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1 Trans. Korean Soc. Mech. Eng. A, Vol. 41, No. 7, pp. 585~590, < 학술논문 > DOI ISSN (Print) (Online) 브러쉬타이어모델의한계점분석및자유도개선 김종민 * 정사무엘 * 유완석 ** * 부산대학교기계설계공학과, ** 부산대학교기계공학부 Analsis of Limitation and Improvement of Degree of Freedom for Brush Tire Model Jong-Min Kim *, Samuel Jung * and Wan-Suk Yoo ** * Mechanical Design Engineering, Pusan Nat l Univ., ** Mechanical Engineering, Pusan Nat l Univ. (Received September 13, 2016 ; Revised April 30, 2017 ; Accepted Ma 10, 2017) Ke Words: Brush Tire Model( 브러쉬타이어모델 ), Tire Force( 타이어힘 ), Self-aligning Torque( 셀프 - 얼라이닝토크 ), Tire-road Friction( 타이어노면마찰계수 ) 초록 : 주행중인차량의거동은지면과타이어의접촉면에발생하는힘과토크에의해결정된다. 타이어에작용하는힘과토크를계산하기위하여다양한타이어모델들이사용되고있으며, 본연구에서사용하는브러쉬모델 (brush model) 은타이어의물리적특성을이용하여지면과의접촉면에발생하는힘과토크를구하는모델이다. 브러쉬모델은힘과토크를계산하는데다른타이어모델에비하여적은수의계수를필요로하지만, 힘과토크를계산하는데낮은자유도를가지기때문에시험데이터를정확하게표현하는데한계가있다. 본논문에서는이러한단점을개선하기위하여타이어마찰계수및특성계수에최소한의변수를추가한개선된모델을제안하고, 이의유효성을검증하였다. Abstract: Vehicle behavior is determined b forces and a torques generated at the ground contact surface of the tire. Various tire models are used to calculate the forces and torques acting on the tire. The brush model calculates the forces and torques with fewer coefficients than other tire models. However, owing to fewer degrees of freedom in calculating the forces, this model has limitations in precisel epressing measured data. In this stud, this problem was addressed b adding the least parameters to the friction coefficient and tire properties of the brush model, and the proposed model was validated. F F F M P - 기호설명 - : 수직하중 : 타이어의종방향힘 : 타이어의횡방향힘 : 셀프얼라이닝토크 : 수직압력 : 종방향마찰계수 k t k : 횡방향마찰계수 : 타이어의수직강성 : 타이어의종강성 Corresponding Author, C 2017 The Korean Societ of Mechanical Engineers k : 타이어의횡강성 a : 접지면길이 b : 접지면폭 σ : 슬립율 S : 슬립각 : 종방향치환상수 : 횡방향치환상수 sat : 셀프얼라이닝토크치환상수 r d r ud a s : 타이어수직변형된반지름 : 타이어반지름 : 타이어점착영역변형 : 타이어미끄럼영역변형

2 586 김종민 정사무엘 유완석 1. 서론 주행중인차량의거동은지면과접촉된타이어에발생하는힘과토크에의해서결정된다. 타이어에발생하는힘과토크를계산하기위하여다양한타이어모델이존재하며, 해석목적에따라가장적합한모델을사용한다. (1) 이러한타이어모델들은 Fiala 타이어모델이나 UA 타이어모델등과같이물리적특성을적용하는타이어모델도있는반면, MF 타이어모델과같이물리적특성없이입출력관계만으로힘과토크를얻는블랙박스모델도존재한다. 본논문에서는물리적특성을적용하여타이어에발생하는힘과모멘트를계산하는브러쉬모델 (brush model) 에대해연구하였다. 브러쉬모델은 1969 년 H.P Willumeit 에의해제안되었으며, (2) 1972 년 H.B Pacejk 는브러쉬모델의트레드부분이변형가능한스프링으로모델링하였다. (3) 1991 년 Sharp 는 combined force 를이용하여힘과토크를정의하였다. (3) 이러한브러쉬모델은가장기본적인타이어모델로적은수의타이어특성계수를이용하기때문에사용자가타이어에작용하는물리적인현상을쉽게이해할수있다. 하지만힘과토크를표현하기위한자유도가부족하여시험데이터를정확하게표현을할수없다. 따라서, 본논문에서는기존브러쉬모델보다힘과토크를정확히표현할수있도록마찰계수에변수를추가하여운동마찰특성이적용될수있도록하였고타이어특성계수에최소한의변수를추가하여기존모델을개선하였다 2. 타이어모델 본연구에서사용된브러쉬모델 (4) 은타이어접지면 (contact patch) 에작용하는힘을물리적특성을이용하여계산한타이어모델이다. 타이어에작용하는힘은순수종방향슬립 (pure longitudinal slip) 또는순수횡방향슬립 (pure lateral slip) 만존재하는경우에대해서만고려한다. 2.1 브러쉬모델브러쉬모델은접지면에슬립이발생하지않는점착 (adhesion) 영역과슬립이발생하는미끄럼 (sliding) 영역으로나누어타이어에발생하는힘을계산한다. 타이어접지면전체길이 2a 는 Fig. 1에서나타낸것과같이접지면중심과접지면끝점이이루는각 φ 와타이어반지름 rud 을이용하여 Fig. 1 Length of contact patch Fig. 2 Schematic of the brush model 구할수있으며, 식 (1) 과같이정의된다. a = r sin( φ) (1) ud 여기서, φ 와수직하중에의해변형된타이어반지름 rd 는각각식 (2), 식 (3) 과같이정의된다. 1 r d φ = cos rud r F (2) d = rud (3) kt UA 모델 (5) 및브러쉬모델과같이이론적타이어모델에서접지면에서작용하는수직압력 p 는식 (4) 와같이파라볼릭 (parabolic) 하게분포한다고가정한다. 2 ( a ) p = p0 1 2 a (4) 접지면에작용하는전체압력과수직하중은동

