promoted.

The novel three degrees of freedom (DOF) coordinate measuring machine driven by four wires is proposed in this paper [START_REF] Ming | Study on Multiple Degree of Freedom Positioning Mechanisms Using Wires (Part I): Concept, Design and Control[END_REF][START_REF] Liu | Kinematic analysis of a 6-dofwire-driven parallel kinematic manipulator[END_REF]. Based on the kinematic analysis of the manipulator, it is confirmed that this manipulator have large workspace, high moment precision and low moment error after the simulation of MATLAB. A primary experiment is presented to show that the moving platform of this manipulator can move smoothly, its measurement error is relatively low and measurement precision is relatively high. The feasibility of wire-driven PKM used as coordinate measuring system is verified by theoretical analysis and experiment. This study will play an important role in pushing forward the application of wire-driven PKM in coordinate measuring field.

Overal design

The basic function of CMMS is to gain the actual shape of a work piece. The actual shape of the work piece is obtained by probing the surface of the work piece at discrete measuring points which can then be analyzed via regression algorithms for the work piece of features. Every measuring point is expressed in terms of its measured coordinates. Based on this principle of measurement, the overall plan design of the measuring system is presented in figure 1.

Fig. 1. Block diagram of the system

The measuring system includes four main components: the main structure, measuring system, touch trigger probe and electrical control system. The main structure is an 1160mm×550mm×1000mm PKMs driven by four wires. The accurate control of the measuring platform is obtained through using step motor to control the lengths of the wires. The step motor controlled by pulse-to-step is generated by a 51 microcontroller. As one of the most important part, the touch trigger probe greatly influences measurement accuracy and precision. The touch trigger probe named TP60 is used in this paper which has a spring loaded ruby ball stylus. The different functions can be realized by the electrical control system through the way of microcontroller programming. The control signal can be import from keyboard or from 51 microcontroller and PC serial communications. After receiving the signal, the microcontroller jump to the corresponding procedure to realize the different functions, such as display coordinates on the LED nixie tube, the measuring platform is programmed to moving up and right. As the probe touched the surface of the component, the stylus deflected and simultaneously the microcontroller sent information of coordinate (X.Y.Z) to the PC computer. Measuring element of the target such as the shape of the component, the shape of the component and position was obtained by fitting the information of coordinate (X.Y.Z).

3

Theories analysis

Structure analysis

The schematic diagram of completely restrained wire-driven PKM is shown in Fig.

figure 2.

Fig. 2. PKM driven of four wires

The measuring platform is connected to static platform through four driving cables, AA', BB', CC', DD'. The length of each cable is denoted as (i = 1, 2, 3, 4). The cable is connected to the static platform at point A, B, C, and D. The cable is connected to the measuring platform at point A', B', C' and D'. The center of ABCD and A'B'C'D' expressed in O and P. Make coordinate system O-xyz become the i L static coordinate system. Let coordinate system O-xyz becomes the moving coordinate system. The unit vector in the direction of the cable is . The manipulator has only 3 Cartesian degrees-of-freedom(x, y, and z). In Figure 2, the design parameters for the PKM driven by 4 wires are 2a (static platform length), 2b (static platform width), 2c (measuring platform length), and 2d (measuring platform width).

is the cable tension applied for the cable. is the position vector from the origin of P to the cable connection. For static equilibrium the sum of external forces and moments exerted on the measuring platform by the cables must equal to the resultant external wrench exerted on the environment. (1) Where T= is the vector of scalar cable forces, F is the resultant external wrench vector exerted on the environment by the measuring platform,

( ) t t 1 4 [ ] A e e = 4
is the statics jacobian matrix.

L= is the vector of cable length. is the cable translational velocity applied for the cable, expressed in V.

1 1 4 4 ( ) T L e L e × × i V T
A is the transpose matrix of A matrix. By means of the method of the vector analysis, the following matrix form can be obtained L= V.

(

T A The analytical expression of cable forces is obtained via the first equation. In other words, to invert (1) we adapt the well-known particular and homogeneous solution T= q.

(3) ( )

+ A F E A A + + -
Where is the Moore-Penrose pseudo inverse of A, is the minimum norm solution of (1), q is an arbitrary 4-vector, E is the 4×4 identity matrix.

A + A F + 3.2
K K K K p , Y p , Z p )
is the position of P relating the static coordinate system. According to the closed vector approach, the following equation can be described

' i i K K P = + . ( 4
) Therefore, the length of each cable can be calculated as follows

1 1 5 0 p p p c a X L K K b d Y Z ⎡ ⎤ - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = - = --⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ , 2 2 6 0 p p p c a X L K K d b Y Z ⎡ ⎤ - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = - = --⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 3 3 7 0 p p p a c X L K K d b Y Z ⎡ ⎤ - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = - = --⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ , 1 4 8 0 p p p a c X L K K b d Y Z ⎡ ⎤ - ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = - = --⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
The inverse position solutions can be expressed as

2 1 1 2 2 2 2 2 2 3 3 2 2 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p p 2 2 2 2 p p p p p p p p p p L L c a X b d Y Z L L c a X d b Y Z L L a c X d b Y Z L L a c X b d Y Z ⎧ = = -- + -- + ⎪ ⎪ = = -- + -- + ⎪ ⎨ = = -- + -- + ⎪ ⎪ ⎪ = = -- + -- + ⎩ .
(5)

Forward position analysis.

The relationship between the length of each cable and ( ' '

p p p X Y Z ) can be obtained via invert equation i i L L =
, so the forward position solutions can be expressed as

2 2 1 4 2 2 2 1 2 2 4 2 2 2 2 1 4 2 3 4 ( ) 4 ( ) ( ) ( p p p p L L X a c L L Y b d Z L a c X b d Y L L L L ⎧ - = ⎪ × - ⎪ ⎪ - ⎪ = ⎨ × - ⎪ ⎪ = - --- --- ⎪ ⎪ -= - ⎩ 2 ) p . ( 6
)
Whatever the position and pose of the mobile platform is, the length of each cable must meet the Eq. ( 6). The position and pose of the measuring platform corresponding to certain cable lengths thus are determined via the forward position analysis of parallel wire-driven robot.

Workspace analysis

Controllable workspace.

The controllable workspace is defined as a set of position and orientation for the platform, which meet equilibrium of force and torque, and the tension of each cable is required to be positive [START_REF] Verhoeven | Works-pace, stiffness, singularities and classification of tendon-driven Stewart platforms[END_REF]. The controllable workspace of the manipulators can be determined by checking the tension of each cable on the basis of the principle of vector closure. Therefore, the manipulators to meet the principle of vector closure should be proved firstly.

