行政院國家科學委員會專題研究計畫 成果報告

行政院國家科學委員會專題研究計畫 成果報告

感測網路分散式機器人研發--子計畫一：救援機器人之設 計與結構分析(3/3)

研究成果報告(完整版)

計 畫 類 別 ： 整合型

計 畫 編 號 ： NSC 98-2218-E-216-001-

執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 中華大學應用統計學系

計 畫 主 持 人 ： 楊立杰

報 告 附 件 ： 出席國際會議研究心得報告及發表論文

公 開 資 訊 ： 本計畫可公開查詢

中 華 民 國 99 年 10 月 31 日

行政院國家科學委員會補助專題研究計畫 ■成果報告

□期中進度報告

感測網路分散式機器人研發--子計畫一：救援機器人 之設計與結構分析(3/3)

計畫類別：□ 個別型計畫 ■ 整合型計畫 計畫編號：NSC 98-2218-E-216-001

執行期間： 98 年 8 月 1 日至 99 年 7 月 31 日

計畫主持人：楊立杰 共同主持人：

計畫參與人員：

成果報告類型(依經費核定清單規定繳交)：□精簡報告 ■ 完整報告

本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

■ 赴大陸地區出差或研習心得報告一份

■ 出席國際學術會議心得報告及發表之論文各一份

□國際合作研究計畫國外研究報告書一份

處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計畫、

列管計畫及下列情形者外，得立即公開查詢

□涉及專利或其他智慧財產權，□一年□二年後可公開查詢 執行單位：中華大學應用數學系

中 華 民 國 99 年 10 月 31 日

行政院國家科學委員會專題研究計畫成果報告

感測網路分散式救援機器人研發─子計畫一：救援機器人之設計與結構分 析(3/3)

Design and Structure Analysis of Rescue Robot 計畫編號：NSC 98-228-E-216-001 執行期限：98 年8 月1 日至99 年7 月31 日

主持人：楊立杰中華大學應用數學所 E-mail: young@chu.edu.tw

一、中文摘要

本計畫用三年的時間，利用邊界 元素法，針對救援型機器人做了有效 的減重分析。首先分析在機器人的設 計上具有許多不同大小的環形工件，

在承受反向壓力後所產生塑性鉸的位 置，塑性鉸的個數將會隨著受力的增 加，以及不同的裂縫位置，而在圓環 的不同位置產生，事實上，當圓環塑 性鉸增加到四個時，此圓環便會崩 塌。另外，也針對救難機器人的支撐 架做應力分析。在救援機器人的救援 過程中，機器人本身的輕巧性對救援 過程的成敗佔有非常重要的影響力，

也因此，本計畫的第二個階段便是應 用邊界元素法，針對目前所設計的救 援機器人中較大的元件加以裁剪，並 做應力分析，進而達到減重的效果，

側面的懸吊板正是本計畫要分析的元 件，結果證明經過裁剪後的機器人重 量只有原來的三分之二，但強度仍在 安全範圍之內 ，因而達到減重的效 果。減重是可以適用在救援型機器人 的任何一個工件的，第三個階段便是 應用邊界元素程式分析，將救援機器 人所使用的齒輪的材料，由鋁合金(楊 式係數 72GPa，密度 2.7 g/cm³)改為塑 鋼 ( 楊 式 係 數 2.5GPa ， 密 度 1.42

g/cm³)，而達到救援機器人的重量再向

下修改的目的。

關鍵詞：邊界元素法

Abstract

Boundary Element Method (BEM) is employed in this project for three years to get a significant result about weight reduction of the rescue robot. First of all we focus on the circular ring which is the most popular specimen in the rescue robot to demonstrate that the location of a crack can strongly affect the sequence of plastic hinge development which in turn affects crack stability of a structure. A specific example of an elastic-plastic ring loaded with diametrically opposite concentrated loads is employed to investigate these effects. In fact, the ring will collapse if the number of plastic hinges is up to four. In addition, the stress analysis in the supporting frame of the rescue robot is also finished. The next step is the weight reduction of the largest portion of the rescue robot, i.e., the lateral plate without causing the strength of it by using the Boundary Element Method. The dynamical loading conditions are performed before and after weight reduction. The numerical results of the stress distribution and the plastic deformation along the center line (interface) of the lateral plate show that the weight of the plate is reduced to two thirds of the original and the endure limits of the plate before and after weight reduction are almost the same and therefore, will not lower the strength of the plate. The final step of the project is to replace the material of the

gears of the rescue robot from Aluminum Alloy (Young’s modulus 72GPa, density2.7 g/cm³ ) by Polyacetal (POM) (Young’s modulus 2.5 GPa, density 1.42 g/cm³ ) and therefore, reduce the weight of the rescue robot to three quarters of the original weight.

Keyword: Boundary Element Method, Polyacetal (POM).

二、Background

The first step in the BEM solution is to divide the specimen into two bodies, referred to below as the interface. The interaction between the two bodies is included through boundary conditions relating the displacements and stresses on either side of the interface, i.e., the 2 displacement components and the 2 stress components must be continuous. Thus, at each pair of points on the interface we have four conditions involving eight quantities.

Four of those are eliminated algebraically using the boundary condition, thus leaving four unknowns at each pair. Two coupled boundary integral equations, written as a function of position on the boundary of a body, enforce all of the field equations of elasticity for that body. The two equations for each of the two artificially divided bodies are applied to each discrtized point on the interface, thus giving four equations and four unknowns at each pair of interface points. If the boundary of either of the artificially divided bodies consists of other than common interface, then at each of these boundary points there are four boundary quantities to be accounted for. The only condition we have used on these external boundaries has been prescribed stress, thus leaving the two displacements as unknowns, with two equations provided by that body’s two boundary integral equations. The BEM consists of the discretization of the boundary surfaces and the numerical approximation of the boundary quantities in the set of equations obtained from boundary integrals as described above. We model the boundary, using straight-line elements, centered about nodes at which the integral equations are

applied in [1]. For straight boundary this introduces no approximations. We assume that the stress and displacement are constants resulting in integrals of the known 2D Green’s function which have been evaluated in closed form in [1]. The final result is a system of simultaneous linear algebraic equations for the unknowns nodal displacements and stresses.

