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Geçki tasarımında yeni eğri yaklaşımları

Geçki tasarımında yeni eğri yaklaşımları

##### Dosyalar

##### Tarih

1997

##### Yazarlar

Tarı, Ergin

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Günümüzde demiryollarında ve karayollarında yüksek hızların gerçekleştirilmesi ve daha yüksek hızlara erişmek için uğraşılması, yol-araç dinamiğine ilişkin özellikleri daha üstün geçiş eğrisi araştırmalarını gündeme getirmiştir. Yeni geçiş eğrisi önerileri yapılırken önerilen eğri, başka eğrilerle yol-araç dinamiği açısından karşılaştırılmakta ve yol-araç dinamiğini belirleyen en uygun kriterin "yanal sademe" olduğu vurgulanmaktadır. Bu çalışmada, öncelikle tüm taşıt hareketi ve yol geometrisi özelliklerini kapsayan bir yanal sademe bağıntısı verilerek geçki elemanı olarak kullanılabilecek iki yeni eğri önerilmiştir. Birinci eğri alinyiman ile dairesel kurbu ikinci dereceden değme koşullarım sağlayarak birleştirmektedir. Böylece farklı iki alinyimanın birleştirilmesi ana problemi, geçiş eğrisi-dairesel kurb-geçiş eğrisi yapısına sahip bileşik bir eğri ile çözülmüştür. Önerilen ikinci eğri, ikinci dereceden değme koşullarım sağlayarak iki alinyimam tek bir eğri ile birleştirmekte, böylece üç elemandan oluşan bileşik eğriler yerine tek bir elemanla geçki tasannu olanağım vermektedir. Yeni geçiş eğrilerini, bilinen bileşik eğriler ile yanal sademe açısından karşılaştırmak için öncelikle bilinen ve önerilen eğrilerin eğrilik ve dever fonksiyonları, yanal sademe fonksiyonunda kullanılabilecek şekilde türetilmiştir. Önerilen her iki eğri için, eğrilik ve dever fonksiyonları yanında aplikasyon elemanları ile ilgili bilgiler de verilmektedir. Önerilen her iki eğriyi bilinen eğriler ile yanal sademe açısından karşılaştırmak için, üç farklı hareket modeli (sabit hızlı, sabit pozitif ivmeli, sabit negatif ivmeli) dikkate alınarak yanal sademe fonksiyonları çıkarılmış ve diyagramları çizilmiştir. Eğrilerin karşılaştırılması amacıyla yanal sademe diyagramlarında sıçrama biçiminde yanal sademe süreksizlikleri, yanal sademenin genliği ve kırılma biçiminde yanal sademe süreksizliklerini dikkate alan üç karşılaştırma kriteri ve bir puanlama sistemi önerilmiş ve uygulanmıştır. Bu çalışmada önerilen Eğri II, hem ayrı ayrı kriterlere göre hem de genel değerlendirmede açık farkla birinci sırayı almıştır. Diğer bir deyişle, Eğri IFnin yol- araç dinamiğine ilişkin özellikleri, karşılaştırma kapsamına alınan dört bileşik eğriye oranla çok üstündür. Bu çalışmada önerilen diğer eğri olan Eğri I, genel değerlendirmede tüm eğriler arasında Eğri IF den sonra ikinci gelmiş, bileşik eğriler arasında ise birinci olmuştur.

