RU2374697C2 - Experimental method for parametrisation of minimum surfaces with complex contour - Google Patents

Abstract

FIELD: technological processes.

SUBSTANCE: method includes operations of flexible element fixation relative to foundation, pulling of net previously made of elastic material on it, measurement of net nodal points coordinates relative to foundation. Peculiarity of the method consists in the fact that spatial frame is made of curvilinear shaping elements, which create quadrangular contour, net of elastic material is pulled onto frame, moreover, each nodal contour point of net is fixed at appropriate points of frame, which are distributed along length of shaping element according to a certain specified law. Frame with net is fixed relative to support surface of foundation. Measured coordinates of net nodal points relative to selected system of coordinates related to support plane, surface metrics components (of the first and second metric tensors) are defined.

EFFECT: generation and mathematical description of minimum surface with complexly shaped contour, which is arbitrarily oriented in space.

3 cl, 8 dwg

Description

The invention relates to industries related to the formation of surfaces of complex geometry and the determination of their metrics, in particular building structures, vehicle bodies, space technology, instrumentation, road surfaces, etc.

Known methods for measuring horizontal irregularities (straightening) and curvature in terms of rail threads, which consists in measuring the current course of the body of the measuring car and the current wobble angle of the measuring carriage relative to the body. Next, measure the distance from the flanges of the corresponding wheels of the trolley to the heads of the rails. Based on the measurement information, the current rates of the rail threads are calculated as the current course of the body minus the current wobble angle of the measuring trolley relative to the body minus the current angle of non-parallelism of the flanges of the wheelsets of the measuring trolley and the heads of the rail threads. Then, the current values of the horizontal irregularities of the rail threads are obtained as the product of the base of the cart for deviations of the current courses of the rail threads from the current average courses of the rail threads calculated on the basis of the cart, calculated from the courses of the rail threads (RF patent No. 2276216, Mcl. E01B 35/00, publ. 05/10/2006).

However, the known methods do not allow the formation of lines of complex geometry and determine the roughness of two-dimensional objects.

Known methods for determining the curvature and slopes of the surface profile of the road surface in two different directions, including measuring the vehicle’s motion parameters, determining and adjusting the reference basis relative to the plane, analyzing the data using a computer, finding the slopes of the road surface in two different directions and displaying measurement -computing complex of design parameters in the process of car movement (RF patent No. 21114391, M.cl. G01C 7/04, publ. 06.27.1998).

However, the known methods do not allow the formation of surfaces of complex geometry, including minimal surfaces of a given shape.

There is also an experimental method for parameterizing minimal surfaces, based on the solution of the two-dimensional Laplace equation (for example, the formulation of the internal Dirichlet problem, when the boundary values of coordinates are known and it is necessary to find the internal ones), in which vertical rods are set to the required height relative to the reference plane according to the requirements of the problem, using measuring device verify the accuracy of the installation of the rods, using the clamp fix the rods at the required height, pull on ikalnye rods preformed web of flexible elastic material by a precision measuring device measure the altitude coordinates at internal points (intersection points) of the elastic flexible network (RF patent №2121166, M.kl. G09B 23/04, publ. 27.10.1998).

The specified method has the following disadvantages:

a) the method does not allow to obtain continuous and smooth contours of arbitrary configuration, freely oriented in space (all contours are discontinuous, discrete in nature);

b) the method is intended only for measuring the geometric parameters of the object and does not allow the formation of minimal surfaces with a complex contour;

c) the method does not provide for the calculation of the surface metric (determination of the components of the first and second metric tensors);

d) the coordinates in the horizontal plane have fixed values (the coordinates of the installation of vertical rods are not changeable).

The objective (goal) of the invention is the formation of a surface with continuous and smooth curved coordinate lines, a description of the surface of complex geometry, arbitrarily oriented in space, and the calculation of the components of the first and second metric tensors (surface metrics).

These tasks are achieved by the fact that in the experimental method of parameterizing minimal surfaces with a complex contour, including the operation of fixing a flexible element relative to the base, pulling on it a prefabricated network of elastic material, measuring the coordinates of the network's nodal points relative to the base, a spatial frame is made of curved forming elements ( for example, pieces of bending wire), forming a given quadrangular contour, pull on the frame initially straight a network, angular in plan, made of elastic material, and each nodal contour point of the network is fixed at the corresponding points of the frame distributed along the length of the forming element either uniformly or according to a certain predetermined pattern. The frame with the network is fixed relative to the support plane of the base. In this case, the surface is formed and parameterized by the nodal points of the network. To do this, measure the coordinates of the nodal points of the network relative to the selected coordinate system associated with the reference plane. Perform the processing of the results with the determination of the components of the surface metric.

