四行程引擎的引擎正時系統用以控制汽門開關時序，使閥桿能夠配合活塞的位置順利完成進氣、壓縮、爆炸與排氣四個動作。當引擎運轉時，動力由汽缸內油氣爆炸帶動主動輪旋轉，經鏈條傳遞至從動輪，使凸輪旋轉並驅動閥桿移動，控制汽門開關。其中由於閥彈簧與機構慣性的影響，從動輪的阻扭矩將隨凸輪軸角位置的改變呈上下波動，此一扭矩波動將造成從動輪的轉速震盪、機構的振動、噪音與耗能等問題。
有鑒於此，本文針對400 cc機車汽門機構建立其數學模型，以得到機構各接頭在不同轉速下的受力情形及從動輪阻扭矩，並利用貝茲曲線(Bézier Curve)參數化閥桿升程，在不改變引擎機構尺寸的前提下，以最佳化方法找出最小化凸輪軸阻扭矩的進排氣凸輪輪廓。
引擎正時系統用以控制汽門開啟與關閉的時機，運轉時動力自曲軸藉由靜音鏈傳遞至凸輪軸，以驅動汽門開關。鏈條張力過小時，將造成運轉時鏈條振動幅度增加而有跳齒的可能，導致引擎時序錯亂，甚至於引擎失效。因此在引擎正時系統中設置了鬆邊導桿與鏈條張力器維持鏈條張力，然而在長時間的運轉下，鬆邊導桿容易斷裂，為了瞭解鬆邊導桿損壞的原因，本研究以基於多體動力學之商用軟體建立550 cc機車引擎正時系統模型，並進行動力分析以瞭解鏈條張力、導桿受力與鏈條張力器受力等動態行為，並藉由調整鏈條張力器頂出力與安裝位置來觀察與討論鏈條張力器對於鏈條張力的影響，並藉由改變鏈條張力器參數來減低鏈條張力。

Design for the Torque Minimization of Engine Camshafts and Dynamic Analysis of a Timing Chain System

Author: Chain-An Wu
Advisor: Chao-Chieh Lan
Department of Mechanical Engineering
National Cheng Kung University

SUMMARY
Design for the Torque Minimization of Engine Camshafts:
This thesis presents the optimal design of a cam profile to minimize the engine camshafts torque. Valve lift can be parameterized with Bezier curve so the cam profile is determinded. Torque of rotary machines causes unwanted vibration that would impair their performance and reliability. The combination of inertia, driving, and static torque on engine crankshafts and camshafts is the major source of vehicle vibration. While previous methods focused on suppressing or isolating vibration motion from engine to chassis, the proposed method seeks to directly reduce the torque on engine shafts without change any size of the engine system.The optimal cam profile can be synthesized such that the camshaft torque cancels itself by antagonism of inertial and static torque at the speed interval which user specified.
Dynamic Analysis of a Timing Chain System:
Engine timing chain systems are used to control engine valves. In timing systems, silent chains are applied to transmit power from crankshafts to camshafts. In practice, ensuring appropriate tension is essential, since small tension may cause unsmooth motion and vibration. Therefore, guides and chain tensioners are used to maintain proper tension. However, the silent chains are prone to damage after long-term operations. In this study, a timing system for a motorcycle engine is modeled by commercial software based on multi-body dynamics. Dynamic analysis is conducted to investigate the dynamic behavior of the timing system, such as chain tension, contact forces and tensioner force.By adjusting the installed position and output force of the tensioner for reducing silent chain tension. To find the maximum chain stress, stress analysis of the silent chain is performed by software based on finite element method after dynamic analysis.
Keywords: Camshafts torque, Bezier curve, Engine timing chain system, dynamic analysis, chain tensioner
INTRODUCTION
Design for the Torque Minimization of Engine Camshafts:
Vibration of rotary machines causes increased wear, unpleasant noise, and imprecise motion that would reduce their performance and reliability. Rotary machine vibration is primarily due to the oscillating rotational speeds that are created by periodic torque fluctuations on rotating shafts. As shown in Figure 1, torque can be categorized into three major types based on the source of fluctuation. The first type is due to the non-negligible inertia force from the acceleration and deceleration of moving parts connected to a rotating shaft. The magnitude of torque fluctuation is in general proportional to the square of shaft speed. Hence the induced speed fluctuation becomes more severe at high speeds. The second type is originated from the uneven input driving actuation on a shaft within one full rotation. For example, the ignition of an internal combustion engine results in highly uneven gas torque on its crankshaft. Switched reluctance motors [3] produce non-constant torque for each phase due to their nonlinear magnetic characteristics. Similarly, air vane motors [4] produce fluctuating torques due to their unmatched vane torques. Unlike the first type, the magnitude of torque fluctuation of the second type is independent of shaft speed. It only depends on the shaft angle and input power (Pin). The third type is caused by the valve spring. This type of fluctuation only depends on the shaft angle and is independent of shaft speed.