It is known theoretically in [2] that the elastic-plastic analysis of statically indeterminate structures of long slender members in bending, from first yield to limit load, involves a sequential formation of plastic hinges at intermediate loads. The total member of plastic hinges becomes the degree of statical indeterminacy plus one at collapse. The objectives of this research to demonstrate that the location of the crack can strongly affect the sequence of plastic hinge development numerically, as well as the crack location, affects crack stability.

With these objectives in mind, the analysis shall proceed to establishing the BEM codes for development of the first, second, and third plastic hinges, respectively.

Earthquakes are so frequently in the world recently which always causing serious damage of the buildings, properties, and injury and death of peoples. Rescue robots with light, thin body are therefore needed in small space of the disaster scene. During weight reduction process the strength of the rescue robot is also evaluated. Several papers show the potential applications of the BEM code to crack problems. Ghorbanpoor and Zhang [1] point out that the accuracy of the BEM prediction is satisfactory when it is compared with results from a finite element solution with very fine mesh and with the analytical solution. Tan and Gao [2] show the powerful application of the BEM in the analysis of biomaterial interface crack problem. Lih-jier Young [3] show the powerful application of the BEM in rough contact mixed mode with the plastic crack tip problem and gives the accuracy up to 98.11%. Boundary element method is used to model the complex resistance to the applied field of the lateral plate of rescue

robot to find the stress distribution along the interface. The outlook and dimension in mm of the plate are shown in Fig. 1. Paper in [4]

also shows the potential application of the BEM determining the effect of crack face roughness in a realistic experimental specimen. The maximum allowable stress of the plate is defined by both the normal and the shear stress of first point along the interface reach the yield stress σy andτy, respectively, whereτy=σy/ 3 . The plastic displacement of the interface is obtained by increasing the loading of the plate after the first point yield. Keep increasing the loading and we can get the plastic displacement of the interface.

In fact, the physical and mechanical properties of Polyacetal (POM) are quiet excellent. Due to these properties, i.e., low wear rate and small friction coefficient (0.1

~ 0.3) POM can be effectively applied in the gear (Figure 10) and bearings in the occasion to be a high degree of wear. POM with high rigidity, high strength, and fatigue resistance has similar properties as of the soft metal which can be applied occasionally to replace the soft metal.

三、Results

The ring and its loading are doubly symmetric and the crack is located by from the horizontal as in Fig. 1. The mean radius of the ring is R and its radial thickness is t, and it is assumed to be of uniform unit thickness perpendicular to its plane. P is the load and locates any section around the ring from the horizontal. It is assumed that the ring is long and slender, R/t10, so that deformation due to in-plane bending is the only significant deformation. The present model is discretized into 260 points and 4 regions (I1-I4) of different types of boundary conditions for the cracked ring shown in Fig.

2. As mentioned in [1], the cracked body is divided into two parts by B(1,2 ) denote the boundary traction and displacement components, respectively; and I₁the crack region and I₂the ligament region of the interface. The boundary condition of

I₂ must satisfy the continuity of the stresses and displacements on either side of the interface, i.e., (2t1)I2 (1t1)I2, and (2t2)I2 (₁t₂)_I2, (₂u₁)_I2 (1u₁)_I2, and (₂u₂)_I2 (1u₂)_I2. The unknowns are (₁t₁)_I2, (₁t₂)_I2, (₁u₁)_I2 and (1u2)I2. The boundary conditions for open crack I1 and the free surface I4 are zero stress, i.e., (_t1)I1,I4 (_t2)I1,I4 0 and the unknowns are ( _u₁)_I1,I4 and ( _u₂)_I1,I4, where 1,2 depending on whether that portion is B₁or B₂. The unknowns boundary components can be obtained by using the Gaussian elimination method after converting boundary conditions into the final matrix form. As a first approximation of the moments in a cracked elastic ring, the internal moments, M₀, in an“uncracked”

ring is used. Table 1 shows the comparison of the theoretical results for M /₀ PR in the uncracked ring with the numerical one (BEM model). It is quiet clear that the numerical accuracy M /₀ PR goes from 91.18% at  60° to 99.13% at  30°. It is noted that according to [2] for a/t＝0.3, the relaxation of moments caused by the crack’s elastic stiffness reduction is 4% or less, but for deeper cracks such as a/t＝0.5, the relaxation is as much as 30%. We can get the same result by inspecting Table 2 and comparing it with Table 1 for the BEM values.

(一) Development of the first plastic hinge As mentioned in [2], if we want to find the load, P₁, at which the first hinge forms and its location, it is denoted that the ring can be analyzed as completely elastic up to that load. As a first approximation of the moments in a cracked elastic ring, the internal moments, M0, in an “uncracked”

ring is used. The elastic solution for the uncracked ring gives the results for

PR

M /₀ in Table 1. However, the crack reduced the stiffness of the ring at the cracked section, causing a redistribution of the moment in the ring and is analyzed by the superposition in [2] and the BEM methods. In finding the first hinge location it

is worthy to be mention that we have to add one more boundary between the outer and the inner boundaries of the ring as shown in Fig. 3. Therefore, we can compute the resulting moment at each nodal point of the new boundary and then get the location of the maximum moment (first hinge). The model is discretized into 256 points and 4 regions (I1- I5) of different types of boundary conditions. As described above I1

is the crack region and I₂, I₅are the ligament regions of the interface. The boundary condition of I2 and I5 must satisfy the continuity of the stresses and displacements on either side of the interface, i.e., (2t1)I2,I5 (1t1)I2,I5 and (2t2)I2,I5 (1t2)I2,I5, (2u1)I2,I5 (1u1)I2,I5, and (2u2)I2,I5 (1u2)I2,I5. The unknowns are (₁t₁)_I2,I5, (₁t₂)_I2,I5, (₁u₁)_I2,I5 and (1u2)I2,I5. The boundary conditions for open crack I1and the free surface I4are zero stress, i.e., (_t₁)_I1,I4 (_t₂)_I1,I4 0 and the unknowns are ( _u1)I1,I4 and ( _u2)I1,I4, where 1,2 depending on whether that portion is B₁or B₂. The unknowns boundary components can also be obtained by using the Gaussian elimination method.