According to the principle of vector closure, in a 3-dimensional real space, the arbitrary vector v is closed only if vector v has at least 4 vectors (w 1 , w 2 , w 3 and w 4 ) satisfying the following two conditions. Condition 1 is that in the four vector (w 1 ... Assume the resultant external wrench exerted on the environment is zero ( the gravity of each rope is ignored), the following matrix can be expressed: AT=0. Therefore, can be obtained through the equation of . Because the mechanisms does not produce singular, in the four vector (w 1 ... w 4 ), any three vectors are linearly independent. The manipulators to meet the principle of vector closure can be proved through the above analysis.

According to the equation of T= q, if the tension solution in the above equation can always be made positive for all positive homogeneous solutions regardless of the external disturbance F, the boundary of the workspace can therefore be generated by letting F=0. Therefore, the tension of each cable can be positive if each element of q is positive. So the controllable workspace bound can be expressed as follow.

( )

+ A F E A A + + - ) 0 (E A A + - ( ) E A A + - > . (7) 3.3.2
Workspace with tension conditions. The workspace with tension conditions is defined as a set of position and orientation under the conditions of the controllable workspace in which the tensions in the cables must range between a pre-tension and a maximum tension and the gravity or elasticity of each rope is ignored. Therefore, the workspace with tension conditions can be determined by the tensions in each cable and the external forces and torques exerted on the environment. Assume the external forces and torques exerted on the environment is zero, in other words F=0. With k=tmax/tmin is defined as critical tension factor, the workspace with tension conditions can be calculated as follows )

i i B K 0 i i B B E A A = = + ⎧ ≤ ⎪ ⎨ ⎪ = - > ⎩ 0 . ( 8
)
Where is the 4×4 matrix. ( )

B E A A + = - >

3.3.3

Workspace with stiffiness conditions. The workspace with tension conditions for n DOF PKMs is defined as a set of position and orientation under the conditions of the controllable workspace in which the pose greater than n positive eigenvalues of stiffness matrix is not less than stiffness coefficients K' [START_REF] Verhoeven | Works-pace, stiffness, singularities and classification of tendon-driven Stewart platforms[END_REF]. Therefore, the stiffness matrix should be calculated firstly.

The stiffness matrix of driven by m (m n+1) wires is obtained by Verhoeven [START_REF] Ming | Study on Multiple Degree of Freedom Positioning Mechanisms Using Wires (Part I): Concept, Design and Control[END_REF].

≥ 0 T K K A A = Λ (9) Where 0 0 i i K k L = ×
is the per unit cable length stiffness, is the cable stiffness applied to the cable, is the original cable length applied to the cable, matrix A must meet the condition of

i k 0 i L F AT =
, with is the transposed matrix of matrix A, is a diagonal matrix. The position and pose of the mobile platform can corresponding to certain cable lengths and tension, and are the certain cable lengths and tension. 1)

) i i diag L K t - Λ = + i L i t Generally, 1 0 i K t - ∞
is small enough which can be ignored [START_REF] Liu | Kinematic analysis of a 6-dofwire-driven parallel kinematic manipulator[END_REF]. The manipulators rigidity performance is determined by geometrical arrangement of wires and the position and orientation of measuring platform. According to formula (9) we can conclude that K is a symmetric positive semidefinite matrix which have n positive eigenvalues i K λ ( ) definitely. Therefore, the workspace with tension conditions must satisfy the following constraints

1 2 i= n 1 4 max ( ) 0 i i K K E A A λ = + ′ ≥ ⎧ ⎪ ⎨ - > ⎪ ⎩ . ( 10
)
3.4 Analysis of kinematic error 3.4.1 Analysis of error motion. The kinematic error for measuring platform moving from one place to another can be obtained via plus the error in each coordinate direction respectively. This article scatters a specific space into different points, and uses the kinematics precision analysis in calculating the variation of the position and pose of measuring platform corresponding to allowable pulse-to-step in different point.

Stepper motor can transform the digital input pulse revolving or the straight line increase activity electromagnetism functional element. When stepping drive to receive a pulse, it drives stepper motor rotate in the direction set by a fixed point of view, then the cable length can be changed in a definite value L Δ . The perimeter of the roller for step motor is 60 mm. So L Δ can be calculated as

1.8 0.3 60 360 L SN SN Δ = × = × m m . (11)
Among them, SN is the subdivision number of stepper motor. So the minimum change of each cable length is ΔL.

The measuring platform with moving left and right is X, moving forward and backward rd is Y and moving up and down is Z. When measuring platform moving up and down, 51 microcontrollers give synchronous control of all cable's speed, the cable length can change L Δ every time. The kinematic error will not produced only in the condition of x=0 and y=0. When measuring platform moving left and right, 51 microcontrollers give synchronous control of the first cable and the second cable's speed and synchronous control of the third cable and the fourth cable's speed. Note is the change of the first and the second cable length, note

1 L m L Δ = * Δ 2 L n L Δ = * Δ
is the change of the third and the fourth cable length. The kinematic error will produced in the condition of y≠0. In this paper, m and n are limited to the same ranges, m, n∈ [0, 20], and let them be the positive integer. The value of m and n can be obtained in the condition of the kinematic error in the Z ordinate direction is the minimum count. When measuring platform moving forward and backward, 51 microcontrollers give synchronous control of the first cable and the fourth cable's speed and synchronous control of the second cable and the third cable's speed. Note is the change of the first and the fourth cable length, note is the change of the second and the third cable length. The method of gain the value of m and n is similar with the analysis of moving left and right.

L m L * Δ Δ = 2 L n L Δ = * Δ
With different position and pose of the measuring platform, the kinematic error and accuracy will change. So the relationship between the kinematic error, accuracy and some parameters should be analyzed by MATLAB simulation. Using computer simulation, we can conclude that 1) Littler the size of measuring platform, lower the error for all movement and the accuracy for moving up and down, higher the accuracy for moving right, left, forward and backward. Therefore, the size of measuring platform can be made as small as possible but it has a lower limit.

2) The position and pose in the direction of z is the critical element for the accuracy of moving up and down, and littler the count of z, higher the accuracy.

3) If the error of moving right, left, forward and backward remained to lower, the original position x, y should be positive and remain low around zero, the position z and moving distance h should be remain low.

The best workspace.