Figure 1 Three types of torque fluctuation
Internal combustion engines exhibit significant vibration from two sources: the speed fluctuation of the crankshaft and the camshaft. The crankshaft speed fluctuation is due to the unsmooth torque from the combination of slider-crank inertia (first type) and piston gas force (second type). The camshaft speed fluctuation is due to the inertia of the valve mechanism (first type) and the fluctuating static torque required to compress and release the valve springs (third type). Engine-mount systems with active or passive vibration control methods [5~7] have been developed to reduce the effect of engine vibration. However, these methods primarily focused on suppressing or isolating the vibration from engine to chassis. The vibration inside engines remains unsolved. It is still a challenge to directly remove the torque fluctuation on crankshafts or camshafts without including extra complexity and cost.
This paper aims at designing a optimal cam profile to reduce the fluctuating inertia (first type) and static (third type) torque of an engine camshaft . The proposed method is based on the camshaft torque cancels itself by antagonism of inertial and static torque at the speed interval which user specified.

Dynamic Analysis of a Timing Chain System:
An engine timing system consists of a silent chain drive and a valve mechanism, as shown in Figure 2. It is applied in a motorcycle engine to control valves. The silent chain drive is used to deliver power from the crankshaft to the camshaft. To suppress vibration of the chain and keep proper tension, guides and a chain tensioner are used in the silent chain drive. However, the silent chain is one of the most likely damaged parts in the timing system due to excessive load from tensioner. To improve the durability of the silent chain, the engine timing system must be analyzed to find out how to reduce silent chain load.

Figure 2. Cad of the engine timing system
In this study, commercial software, RecurDyn, is utilized to model the timing system and analyze the dynamic behavior of silent chain drive, such as chain tension, contact forces and tensioner force. After dynamic analysis of the timing system, tensioner forces can be obtained. Then, adjusting the installed position and output force of the tensioner for reducing chain tension. To find the maximum silent chain stress, stress analysis of the silent chain is performed by software based on finite element method after dynamic analysis.
MATERIALS AND METHOD
Design for the Torque Minimization of Engine Camshafts:
First of all, this study presents a 400 cc engine timing system model. Figure 3 shows the schematic front view of the valve mechanism. Given input rotation θ from the camshaft, the exhaust cam causes the exhaust rocker to rotate with angle ϕe whereas the inlet cam causes the inlet rocker to rotate with angle ϕi. The rotation of the rockers opens and closes their respective valves. By assuming constant camshaft speed, we can obtain force equations from free-body diagram of valve mechanism, hence the input torque of valve mechanism can be solved. Figure 3 shows the input torque of valve mechanism at different crank speeds, the speed ratio of crankshaft to camshaft is 2:1.

Figure 3 Schematic front view of the valve mechanism [12]

Second, in order to minimize the torque fluctuation of valve mechanism, valve lift can be parameterized with Bezier curve so the cam profile is determinded. Using optimal method to design a new cam profile without changing any size of the engine system.

Figure 4 Valve torque at different crank speed [12]

Dynamic Analysis of a Timing Chain System:
In the RecurDyn model, the chain links are connected by bushing force elements which consist six direction spring-dampers. Because of large link stiffness, clearances in a chain are also considered, even though small clearances have a considerable effect on chain tension. To determine the stiffness and clearances of the silent chain, a tensile test was performed as shown in Figure 5. The results of the tensile test are shown in Figure 6. Then, set the clearance and radial stiffness of bushing force elements in the simulation to fit the experimental result.