Table 2 shows the comparison between the theoretical and the BEM values of M/PR for different a/t and  0°. It has been seen that the accuracy is in the range of 93.24%

to 99.99%. It has been assumed in [2] that the crack is into the inside or outside of the ring, so that it is on the tension side for each location. Also notice that the first hinge occurs at the crack location, when that location is at a relatively high moment in the uncracked ring, that is 0° to 15°and

 75° to 90°. However, for locations with relatively low moment, the first hinge forms at the maximum moment location,  90°. Finally, it is noted that the first hinge load P1, is affected appreciably only when the first hinge forms at the crack location. Upon comparing the values of M/PR in Table 2 with Table 1, the effects of changes of elastic moment stiffness of the “ cracked element”are also noted to be appreciable only when the crack is placed at a relatively high moment position. Of course, this effect

would increase with larger a/t values (a/t0.3 in Table 2), but for this example it is really quite small.

(二)Development of the second plastic hinge In an uncracked ring, the second hinge development at the same time as the first hinge, at the load points or points of maximum moment. Again the elastic superposition in [2] and the BEM methods can be used even though a first hinge is already formed. The boundary conditions of the hinge point are just set the values of the two stress components which leave the two displacements unknown. Table 3 has been prepared theoretically and numerically. The accuracy is in the range of 70.06% to 99.99%.

(三) Development of the third plastic hinge Once again, the formation of the full analysis at the instant of development of the third plastic hinge can be done using the 2D BEM method as above. For the border perspective, it is more relevant to combine Table 2-4 into a composite of the sequence of hinge formation as affected by the crack location. This is given as Table 5. The notations, 1 through 4, indicate the first through fourth hinges formed, respectively.

The notations, 2, 3, and 3, 4 indicate simultaneous formation of two hinges, symmetric case of no crack. Indeed, the widely varied pattern of the sequence of hinge formation in Table 5 is quite surprising. Simply, the change of the crack location (for a given crack size, a/t=0.3, and ring slenderness ratio, R/t=10) causes this wide variation in the pattern. Moreover, expect for having the crack at the load point,

90

 °, the relative loads for hinge formation do not vary greatly (less than 10%

except for the first hinge for 75°, near the load point). This seems quite surprising in view of the wide variety of hinge sequences.

(四)Development of the fourth plastic hinge Theoretically as in [2], the fourth (last) hinge forms either 180°or 0°, which means that an elastic path connects the load points up until the fourth hinge forms. Since

this method computes the elastic bending moments on that elastic path, the relative load point displacements can easily be computed for each successive hinge formation load, including the fourth hinge.

Thus, a complete load-displacement diagram may be constructed, since that diagram is linear between successive hinge formation loads. Figure 5 shows the supporting frame of the rescue robot which is discretized into 222 nodal points and 6 regions (I₁- I₇) of different types of boundary conditions. As mentioned before, The boundary conditions of I₂are (₂t₁)_I2 (1t₁)_I2, and (₂t₂)_I2 (1t₂)_I2, (₂u₁)_I2 (₁u₁)_I2, and (₂u₂)_I2 (₁u₂)_I2 and the unknowns are (1t1)I2, (1t2)I2, (1u1)I2 and (1u2)I2. The upper and right pins are assumed fixed. A compressive load is applied through the lower pin in the positive x1direction by assuming a uniform distribution of normal traction over 90° of lower pin hole surface, i.e., points 214, 215, 216, 217, and 218 (I4).

Therefore, the boundary conditions of these points are (2t1)I2pcosθ¹and (2t2)I2psinθ¹. The unknowns are (2u1)I4and (2u2)I4, where p is applied normal stress on the lower hole andθ1 is the angle between the direction normal of each node and x₁-axis. The boundary conditions of the free surface I₅are (_t1)I5 ( _t2)I5 0, and the unknowns are(_u1)I5 and (_u2)I5. In order to have zero shear stress on the hole, the horizontal displacement component of points 110, 111, 98, 99, 220, 137, 138, 139 and 140 (I6) is taken to be zero and their two traction components are related by tanθ2, whereθ2is the angle between the direction normal and x1-axis, in order to have zero shear stress on the hole. Hence the boundary conditions for points on I6 in the upper half plane are(₁u₁)_I60, (₁t₂)_I6(1t₁)_I6,tanθ2 and the unknowns are (₂u₂)_I6, (₂t₁)_I6. Point 97 (I₇) is totally fixed point. The boundary conditions of this point are (1u1)I7(1u2)I70 and the unknowns are (₁t₇)_I7, (₁t₂)_I7. This combination of boundary conditions results in a free body diagram of the form given in

Figure 5(b). The unknowns boundary components can be obtained by using the Gaussian elimination method. Figure 6 shows the displaced position of the interface of the specimen. The majority of the displacement shown is normal to the interface and the motion parallel to the interface cannot be seen on the scale shown in this figure. It can be seen, and expected from Figure 5(b), that the ligament portion of the interface is rotated about 0.01°

counterclockwise from horizontal.