A best workspace B which covering x, y∈(-100,100) and z (-924, -724) is obtained by computer simulation. In this workspace, the kinematic accuracy in the direction of coordinate x range from 0.0167 mm to 0.4165 mm; the kinematic accuracy in the direction of coordinate y range from 0.0513 mm to 0.7244 mm; the kinematic accuracy in the direction of coordinate z range from 0.0105 mm to 0.0116 mm. Therefore, the kinematic accuracy of measuring platform is relatively high. If the moving distance is 100 mm, the kinematic error in the direction of coordinate x range from 0mm to 1.82955 mm; the kinematic error in the direction of coordinate y range from 0 mm to 1.85239 mm; the kinematic error in the direction of coordinate z range from 0.428648 mm to 7.17298 mm. Therefore, the kinematic error of measuring platform is relatively low. The above analyzed results show that the manipulators have high kinematics precision and low kinematics error in the workspace B, these advantages just meet the measure requirements of coordinate measuring systems.

experiment

This part attempts to verify the rationality and feasibility of adopting wire-driven PKM as coordinate measuring system through experiments. As some limitations, the designed probe is replaced by some parts in the following experiment. Therefore, checking the probe contacts with the work piece is determined by visual inspection. Besides, the kinematics error of the measuring platform is discussed, it includes the errors of the probe, the distortion of each rope, external conditions and so on. The material object photography of CMM is shown in figure 3. In the experiment, the size of the two rectangular objects is measured in the best working space. The experimental data are in the following table, the error can be obtained via the comparison of measure and actual size.

1

 1

w 4 )

 4 , any three vectors are linearly independent; Condition 2

1

 1

∈

Fig. 3 .

 3 Fig. 3. The material object photography of the designed CMMS

Experimental data show that the measurement error of the designed CMMS is up to a few millimeters, which is somewhat of big. The main factors for the measurement error are human factors, such as the visual error in checking the probe contacts with the work piece and reaction time of human. If the designed probe is used in the experiment, the measurement error should be greatly reduced and it would meet the measure requirements of coordinate measuring systems. Through the observation of the experiment we can conclude that the measurement platform can moved smoothly even in the condition of fast moving. The feasibility of adopting wire-driven PKM as coordinate measuring system is verified once again.

Conclusion

The forward and inverse position solutions are firstly obtained by means of the static analysis and the controllable workspace, workspace with tension and stiffness conditions are presented in this paper. Then the kinematics precision and error analysis of the wire-driven PKM is given by MATLAB simulation based on the kinematic analysis of the manipulator. The best workspace in which the manipulators have high kinematics precision and low kinematics error is obtained. These advantages meet the measure requirements of coordinate measuring systems exactly. Therefore, the theories analysis verifies the feasibility of adopting wire-driven PKM as coordinate measuring system. Furthermore, the overall scheme of CMMS based on wire-driven PKM is put forward in details. Finally, a primary experiment is presented to show that the measuring platform of this manipulator can move smoothly and its measurement error is relatively low. The feasibility of adopting wire-driven PKM as coordinate measuring system is verified both by theoretical and experimental study.


Laipeng Luo

Ergen Liu

A Compression Method of Decision Table Based on Matrix Computation

Keywords: Rough Set, equivalence matrix, attribute reduction, information compression

A new algorithm of attribute reduction based on boolean matrix computation is proposed in this paper. The method compresses the valid information stored in table into a binary tree, at the same time deleting the invalid information and sharing a branch about the same prefix information. Some relative concepts such as local core attributes, local attribute reduction and global core attributes, global attribute reduction are introduced. The conclusions that the global core set is the union of all local core sets and the global attribute reduction sets are the union of respective local attribute reduction sets are proved. The attribute reduction steps of the algorithm are presented. At last, The correctness and effectiveness of the new algorithm are also shown in experiment and in an example.

Introduction

Rough set theory, introduced by Zdzislaw Pawlak in the early 1980s [START_REF] Pawlak | Rough Sets[J][END_REF][START_REF] Pawlak | Rough Sets:Theoretical Aspects of Reasonimg about data[END_REF], is a new mathematical tool to deal with vagueness and uncertainty. This approach seems to be of fundamental importance to artificial intelligence and cognitive sciences, especially in the areas of machine learning, knowledge acquisition,deci-sion analysis, knowledge discovery from databases, expert systems, decision support systems,inductive reasoning, and pattern recognition [3,4,5] .

Attribute reduction is one of the main applications of Rough set.The general problem of finding all reductions are NP-hard, thus it is important for attribute reduction of Rough set to design algothms with lower price and investigate new computation method.A matrix computation method of Rough set was proposed by the author in [START_REF] Guan | Matrix computational method for information systems[END_REF][START_REF] Guan | Matrix computation for information systems[END_REF]for information system.A matrix was seen as an internal representation of equivalence relations.By defining the operation of the equivalence matrix ,matrices are applied to define dependencies between two subsets of attributes,significance of an attribute ect.The approach presents a series of algorithms and their time complexity of attribute reduction.Howevre,there are still several problems to be solvered for the method:(1)The number of objects has great influence on time complexity of these algorithms;(2)These algorithms need too many computations of matrices;(3)The method only discusses matrix computation for information system;( 4 ) How to apply the method to variable precision Rough set model.

With the above-mentioned motivation, in another paper [START_REF] Laipeng | Matrix approach to the study of rough set theory[END_REF],aiming at the problem that how to apply the matrix computation to variable precision Rough set model,we have proposed a measure and computation approach based on matrix about studying Rough set theory.In this paper,we will provide a new insight into attributes reduction in decision table.A compression method of decision table based on matrix computation is proposed.We shall prove the feasibility of the compression method in theory and show its effectiveness in experiment and in an agriculture example.

2.Equivalence matrix of decision table

Definition2.1.Let U be a non-empty finite set of objects and R be equivalence relation on U,We denote the partition gived by the R as follows:U/R= } ,... , {

D T M M ≤ ′ ;(2)if , T T ′ ⊂ ′ ′ then D T M M > ′ ′ : .

3.Theory analysis of matrix computation

Put [R] ij express the element in row i and in column j of M R ,where R={r 1 ,r 2 ,…,r m }.Obviously, [R] ij has the following properties:

(1) [R] ij =[r 1 ] ij ∧[r 2 ] ij ∧…∧[r m ] ij (i=1,2…n,j=1,2…n);(2)if [R] ij =1,then for arbitrary r k ∈R, [r k ] ij =1;(3) if [R] ij =0
,then there at least exist an attribute r k ∈R,such that [r k ] ij =0.Referring to properties,we immediately derive the following facts:

Theorem 3.1 Let S=(U,R=C∪D,V,f, D C M M ≤ ) be a decision table. A subset T ′ of C is said to be a attribute reduction if and only if for any [D] ij =0 in M D , [T ′ ] ij satisfies:(1) [T ′ ] ij =0;(2)There no exist , T T ′ ⊆ ′ ′ such that [ T ′ ′ ] ij =0. Theorem 3.2 Let S=(U,R=C∪D,V,f, D C M M ≤ ) be a decision
D] ij =0 ,[C] ij =min {[c 1 ] ij ,[c 2 ] ij ,…[c m ] ij }=0
,then after deleting attribute b in C,we have

D c C M M ≤ -} {
.Thus there exist attribute set ⊆ ′ C {C-c},such that C′ is attribute reduction of S which contradict that c is core attribute in S.