Figure 5 Configuration of tensile test

Figure 6 Simulation and experimental results of the tensile test
Chain tensioner output force model consists of three types force, which displayed in Figure 7. The first type is outer and inner spring force (Fsout, Fsin), the second type is oil pressure force (FLPC) and the third type is high pressure chamber force (FHPC), sum of these force is no-return force (Fp). According to the definition of bulk modulus, there exists a equivalent oil stiffness (koil) which presents high pressure chamber force (FHPC) due to the compressibility of oil. As showed in Figure 8, equivalent oil stiffness (koil) and outer inner spring properties are imported in RecurDyn tensioner force model.

Figure 7 Tensioner force

Figure 8 RecurDyn HAT Type A

RESULT AND DISCUSSION
Design for the Torque Minimization of Engine Camshafts:
After optimizing, new cam profile(camshaft 7500 rpm, SL = 8000 rpm) is displayed in Figure 9, the camshaft torque minimization result as Figure 10 and Figure 11 shown. Figure 10 shows the camshaft torque RMS value when camshaft speed is under 4000~8000 rpm and engine speed limit SL = 8000, 9000 and 10000 rpm, RMS value can present the fluctuation of the curve. Compared to different situations, the optimal result show that when camshaft is under 7500 rpm with SL = 8000 rpm, RMS torque decreased by 21.45% at most. Figure 11 shows that the camshaft torque peak value with camshaft speed under 4000~8000 rpm and engine speed limit SL = 8000, 9000 and 10000 rpm. With camshaft speed under 6200 rpm amd SL = 8000 rpm, peak1 torque value decreased by 44.17% at most; with camshaft speed under 5800 rpm and SL = 8000 rpm, peak2 torque value decreased by 37.75% at most.

Figure 9 Exhaust and inlet cam profile (camshaft 7500 rpm, SL = 8000 rpm)

Figure 10 Camshaft 4000~8000 rpm different SL torque rms value

(a) Peak1 (b) Peak2
Figure 11 Camshaft 4000~8000 rpm different SL torque peak value
Dynamic Analysis of a Timing Chain System:
The engine timing system model is used to investigate the dynamic behavior of the silent chain drive. Figure 12 shows the simulation results at driving sprocket under 1000, 6000 rpm, including tensioner displacement and force, camshaft torque generated by the valve mechanism and tension force of a chain link. Divided into six parts, the chain links 0~44 is slack span, 44~54 is inlet driven sprocket, 54~68 is between two driven sprockets, 68~78 is exhaust driven sprocket, 78~120 is tight span, 120~132 is driving sprocket. The tension of tight and slack span is dominated by the camshaft torque, and the slack-span tension affects the tensioner force due to the contact between chain and guide.

(a) Driving sprocket 1000 rpm (b) Driving sprocket 6000 rpm
Figure 12 Camshaft torque with tensioner displacement and force
When tensioner force consists of outer spring force, The output force will cause chain tension so high that the chain damaged, since tensioner force consists of outer spring force .independent with installed position as Figure 13 shown. Figure 14 shows outer spring removed, chain tension will decrease when installed position change.

(a) Chain tension peak value (b) Chain tension RMS value
Figure 13 Chain tension (w/ outer spring)

(c) Chain tension peak value (d) Chain tension RMS value
Figure 14 Chain tension (w/o outer spring)

CONCLUSION
Design for the Torque Minimization of Engine Camshafts:
This thesis presentes a optimal cam profile applied to reduce valve torque at the speed interval we specified. A torque model of camshaft and valve mechanism showed that the inertia and staitc torque fluctuation were a result of unavoidable mechanical inertia and large stiffness of the valve springs. With optimal specifically designed cam profile, the torque fluctuation on camshafts could be reduced. The RMS value of the fluctuating torque curves decreased by 21.45% at most, the value of fluctuating torque curves of peak1 decreased by 44.17% at most, the value of fluctuating torque curves of peak2 decreased by 37.75% at most.
Dynamic Analysis of a Timing Chain System:
The RecurDyn model has been developed to conduct dynamic analysis of the engine timing system. In this model, each link of the chain is connected by bushing force elements. A continuous contact force model has been applied to calculate the contact among every part. It has been shown that the chain tension and tensioner force are affected by the camshaft torque. After dynamic analysis, the chain tension is reduced by adjusting the installed position and output force of tensioner. Finally, the result of simulation show that the absence of outer spring obviously reduces the chain tension.