(五) Dynamical Loading Conditions before Weight Reduction of Lateral Plate As discussed in [5], the first step in the BEM solution is to divided the homogeneous medium into two bodies Bγ

(1,2) along the center line which we call the interface as shown in Fig. 8(a). The interaction between the two bodies included through boundary conditions relating the displacements and stresses on either side of the interface. Let _t_i and _u_i ( 1,2 andi1,2) denote the ith boundary traction and displacement components, respectively, on the boundary of Bγ. The present model is discretized into 648 nodal points and 4 regions (I2, I4, I5, and I6) of different types of boundary conditions shown in Fig. 8(a). The interface is denoted by I₂. At points on this region the 2 displacement components and 2 stress components must be continuous.

Therefore, the boundary conditions are

2

2 ( )

)

(₂t_{1 I} ₁t_{1 I} and

2

2 ( )

)

(₂t_{2 I} ₁t_{2 I} ,

2

2 ( )

)

(₂u_{1 I} ₁u_{1 I} and

2

2 ( )

)

(₂u_{2 I} ₁u_{2 I} . This leaves

) 2

(₁t_{1 I} , ) 2

(₁t_{2 I} , ) 2

(₁u_{1 I} and ) 2

(₁u_{2 I} , as the unknowns. Compressive loads are applied through the six pins as in Fig. 9(a) by assuming a uniform distribution of normal traction over 90 ﾟ of pin hole surfaces (I6). Therefore, the boundary conditions of these points are (_ ) cos

1 6 p

t _I  and (_ ) sin

2 6 p

t _I  .

The unknowns are ) 6

(_u_{1 I} and ) 6

(_u_{2 I} , where p is applied normal stress on the six holes andis the angle between the direction normal of each node and x₁-axis as in Fig.

8(a). A concentrate load is also applied through lower half plane point 110 (I4) The boundary conditions are t _I p

) 4

(_{1 2} , 0

) ( ) ( )

(2t1 _I4 2t2 _I4 1t1 _I4  and the unknowns are

) 4

(_u_{1 I} and ) 4

(_u_{2 I} . The boundary conditions of the free surfaces (I₅) are ( ) ( ) 0

5

5 2

1 _I  t _I 

t _

 , and the unknowns

are (_u_{1 I})5 and ) 5

(_u_{2 I} , where 1 or 2 depending on whether that portion of I5is in B₁ or B₂. Once the points on the interface yield (I₈) in shear direction the boundary conditions are 

) 8

(₁t_{1 I} σy/ 3 ,  ) 8

(₂t_{1 I} σ_y/ 3 ,

8

8 ( )

)

(₁t_{2 I} ₂t_{2 I} and

8

8 ( )

)

(₁u_{2 I} ₂u_{2 I} . The unknowns are

)8

(₂t_{2 I} , )8

(₁u_{1 I} , ) 8

(₁u_{2 I} and (₂u₁)_I8. For the case both the normal and shear directs yield the boundary

conditions are 

) 8

(₁t_{1 I} σy/ 3 ,



8  )

(₂t_{1 I} σy/ 3 ,  ) 8

(₁t_{2 I} σy and



8  )

(₂t_{2 I} σy.

The BEM consists of the discretization of the boundary surfaces and the numerical approximation of the boundary quantities in the set of equation obtained from the boundary integrals. We model the boundary, using straight-line elements, centered about nodes at which the integrals of the 2D Green’s function as in [6]. The final system of simultaneous linear algebraic equations for the unknown nodal displacements and stresses, can be obtained by using Gaussian elimination method.

Figure 10(a) shows the shear stress distribution along the interface. It can be seen that the shear stress of point 100 reaches the yielding criteria in (300 MPa) with the applied load p5.4 MPa.

However, the normal stress as shown in Fig.

11(a) is not yield yet. We can keep increasing the applied load top7.1Mpa until the normal stress of point 100 reaches the yielding criteria (520 MPa) as in Fig.

12(a). There are three points, i.e., 100, 99, and 98 yield in shear direction at this moment as shown in Fig. 13(a). Both the shear and normal stresses reach the yielding

criteria of points 100, 99, 98, and 97 as in Figs 14(a) and 15(a), respectively, when applied load is up to7.3MPa. Figures 16(a) and 17(a) show the plastic displacements both in shear and normal direction of the four yielding point mention above.

( 六 ) Dynamical Loading Conditions after Weight Reduction of Lateral Plate

The model for the weight reduction lateral plate model is discretized into 904 nodal points and 4 regions (I2, I4, I5, and I6) of different types of boundary conditions shown in Fig. 8(b). All boundary conditions remain the same as described above. In addition, two rectangular regions with the size 8046mm²have been cut as shown in Fig. 8(b) which reduce the mass of the plate from 3 kg to 2 kg. The boundary conditions of the inner free surface are the same as the outer one (I₅). The final system of simultaneous linear algebraic equations can also be obtained by using Gaussian elimination method.

Figures 10(b) and 11(b) show the stress distribution along the interface. It can be seen from 10(b) that the shear stress of point 228 reaches the yielding criteria in (300 MPa) with the applied load p4.4MPa.

However, the normal stress as shown in Fig.

11(b) is not yield yet. We can keep increasing the applied load to p5.7Mpa until the normal stress of point 111 reaches the yielding criteria (520 MPa) as in Fig. 12 (b). There are three points, i.e., 111, 112, and 228 yield in shear direction at this moment as shown in Fig. 13(b). Both the shear and normal stresses reach the yielding criteria of points 111, 112, and 228, as in Figs 14(b) and 15(b), respectively, when applied load is up to6.0MPa. Figures 16(b) and 17(b) show the plastic displacement both in shear and normal direction of the four yielding point mention above.