Conversely,if there exist positive integer i,j,(i=1,2…n,j=1,2…n), such that [D] ij =0 , [c] ij =0,and [b] ij =1 for every b∈C-c,then after deleting attribute b in C,we have

[C] ij =0≠[C-c] ij =1.Thus c is core attribute in S by theorem 3.1 Definition3.1 Let S=(U,R=C∪D,V,f, D C M M ≤ ) be a decision table.If there exist positive integer i,j,such that attribute c satisfies [c] ij =0, [C-c] ij =1 when [D] ij =0 ,[C] ij =0,then attribute c is called local core attribute of decision table S. Theorem 3.3 If c 1 ,c2…,c k be all local core attribute of S=(U,R=C∪D,V,f, D C M M ≤ ),then core attribute set C′ of decision table S is U k i i c 1 = .That is, U k i i c C 1 = = ′ . Definition3.2 Core attribute set C′ of decision
D C M M ≤ ),where ) 1 }( , , , { 2 1 k i A A A B i ik i i i ≤ ≤ = L and ) 1 ( k ij i j A ≤ ≤
is a local reduction.If T t is attribute reduction,then T t satisfies:(1)If for arbitrary positive integer i,j, B i ∩B j =φ,then

k k i ip t i p A T ≤ ≤ = = 1 , 1 U ;(2)If there exist positive integer i,j, B i ∩B j φ,then k ip t i p A T ≤ ≤ = 1 ,

U

where for any A im ,A jn ∈T t , A im ,A jn must satisfy

j im B A ∉ or i jn B A ∉ . Proof (1)Put U k i ip t A T 1 = =
and let C ′ be core attribute set.There at least exist an attribute

t T b ∈ for any [D] ij =0,[C] ij =0,such that [b] ij =0.That is [ t T ] ij =0. On the other hand,for arbitrary t T B⊂ , Suppose t k T c B = ∪ and m k B c C ∈ ∪ ′ 。By definition3.2 there exist positive integer p,q,such that 0 ] [ , 1 ] [ = = ′ pq k pq c C .While for arbitrary i,j,if B i ∩B j =φ,then 0 ] [ ] [ ≠ - = k t pq c T B
By theorem 3.1,attribute set B is not attribute reduction of system S.

(2)If there exist positive integer i,j,B i ∩B j ≠φ, then

k k i ip t i p A T ≤ ≤ = = 1 , 1 U
is not the optimal attribute reduction of system S.It is fact that if B i ∩B j A′ = and

j jm B A A ∈ ≠ ′ ,then ′ ⊆ ∪ ′ t jm T A A
is not the optimal attribute reduction.

Hence ,by above (1), if for any

j jn i im B A B A ∈ ∈ , and t jn im T A A ∈ , , A im ,A jn satify j im B A ∉ or i jn B A ∉ then k ip t i p A T ≤ ≤ = 1 ,

U

is the optimal attribute reduction.

4.Compression of decision table

Because equivalence matrices of attribute set are symmetric,in practical application ,we pay attentin to the upper-triangle above the diagonal or the lower-triangle below the diagonal.Let C={c 1 , c 2 ,…, c r } be condition attribute set and D be decision attribute set. For arbitrary positive integer i,j,we obtain: Please refer to below for details.First,create the root of the tree,labeled with "null".Scan the elements of M D a time and find all positive integer i,j which satisfy [D] ij = 0.The corresponding elements of each equivalence relation matrix of condition attributes orderly sorted lead to the construction of the first branch of the tree with r nodes where [c 1 ] ij is linked as a child of the root, [c 2 ] ij is linked to [c 1 ] ij .The rest may be deduced by analogy.

[C] ij =[c 1 ] ij ∧[c 2 ] ij ∧…∧ [c
The second [c 1 ] mn ,[c 2 ] mn ,…∧[c r ] mn would result in a branch where [c 1 ] mn is linked as a child of the root, [c 2 ] mn is linked to [c 1 ] mn .the rest may be deduced by analogy.Howeve,this branch would share an existing path with other branchs if along the root node ,some branch has the common prefix.For example,if

[c 1 ] ij =[c 1 ] mn ,[c 2 ] ij =[c 2 ] mn ,[c 3 ] ij ≠ [c 3 ] mn ,then the first two nodes of the branch which contains [c 1 ] ij , [c 2 ] ij ∧…∧[c r ] ij is the same as the branch which contains[c 1 ] mn , [c 2 ] mn ∧…∧[c r ] mn.
The rest branches may be constructed by analogy.

By theorem 3.2 ,definition 3.2,all local core attributes and all local attribute reducion are derived from these branches.By theorem 3.3,3.4, we get global core attributes and global attribute reduction.

5.Description of attribute reduction algorithm

Let S=(U,R=C∪D,V,f,

D C M M ≤
) be a decision table.

Step1:Compute the equivalence matrices of decision attribute set and each of condition attributes ,and arrage the equivalence matrices of each of condition attributes in ordor; Step3:Scan the tree and find the only zero value node in every branch.We get local core of every branch. Core set of system S is union of all local core. Step4:Prune the branch that includes local core and at the same time, retain shareable prefix part.

Step2:According to [D] ij = 0,compress [c 1 ] ij ,[c 2 ] ij ,…∧[c r ] ij
Step5:Travel every branch of binary tree pruned and find local attribute reduction sets of all branchs

Step6:Compute attribute reduction of system S according to

} 1 , { k ip t t i p A T T ≤ ≤ = U

6.Algorithm analysis

The time complexity of algorithm in [START_REF] Guan | Matrix computation for information systems[END_REF] for finding all core attributes is ) (

C

U >> ,hence,the method presented in this paper has an advantage over the method in [START_REF] Guan | Matrix computation for information systems[END_REF].

Next,to compare the two methods,We made the relevant experiments on monks datasets in UCI database.The datasets have one decision attribute ,six conditon attributes and 423 records.We did four experiments with the first 100,150,300,423 recordes of standard datasets monks datas. The experiment environment is Petium4 2.1GMHZ,RAM512M, windows XP .The results are as follows:

Table 1 Run time of two algorithms

The chart shows that the method presented in this paper is efficient and scalable for finding core attributes set and attribute reduction sets ,and is faser than the method in [START_REF] Guan | Matrix computation for information systems[END_REF].

7.An Application Example in Agriculture

The following is knowledge representation system of cotton diseases.The condition attributes are a-"diseased spot color", b-"disease site",c-"disease shape",d-"feature" and the decision attribute is e-"the type of disease".{1,2,3} represent the different value of each attribute.

