多轴NC机床上的雕塑曲面CADCAM技术

CADCAM

Multi-Axis

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"CADCAM of Sculptured Surfaces on Multi-Axis NC Machine 主要探讨了在多轴数控（NC）机床上对雕塑曲面进行CAD/CAM处理的方法，特别是基于DG/K（Discrete Geometric and Kinematic）方法的途径。该书由Stephen P. Radzevich撰写，详细阐述了如何生成用于多轴数控机床的G代码，以精确控制复杂曲面的加工过程。" 在多轴数控机床中，CADCAM（计算机辅助设计与计算机辅助制造）系统对于制造具有雕塑曲面的产品至关重要。这些曲面通常用于提升产品的美学价值和功能性。G代码，或称为RS-274，是NC机床理解的编程语言，用于指示机器执行各种操作，如移动、切割和钻孔。生成有效的G代码是确保复杂几何形状准确无误加工的关键。本书《CAD/CAM of Sculptured Surfaces on Multi-Axis NC Machine: The DG/K-Based Approach》深入讨论了DG/K方法，这是一种将离散几何与运动学相结合的技术。DG/K方法允许精确建模和分析非均匀有理B样条（NURBS）曲线和曲面，这是现代CAD系统中广泛使用的参数化几何表示方法。通过这种方法，设计师可以创建和修改复杂的三维形状，并生成能够驱动多轴机床的G代码。在多轴加工中，机床可以在多个轴上同时运动，提供更高级别的自由度和精度，尤其适合处理雕塑表面。这些表面可能出现在汽车、航空、医疗器械以及艺术创作等众多领域。使用DG/K方法生成G代码时，需要考虑的因素包括刀具路径规划、工件定位、切削参数以及机床的动态特性。书中可能涵盖了以下几个关键知识点： 1. NURBS曲面的建模与编辑：如何使用NURBS技术来表示和编辑复杂的曲面形状。 2. 多轴机床的工作原理：理解不同轴的协调运动以实现精确的曲面加工。 3. G代码生成算法：探讨如何将三维模型转化为机床可执行的指令序列。 4. 刀具路径优化：如何设计路径以减少加工时间、提高表面质量并避免工具损坏。 5. 实际应用案例：展示DG/K方法在实际产品开发中的应用，可能包括汽车内外饰件、航空部件或其他具有雕塑感的设计。 6. 模拟与验证：利用仿真软件验证G代码的正确性和加工效果。 7. 误差分析与补偿：探讨如何处理机床误差和材料变形，以提高最终零件的精度。通过学习本书，读者将能掌握针对多轴NC机床进行雕塑曲面加工的高级技能，包括理解G代码生成的全过程，以及如何运用DG/K方法来优化设计和制造流程。这对于希望提升产品设计和制造能力的工程师和技术人员来说是一份宝贵的资源。

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6 CAD/CAM OF SCULPTURED SURFACES ON MULTI-AXIS NC MACHINE

A sculptured surface can be uniquely speciﬁed in terms of two independent variables. There-

fore, we give a surface (P) (Figure 2.1), in most cases, by expressing its rectangular coordinates X

, and Z

, as functions of two Gaussian coordinates U

and V

in a certain closed interval [48]:

= r

, V

) =

⎡

⎢

⎣

, V

)

, V

)

, V

)

⎤

⎥

⎦

, (U

1.P

≤ U

2.P

; V

1.P

≤ V

2.P

)

, (1)

where:

is the position vector of a point of the sculptured surface P

, V

are curvilinear (Gaussian) coordinates of the point of the surface P

, Y

, Z

are Cartesian coordinates of the point of the surface P

1.P

, U

2.P

are boundary values of the closed interval of the U

parameter

1.P

, V

2.P

are boundary values of the closed interval of the V

parameter.

The parameters U

and V

must enter independently, which means that the matrix

M =

⎡

⎢

⎣

⎤

⎥

⎦

(2)

FIGURE 2.1: Principal elements of the local topology of a sculptured surface.

8 CAD/CAM OF SCULPTURED SURFACES ON MULTI-AXIS NC MACHINE

and v

are valid. Vector u

speciﬁes the direction of the tangent to the U

coordinate curve

through the given point M on the surface P. Similarly, vector v

speciﬁes the direction of the tangent

to the V

coordinate curve through that same point M on P.

Fundamental magnitudes E

, F

, and G

of the ﬁrst order are scalar functions of U

and V

parameters of the surface P. In general form, these relationships can be expressed as E

= E

, V

= F

, V

), and G

= G

, V

Fundamental magnitudes E

and G

are always positive (E

> 0, G

> 0), whereas the funda-

mental magnitude F

can be equal to zero (F

³ 0). Because the ﬁrst fundamental form represents

the length of a curved-line segment, it is thus always nonnegative, i.e., the inequality

³ 0 is

always observed.

Discriminant H

of the ﬁrst fundamental form

can be computed from the equation

= ÖE

- F

. It is assumed that the discriminant H

is always nonnegative, i.e.,

= + ÖE

- F

The second fundamental form of the surface P is one of the two aforementioned important

issues.

The second fundamental form

describes the curvature of a sculptured surface P. Usually,

it is represented in the quadratic form

2.P

Þ - dr

× dn

= L

+ 2M

+ N

(6)

Equation (

6) is known from many advanced sources.

In the theory of surface generation, the matrix form of the analytical representation of the

second fundamental form

2.P

is proven be useful:

2.P

0 0

0 0 1 0

0 0 0 1

│

┐

┘

│

┐

┘

│ │

│

┐

┘

│

┐

┘

│ │

(7)

This type of analytical representation of the second fundamental form

is proposed by

Radzevich [22]. Similar to Eq. (4), the practical advantage of Eq. (7) is that it can be easily incor-

porated into computer programs using multiple coordinate system transformations that are vital for

CAD/CAM applications.

Here, in Eq. (

7), the parameters L

, M

, N

designate fundamental magnitudes of the second

order.

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多轴NC机床上的雕塑曲面CADCAM技术

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