(七) Stress Analysis of POM Gear with a Crack

There are mainly six different kinds of gear in the rescue robot, i.e., 15T, 22T, 40T, 46T, 65T and 723 mm 0T. The results shown here is the 46T with pitch circle radius, 24 mm addendum circle radius, 11mm root

radius and 20ﾟpressure angle as shown in Fig. 18. The present model is discretized into 600 nodal points and 4 regions (I₁, I₂, I₃, and I₅) of different types of boundary conditions shown in Fig. 19. The interface is denoted by I2. At points on this region the 2 displacement components and 2 stress components must be continuous. Therefore, the boundary conditions are

2

2 ( )

)

(₂t_{1 I} ₁t_{1 I} and

2

2 ( )

)

(₂t_{2 I} ₁t_{2 I} ,

2

2 ( )

)

(₂u_{1 I} ₁u_{1 I} and

2

2 ( )

)

(₂u_{2 I} ₁u_{2 I} . This leaves

) 2

(₁t_{1 I} , ) 2

(₁t_{2 I} , ) 2

(₁u_{1 I} and ) 2

(₁u_{2 I} , as the unknowns. Compressive load p is action in the direction of line of action shown in Fig. 18. Therefore, the boundary conditions of these points are ( ) cos

1 3

1t _I p and( ) sin

2 3

1t _I p , whereis the pressure angle of the gear. The unknowns are (₁u₁)_I3 and

)3

(₁u_{2 I} . The boundary conditions for open crack I₁ and the free surface I5 are zero stress, i.e., (_t₁)_I1,I5 (_t₂)_I1,I5 0 and the unknowns are (_u1)I1,I5 and (_u2)I1,I5, where for I1 1

and 1,2for I5depending on whether that portion is B1or B2. The unknowns boundary components can also be obtained by using the Gaussian elimination method.

Both Figs 20 and 21 show the opening and shear displacements of the crack of the gear with different materials, respectively.

It is obviously that the POM displacement of the crack is much larger than the Aluminum Alloy. Stress singularity at the crack tip is clear shown on Figs 22 and 23 for normal and shear stresses, respectively nevertheless the materials are different.

The mode I and mode II stress intensity factors can be calculated according to the formula

t t

II

I x

G x

K u 



2 1 ) ( 2

,   (1)

where u(x_t) is the displacement of the crack tip element, G is the shear modulus, xt is the coordinate of the crack tip element.The value of is4(1).for plane strain and



 1

4 for plane stress,is the Poisson Ratio. The results shown on Figs 24 and 25 show both the values of K_I and K_II are much larger for POM gear than Aluminum Alloy gear even the weight of POM gear is much lighter.

四、Conclusions

The sequence of plastic hinge formation of a cracked circular ring is investigated by using 2-D boundary element method as the numerical portion to support the theoretical one in [2]. In addition, the stress analysis of the supporting frame of the rescue robot is also finished. It can be seen from the results mentioned above that the weight of one lateral plate of the robot have be reduced 1 kg of mass (from 3 kg to 2 kg) but won’t lower the strength that much.

(allowable stress is reduced from7.1MPa to5.7MPa). The total mass of one rescue robot may be reduced at least 2 kg after weight reduction and therefore, can moves nimbly during the rescue process. The specimen made of POM is much lighter than Aluminum Alloy and of course will provide a lighter robot. However, the strength of the specimen may be lowered and therefore, some portion of the specimen with heavy loadings will damage very easily. In fact, how to replace the heavy loading portion of POM specimen by Aluminum Alloy is the next topic of this research.

五、References

[1] Young, L.J. and Tsai, Y.P., 1999. A boundary element application for mixed mode loading idealized sawtooth fracture surface, Int’l J. of Solids and Structures, 36, pp. 3239-3252.

[2] Young, L.J. Plastic hinge development and crack stability analysis in a circular ring, Int’l J. of Solid and Structure, 38, pp. 1355-1367

[3] A. Ghobanpoor and J. Zang, Boundary Elemant Analysis of Crack Growth for Mixed Mode Center Slant Crack Problems, Engineering Fracture

Mechanics, Vol. 36, No.5 pp. 661-668, 1990.

[4] C.L. Tan and Y.L. Gao, Treatment of Bimaterial Interface Crack Problems Using The Boundary Element Method, Engineering Fracture Mechanics, Vol. 36, No. 6, pp. 919-932, 1990.

[5] L.J. Young and Y.P. Tsai, A Boundary Element Application for Mixed Mode Loading Idealized Sawtooth Fracture Surface, Int'l J. of Solids and Structure, 36, 3239-3252, 1999.

[6] L.J. Young, An Analysis of a Fracture Specimen for Rough Crack in Shear, Int’l J. of Fracture, 115, 3, 273~285, 2002.

[7] C.A. Brebbia , J.C.F Telles and L.C.

Wrobel, Boundary Element Techniqes , Springer - Verlag,New York,1970.

Figures and Tables

Fig. 1 Cracked ring with diametrically opposite concentrated loads.

(a)

(b)

Fig. 2 Schematic of (a) BEM mesh, (b) different regions of cracked ring.

(a)

(b)

Fig. 3 Schematic of (a) BEM mesh, (b) different regions of cracked ring with hinges.

Fig. 4 Supporting frame of rescue robot.

(b)

Fig. 5 Schematic of (a) BEM mesh, (b) free body diagram of supporting frame.

Fig. 6 Displace position of interface of supporting frame.

(a)

(b)

Fig. 7 (a) Outlook (b) dimension in mm of the lateral plate of rescue robot.

(a)

(b)

Fig 8 Schematic of BEM mesh of the lateral plate of rescue robot (a) before (b) after weight reduction.

(a)

(b)

Fig. 9 Loading conditions of the lateral plate of rescue robot (a) before (b) after weight reduction.

(a)

(b)

Fig. 10 Shear stress distribution of the interface of the lateral plate (a) before weight reduction with

4 .

5



p MPa (b) after weight reduction with p4.4MPa.

(a)

(b)

Fig. 11 Normal stress distribution of the interface of the lateral plate (a) before weight reduction with

4 .