According to the method proposed in this paper,we can obtain that the core attribute is {a} and the attribute reduction set are {a,b} and {a,c,d}.The conclusion is the same as the other method.That is to say that diseased spot is chief factor to judge the type of disease and diseased spot color, disease site or diseased spot color, disease shape, feature may judge exactly the type of disease.

8.Conclusion

In this paper,We further discuss the approach of matrix computation about Rough set and applied it to dicision table.In theory,we have proved the relation between the elements of equivalence matrice and core attributes,attribute reduction .At the same time,we suggest an attribute reduction algorithm based on a storage structure of binary tree which can compress the invalid and the same prefix information.The algorithm designed is lower price.we also find that by changing the order of condition attributes sorted,the algorithm is more efficient.

  r ] ij .By theorem 3.1,in practical application,we only devote our attention to [D] ij = 0 and [C] ij = 0 in the upper-triangle above the diagonal or the lower-triangle below the diagonal. In this paper,we compress information of [D] ij = 0 and [C] ij = 0 in the equivalence matrix into a binary tree where [c 1 ] ij ,[c 2 ] ij ,…∧[c r ] ij are orderly arranged and nodes of the tree.

  into a binary tree where [c k ] ij ,(k=1,2,…r)is node of the PDF 文件使用 "pdfFactory Pro" 试用版本创建 www.fineprint.

.

  The time complexity of algorithm in this paper for finding all core attributes is

3.Let P=

  (p ij ) n×n ,Q=(q ij ) n×n be two binary n×n matrices.The intersection P∩Q of matrices P and Q is defined as follows:P∩Q=[t ij ] n×n ,where t ij =min{p ij ,q ij } PDF 文件使用 "pdfFactory Pro" 试用版本创建 www.fineprint.cn

	2 Definition2.2.Let R be equivalence relation on U,then R is expressed in terms of a binary n×n requivalence matrix 1 n X X X M R =[m ij ] n×n where n= U ,    = or Rx x m j i ij 0 1 . Definition2.Definition2.4.Let R={r 1 ,r 2 ,…,r m },then I m i r R 1 = i M M =

Definition2.5.LetC,D be

  equivalence relation and M C =(c ij ) n×n , M D =(d ij ) n×n be their matrices respectively.If for arbitrary positive integer i,j, c ij ≤d ij, then ) be a decision table.A subset T ′ of C is said to be a attribute reduction of C if and only if T ′ satisfies:[START_REF] Pawlak | Rough Sets[J][END_REF]

		C M ≤	M	D
	Definition2.6.Let S=(U,R=C∪D,V,f) be a decision table,where C is condition attributes and D is decision
	attributes,Then S is consistent if and only if	C M ≤	M	D	.
	Definition2.7. Let S=(U,R=C∪D,V,f,	D C M M ≤ ) be a decision table,Given an attribute a∈C,then attribute a is
	nonsignificant in C if Definition2.8.Let S=(U,R=C∪D,V,f, D a C M M ≤ { . -}	D C M M ≤ ) be a decision table.The set of all attributes	C ⊆ ′ ′	C	which are
	significant in S is called the core set of C.
	Definition2.9. Let S=(U,R=C∪D,V,f,	D C M M ≤

  table S is called global core attributes ) be a decision table and C′ be core attribute set.If a k ∈C-C′ satisfies that there exist positive integer i,j,such that [C′ ] ij =1, [C] ij =0 and [a k ] ij =0,then attribute set C′ ∪{a k } is called a local attribute reduction of decision table S. Obviously, local attribute reduction derived by [C] ij =0 can has not only one. All local attribute reduction derived by [C] ij =0 is called a local attribute reduction set. Theorem 3.4Let B 1 ,B2…,B k be all local attribute reduction sets of decision table S=(U,R=C∪D,V,f,

	Definition3.3 Let S=(U,R=C∪D,V,f, M ≤ Definition3.4Attribute reduction set C M D

C T ⊂ of decision

table is called global attribute reduction. PDF 文件使用 "pdfFactory Pro" 试用版本创建 www.fineprint.cn


Tangqing Kuang
email: kuangtq@yahoo.cn

Wenjuan Gu

Optimum Design of Runner System for Router Cover Based on Mold Flow Analysis Technology

Keywords: Mold flow analysis, Gate location, optimum design

During the design of a two-cavity injection mould for router covers, two plans, single-gate plan and dual-gates plan, were setup based on the results of best gate location analysis combined with other aspects. Then fill analysis for the two plans were carried out. The dual-gates plan was selected as an optimal choice for its smaller clamping force, wall shear stress and injection pressure in comparison with the single-gate plan. Although weld lines and air trap exists between the two gates, but they can be eliminated or just have a little effect on the quality of parts. It indicated that the technology of mold flow analysis can be used to optimize the design of gate location and gates number and assist the engineers to improve the quality of their designs of mould.

Introduction

The use of plastic has been spread to all walks of life, about 80% plastic production used injection molding method. Therefore, improving the design quality is the key to improve the products quality, productivity and reduce production cost. For a given part design, the runner system design is one of the key issues of its mold design. The complexity of mold structure and cost of mold are determined by the form of gating system directly [START_REF] Beaumont | Runner and Gating Design Handbook: Tools for Successful Injection Molding[END_REF]. Section size and length of the runner system affect the pressure loss of melt flow directly [START_REF] Lee | Design of the runner and gating system parameters for a multicavity injection mould using FEM and neural network[END_REF][START_REF] Zhai | Runner sizing and weld line positioning for plastics injection moulding with multiple gates[END_REF]. Gates location and its number affect the flow behavior of melt in the cavity and eventually affect the quality of product [START_REF] Lee | Gate positioning design of injection mould using bi-objective micro genetic algorithm[END_REF]. Using traditional methods which rely on the experience to design gating system can not gain a better runner system, and if we encounter a complex part or not covered before, the gate design can not success in once, we often have to go through a trial mode-modify-try mode-and the modify the phase, that is a great waste for time and cost. With the CAE technology for injection molding developing, mold flow analysis has been successfully used in injection mold design process [START_REF]Simulation Fundamentals-An Introduction for MPI 6.0[END_REF]. By means of mold flow analysis, the optimized gating system becomes a reality. By the flow analysis for different proposals of gating system and comparison of the analysis results such as filling status, weld line location, air trap location, injection pressure, clamping force and other results, the optimal proposal achieved. That can aid the engineers to optimize their design. Take a router cover for an example, the procedure of optimization runner system of injection mold by mold flow analysis techniques was illuminated in this paper.

Modeling and Meshing

The router cove is 430 mm long, 81 mm wide and 28 mm height and shown in Figure 1. And the model was output in iges surface format for the convenient of meshing in Moldflow Plastic Insight (see Ref. 1), software of mold flow analysis for injection molding.