One of the main problems in the design of route alignment is to join two straight lines with a curve. This problem was simply solved by using an arc of circular curve with a radius R. However, this solution failed as the speed increased, and it causes significant troubles related to vehicle-road dynamics. Therefore, it was necessary to find out new solutions to increase the traveling safety and comfort and to reduce the cost for maintenance and repair, especially on railways. Better solutions to the vehicle-road dynamics, can be summarized as:. changes made on vertical geometry of roads (e.g. superelevation). changes made on horizontal geometry of roads (e.g. transition curve). Transition curves are used for joining a straight line with a circle or for joining a circle of a radius, Ri with a circle of a radius R2. The task of these curves is to provide a regular transition by removing (or minimizing) the effects of instantaneous centrifugal force due to the reasons of sudden changes in curvature. Thus, horizontal geometry of a route becomes more suitable for vehicle-road dynamics. The conformity between horizontal and vertical geometry of route is also satisfied due to the use of superelevation along the transition curve. In the past, circle with a radius 2R and cubic parabola were used as transition curves. As the speed increases it was seen that these curves were insufficient and more suitable new curves theoretically developed were not able to be applied due to the difficulties in the calculation techniques. Spiral curve (clothoid), which has better characteristics for vehicle-road dynamics, has been used widely, due to the developments in the computer technology. The ideal case regarding the vehicle-road dynamics is that there is no breaking points on the diagrams of curvature belongs to the curved sections (formed by the arcs of transition curves and/or circular curves) and diagrams of superelevation which should have the same functional structure with curvature, and lateral change of acceleration should have a continuos diagram. There are discontinuities due to the breaking points on the diagrams of curvature and superelevation of combined curves in which spiral is used. In addition, the diagrams of lateral change of acceleration have discontinuities as jumps. As the speed increases, these discontinuities reduce travel comfort and cause wear and tear on vehicle wheels and rails on railways. The discontinuities also change horizontal geometry of rails. In the recent years, considerations of 400 km/h speeds on railways and extremely high speeds on motorways have improved the importance of relationship between horizontal and vertical geometry of roads with respect to road dynamics. Therefore many investigations have been carried out on developing new curves, which can be used for the design of horizontal geometry of roads. The advantages of these new curves related to road dynamics are more than spiral curves. In those investigations, different curves are generally compared with each other and the following criteria are considered:. difficulties on computational techniques,. possibility of any discontinuation caused by breaking point on the diagrams of curvature and superelevation,. lateral change of acceleration. Developments in computer technology have resolved most of the difficulties regarding the computational techniques. Computations, which were not imagined to be done 15-20 years ago, can now easily be carried out. Discontinuities in the diagrams of curvature are eliminated by the conformity of the functions of curvature being selected. In addition, superelevation that has the same functional structure with the curvature is applied to the roads. In this way, breaking points on the diagrams of superelevation are also controlled. In conclusion, lateral change of acceleration becomes important as a criteria in the comparisons of curves. Lateral change of acceleration is the change of the resultant acceleration occurring along the curve normal with respect to the time. The resultant acceleration is formed by the free forces (unbalanced) acting on a vehicle with a mass (m) and an instantaneous velocity (v), moving on a curved orbit. It can be expressed as da _ z = - n (1) dT Where, z : lateral change of acceleration (m/s3), a : resultant acceleration formed by free forces (m/s2), T : time (s), n : unit vector along the curve normal. Forces acting on a vehicle moving on a superelevated road are the gravitational force (P=mg), the centrifugal force (F=mkv2) and the motor force (Ft=mat) (Fig. la, lb). According to Fig la, the resultant force (R) of the gravitational and centrifugal forces can be divided into two: D and Fn perpendicular and parallel to the road platform respectively. D component is balanced with the reaction of the road platform and becomes effective in only vertical change of acceleration. Fn and Ft are the forces which form the lateral change of acceleration (Fig lb). From Fig la, Fn =s m(kv 2 - gtanx)cosa (2) can be derived. XI F=mkv2 u ?\ (a) Cross Section (b) Plane View Figure 1. Forces acting on vehicle moving on superelevated road. Introducing into (2) tana=- and cosa = Vu2 +1 2 g b 5n=r -bu)7u^FJn (3) can be derived. Where, a" : acceleration along the direction of the curve normal on an inclined road platform (m/s^), k = - : curvature of orbiting curve defined on a horizontal plane ( 1 /m), r g : gravitational acceleration (9.81 m/s^), b : horizontal width of the road platform (m), u : superelevation, inclination of the outer edge of the road with respect to the inner edge (m). Tangential acceleration caused by the motor force is _ dv- a.