Figure 1 schematically shows a General view of the installation for experimental parameterization of the surface; figure 2 shows the auxiliary coordinate lines plotted on the surface of the base; figure 3 presents a fragment of the formed surface when dividing the contour into a given number of points; figure 4 shows a diagram of a network of elastic material in the form of a single square, which is designed to align with the contour of the fabricated frame; figure 5 is a three-dimensional image of the original surface formed by the network of a single square when superimposed on a frame with a complex contour; figure 6 is a three-dimensional image obtained after building the surface with a complex contour; 7 is a three-dimensional image obtained after building the surface of the first derivatives with respect to the first parameter; on Fig - three-dimensional image obtained after building the surface of the first derivatives of the second parameter.

In the figures indicated:

x, y, z - coordinates in the Cartesian system;

G ₁ , G ₂ , G ₃ , G ₄ - contour lines of the formed surface (contours of the frame);

Ω is the area occupied by the formed surface;

t ¹ , t ² - coordinates (parameters) of the unit square (upper indices 1 and 2 - indices showing the direction of coordinates);

Ω _f - the area occupied by the unit square in the coordinates t ¹ , t ² (parameters t ¹ and t ² vary in a particular case from 0 to 7);

M is an arbitrary point on the formed surface;

M _f - a point on the region Ω _f corresponding to an arbitrary point M of the formed surface;

- радиус-вектор произвольной точки М на формируемой поверхности.

is the radius vector of an arbitrary point M on the formed surface.

The method is as follows.

A spatial framework of the contour of the formed surface is made. In this case, the frame is made up of four spatial curvilinear forming elements 1, 2, 3 and 4 (for example, pieces of bending wire) with the marks applied to them according to the given conditions. The ends of the forming elements 1, 2, 3 and 4 are rigidly joined to form a closed quadrangle of a given shape, which is the contour of the formed surface.

An elastic network 5 is pulled onto the frame obtained. The network 5 consists of longitudinal 6 and transverse 7 elastic, for example, rubber threads, which initially in the initial state (in the undeformed state of the elastic network) form rectangular cells. The longitudinal 6 and transverse 7 rubber threads at the intersection are bonded to each other with the formation of nodal points 8. In theoretical processing, the elastic network 5 is conventionally taken as a unit square with parameters t ¹ and t ² varying from 0 to 7.

Attach the outer threads of the network 5 to the corresponding parts 1, 2, 3 and 4 of the frame according to the marked marks (points) distributed along the length of the forming element either uniformly, or according to a specific predetermined pattern. In this case, the nodal points 8 of the network 5 during the deformation of the network 5 determine the formed surface, that is, the nodal points 8 form the surface.

Next, proceed to the surface parametrization procedure. To do this, the frame is fixed on the installation containing the support plane of the base 9 with the coordinate lines deposited on it and supports 10 for fixing the frame.

Then, using the appropriate tools (instruments), the coordinates of the nodal points 8 of the deformed network 5 are measured along the x, y, and z axes with the corresponding parameters t ¹ , t ^{2 of a} unit square with the region Ω _f , that is, coordinates x (t ¹ , t ² ) are obtained , y (t ¹ , t ² ), z (t ¹ , t ² ) and determine the radius vector at the grid nodes by the formula:

,

,

- единичные орты в декартовой системе координат.Where

,

,

- unit vectors in the Cartesian coordinate system.

Data can be presented in tabular form.

The algorithm for constructing the surface and calculating its parameters is carried out in the following sequence:

1. One-dimensional Hermitian splines are constructed along the coordinate t ¹ with a fixed value of the coordinate t ² for all coordinate functions x (t ¹ , t ² ), y (t ¹ , t ² ), z (t ¹ , t ² ), sequentially running over all values corresponding to the longitudinal threads (lines) 6 of the network 5.

2. One-dimensional Hermitian splines are constructed along the coordinate t ² for a fixed value of the coordinate t ¹ for the functions x (t ¹ , t ² ), y (t ¹ , t ² ), z (t ¹ , t ² ), sequentially running over all values corresponding to the transverse threads (lines) 7 of the network 5.

Thereby, the coordinates of the formed surface σ are obtained for any value of the parameters t ¹ and t ² .

3. Using the values of the first derivatives, the second mixed derivatives are determined for the coordinate functions x (t ¹ , t ² ), y (t ¹ , t ² ), z (t ¹ , t ² ).

4. Build Hermitian cubic splines for the coordinate functions x (t ¹ , t ² ), y (t ¹ , t ² ), z (t ¹ , t ² ).

и

:5. Differentiating expression (1) with respect to t ¹ and t ² , coordinate vectors are determined

and

:

и контравариантные a¹¹, a¹², a²² компоненты первого основного метрического тензора и фундаментальный определитель a:6. Determine covariant

and contravariant a ¹¹ , a ¹² , a ²² components of the first basic metric tensor and fundamental determinant a:

,

,

,

,

,

:7. Determine the Christoffel symbols of the second kind

,

,

,

,

,

:

,

,

,

,

,

,

,

,

.