Gate location analysis and programming

Gate location analysis was conducted for the model and the result shown in Figure 3. The blue area shows the suggested best gate location for the cover. The gate location analysis does not take into account some practical considerations for the gate location such as the type of tool, or limitations on the gate location due to part aesthetics etc. The router cover is limited to a 2-plate tool with a cold runner system. As a result of that tooling restriction, the suggested best gate location for the cover, in the middle of the part, is not possible. These results can only be used as a guide to place a gate on the parting line. For a long kind of part, two-cavity mold layout is a rather good choice. Considering parting location of the router cover, cavity layout of the mold, appearance requirements of the product, simplification and reliability of the mold structure and the best gate location result, single side gate was adopted and it locates at the middle of the part in length direction, close to the best gate location result. Considering the larger size of the part in length direction, another plan which dual side gates locate at the sides of the best gate location result is proposed. Single-gate and dual-gates runner system are shown in Figure 4 and 5 respectively.

Comparison and analysis of plans 4.1 Process parameters setting

ABS material was used to meet the requirement of strength and hardness of the part. Lanxess ABS 1146 of Lanxess Corporation is selected here. For comparison, the two programs are used same process parameters, mold temperature 80 ℃ , material temperature 260℃, flow rate 200cm3/s; speed / pressure switch is set to 100% of the filling volume. The better program is chosen by comparing the analysis results of the two programs such as filling time, the pressure distribution at the end of filling time, the distributions of weld line and air trap, the maximum injection pressure and clamping force, the maximum wall shear stress and so on.

Comparison of the analysis results of the two programs

The fill time result shows the progression of the flow front. Figure 6 Pressure at the end of fill will affect the residual stress in the molding parts, the greater the pressure, the greater the residual stress. If the pressure is uneven, the larger the difference, the more uneven the residual stress, the larger the warpage consequently. So the pressure at the end of fill needs to be uniform as far as possible. Pressure distribution at the end of filling time of single-gate plan and dual-gates plan are shown in figure 7 (a) and (b) respectively. The pressure difference is about 85MPa in single-gate plan while it's 63MPa in dual-gates plan. Pressure distribution in dualgates plan is more uniform than that in single-gate plan. Weld lines occur where two or more flow fronts converge. The presence of weld lines may indicate a structural weakness and/or a surface blemish. Weld line strength is influenced by the temperature at which the weld line is formed and the pressure exerted on the weld while the material is within its recommended processing temperature range. Weld line results with the flow front temperature highlightened of single-gate plan and dual-gates plan are shown in figure 8 (a) and (b) respectively. In the two plans, weld lines exist near the rectangular hole and small radial holes. But all the flow front temperature at weld lines are about 260℃ which is melt injected temperature. In comparison with single gate plan, an extra long weld line exists between the two gates in dual-gates plan, but the flow front temperature and the pressure exerted on them are high, so the weld line strength is ensured and it has less negative impact on part quality. The selected injection molding machine must meet the requirements of the maximum injection pressure and clamping force. The higher the maximum injection pressure and clamping force are, the more the energy consumption is and the higher the injection molding machine specifications is. Pressure trace at the injection location of single-gate plan and dual-gates plan are shown in figure 10 (a) and (b) respectively. The maximum injection pressure in single-gate plan is 114MPa while it's 106 MPa in dual-gates plan. Figure 11 Shear flow is the main flow behavior during the filling stage of injection molding and the maximum shear stress exist near the wall. The wall shear stress needs to be below the maximum allowable shear stress of the used material as far as possible to avoid material degradation for the excessive shear stress. The maximum allowable shear stress of selected material is 0.25MPa.

The analysis log result shows that the maximum shear stress and average shear stress in single-gate plan are 0.922 MPa and 0.185 MPa respectively. The maximum shear stress and average shear stress in dual-gates plan are 0.816MPa and 0.173MPa respectively and they all less than the values of single-gate plan. Shear stress at wall of single-gate plan and dual-gates plan are shown in figure 12 (a) and (b). As we can see, most of the shear stress at wall of dual-gates plan are below the allowable shear stress 0.25 MPa while shear stress at wall at a large area of single-gate plan exceeds the allowable value.

Choice

In comparison with the single-gate plan, the dual-gates plan has smaller clamping force, wall shear stress and injection pressure. Although weld lines and air trap exists between the two gates, but they can be eliminated or just have a little effect on the quality of parts. So the dual-gates plan is selected as an optimal choice.

Conclusion

In this paper, comparison and selection of different gate settings is conducted by mold flow analysis. Finally we get the best gate location and gate panel number for the router which is the author's major work, The length and cross section size can be determined by the following choice of mold plate and the mold specifically designs. The runner section size can be optimized by runner balance analysis. And the gate section size can be optimized to ensure the full compensation by packing analysis and cooling analysis. The resulting optimum runner system can be used to improve product quality.

From this example, we know that the successful application of mold flow analysis for injection mold design provides the scientific basis and reference for mold designers and aids designers to optimize the design programs, reduce testing time, and improve qualities of mold and products.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Router cover model

Fig. 3 .

 3 Fig. 3. Best gate location result

Fig. 4 .

 4 Fig.4. Single-gate plan

  (a) and (b) show the fill time results of single-gate plan and dual-gates plan respectively. They indicate that the filling time of the two plans are all about 1.45s and they all filling completely. The flow fronts meet at 0.59s in dual gates plan.

Fig. 6 .

 6 Fig.6. Fill time of single-gate plan (a) and dual-gates plan (b)

Fig. 7 .

 7 Fig.7. Pressure distribution at the end of filling time of single-gate plan (a) and dual-gates plan (b)

Fig. 8 .

 8 Fig.8. Weld line distribution of single-gate plan (a) and dual-gates plan (b) Air trap occurs at the ends of flow paths or where the melt stops at a convergence of at least two flow fronts. The presence of weld lines may indicate a structural

  Figure 9 (a) and (b) show the air trap distribution of single-gate plan and dual-gates plan respectively. In comparison with single gate plan, extra air traps exist between the two gates in dual-gates plan. The air trap existing where the melt stops at a convergence of several flow fronts can be eliminated by setting knockout pins or adopting the mosaic of core there. And the air trap existing at the ends of flow paths can be eliminated by the venting at the parting plane.

Fig. 9 .

 9 Fig.9. Air trap distribution of single-gate plan (a) and dual-gates plan (b)

  (a) and (b) show the clamp force development of single-gate plan and dual-gates plan respectively. The maximum clamp fore in single-gate plan is 344 ton while it's 259 ton in dual-gates plan.

Fig. 10 .Fig. 11 .

 1011 Fig.10. Pressure trace at the injection location of single-gate plan (a) and dual-gates plan (b)

Fig. 12 .