= - t ' dT (4) Xll where, T is the unit vector along the direction of the tangent of the curve, and the resultant acceleration of the motion is If v = v(l) = function of velocity related to the road, k = k(l) = function of curvature related to the road u = u(l) = function of super elevation related to the road, and g, b = constant values are known, lateral change of acceleration function given by (1) can be derived. The equation of lateral change of acceleration is obtained as follows: da bv dk kv2u + gb du (ç\ This equation (6) of lateral change of acceleration can be used for all conditions related to road geometry and vehicle motion. In the literature, it is suggested that this equation should be used when comparing curves based on the lateral change of acceleration. Thus, all the curves taken into account in this thesis are compared with a manner that based on lateral change of acceleration function given by (6). Clothoid (spiral), Bloss and sinusoidal curves, which are located in combined curves formed by straight line- 1st transition curve-circular curve-2nd transition curve-straight line (Figure 2), are examined by their functions used in lateral change of acceleration. Three motion models (motion with constant velocity, motion with positive acceleration and motion with negative acceleration) are taken into account. yQCircular Jf,^^i\ curve/^^^ i M Figure 2. Horizontal Geometry of The Combined Curves Two new transition curves are suggested in this thesis and they are named as curve I and curve II. The curve I which is located in combined curves (Figure 2) and curve II which joins two straight lines with only one curve (Figure 3) are examined by their functions used in lateral change of acceleration as well as their setting out parameters. Three motion models which are the same as the above mentioned models are taken into account. xui Figure 3. Horizontal Geometry of The Curve II The curvature and superelevation functions of the curve I are as follows: kl(t) = j(6t2-l5t + \0) «,(0*«^3(6r2-15/ + 10) The curvature and superelevation functions of the curve II are as follows: 823543 *"(') = 6912İ? 823543m, 6912 (t1 -4t6+6t5-4t* +t3) -(/7-4/6+6f5-4r4+f3) (7) (8) (9) (10) In order to compare the known curves with the suggested curves, I and II, the diagrams of lateral change of acceleration in three motion models are illustrated. Then, these diagrams are examined by means of three criteria. Criterion 1: Continuity of diagram of lateral change of acceleration is the most important criterion in the comparisons of curves because discontinuities in the form of jumps a) affect travel comfort in railways and motorways. b) cause change of horizontal geometry of rail c) cause wear on vehicle wheels and rails in railways. Therefore, it is clear that any curve, which does not have discontinuities in the form of jumps in the diagrams of lateral change of acceleration, is superior to the others which have discontinuities. In order to compare the curves according to the criterion 1, curve which has discontinuities is given a rank of -50 for each discontinuity point, and given 100 times of negative remark of sum of the absolute values of the discontinuities. xiv Criterion 2: The amplitudes of lateral change of acceleration function can be considered as a second criterion. In the literature, only transition from straight line to circular curve with spiral curve and motion model with constant velocity are examined. The values given in the literature (0.3 m/sec3-0.6m/sec3) are the maximum values of lateral change of acceleration with respect to the travel comfort. Whichever curve is used, it is possible to be successful not to exceed the boundary values by selecting geometric parameters (L, R, Umax etc.) properly or by velocity limitations. It should be noticed that discontinuities of lateral change of acceleration in the form of jumps have more effects on travel comfort than the values of large amount and continuous lateral change of acceleration. In order to compare the curves, "the amplitude of the lateral change of acceleration function" is suggested as the second criterion. The amplitude of the lateral change of acceleration function is defined as "the difference between the maximum and minimum lateral change of acceleration function values". All curves are given 100 times of negative remark of the amplitude of the lateral change of acceleration function. Criterion 3: The discontinuities in the form of break affect the traveling comfort and cause irregular change of the lateral change of acceleration, even though they do not cause as much as the ones in the form of jump. These type of discontinuities occur at the points on which two different alignment elements join by means of the structures of the functions of the curvature, superelevation and motion models. The discontinuity in the form of break is defined as the difference between the slope values of the two tangents on the diagram of lateral change of acceleration at the points mentioned above. In order to compare the curves according to the criterion 3, curve which has discontinuities in the form of break is given a rank of -5 for each discontinuity point, and given 10 times of negative remark of sum of the absolute values of the discontinuities. When the different criteria are taken into account, different orders are obtained. When the curves are sorted according to the criterion 1, all curves place in the same order except the clothoid. When the sequence according to the criterion 2 is obtained, the best curve is the curve II, the second curve is the clothoid and the third curve is the Bloss, the fourth curve is the curve I and the fifth curve is sinusoidal curve. When the sequence without clothoid according to the criterion 3 are obtained, all curves place in the same order except the Bloss. Therefore, a general evaluation in which all criteria are taken into account should be done in order to obtain the optimal sequence of curves. The ranks of all curves according to the each criterion and total ranks of all curves according to the all criteria are summarized in Table 1. Table 1. Individual and Total Ranks of All Curves According to The Each Criteria XV When evaluations related with the criteria are taken into consideration, it can be easily stated that the curve II is extremely suitable curve on the base of road-vehicle dynamics. Using the curve II in rail transportation system gives an opportunity to design routes with only two elements and using it consecutively in highways gives an opportunity to design routes with a single transition curve. The curve I takes place in the second order among all curves. The combined curve I, formed by the curve I and circular arc (Figure 2), can be used as an appropriate alternative to the known combined curves in highways and rail systems.