5



p MPa (b) after weight reduction with p4.4MPa.

(a)

(b)

Fig. 12 Normal stress distribution of the interface of the lateral plate (a) before weight reduction with

1 .

7



p MPa (b) after weight reduction with p5.7MPa.

(a)

(b)

Fig. 13 Shear stress distribution of the interface of the lateral plate (a) before weight reduction with

1 .

7



p MPa (b) after weight reduction with p5.7MPa

(a)

(b)

Fig. 14 Shear stress distribution of the interface of the lateral plate (a) before weight reduction with

3 .

7



p MPa (b) after weight reduction with p6.0MPa.

(a)

(b)

Fig. 15 Normal stress distribution of the interface of the lateral plate (a) before weight reduction with

3 .

7



p MPa (b) after weight

reduction with p6.0MPa.

(a)

(b)

Fig. 16 Plastic deformation in shear direction of the lateral plate (a) before weight reduction with

3 .

7



p MPa (b) after weight reduction with p6.0MPa.

(a)

(b)

Fig. 17 Plastic deformation in normal direction of the lateral plate (a)

before weight reduction with 3

.

7



p MPa (b) after weight reduction with p6.0MPa.

Fig. 18 46T spur g pitch circle radius, 24 mm addendum circle radius, and 20ﾟ pressure angle.

Fig. 19 Schematic of BEM mesh and different regions of cracked ring.

-15 -10 -5 0 5 10 15

0 0.1 0.2 0.3 0.4 0.5 0.6

cm

cm

10exp(-2)

POM

POM AL

Fig. 20 Comparison between the crack opening displacements of the crack of the gears made of POM and Aluminum Alloy.

-30 -20 -10 0 10 20 30 40

0 0.1 0.2 0.3 0.4 0.5 0.6

cm

cm

POM POM

AL 10exp(-2)

Fig.21 Comparison between the crack shear displacements of the crack of the gears made of POM and Aluminum Alloy.

-100 0 100 200 300 400 500 600 700 800 900

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

AL POM

cm MPa

Fig. 22 Comparison between normal stress distributions in the ligament portion of cracked tooth of the gears made of POM and Aluminum Alloy.

0 500 1000 1500 2000 2500

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

MPa

cm

AL POM

Fig. 23 Comparison between shear stress distributions in the ligament portion of cracked tooth of the gears made of POM and Aluminum Alloy.

0 50 100 150 200 250 300

0 10 20 30 40 50 60

POM

AL

Mpa KI

Fig. 24 Comparison between mode I stress intensity factors of the crack the gears made of POM and Aluminum Alloy.

0 50 100 150 200 250 300 350 400 450

0 10 20 30 40 50 60

POM

AL

Mpa KII

Fig. 25 Comparison between mode II stress intensity factors of the crack the gears made of POM and Aluminum Alloy.

行政院國家科學委員會補助國內專家學者出席國際學術會議報告

99 年 4 月 8 日 報告人姓名

楊立杰 服務機構

及職稱 中華大學教授

時間 會議

地點

99 年 3 月 27 日至 99 年 4 月 4 日

成都四川大學

本會核定 補助文號 會議

名稱

(中文)2010 兩岸五校數學研討會 (英文)

發表 論文 題目

(中文)一些邊界元素在工業上的應用

(英文)Some Boundary Element Applications in Industry

報告內容應包括下列各項：

一、參加會議經過

3/27 日我一大早便由桃園出發，至香港轉成中國國際航空的班機至成都，本來 11:10 的班機，因為擋風玻璃破裂，必須由北京運送下來，以至於飛機到 19:30 才 由香港起飛，到成都已是晚上 22:00，實在有些不敢恭維，還好後面的行程運作順 利，總算是不虛此行。

二、與會心得

這次研討會是由兩岸五校(中山、義守、淡江、中華、四川)的十位學者，針對 目前的研究領域發表成果，由於時間安排的非常寬鬆，讓我們都能充分的了解每位 學者的研究精隨，對自己所從事的研究也有改進的方向，與成長的空間。

三、考察參觀活動(無是項活動者省略)

考察參觀活動是 4/3 日誌成都市區的參觀，整個市區可能有台北市的三倍大，

外環道共分三環，市內也有很多的名勝古蹟，一年前的汶川地震在此也看得到所遺 留下的痕跡，整個城市看起來各方面都在建設之中，惜市民不遵守交通規則，衛生 習慣不好，讓我對成都的觀感打了折扣。

四、建議

此次活動讓我們增廣見聞，不但了解對岸學者在大學中的學術定為有了更進一 步的了解，更對成都居民的生活方式感到好奇，建議以號應多多辦理這方面的學術 活動，以增加彼此的了解程度。

五、攜回資料名稱及內容

此次攜回 2010 年兩岸五校數學研討會演講摘要一份，其中包括二十位學者的近期 研究方向與成果，頗具參考價值。

六、其他

附件三

行政院國家科學委員會補助國內專家學者出席國際學術會議報告

99 年 6 月 11 日

報告人姓名 楊立杰 服務機構

及職稱

中華大學教授

時間 會議

地點

99 年 6 月 7 日至 99 年 6 月 9 日

馬來西亞，吉隆坡

本會核定 補助文號

NSC 98-2218-E-216-001

會議名稱 (中文)

(英文)The 8^thInternational Conference on Fracture and Strength of Solids 發表論文題目 (中文)

(英文)NUMERICAL APPLICATION IN WEIGHT REDUCTION OF LATERAL PLATE OF RESCUE ROBOT

報告內容應包括下列各項：

七、參加會議經過

6/6 日下午 3:05 由桃園出發，搭乘馬來西亞航空公司的班機至吉隆坡，到達住 宿的飯店(Istana Hotel)已將近晚上 10 點，原本感覺沒有多遠的行程，前後也花 了 7 個多小時。