[1] 林登穎(2015)。引擎正時系統受力最小化之分析與實驗(碩士論文)。國立成功大學機械工程學系，台南市，台灣。
[2] Williams, J. (2012). Introduction to analytical methods for internal combustion engine cam mechanisms: Springer Science & Business Media.
[3] Sun, W., Li, Y., Huang, J., & Zhang, N. (2015). Vibration effect and control of in-wheel switched reluctance motor for electric vehicle. Journal of Sound and Vibration, 338, 105-120.
[4] Cheng, C.-W., Lan, C.-C., & Tseng, C.-Y. (2012). Modeling and design of air vane motors for minimal torque ripples. Journal of Mechanical Design, 134(5), 051003.
[5] Lee, B.-H., & Lee, C.-W. (2009). Model based feed-forward control of electromagnetic type active control engine-mount system. Journal of Sound and Vibration, 323(3), 574-593.
[6] Hausberg, F., Scheiblegger, C., Pfeffer, P., Plöchl, M., Hecker, S., & Rupp, M. (2015). Experimental and analytical study of secondary path variations in active engine mounts. Journal of Sound and Vibration, 340, 22-38.
[7] Peng, Z., & Lang, Z. (2008). The effects of nonlinearity on the output frequency response of a passive engine mount. Journal of Sound and Vibration, 318(1), 313-328.
[8] Gauthier, J.-P., & Micheau, P. (2008). Extremal harmonic active control of power for rotating machines. Journal of Sound and Vibration, 318(4), 663-677.
[9] Fan, H., Jing, M., Wang, R., Liu, H., & Zhi, J. (2014). New electromagnetic ring balancer for active imbalance compensation of rotating machinery. Journal of Sound and Vibration, 333(17), 3837-3858.
[10] 查宇衡(2007)。應用具彈簧之倒置凸輪機構平衡機構輸入扭矩的最佳設計(碩士論文)。國立成功大學機械工程學系，台南市，台灣。
[11] Norton, R. (2009). Cam design and manufacturing handbook: Industrial Press.
[12] 侯柏均(2016)。引擎凸輪軸扭矩平衡凸輪機構之設計與實驗(碩士論文)。國立成功大學機械工程學系，台南市，台灣。
[13] Chang, W. T., & Fang, H. L. K. Y. Y. (2015, November). Optimal Synthesis of Disk Cam Mechanisms with a Roller Follower Considering Cam Size Minimization and Rotational Balancing. In Proceedings of the 14th IFToMM World Congress (pp. 150-159).
[14] Macfarlane, S., & Croft, E. A. (2003). Jerk-bounded manipulator trajectory planning: design for real-time applications. IEEE Transactions on Robotics and Automation, 19(1), 42-52.
[15] Gyorfi, J. S., & Wu, C. H. (2006). A minimum-jerk speed-planning algorithm for coordinated planning and control of automated assembly manufacturing. IEEE transactions on automation science and engineering, 3(4), 454-462.
[16] Li, H., Le, M. D., Gong, Z. M., & Lin, W. (2009). Motion profile design to reduce residual vibration of high-speed positioning stages. IEEE/ASME Transactions On Mechatronics, 14(2), 264-269.
[17] Bianco, C. G. L. (2013). Minimum-jerk velocity planning for mobile robot applications. IEEE Transactions on Robotics, 29(5), 1317-1326.
[18] Gasparetto, A., Lanzutti, A., Vidoni, R., & Zanotto, V. (2011). Validation of minimum time-jerk algorithms for trajectory planning of industrial robots. Journal of Mechanisms and Robotics, 3(3), 031003.
[19] Qiu, H., Lin, C. J., Li, Z. Y., Ozaki, H., Wang, J., & Yue, Y. (2005). A universal optimal approach to cam curve design and its applications. Mechanism and Machine Theory, 40(6), 669-692.