One of the main problems in the design of route alignment is to join two straight lines with a curve. This problem was simply solved by using an arc of circular curve with a radius R. However, this solution failed as the speed increased, and it causes significant troubles related to vehicle-road dynamics. Therefore, it was necessary to find out new solutions to increase the traveling safety and comfort and to reduce the cost for maintenance and repair, especially on railways. Better solutions to the vehicle-road dynamics, can be summarized as:. changes made on vertical geometry of roads (e.g. superelevation). changes made on horizontal geometry of roads (e.g. transition curve). Transition curves are used for joining a straight line with a circle or for joining a circle of a radius, Ri with a circle of a radius R2. The task of these curves is to provide a regular transition by removing (or minimizing) the effects of instantaneous centrifugal force due to the reasons of sudden changes in curvature. Thus, horizontal geometry of a route becomes more suitable for vehicle-road dynamics. The conformity between horizontal and vertical geometry of route is also satisfied due to the use of superelevation along the transition curve. In the past, circle with a radius 2R and cubic parabola were used as transition curves. As the speed increases it was seen that these curves were insufficient and more suitable new curves theoretically developed were not able to be applied due to the difficulties in the calculation techniques. Spiral curve (clothoid), which has better characteristics for vehicle-road dynamics, has been used widely, due to the developments in the computer technology. The ideal case regarding the vehicle-road dynamics is that there is no breaking points on the diagrams of curvature belongs to the curved sections (formed by the arcs of transition curves and/or circular curves) and diagrams of superelevation which should have the same functional structure with curvature, and lateral change of acceleration should have a continuos diagram. There are discontinuities due to the breaking points on the diagrams of curvature and superelevation of combined curves in which spiral is used. In addition, the diagrams of lateral change of acceleration have discontinuities as jumps. As the speed increases, these discontinuities reduce travel comfort and cause wear and tear on vehicle wheels and rails on railways. The discontinuities also change horizontal geometry of rails. In the recent years, considerations of 400 km/h speeds on railways and extremely high speeds on motorways have improved the importance of relationship between horizontal and vertical geometry of roads with respect to road dynamics. Therefore many investigations have been carried out on developing new curves, which can be used for the design of horizontal geometry of roads. The advantages of these new curves related to road dynamics are more than spiral curves. In those investigations, different curves are generally compared with each other and the following criteria are considered:. difficulties on computational techniques,. possibility of any discontinuation caused by breaking point on the diagrams of curvature and superelevation,. lateral change of acceleration. Developments in computer technology have resolved most of the difficulties regarding the computational techniques. Computations, which were not imagined to be done 15-20 years ago, can now easily be carried out. Discontinuities in the diagrams of curvature are eliminated by the conformity of the functions of curvature being selected. In addition, superelevation that has the same functional structure with the curvature is applied to the roads. In this way, breaking points on the diagrams of superelevation are also controlled. In conclusion, lateral change of acceleration becomes important as a criteria in the comparisons of curves. Lateral change of acceleration is the change of the resultant acceleration occurring along the curve normal with respect to the time. The resultant acceleration is formed by the free forces (unbalanced) acting on a vehicle with a mass (m) and an instantaneous velocity (v), moving on a curved orbit. It can be expressed as da _ z = - n (1) dT Where, z : lateral change of acceleration (m/s3), a : resultant acceleration formed by free forces (m/s2), T : time (s), n : unit vector along the curve normal. Forces acting on a vehicle moving on a superelevated road are the gravitational force (P=mg), the centrifugal force (F=mkv2) and the motor force (Ft=mat) (Fig. la, lb). According to Fig la, the resultant force (R) of the gravitational and centrifugal forces can be divided into two: D and Fn perpendicular and parallel to the road platform respectively. D component is balanced with the reaction of the road platform and becomes effective in only vertical change of acceleration. Fn and Ft are the forces which form the lateral change of acceleration (Fig lb). From Fig la, Fn =s m(kv 2 - gtanx)cosa (2) can be derived. XI F=mkv2 u ?\ (a) Cross Section (b) Plane View Figure 1. Forces acting on vehicle moving on superelevated road. Introducing into (2) tana=- and cosa = Vu2 +1 2 g b 5n=r -bu)7u^FJn (3) can be derived. Where, a" : acceleration along the direction of the curve normal on an inclined road platform (m/s^), k = - : curvature of orbiting curve defined on a horizontal plane ( 1 /m), r g : gravitational acceleration (9.81 m/s^), b : horizontal width of the road platform (m), u : superelevation, inclination of the outer edge of the road with respect to the inner edge (m). Tangential acceleration caused by the motor force is _ dv- a.= - t ' dT (4) Xll where, T is the unit vector along the direction of the tangent of the curve, and the resultant acceleration of the motion is If v = v(l) = function of velocity related to the road, k = k(l) = function of curvature related to the road u = u(l) = function of super elevation related to the road, and g, b = constant values are known, lateral change of acceleration function given by (1) can be derived. The equation of lateral change of acceleration is obtained as follows: da bv dk kv2u + gb du (ç\ This equation (6) of lateral change of acceleration can be used for all conditions related to road geometry and vehicle motion. In the literature, it is suggested that this equation should be used when comparing curves based on the lateral change of acceleration. Thus, all the curves taken into account in this thesis are compared with a manner that based on lateral change of acceleration function given by (6). Clothoid (spiral), Bloss and sinusoidal curves, which are located in combined curves formed by straight line- 1st transition curve-circular curve-2nd transition curve-straight line (Figure 2), are examined by their functions used in lateral change of acceleration. Three motion models (motion with constant velocity, motion with positive acceleration and motion with negative acceleration) are taken into account. yQCircular Jf,^^i\ curve/^^^ i M Figure 2. Horizontal Geometry of The Combined Curves Two new transition curves are suggested in this thesis and they are named as curve I and curve II. The curve I which is located in combined curves (Figure 2) and curve II which joins two straight lines with only one curve (Figure 3) are examined by their functions used in lateral change of acceleration as well as their setting out parameters. Three motion models which are the same as the above mentioned models are taken into account. xui Figure 3. Horizontal Geometry of The Curve II The curvature and superelevation functions of the curve I are as follows: kl(t) = j(6t2-l5t + \0) «,(0*«^3(6r2-15/ + 10) The curvature and superelevation functions of the curve II are as follows: 823543 *"(') = 6912İ? 823543m, 6912 (t1 -4t6+6t5-4t* +t3) -(/7-4/6+6f5-4r4+f3) (7) (8) (9) (10) In order to compare the known curves with the suggested curves, I and II, the diagrams of lateral change of acceleration in three motion models are illustrated. Then, these diagrams are examined by means of three criteria. Criterion 1: Continuity of diagram of lateral change of acceleration is the most important criterion in the comparisons of curves because discontinuities in the form of jumps a) affect travel comfort in railways and motorways. b) cause change of horizontal geometry of rail c) cause wear on vehicle wheels and rails in railways. Therefore, it is clear that any curve, which does not have discontinuities in the form of jumps