八、與會心得

這次研討會是由馬來西亞的 Professor M. N. Tamin 主辦，與會者不乏力學界 知名學者，針對目前的研究領域發表成果，由於時間安排洽當，讓我們都能充分的 利用自己的時間去了解每位學者的研究精隨，對自己所從事的研究也有改進的方 向，與成長的空間。

九、考察參觀活動(無是項活動者省略)

馬來西亞的種族是非常複雜的，在最後一天下午的參訪活動中，我充分的了解到 這三大民族(馬來、華人、印尼)的融合性與各種不童的發展色彩，吉隆坡是觀光城市，

捷運四通八達但稍嫌老舊，整個城市看起來各方面都仍在建設之中，石油雙塔位於吉 隆坡市中心，為整個城市增添了不少可看性。

十、建議

此次活動讓我們增廣見聞，不但對別的國家學術發展更有進一步的了解，更對居 民的生活方式感到好奇，建議以號應多多參加這方面的學術活動，以增加彼此的了解 程度。

十一、 攜回資料名稱及內容

攜回 The 8^thInternational Conference on Fracture and Strength of Solids 演講摘要及 CD 各一份，其中包括與會所有學者近期研究方向與成果，頗具參考價值。

十二、 其他

附件三

NUMERICAL APPLICATION IN WEIGHT REDUCTION OF LATERAL PLATE OF RESCUE ROBOT

Lih-jier Young

Department of Applied Mathematics, Chung Hua University

No. 707 Section II, Wu Fu Road, Hsin Chu City Taiwan 30012, R.O.C.

young@chu.edu.tw

Keywords: Boundary Element Method

Abstract. In general, weight reduction will always lower the strength of the specimen. The primary purpose of this paper is weight reduction of lateral plate of rescue robot without causing the strength of it by using the Boundary Element Method (BEM). The dynamical loading conditions are performed before and after weight reduction. The numerical results of the stress distribution and the plastic deformation along the center line (interface) of the lateral plate show that the endure limits of the plate before and after weight reduction are almost the same and therefore, will not lower the strength of the plate.

Introduction

Earthquakes are so frequently in the world recently which always causing serious damage of the buildings, properties, and injury and death of peoples. Rescue robots with light, thin body are therefore needed in small space of the disaster scene. During weight reduction process the strength of the rescue robot is also evaluated. Several papers show the potential applications of the BEM code to crack problems. Ghorbanpoor and Zhang [1] point out that the accuracy of the BEM prediction is satisfactory when it is compared with results from a finite element solution with very fine mesh and with the analytical solution. Tan and Gao [2]

show the powerful application of the BEM in the analysis of biomaterial interface crack problem. Lih-jier Young and Tsai [3] show the powerful application of the BEM in rough contact mixed mode with the plastic crack tip problem and gives the accuracy up to 98.11%.

Boundary element method is used to model the complex resistance to the applied field of the lateral plate of rescue robot to find the stress distribution along the interface. The outlook and dimension in mm of the plate are shown in Fig. 1. Young [4] also shows the potential application of the BEM determining the effect of crack face roughness in a realistic experimental specimen. The maximum allowable stress of the plate is defined by both the normal and the shear stress of first point along the interface reach the yield stress σyandτy, respectively, whereτy=σy/ 3 . The plastic displacement of the interface is obtained by increasing the loading of the plate after the first point yield. Keep increasing the loading and we can get the plastic displacement of the interface.

Dynamical Loading Conditions before Weight Reduction

As discussed in [3], the first step in the BEM solution is to divided the homogeneous medium into two bodies Bγ(1,2) along the center line which we call the interface as shown in Fig. 2(a). The interaction between the two bodies included through boundary conditions relating the displacements and stresses on either side of the interface.

Let_t_iand_u_i (1,2 andi1,2 ) denote the ith boundary traction and displacement components, respectively, on the boundary of Bγ. The present model is discretized into 648 nodal points and 4 regions (I2, I4, I5, and I6) of different types of boundary conditions shown

in Fig. 2(a). The interface is denoted by I₂. At points on this region the 2 displacement components and 2 stress components must be continuous. Therefore, the boundary conditions are (₂t₁)_I2 (₁t₁)_I2 and

2

2 ( )

)

(₂t_{2 I} ₁t_{2 I} ,

2

2 ( )

)

(₂u_{1 I} ₁u_{1 I} and

2

2 ( )

)

(₂u_{2 I} ₁u_{2 I} . This leaves ) 2

(₁t_{1 I} , ) 2

(₁t_{2 I} , ) 2

(₁u_{1 I} and ) 2

(₁u_{2 I} , as the unknowns. Compressive loads are applied through the six pins as in Fig. 3 by assuming a uniform distribution of normal traction over 90 ﾟof pin hole surfaces (I6). Therefore, the boundary conditions of these points are(_ ) cos

1 6 p

t _I  and

 ) sin

( t2 _I6 p . The unknowns are ) 6

(_u_{1 I} and ) 6

(_u_{2 I} , where p is applied normal stress on the six holes andis the angle between the direction normal of each node and x1-axis as in Fig. 2(a). A concentrate load is also applied through lower half plane point 110 (I₄) The boundary conditions are t _I p

) 4

(_{1 2} , ( ) ( ) ( ) 0

4 4

4 2 2 11

1

2t _I  t _I t _I  and the unknowns are(_u_{1 I}) 4and

) 4

(_u_{2 I} . The boundary conditions of the free surfaces (I₅) are( ) ( ) 0

5

5 2

1 _I  t _I 

t _

 ,

and the unknowns are )5

(_u_{1 I} and )5

(_u_{2 I} , where or 2 depending on whether that portion of1 I5 is in B1 or B2. Once the points on the interface yield (I8) in shear direction the boundary conditions are 

) 8

(₁t_{1 I} σ_y/ 3 ,  ) 8

(₂t_{1 I} σ_y/ 3 ,

8

8 ( )

)

(₁t_{2 I} ₂t_{2 I} and

8

8 ( )

)

(₁u_{2 I} ₂u_{2 I} . The unknowns are

)8

(₂t_{2 I} , )8

(₁u_{1 I} , )8

(₁u_{2 I} and ) 8

(₂u_{1 I} . For the case both the normal and shear directs yield the boundary conditions are 

) 8

(₁t_{1 I} σ_y/ 3 ,  ) 8

(₂t_{1 I} σ_y/ 3 ,  ) 8

(₁t_{2 I} σ_y and



8  )

(₂t_{2 I} σy.