3 브러쉬타이어모델의한계점분석및자유도개선 587 일하므로식 (5) 에서압력상수 p 0 를정의할수있다. 식 (6) 에서접지면위치 에따른분포하중을구할수있다. Fig. 2에브러쉬모델의하중분포를나타냈다. ( ) 2 a 3F F = bp d p = 2a 2 0 1, 2 0 (5) 8ab 0 3F p ( ) = 3 { ( 2a ) } (6) 8ab 본논문에서는순수슬립 (pure slip) 상태만을고려하여주행중인차량에작용하는종방향힘은슬립율 σ 에따라발생하며, 횡방향힘은슬립각 S 에따라발생한다. 타이어에발생한힘에의해접지면변형이발생하며, 횡방향힘에의한점착 영역의변형 a 과미끄럼영역의변형눠식 (7), (8) 과같이정의된다. s 으로나 a ( ) = S (7) s ( ) = p( ) k 1 = 2a [ (2 a ) ] 2 여기서, 타이어의횡강성 k ( N / m ) 및치환상수 는식 (9) 와같이정의된다. k (8) 2 c 2a c =, = (9) 2b 3 F 타이어접지면에서점착영역과미끄럼영역이분할되는지점 s 에서의타이어변형을식 (10) 과같이정의하면, s 는식 (11) 과같이정의된다. ( ) 3 1 F 1 1 S, S F 1 F, S > (12) 식 (12) 와 combined force (3) 의정의로부터슬립에따른종방향힘은식 (13) 과같이정의된다. F 1 1 σ, σ F 1 F, σ > ( ) 3 1 (13) 셀프얼라이닝토크 M 는타이어의횡방향힘의합이타이어중심점뒤에존재하여발생하며, 횡방향힘과뉴매틱트레일 t의곱으로정의된다. M 는식 (14) 와같이정의된다. M = t(s) F (14) ( )( 1 ) 3, 1 Fa σ σ σ 1 0, σ > 2.2 브러쉬모델의한계브러쉬모델은적은수의타이어특성계수를사용하여물리적특성을이해하기쉽지만, 힘과토크를표현하기위한자유도가낮아정확한표현을할수없다. 브러쉬모델에서종방향힘은완전미끄럼영역에서 F 으로일정하게유지된다. 반면, 시험데이터에서는완전미끄럼영역에서종방향힘은슬립율에따라감소한다. 또한, 브러쉬모델에서횡 (s ) = (s ) (10) a s ( ) s = 2a 1 S (11) 타이어에작용하는횡방향힘 F 는점착영역과미끄럼영역에서작용하는힘을합하여구할수있다. F 는 σ = 1/ 일때, 타이어에작용하는힘은최대가되며, σ 1/ 에서는접지면에서완전미끄럼 (full sliding) 이발생하여슬립이증가하여도힘은일정하게유지된다. 슬립에따른횡방향힘은식 (12) 와같이정의된다. Fig. 3 Longitudinal force of Brush model