in the diagrams of lateral change of acceleration, is superior to the others which have discontinuities. In order to compare the curves according to the criterion 1, curve which has discontinuities is given a rank of -50 for each discontinuity point, and given 100 times of negative remark of sum of the absolute values of the discontinuities. xiv Criterion 2: The amplitudes of lateral change of acceleration function can be considered as a second criterion. In the literature, only transition from straight line to circular curve with spiral curve and motion model with constant velocity are examined. The values given in the literature (0.3 m/sec3-0.6m/sec3) are the maximum values of lateral change of acceleration with respect to the travel comfort. Whichever curve is used, it is possible to be successful not to exceed the boundary values by selecting geometric parameters (L, R, Umax etc.) properly or by velocity limitations. It should be noticed that discontinuities of lateral change of acceleration in the form of jumps have more effects on travel comfort than the values of large amount and continuous lateral change of acceleration. In order to compare the curves, "the amplitude of the lateral change of acceleration function" is suggested as the second criterion. The amplitude of the lateral change of acceleration function is defined as "the difference between the maximum and minimum lateral change of acceleration function values". All curves are given 100 times of negative remark of the amplitude of the lateral change of acceleration function. Criterion 3: The discontinuities in the form of break affect the traveling comfort and cause irregular change of the lateral change of acceleration, even though they do not cause as much as the ones in the form of jump. These type of discontinuities occur at the points on which two different alignment elements join by means of the structures of the functions of the curvature, superelevation and motion models. The discontinuity in the form of break is defined as the difference between the slope values of the two tangents on the diagram of lateral change of acceleration at the points mentioned above. In order to compare the curves according to the criterion 3, curve which has discontinuities in the form of break is given a rank of -5 for each discontinuity point, and given 10 times of negative remark of sum of the absolute values of the discontinuities. When the different criteria are taken into account, different orders are obtained. When the curves are sorted according to the criterion 1, all curves place in the same order except the clothoid. When the sequence according to the criterion 2 is obtained, the best curve is the curve II, the second curve is the clothoid and the third curve is the Bloss, the fourth curve is the curve I and the fifth curve is sinusoidal curve. When the sequence without clothoid according to the criterion 3 are obtained, all curves place in the same order except the Bloss. Therefore, a general evaluation in which all criteria are taken into account should be done in order to obtain the optimal sequence of curves. The ranks of all curves according to the each criterion and total ranks of all curves according to the all criteria are summarized in Table 1. Table 1. Individual and Total Ranks of All Curves According to The Each Criteria XV When evaluations related with the criteria are taken into consideration, it can be easily stated that the curve II is extremely suitable curve on the base of road-vehicle dynamics. Using the curve II in rail transportation system gives an opportunity to design routes with only two elements and using it consecutively in highways gives an opportunity to design routes with a single transition curve. The curve I takes place in the second order among all curves. The combined curve I, formed by the curve I and circular arc (Figure 2), can be used as an appropriate alternative to the known combined curves in highways and rail systems.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1997

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1997

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1997

##### Anahtar kelimeler

Bilgisayar destekli tasarım,
Eğriler,
Computer aided design,
Curves