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R&D Center for Composite Industry Automation，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

（Received 16 April 2019；revised 23 December 2019；accepted 17 December 2020）

Abstract:There are amounts of issues to be resolved in the process of designing the fiber placement trajectory of the cylindrical component，such as the interference between the machine and the component and the over-travel of the axis of rotation on the fiber placement head.When the pressure on the cylinder surface inclines in a certain direction or at an angle within the normal plane，the motion characteristics of the rotation axis will be different.This paper analyzes the pressure angle effect of the concave cylinder surface on the motion features of a fiber placement machine.The placement area is enlarged by tilting pressure with the same lifting stroke，which is significant in preventing interference and selecting post-processing algorithm.

Key words：composite materials；fiber placement；inclination angle pressure；cylinder；postprocess

0 Introduction

Cylindrical component is one of the most common structural components in the field of automatic fiber placement.In the field of the trajectory design and post-processing technology of the cylindrical component，numerous scholars have carries out a large number of valuable research work［1-3］.However，as for the real production and processing，it was seldom mentioned that there is an interference problem between the fiber placement head and the concave cylinder with a small radius.Due to the differences of effective approach angle and departure angle of fiber placement head，on the condition that the fiber placement head is vertically pressed on the concave cylinder，the contact interference is likely to emerge between parts of the fiber placement head and the mold［4-5］based on the larger fiber placement head and the smaller cylindrical radius in the circumferential direction［6］.In order to ensure the fiber placement to proceed smoothly，during trajectory design，inclination pressure method is proposed to replace the vertical pressure.Depending on the interference situation，fiber placement head would lean forward or backward at a certain angle.However，on the basis of the inclination angle，rotation range of the fiber placement head in real-processing is much greater than before［7-8］.In this paper，we analyzed the problem and put forward a method to make a better alternative between normal vector pressure and inclination angle pressure.

As to the trajectory post-process of cylindrical concave surface，the vectors are involved along the 45°trajectory are shown in Fig.1.At any trajectory pointpi，F(xiàn)represents the normal vector，Qthe tangent vector，F(xiàn)Fthe auxiliary normal vector ，andQFthe inclination angle vector.For the convenience of the following analysis，αrepresents the cylinder circumferential angle，βthe laying angle，andθthe inclination angle［9-11］.

Fig.1 Schematic diagram of cylinder pressure

1 Rotation Range Analysis with Inclination Angle Pressure

At the trajectory pointP，the normal vectorFis（nx，ny，nz）.Its detailed expression of the normal vector can be achieved easily

The expression of tangential vectorQis（tx，ty，tz）

Auxiliary normal vector（fx，fy，fz）could be derived from the multiplication cross ofFandQ

The inclination vector is（qx，qy，qz）.According to Fig.1，ifθis the angle of inclination，the module of the normal vector can be described ascosθ，and the module of tangent vector issinθ.Then normal component could be expressed by(0，cosθ·ny，cosθ·nz)， while the Tangent vector is(sinθ·tx/r，sinθ·ty/r，sinθ·tz/r)，And inclination vector can be expressed as

When plugging Eq.（1）and Eq.（2）into Eq.（3），we can get

The fiber placement head comprises three rotation axesA/B/Cand three translation axesX/Y/Z.

The rotation center of theC-axis is vertically downward.TheB-axis rotation center line intersects vertically with theC-axis rotation center line.These axesBandCdetermined fiber placement head located along the normal vector of trajectory points［12-14］.In this paper，in order to avoid interference，instead of normal vector pressure on the mold，we choose the placement head along inclination vector.Therefore，it is necessary to justify the fiber placement head parallel to the inclination vector.∠Crepresents rotating angle，and∠Bthe swing angle.Theoretically the combined movement ofBandCmakes the posture of fiber placement head universal available［15-16］.The head structure is shown in Fig.2.

Fig.2 Mechanical structure schematic diagram of rotating shaft

For theC-axis，it is installed at the end of translational axisZand rotates aroundZ-axis.So its rotation angle is determined byqxandqy.

∠Cvaries with the cylinder circumferential angleα，as shown in Fig.3.

Fig.3 Angle variation of C-axis with inclination pressure method

For theB-axis，it is a swing axis，whose swing angle is determined byqz.Based on the above analysis，we can get

The angleBalso varies with the cylinder circumferential angleα，as shown in Fig.4.

Fig.4 Angle variation of B-axis with inclination pressure method

2 Rotation Range Analysis with Vertical Normal Pressure

The analysis of vertical pressure is based on the normal vector（nx，ny，nz）.The specific expression is shown as follows.

In the case of vertical pressure，theC-axis is determined bynxandny，which are normal vector components.Then it is easy to get the rotation angle expression ofC-axis，as shown in Eq.（12）.

Divided by the lowest line of the mold，theC-axis angle is 90°on one side of the cylinder mold，while theC-axis angle is-90°on the other side.TheC-axis angle varies with the cylinder circumferential angleα.The trend is shown in Fig.5.

Fig.5 Angle variation of C-axis by vertical pressure method

TheB-axis is the swing axis，which determined by thenzcomponent of the normal vector.The coordinate ofB-axis could be got by

Swing trend ofB-axis is shown in Fig.6.

Fig.6 Angle variation of B-axis with vertical pressure method

3 Analysis and Discussion

From Figs.5 and 6，we can get that the rotation ranges ofC-axis andB-axis are very large，which is the same with the case of inclination angle pressure.But different from inclination angle pressure，the axis changes its rotation direction at one limit point，which may be the highest point or the lowest point，without any transient process.That provides operation space for post process.We can realize the placement head adjustment by keepingC-axis still and reversingB-axis.For cylinder surface，whose concave is generally upwards，theB-axis rotation angle would vary like the trend shown in Fig.7.

Fig.7 Trend of B-axis after adjustment

For inclination angle pressure，rotating axis of fiber placement head rotates gradually，but not finish its rotation at one point.Therefore，the abovementioned adjustment method of vertical pressure is not applicable to the case of the inclination pressure.Once introduced，the fiber placement head may run just like drawing arc，which does exactly the opposite.The phenomenon will be discussed in detail in the later work.

4 Conclusions

For inclination pressure，large rotate angle ofC-axis andB-axis is inevitable.However，as for vertical pressure，the rotation angle ofC-axis andB-axis can be controlled within a small range.Therefore，for equipment design，in order to meet small rotation angle of the vertical pressure，it is necessary to achieve theB-axis rotation range up to［-90°，90°］.And in order to meet the rotating angle requirements of inclination pressure，it is necessary to designC-axis with a large rotation range，such as［-180°，180°］or more.