 12 Fig.12. Shear stress at wall of single-gate plan (a) and dual-gates plan (b)

Acknowledgments

This work forms part of a project supported by the Education Department of Jiangxi Province with grant GJJ10126.


Leping Liu

Mingdong Zhong

Qizheng Dong
email: dongqizheng205@yahoo.com.cn

Design of Automatic Cutting and Welding Machine for Brake Beam-Axle

Keywords: Brake beam-axle, Maintenance, Automatic cutting, Automatic welding

Brake beam is one of the important components working in railway vehicles braking process, which directly affects the security and stability of the high-speed running railway vehicles. Through the research of the cutting and welding technology for the 209T straight plate type brake beam, this paper presents an advanced maintenance method, and designs an automatic cutting and welding machine for the brake beam-axle, then studies the basic structure and working principle for the machine in detail. The automatic cutting and welding machine has been operating properly since been used in the maintenance workshop. What is more, the maintenance of each brake beam only spends about 15 minutes, so this advanced maintenance method can improve the efficiency of the maintenance appropriately, and ensure the reliability of maintenance quality.

Introduction

Brake beam is an important component of the foundation brake rigging in railway vehicle bogie. And it is the essential device to ensure the security of the high-speed running train [START_REF] Wang | Strength character research on brake beam for type of 209 bogie[END_REF]. The 209T type brake beam withstands braking force repeatedly, and its end axles are abraded seriously, which connect with the railway vehicle's brake shoes [START_REF] Song | Technological measures to guarantee the coaxiality for brake beam[END_REF]. Due to the 209T type brake beam's being widely used in the raising speed passenger train in china, its maintenance quality is crucial to the security of passenger train. In the traditional process of brake beam maintenance, the brake beam-axle is replaced by manual cutting and welding [START_REF] Wang | Investigation and analysis on problems existing in inspection and repair of brake beams[END_REF]. In this case, the welding fume, dust and radiation endanger health of the workers severely [START_REF] Li | Design and practice of new type pneumatic control system for trunk brake beam electric welding workshop[END_REF]. What is more, the cutting gap is not smooth, and it is difficult for workers to control the welding seam track.

Based on the theoretical calculations, J.N. Chen [START_REF] Chen | Finite Element Analysis for Brake Beams[END_REF] has analyzed the brake beam by using finite element software, and the simulation result has accorded with the theory calculations. Other authors, such as Y.P. Tang [START_REF] Tang | Automatic welder based on DSP motion control card for van brake beam[END_REF] and his colleagues, have focused on designing the numerical control system for automatic welder, which is used for brake beam of railway vehicle. Over the past years, welding robots are employed as substitutes for workers even in some developed vehicle depots, but it means high cost for most vehicle depots in china [START_REF] Xin | Application of Welding robot in Manufacture of Bogie frame[END_REF]. So we design an automatic cutting and welding machine for the brake beam-axle, which aims at solving the problem that the traditional maintenance method can not ensure the reliability of maintenance quality for break beam. Previous relevant researches [START_REF] Fu | Application Research on Automatic Cutting and Welding process for Brake Beam-Axle[END_REF][START_REF] Zhou | Design and analysis of multi-function auto-welding machine tool control system[END_REF] have provided a certain reference for the design of automatic cutting and welding machine.

The advanced machine improves the efficiency of the maintenance appropriately, and gives full play to the advantages of the computer numerical control system with high precision, good stability, and simple operation.

Maintenance process analysis for end axle 2.1 Maintenance process requirements of brake beam

The 209T type brake beam is composed of two end axles, two groups of pulling boards and a brake beam body. And the end axle consists of cylinder and cone. The cylinder's diameter is φ55 -0.19 -0.49 mm. The large end of the cone's diameter is φ70 +1 0 mm. In addition, the cone is welded with the brake beam body, which thickness is 20 +1 0 mm. The overall structure of 209T type brake beam is shown in Fig. 1. Due to sleeve gasket working surface of end axles being protected effectively by the sleeve gaskets, they are not abraded. However, the brake shoes working surface are abraded seriously. Then the maintenance process of 209T type brake beam must meet the following requirements:

• Dimensional Precision (as shown in Fig. 1).

• Position Precision

The symmetry error of two end axles should be less than 0.5 mm in the direction of thickness of the brake beam body, and their perpendicularity error that takes for reference with plane A should also be less than 0.5 mm. In addition, the permissible error of coaxiality for two end axles should be less than 1mm. Finally, the two groups of pulling boards take center line L1 for their symmetrical line.

• Tensile Strength

The repaired brake beam must take repeated tensile stress test, and permanent deformation can not emerge.

• Abrasive Hardness

The carburized depth of end axle must reach 0.8~1.5 mm, and Rockwell hardness of the cylindrical surface should reach HRC40~55.

Key technology for maintenance process

First, the maintenance process is not only the welding process, it must be in accordance with following technological process: "locating and clamping the worn brake beam → cutting the worn end axles → locating and clamping the new end axles → welding. " As is shown in Fig. 2, the brake beam body should be welded with the new end axles in double-side; furthermore, their welding trace must follow the cutting trace appropriately. Therefore, it is significant to apply the numerical controlling method to control the welding trace.

welding seam cutting thickness slag inclusion Fig. 2. The maintenance process of 209T type brake beam

In addition, there are many serious technological problems in traditional process, such as lack of fusion, incomplete penetration and slag inclusion. In this case, there will exist potential danger in the high-speed running passenger trains. Accordingly, reasonable cutting and welding process parameters must be adopted to reduce this sort of defects.

Finally, serious welding heat deformation occurs frequently in the traditional maintenance process. And its coaxial error can not meet the process requirement.

Thus, a locating and clamping device must be adopted to prevent the welding heat deformation efficiently.

Scheme design of automatic cutting and welding machine

Technical scheme of automatic locating and clamping

Locating the brake beam body

As is shown in Fig. 1, the brake shoes' working surface is abraded seriously.

However, sleeve gasket working surface of end axles are unworn. In this case, sleeve gasket working surface and end-face of the cone can be used as automatic cutting and welding location datum. Therefore, the locating device of automatic cutting and welding machine can find the center line L1, L2 and center plane B rapidly, and locate the brake beam body accurately [START_REF] Di | Mechanical Manufacture Engineering[END_REF].

Clamping the brake beam body

To cut the worn end axles and weld the new end axles accurately, the brake beam must be clamped stably. As is shown in Fig. 2, the worn end axles will be cut after locating the brake beam body accurately, so they can not be used as clamping parts. In contrast, the plane B exists as the center plane of the brake beam body in the direction of thickness during the whole maintenance process. Thus, the plane B can exist as datum plane for locating and clamping the brake beam body. And the automatic centering mechanism can achieve this function appropriately.