The BEM consists of the discretization of the boundary surfaces and the numerical approximation of the boundary quantities in the set of equation obtained from the boundary integrals. We model the boundary, using straight-line elements, centered about nodes at which the integrals of the 2D Green’s function as in [4]. The final system of simultaneous linear algebraic equations for the unknown nodal displacements and stresses, can be obtained by using Gaussian elimination method.

Figure 4(a) shows the shear stress distribution along the interface. It can be seen that the shear stress of point 100 reaches the yielding criteria in (300 MPa) with the applied loadp5.4MPa. However, the normal stress is not. We can keep increasing the applied load top7.1Mpa until the normal stress of point 100 reaches the yielding criteria (520 MPa). There are three points, i.e., 100, 99, and 98 yield in shear direction at this moment as shown in Fig. 5(a). Both the shear and normal stresses reach the yielding criteria of points 100, 99, 98, and 97 as in Figs 6(a) and 7(a), respectively, when applied load is up to7.3MPa.

Figures 8(a) and 9(a) show the plastic displacements both in shear and normal direction of the four yielding point mention above.

Dynamical Loading Conditions after Weight Reduction

The model for the weight reduction lateral plate model is discretized into 904 nodal points and 4 regions (I2, I4, I5, and I6) of different types of boundary conditions shown in Fig.

2(b). All boundary conditions remain the same as described above. In addition, two rectangular regions with the size 8046mm² have been cut as shown in Fig. 2(b) which reduce the mass of the plate from 3 kg to 2 kg. The boundary conditions of the inner free surface are the same as the outer one (I5). The final system of simultaneous linear algebraic equations can also be obtained by using Gaussian elimination method.

Figures 4(b) and 5(b) show the stress distribution along the interface. It can be seen from 4(b) that the shear stress of point 228 reaches the yielding criteria in (300 MPa) with the applied load p4.4MPa. However, the normal stress is not. We can keep increasing the applied load top5.7Mpa until the normal stress of point 111 reaches the yielding criteria

(520 MPa). There are three points, i.e., 111, 112, and 228 yield in shear direction at this moment as shown in Fig. 5(b). Both the shear and normal stresses reach the yielding criteria of points 111, 112, and 228, as in Figs 6(b) and 7(b), respectively, when applied load is up to6.0MPa. Figures 8(b) and 9(b) show the plastic displacement both in shear and normal direction of the four yielding point mention above.

Conclusions

It can be seen from the results mentioned above that the weight of one lateral plate of the robot have be reduced 1 kg of mass (from 3 kg to 2 kg) but won’t lower the strength that much. (allowable stress is reduced from7.1MPa to5.7MPa). The total mass of one rescue robot may be reduced at least 2 kg after weight reduction and therefore, can moves nimbly during the rescue process.

References

[1] A. Ghobanpoor and J. Zang, “Boundary Elemant Analysis of Crack Growth for Mixed Mode Center Slant Crack Problems,” Engineering Fracture Mechanics, Vol. 36, No.5 pp.

661-668, 1990.

[2] C.L. Tan and Y.L. Gao, “Treatment of Bimaterial Interface Crack Problems Using The Boundary Element Method,” Engineering Fracture Mechanics, Vol. 36, No. 6, pp.

919-932, 1990.

[3] L.J. Young and Y.P. Tsai, “A Boundary Element Application for Mixed Mode Loading Idealized Sawtooth Fracture Surface, ” Int'l J. of Solids and Structure, 36, 3239-3252, 1999.

[4] L.J. Young, “An Analysis of a Fracture Specimen for Rough Crack in Shear, ” Int’l J. of Fracture, 115, 3, 273~285, 2002.

[5] C.A. Brebbia , J.C.F Telles and L.C. Wrobel, “Boundary Element Techniqes ,”Springer - Verlag,New York,1970.

Figures

(a)

(b)

Fig. 1 (a) Outlook (b) dimension in mm of the lateral plate of rescue robot.

(a)

(b)

Fig 2 Schematic of BEM mesh of the lateral plate of rescue robot (a) before (b) after weight reduction.

Fig. 3 Loading conditions of the lateral plate of rescue robot.

(a) (b)

Fig 4 Shear stress distribution of the interface of the lateral plate (a) before weight reduction with p 5.4MPa (b) after weight reduction with p4.4MPa.

(a) (b)

Fig 5 Shear stress distribution of the interface of the lateral plate (a) before weight reduction with p 7.1MPa (b) after weight reduction with p5.7MPa

(a) (b)

Fig 6 Shear stress distribution of the interface of the lateral plate (a) before weight reduction with p 7.3MPa (b) after weight reduction with p6.0MPa.

(a) (b)

Fig 7 Normal stress distribution of the interface of the lateral plate (a) before weight reduction with p 7.3MPa (b) after weight reduction with p6.0MPa.

(a) (b)

Figure 8 Plastic deformation in shear direction of the lateral plate (a) before weight reduction with p 7.3MPa (b) after weight reduction with p6.0MPa.

(a) (b)

Figure 9 Plastic deformation in normal direction of the lateral plate (a) before weight reduction with p7.3MPa (b) after weight reduction with p6.0MPa.