The art design of industrialised manufacturing furniture products based on the simulation of mathematical curves

Furniture can be categorised into beds, cabinets, tables, chairs, etc., according to varieties. Among them, chair and bed are the more representative types of furniture [2]. Chair is composed of chair brain, front foot, back foot, armrest, backrest, seat frame and other parts. The more typical curved parts are the chair brain, rear legs, armrests and front legs, as shown in Figure 1. The chair brain is the top crossbar on the back of the chair, and its curve forms are mainly curved, corrugated and arched. The back foot has generally three stages, and there are two main types of curves: one is a trapezoid, the other is based on the backrest and the upper and lower sides are arcs with a certain angle to the vertical. The armrests are two crossbars parallel to the seat surface on both sides of the chair. Most of the curves are wavy, and some are similar to ‘S’-shaped curves. The front legs are two vertical planks of wood connected to the chair frame. There are three main types of curves: (1) The upper part of the front foot is rectangular, and the lower part is an ‘S’ curve; (2) the upper part of the front foot is rectangular, and the lower part is sloping outward. Saber shape; (3) Improved saber leg.

Fig. 1

Bed furniture is mainly composed of four parts: headboard, footboard, bed support and mattress. The curved parts mainly include the head (tail) top line and the head (bottom) foot. The curve of the full line of the head (bottom) of the bed is symmetrical, mainly including arc, arch, wave etc. The curvilinear shape of the head (tail) foot of the bed is an extension of the sleigh bed’s head (tail) foot style.

The curve shapes of furniture parts mainly include arcs, waves, ‘S’ shapes, saber shapes etc. These curve shapes are similar to some mathematical curves. By analysing the correlation that the curve shapes of furniture parts bear with mathematical curves, it is possible to simulate these parts with mathematical curves. The curvilinear modelling shown in Figure 2 is partly related.

Fig. 2

It can be seen that these mathematical curves are similar to the curves of certain parts of furniture. These curve shapes can be simulated through the setting of mathematical curve variables and intervals. For example, a cosine curve can be modified by determining its period and amplitude and using symmetric intervals [3]. We adopt the method of mathematical curve simulation and use its calculation formula to generate the curve characteristics of the required part quickly, thereby further generating the three-dimensional model of the position. Since the mathematical curves are relatively rigid, different angles can only be obtained by changing the variables and value ranges. However, designers often have many random variations when drawing these part curves; moreover, it is not easy to achieve a complete match in the simulation process. We can use the approximation method to fit to find a closer angle continuously.

We assume that the control points V₀,V₁,...,V_m must reach the corresponding weight mountain ω₀,ω₁,...,ω_m and the node vector u₀,u₁,...,u_m−k+1; then the kth non-uniform rational B-splines curve is defined as: (1) C(u)=∑i=CnωiViNi,p(u)∑t−CnωiViNi,k(u) C\left( u \right) = {{\sum\limits_{i = C}^n {{\omega _i}{V_i}{N_{i,p}}\left( u \right)} } \over {\sum\limits_{t - C}^n {{\omega _i}{V_i}{N_{i,k}}\left( u \right)} }}

The primary objective is to find a unified mathematical method with other forms through which free-form curves and surfaces can be described and accurately represent elementary analytical curves and surfaces. From Eq. (1), it can be seen that non-uniform rational B-splines has not only more denominators but also more weight factors than non-rational B-spline curves. We can use it to adjust the fullness of the curve, provide sufficient flexibility for geometric calculations and make it an additional modelling parameter.

Similar to the non-uniform rational B-splines curve, a non-uniform rational B-splines surface of degree k × 1 can be expressed as (2) ∑i=0m∑j=(t)nωidiNi,k(u)Nj,(un \sum\limits_{i = 0}^m {\sum\limits_{j = \left( t \right)}^n {{\omega _i}{d_i}{N_{i,k\left( u \right)}}{N_{j,(un}}} }

The control vertex d_i(i = 0.1,L, m, j = 0,1,L, n) presents a topological rectangular array and then forms a control network; ω_i is the weighting factor connected with the vertex. We specify the positive weighting factor ω_1,m,ω_2,m,ω_3,m,ω_i,m > 0 at the vertices of the four corners and the remaining ω_i ≥ 0,N_i,k(u)(i = 0,1,L, m) and N_i,k(j = 0,1,L, n) are the canonical B-spline bases of k times in u direction and l times in the v direction, respectively [5]. They are, respectively, determined by the node vector U = [u₀,u₁,...,u_m+k−1],V = [v₀,v₁,...,v_m+l−1] of the u and v directions, according to the De Boer recurrence formula.