.

:8. Determine the unit normal vector

:

и

образует основной базис в каждой конкретной точке области Ω, принадлежащей формируемой поверхности σ.which together with coordinate vectors

and

forms the main basis at each specific point of the domain Ω belonging to the formed surface σ.

,

,

,

второго основного метрического тензора:9. Determine covariant components b ₁₁ , b ₁₂ , b ₂₂ and mixed components

,

,

,

second main metric tensor:

, где

;

where

;

, где

;

where

;

, где

;

where

;

Thus, for the formed surface, the coordinate values and components of the first and second derivatives are obtained for any values of the unit square parameters t ¹ and t ² .

Further, the obtained values can be used to select the necessary surfaces when the architect searches for the desired shape, to calculate the stress-strain state of shells with a middle surface σ, etc.

An example of constructing a surface of complex geometry.

An experimental model was constructed according to the method described, which is a square in plan (with dimensions of 40 cm by 40 cm) frame with curved contours. A net of elastic material was pulled onto the frame with eight nodal points in both directions of a unit square. Thus, the network was divided into 7 cells along the longitudinal and transverse coordinates. The frame with the network was fixed relative to the support plane of the base on four supports made in the form of vertical rods.

On the experimental model, the coordinates of the nodal points were measured. The values of the measured coordinates z _ij for the studied formed surface of complex shape are given in the table.

The values of the measured vertical coordinates z of the nodal points for the network in question according to the parameters t ¹ and t ² , cm Numbers of nodal points in t ² Numbers of nodal points in t ¹ one 2 3 four 5 6 7 8 one 16.8 16.8 14.8 10.3 4.0 0.004 0,0 0,0 2 14.6 13.5 12.0 9.5 6.5 4.7 4.3 4.1 3 12,2 11.3 10.1 8.8 7.5 7.2 7.1 6.5 four 9.6 9.6 9.0 8.2 8.2 8.2 8.7 8.4 5 7.7 7.8 8.0 8.2 8.4 8.8 9.6 10,2 6 5,6 5.8 6.5 8.1 9.6 11,2 12.0 12.6 7 3.0 3,5 4.7 7.6 9.9 12.6 13.5 14.7 8 0,0 0,0 0,0 4.8 11.6 15.4 16.8 16.8

After measuring the coordinates x (t ¹ , t ² ), y (t ¹ , t ² ), z (t ¹ , t ² ), a surface was constructed at the nodal points using the apparatus of spline functions according to the described method and the geometric parameters of the formed surface were calculated. In particular, graphical images of a certain surface were constructed from the calculated values. For example, figure 5 presents the image of the surface of the original geometry, built on the basis of the measured coordinates x, y and z. Figure 6 shows the geometry of a fragment of the constructed surface obtained using the above algorithm. Figures 7 and 8 show the surfaces of the first derivatives in the directions t ¹ and t ^2, respectively.

The proposed method allows to form surfaces of complex geometry with continuous and smooth curved coordinate lines, to describe surfaces of complex geometry, arbitrarily oriented in space, and to calculate the components of the first and second metric tensors (surface metric). The method allows to obtain continuous and smooth contours of arbitrary configuration, freely oriented in space, to form minimal surfaces with a complex contour. The method is an effective means of solving a wide variety of applied problems and can be widely used in design organizations and educational institutions in the design and study of various surfaces of complex geometry, as well as in the calculation of the stress-strain state of thin-walled structures of complex geometry, when in the first stage it is necessary to solve the parameterization problem (tasks) of the middle surface of complex geometry and calculate the corresponding components of the first and second metric tensors, symbol ly Christoffel, etc.

Claims (3)

1. An experimental method for parameterizing minimal surfaces with a complex contour, including the operations of fixing a flexible element relative to the base, pulling on it a prefabricated network of elastic material, measuring the coordinates of the nodal points of the network relative to the base, characterized in that a spatial frame with a quadrangular contour is made of curvilinear forming elements, pull on the frame initially rectangular mesh plan of elastic material, with each nodal to the grid point of the network is fixed at the corresponding points of the frame, distributed along the length of the forming element according to a certain regularity, while the surface is formed and parameterized by the nodal points of the network taking into account the coordinates of the nodal points of the network relative to the selected coordinate system, the results are processed to determine the components of the surface metric. 2. An experimental method for parameterizing minimal surfaces with a complex contour according to claim 1, characterized in that the spatial curvilinear forming elements are made from pieces of bendable wire. 3. An experimental method for parameterizing minimal surfaces with a complex contour according to claim 1 or 2, characterized in that the attachment points of the nodal contour points of the network are distributed uniformly along the length of the forming element.

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