Locating, clamping and rotatiing device

The end axle can be located on the V-shape block by outside column surface accurately. What is more, the device can achieve several functions for the new end axle, such as locating, clamping and rotating. Accordingly, the welding process of new end axle is carried out with the following steps. First, the new end axles must be

Technical scheme of automatic cutting

There are many kinds of thermal cutting methods for practical application, such as plasma cutting, oxyacetylene cutting and laser cutting. Unfortunately, due to the high cutting cost, laser cutting is not suitable to be adopted in domestic vehicle depots. In this case, the appropriate cutting method must be chosen between plasma cutting and oxyacetylene cutting.

Plasma cutting is applied extremely widely in the steel plate cutting industry, such as cutting carbon steel, stainless steel, aluminum, copper and other metal materials [START_REF] Hang | Position of plasma cutting in thermal cutting methods[END_REF]. The local melt of steel is carried out by the high temperature plasma arc, and molten metal is blown away by the high speed plasma jet.

Oxyacetylene flame is suitable for cutting low-carbon steel materials. First, the oxyacetylene flame preheats the metal to its igniting temperature, and then the metal is burned violently in the high temperature oxygen stream. The molten slag is blown away by the high pressure oxygen stream. Afterwards, the metal is preheated under the combined action of the heat of oxyacetylene flame and combustion heat.

Obviously, this process continues until the steel plate is cut down.

As the material of brake beam and welding seam is low-carbon steel, in this case, both oxyacetylene flame and Plasma can be adopted to cutting the worn end axles.

However, the thickness of brake beam body is 20 mm, and the single-side thickness of welding seam is about 5~10 mm. Accordingly, the actual cutting thickness is almost about 30~40 mm. As is shown in Fig. 4, when the cutting thickness is more than 30mm, the cost of plasma cutting will be increasing rapidly, especially compared with oxyacetylene cutting. This is because that the expensive dedusting system must be adopted during the plasma cutting. From the economic aspect, the oxyacetylene cutting takes priority over plasma cutting. As is shown in Fig. 5, when the cutting thickness is less than 30 mm, the speed of plasma cutting is significantly faster than that of oxyacetylene cutting, otherwise, it decreases dramatically. In contrast, the cutting speed curve of oxyacetylene is nearly level. In view of the cutting efficiency and cutting cost, low-carbon steel thickness below 30~40mm is suitable for oxyacetylene cutting. Consequently, oxyacetylene flame cutting is adapted appropriately to cutting the worn end axles.

Technical scheme of automatic welding

The brake beam is located at the bottom of railway vehicle, although it is cleaned before maintenance, its surface is still covered with rust and dirt. However, the carbon dioxide arc welding is insensitive to rust. What is more, it is very suitable for automatic welding with the excellent welding process, including high welding speed and excellent weld appearance quality.

In addition, the trapezoid slot of brake beam body has a low forming precision, because of the oxyacetylene flame cutting. And the preset welding gap between the brake beam body and the new end axle is almost about 3~4 mm. In this case, the welding trace must follow a short and tortuous path (as shown in Fig. 6) according to our experiment results.

Main cutting and welding parameters

The cutting parameters have a great influence on cutting quality, such as heat affected zone, cutting width, cutting dross and roughness. And the optimal cutting parameters can be determined by orthogonal experimental design, and suitable for the balanced pressure cutting nozzles, as shown in Table 1. Similarly, the main welding parameters such as welding current, arc voltage and welding speed have been optimized by the orthogonal test. The inverter CO 2 welding machine's type is NB-350K. And its optimal parameters are shown in Table 2.

System design of automatic cutting and welding machine

System design

The automatic cutting and welding machine is shown in Fig. 7. It is composed of machine tool bed, drive boxes, cross-type moving workbench, automatic cutting and welding device, clamping devices, brackets, hydraulic system, numerical control system and welding controller. And the cutting gun and welding torch are installed on the automatic cutting and welding device, which is located at moving workbench. • Welding speed: 400 mm/min.

• The total power: 24 KW.

Experimental verification

There is no permanent deformation after the repaired brake beam takes repeated tensile stress test. As is shown in Fig. 9, the pulling force acting at each group of pulling boards is 35 kN [START_REF]Railway Ministry of the PRC, Load Test Methods for TB/T 1978-2007 Brake Beams[END_REF]. In addition, tensile stress tests show that the tensile strength of repaired brake beam has completely reached the process requirement. indicate that the advanced maintenance method is feasible to control the error of coaxiality, and the improved process takes obvious priority over original process.

Conclusions

An automatic cutting and welding machine for the brake beam-axle, to be employed in a vehicle depot, was designed. This maintenance process has the advantages of convenient clamping and accurate locating. This designed machine was set up and tested, and experimental results show that the designed machine is appropriate for practical application. After refitting the clamping device and changing the control program, this automatic cutting and welding machine could be applied to the field of maintenance for the similar type brake beam. The application of such low-cost automatic cutting and welding machine, which is urgently needed in many vehicle depots in China, not only improves the efficiency and maintenance quality, but also protects the operators from hazardous circumstances.

  ; 2 pulling board; 3 brake beam body; 4 sleeve gasket working surface; 5 brake shoe working surface.

Fig. 1 .

 1 Fig.1. The structure diagram of 209T type brake beam

Fig. 3 .

 3 Fig.3.

Fig. 4 .

 4 Fig.4. Cutting cost curve of plasma and oxyacetylene

Fig. 5 .

 5 Fig.5. Cutting speed curve of plasma and oxyacetylene

Fig. 6 .

 6 Fig.6. The welding path of new end axles

Fig. 7 . 5 . 2

 752 Fig.7. The system design of automatic cutting and welding machine

Fig. 8 . 5 . 3

 853 Fig.8. Automatic welding device in operation

Fig. 9 .

 9 Fig.9. Repeated tensile stress test The curves of coaxiality under different process conditions are shown in Fig. 10, and coaxiality of improved process is less than the permissible error. These results

Fig. 10 .

 10 Fig.10. Curve diagram of coaxiality under different process conditions

Table 1

 1 The optimal cutting parameters

		Thickness	Speed	Gas pressure	Gas consumption
	Nozzles	( mm )	( mm/min )	( Mpa ) Oxygen Acetylene Oxygen Acetylene ( m3/h )
	1	30~40	320	0.5	0.07	2.5	0.34
	2	30~40	280	0.65	0.07	3	0.47

Table 2

 2 The optimal welding parameters

	Weld modes	Current ( A )	Voltage ( V )	Welding spot size ( mm )	Speed ( mm/min )
	Spot weld	190	21	15~20	-
	First weld layer	210	23	continuous	400
	Second weld layer	250	27	continuous	400