[20] Tsay, D. M., & Huey, C. O. (1993). Application of rational B-splines to the synthesis of cam-follower motion programs. Journal of Mechanical Design, 115(3), 621-626.
[21] Yoon, K., & Rao, S. S. (1993). Cam motion synthesis using cubic splines. Journal of Mechanical Design, 115(3), 441-446.
[22] Lampinen, J. (2003). Cam shape optimisation by genetic algorithm. Computer-Aided Design, 35(8), 727-737.
[23] Nguyen, V. T., & Kim, D. J. (2007). Flexible cam profile synthesis method using smoothing spline curves. Mechanism and machine theory, 42(7), 825-838.
[24] Jeon, H. S., Park, K. J., & Park, Y. S. (1989). An optimal cam profile design considering dynamic characteristics of a cam-valve system. Experimental Mechanics, 29(4), 357-363.
[25] Sandgren, E., & West, R. L. (1989). Shape optimization of cam profiles using a B-spline representation. Journal of Mechanisms, Transmissions, and Automation in design, 111(2), 195-201.
[26] Acharyya, S., & Naskar, T. K. (2008). Fractional polynomial mod traps for optimization of jerk and hertzian contact stress in cam surface. Computers & structures, 86(3-5), 322-329.
[27] Kwakernaak, H., & Smit, J. (1968). Minimum vibration cam profiles. Journal of Mechanical Engineering Science, 10(3), 219-227.
[28] Xiao, H., & Zu, J. W. (2009). Cam profile optimization for a new cam drive. Journal of mechanical science and technology, 23(10), 2592-2602.
[29] Takagishi, H., Muguruma, K., Takahashi, N., & Nagakubo, A. (2008). Analysis of effect of tensioner on chain system (No. 2008-01-1496). SAE Technical Paper.
[30] Krueger, K., Engelhardt, T., Ginzinger, L., & Ulbrich, H. (2007). Dynamical analysis of hydraulic chain tensioners-experiment and simulation (No. 2007-01-1461). SAE Technical Paper.15
[31] Krueger, K., Ginzinger, L., & Ulbrich, H. (2008). Influences of Leakage Gap Variations on the Dynamics of Hydraulic Chain Tensioners–Experiment and Simulation (No. 2008-01-0294). SAE Technical Paper.
[32] Huber, R., & Clauberg, J. (2017). Physically Motivated Model for Efficient Dynamic Simulation of Chain Tensioners with Labyrinth Seals. SAE International Journal of Engines, 10(2017-01-1073), 656-667.
[33] Hatakeyama, R., & Niino, T. (2018). Analysis of Rotational Vibration Mechanism of Camshaft at High Engine Speed in Engines with In-Line Four-Cylinder DOHC Configuration (No. 2018-32-0072). SAE Technical Paper.
[34] 林冠儒(2014)。機車引擎正時鏈系統之動態分析與實驗驗證(碩士論文)。國立成功大學機械工程學系，台南市，台灣。
[35] Waldron, K. J., Kinzel, G. L., & Agrawal, S. K. (2016). Kinematics, dynamics, and design of machinery: John Wiley & Sons.
[36] Shigley, J. E. (2011). Shigley's mechanical engineering design: Tata McGraw-Hill Education.
[37] Function Bay, Inc., 2012, “RecurDyn Help.”
[38] L. Xu, Y. Yang, Z. Chang and J. Liu, 2010, “Dynamic modeling of a roller chain drive system considering the flexibility of input shaft,” Chinese Journal of Mechanical Engineering, 23(3), pp. 367-374.
[39] K. Dan and T. Kawakami, 2009, “Research on dynamic behavior simulation technology for cam-drive mechanism in single-cylinder engines,” SAE paper 2009-32-0089.
[40] 陳佑典(2018)。引擎正時系統張力器之分析模擬, 第21屆全國機構與機器設計學術研討會。國立成功大學機械工程學系，台南市，台灣。