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A mechanical system is often modeled as a multi-body system with non-smoothness. Typical examples are the noises and vibrations produced in railway brakes, impact print hammers, or chattering of machine tools. These effects are due to the non-smooth characteristics such as clearance, impact, intermittent contact, dry friction, or a combination of these effects. In a non-smooth system, neither of the time evolutions of the displacements and the velocities are requested to be smooth. Beam-cam device is an important kind of impacting system which has a wide range of applications. The rotation of the cam at some constant speed provides a force to operate the beam. The most common example is the valve trains of internal combustion engines, where the rotation of the cam imparts the proper motion to the engine valves through the follower while a spring provides a restoring force necessary to maintain contact between the components. The impact on beam-cam is a typical oblique-impact. It has been observed that under variations of the cam rotational speed and other parameters, the follower can exhibit a complex behavior including bifurcations and chaos. We study a rigid flexible coupling system, which moves in the horizontal plane, and is composed of the hub, the flexible beam and a cam with constant rotating speed. Considering the second-order coupling of axial displacement which is caused by the transverse deformation of the beam, the kinetic energy and the potential energy of the whole system are calculated. The governing equations of the flexible beam-cam oblique-impact system are derived from Hamilton theory, when taking into account the second-order coupling quantity of axial displacement caused by the transverse displacement of the beam. Hertz contact theory and nonlinear damping theories are used to establish the contact model. By the equivalent conversion method in structural mechanics, the deflection curve of flexible beam is calculated. The acceleration at the contact point of beam and cam is used to judge whether they are separate, or contacted, or impacting. Due to the flexibility of beam, the impact point of beam-cam always changes with time and speed. We propose a method, which is a trial calculation method, to determine the impact point of flexible beam-cam. Simulation results show that there is transverse vibration at the free end of flexible beam. There is inter-coupling among the flexible of beam, the large range of motion, and the impact. After the impact, the rotation angle of the flexible beam changes with time and the angle amplitude mainly decreases with the increase of the time, but the regularity is poor.
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Keywords:
- oblique-impact /
- Hamilton theory /
- flexible bar /
- rigid-flexible coupling
[1] Pfeiffer F, Glocker C 2000 PMM-J. Appl. Math. Mech. 64 773
[2] Han Y, Yang J K, Koji M 2011 Chin. J. Mech. Eng. 24 1045
[3] Li H G, Cao Z Q, Lu H F, Shen Q S 2003 Appl. Phys. Lett. 83 2757
[4] Alzate R, di Bernardo M, Montanaro U, Santini S 2007 Nonlinear Dyn. 50 409
[5] Wu X, Wen G L, Xu H D, He L P 2015 Acta Phys. Sin. 64 200504 (in Chinese)[伍新, 文桂林, 徐慧东, 何莉萍2015物理学报 64 200504]
[6] He X S, Deng F Y, Wu G Y, Wang R 2010 Acta Phys. Sin. 59 25 (in Chinese)[和兴锁, 邓峰岩, 吴根勇, 王睿2010物理学报 59 25]
[7] Ding H, Chen L Q, Yang S P 2012 J. Sound Vib. 331 2426
[8] Ding H, Chen L Q 2010 J. Sound Vib. 329 3484
[9] Hua W J, Zhang D G 2007 J. Mech. Eng. 43 222(in Chinese)[华卫江, 章定国2007机械工程学报 43 222]
[10] Shen Y N, Stronge W J 2011 Eur. J. Mech. A:-Solid. 30 457
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[1] Pfeiffer F, Glocker C 2000 PMM-J. Appl. Math. Mech. 64 773
[2] Han Y, Yang J K, Koji M 2011 Chin. J. Mech. Eng. 24 1045
[3] Li H G, Cao Z Q, Lu H F, Shen Q S 2003 Appl. Phys. Lett. 83 2757
[4] Alzate R, di Bernardo M, Montanaro U, Santini S 2007 Nonlinear Dyn. 50 409
[5] Wu X, Wen G L, Xu H D, He L P 2015 Acta Phys. Sin. 64 200504 (in Chinese)[伍新, 文桂林, 徐慧东, 何莉萍2015物理学报 64 200504]
[6] He X S, Deng F Y, Wu G Y, Wang R 2010 Acta Phys. Sin. 59 25 (in Chinese)[和兴锁, 邓峰岩, 吴根勇, 王睿2010物理学报 59 25]
[7] Ding H, Chen L Q, Yang S P 2012 J. Sound Vib. 331 2426
[8] Ding H, Chen L Q 2010 J. Sound Vib. 329 3484
[9] Hua W J, Zhang D G 2007 J. Mech. Eng. 43 222(in Chinese)[华卫江, 章定国2007机械工程学报 43 222]
[10] Shen Y N, Stronge W J 2011 Eur. J. Mech. A:-Solid. 30 457
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