4 588 김종민 정사무엘 유완석 기존모델에서는최대정지마찰만을이용하기때문에운동마찰이적용되는완전미끄럼영역에서종방향힘을표현하는데한계를가진다. 운동마찰특성을적용하기위해완전미끄럼영역에서식 (16) 과같이마찰계수에변수 P 1 를추가하였다. Fig. 4 Lateral force of Brush model ma 1, σ ( σ) ma P1 σ, σ > (16) 순수횡방향슬립상태에서셀프얼라이닝토크의크기와기울기는횡방향힘및횡강성에의해결정된다. 셀프얼라이닝토크표현의자유도를증가시키기위하여식 (17) 과같이횡강성 k 에수직하중 F 에따른변수 P2, P3로정의하였다. k = P + P F (17) 2 3 Fig. 5 Self-aligning torque of Brush model 방향힘은정확하게표현이되나, 셀프얼라이닝토크는슬립각이커짐에따라시험데이터를정확하게표현할수없다. 하중에따른시험데이터를브러쉬모델의결과와비교하기위하여, 슬립율에따른종방향힘을 Fig. 3 에나타냈으며, 슬립각에따른횡방향힘및셀프얼라이닝토크를 Fig. 4, 5 에나타냈다. 3. 브러쉬모델개선 브러쉬모델의장점인적은타이어특성계수를유지하고힘과토크표현의자유도를높이기위하여, 타이어마찰계수및특성계수에최소한의변수를추가하여모델을개선하였다. 기존브러쉬모델에서마찰계수는타이어에작용하는힘이최대가되는지점에서발생하는최대 정지마찰을적용한다. 최대정지마찰계수식 (15) 로정의된다. ma 은 ma ma F = (15) F 또한, 접지면길이 a 는식 (1) 에서기하학적으로구할수있다. 하지만, 하중변화에따른실제접지면길이와는차이가있기때문에, 타이어힘의표현에한계를가진다. 따라서, 하중에따른접지면길이를추정하기위해서, 접지면길이 a 에수직하중 F 에따른변수 P4, P5 로정의하여식 (18) 과같이나타냈다. a = P4 + P5 F (18) 그리고실제접지면에서작용하는하중분포는완전한파라볼릭형태가아닌중심에서약간편향된형태를가지기때문에셀프얼라이닝토크에서한계슬립영역결정에오차가발생하여결과값이빠르게 0으로수렴한다. 이를고려하기위하여횡방향치환상수에변수 P 6 를추가하여셀프얼라이닝토크치환상수 sat 를식 (19) 와같이정의하였다. = (19) sat P 6 개선된모델의유효성을검증하기위하여, 식 (16~19) 를브러쉬모델에적용하여참조문헌 (6) 에서의시험데이터와비교하였다. 본논문에서제안한브러쉬모델의개선된결과를아래에나타냈으며, Fig. 6, 7 은슬립율에따른종방향힘및마찰계수를나타냈다. Fig. 8, 9 는슬립각에따른횡방향힘및셀프얼라이닝토크를나타냈다.

5 브러쉬타이어모델의한계점분석및자유도개선 589 Fig. 6 Longitudinal force of improved brush model Fig. 7 Friction over slip ratio Fig. 9 Self-aligning torque of improved brush model 4. 결론 본연구에서는브러쉬모델의힘과토크표현을개선하기위하여, 타이어마찰및특성계수에변수를추가하였다. 브러쉬모델의장점을유지하기위하여, 타이어특성계수에최소한의변수만을추가하였다. 최대정지마찰만을적용하는모델에서운동마찰을적용할수있도록마찰계수에변수를추가하였다. 그리고셀프얼라이닝토크의크기와기울기를개선하기위하여, 횡강성과접지면길이에변수를추가하여브러쉬모델에적용하였다. 개선된브러쉬모델의유효성을검증하기위하여시험데이터와커브피팅을수행하였다. 그결과, 기존모델에서표현할수없었던완전미끄럼영역에서운동마찰의적용으로종방향힘이감소되는것을표현할수있게되었다. 또한셀프얼라이닝토크의크기와기울기표현이개선되어기존모델대비 RMS 오차가 41.41% 가감소하였다. 후기 본연구는국방과학연구소주관으로수행중인 경전투용다중로봇통합운용 / 제어기술개발 과제의일환으로수행되었음. Fig. 8 lateral force of improved brush model 개선된브러쉬모델의결과로부터, 종방향힘은완전미끄럼영역에서운동마찰이적용되어슬립율이증가함에따라종방향힘이감소하는것을표현할수있게되었으며, 횡강성및접지면길이의변수추가로셀프얼라이닝토크표현의자유도가증가하여 RMS 오차는기존브러쉬모델보다 44.57% 에서 3.16% 로감소하였다. 참고문헌 (References) (1) Son, J. H., 1999, Comparison of Several Tire Models for the Vehicle Simulation, KSME, Vol. 2, pp. 922~927. (2) H. P. Willumeit, 1997, Vehicle Dnamics, Dong Meong. (3) Pacejka, H. B. and Sharp, R. S., 1991, Shear Force Development b Pneumatic Tres in Stead State Condtion : A Review of Modeling Aspects, Vehicle

6 590 김종민 정사무엘 유완석 Sstem Dnamics, pp. 121~176. (4) Rajesh Rajmani, 2006, Vehicle Dnamics and Control, Springer, New York, pp. 387~433. (5) Kim, G. H., 1997, A Tire Model for Practical Cornering Simulations of Vehicles, KSAE, Vol. 4, pp. 32~36. (6) MSC.sofware, 2013, User Manual: Using the PAC2002Tire Model. (7) Lee, M. S., 1996, Development of a UA Tire Model for the Automobile Dnamics Program AUTODYN7, KSAE, Vol. 1, pp. 100~106.

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