The system is operated under the Windows environment using powerful, object-oriented Visual Basic visual-programming software. The system uses SolidWorks as the primary platform for its secondary development and realises the parametric design of the curve of American furniture parts. We use mathematical curves to simulate part curves to achieve a part curve parametric design system that integrates part curve selection, geometric curve design, parametric modelling and engineering drawings.

SolidWorks is a feature-based solid modelling software. The system can give different design schemes to avoid or reduce errors in the design process. It uses intuitive design technology, Windows OLE technology, Para-solid kernel and excellent technology integration with third-party software. SolidWorks is currently a commonly used 3D CAD software.

SolidWorks has an open interface, and provides convenient user-development tools and rich data-exchange methods [8]. The secondary development of SolidWorks is mainly to use its embedded Application Programming Interface (API) objects. Solid Works API provides users with free, open and fully functional development tools. Reliable Works API provides users with many functions that Visual Basic can call, such as Delphi, VBA and Visual C++ through OLE technology. These functions are related to the methods and properties of the object, which improve the programmer’s ability to develop SolidWorks directly. By calling these methods and setting object properties, users can control various operations on SolidWorks in their applications.

According to the characteristics and goals of the system, the system we designed is mainly composed of five parts: curve-type selection module, geometric curve design module, part 3D modelling module, database query module and menu plug-in. Simultaneously, we have established a SolidWorks menu plug-in, from which it is more convenient for users to query and call the required part curves. The overall plan of the system is shown in Figure 4.

Fig. 4

5.2.5

SolidWorks menu plug-in

The system developed the EXE program of the part curve and developed the SolidWorks menu plug-in (DLL program) for the user’s convenience. The DLL program hangs different part curves on the main menu of the SolidWorks software, making the two systems integrated and more flexible and convenient. The procedure flow of the whole system is shown in Figure 5.

Fig. 5

We can use the parametric design method to build the model for the furniture curve and surface modelling with relatively finalised structural shapes. Through parametric drawing design, we can quickly realise the design of serialised products or make partial changes to the design works, avoiding designers’ tedious and repetitive work [11]. The flow chart of furniture leg parameter modelling is shown in Figure 6. The paper adopts the parametric design method based on the characteristic shape line to construct the leg type mathematical model. Since the development of the system adopts a modular programming method, it has an excellent interface to facilitate subsequent development.

Fig. 6

After constructing the initial model, we can quickly and efficiently modify the model according to our requirements. The system uses two commonly used non-uniform rational B-splines curve and surface modification methods: changing the control vertices and modifying the weight factors of the control vertices. Since we modify the control vertex coordinates to make the range of the non-uniform rational B-splines curve and surface repair more significant, we are able to use it for rough repair of the non-uniform rational B-splines curve and surface. Modifying the weight factor of the control vertex makes the modification range of the non-uniform rational B-splines curve and surface limited to the convex hull of the control vertex, so it is used to realise the refinement of the non-uniform rational B-splines curve and surface. The actual matching of control vertices and weight factors makes the user’s control of curves and surfaces more flexible and the modelling function is enhanced.

We use the example of a cosine curve to simulate the angle of furniture parts to illustrate the operation method of the whole system, the operation process and the programming code to realise the process.

5.4.1

System start-up interface

In the Windows environment, we open the executable program of this system and start the login interface of the furniture part curve parametric design system. We enter the corresponding account and password to enter the following window interface, as shown in Figure 7.

Fig. 7

5.4.2

Curve-type selection interface

After we enter the login interface, an interface window–shown in Figure 8–will pop up, and we can use it to select the desired curve type. Select the cosine curve here, and then click the button to the right to pop up a legend and text description of the cosine curve. If you are sure of the choice made, click the ‘OK’ button to enter the design interface of the geometric cosine curve.

Fig. 8

5.4.3

Geometric curve design interface

After viewing the cosine curve data table, enter the name of the required curve part, and we obtain the geometric parameters and enter them in the corresponding positions. The relevant parameters cover four parameters: amplitude, period, left interval and suitable interval. We can calculate the value of through four parameters [12]. In this way, a geometric curve model can be built in SolidWorks and saved to the corresponding folder. After completion, click the ‘Enter Part 3D Modelling Interface’ button to enter the following interface, as shown in Figure 9 [13].

Fig. 9

5.4.4

Part 3D modelling interface

The primary purpose of the three-dimensional modelling design is to generate the three-dimensional model of the part and the creation of the engineering drawing interface, as shown in Figure 10.

Fig. 10

After ascertaining the corresponding part design parameters from the data table, we input them into the corresponding position. There are three parameters of straight-line length, part width and stretched thickness [14]. After inputting the parameters, click ‘Part Model’ to build the necessary curve part in SolidWorks, as shown in Figure 11.

Fig. 11

According to the actual needs, we can click the ‘engineering drawing’ button to generate the corresponding engineering drawing, as shown in Figure 12.

Fig. 12