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In order to quickly and accurately analyze the complex behavior of cam-follower oblique-impact system, a mathematical model which can describe separation, impact, and contact was established in this paper. The transient impact hypothesis was extended, and the oblique collision model was established by considering the tangential slip. Moreau time-stepping method was employed to solve the linear complementarity problem which transformed by the oblique-impact equations. The simulation results show that the cam and follower kept permanent contact when the cam rotational speed was low. With the increase of the cam rotational speed, the cam and follower would be separated and then impact under the gravity action. The system performance shows very complex nonlinear characteristics.

Cam-follower devices are important classical impacting systems which are used in a wide range of applications. The rotation of the cam at some constant speed provides the force to operate the follower. The most common example is valve trains of internal combustion engines, where the cam rotation imparts through the follower the proper motion to the engine valves while a spring provides the restoring force necessary to maintain contact between the components [

Alzate et al. [

A linear complementarity problem (LCP) is a problem of the form

LCP equations may have unique solution, no solution, or multiple solutions. Many problems in scientific computing and engineering applications may lead to solutions of LCP. For example, the LCP may arise from application problems such as the linear and quadratic programming, the economies with institutional restrictions upon prices, the optimal stopping in Markov chain, and the free boundary problems. Several computational methods have been developed for solving LCPs. The two basic functions of the LCP are Upr function and Sgn function:

In mechanics, Upr function is used to simulate geometry and dynamics of unilateral constraint, and Sgn function is used to simulate various kinds of dry friction.

A cam-follower with oblique-impact on the contact point is studied in Figure

Sketch of cam-follower oblique-impact model.

In the present work, the normal contact between rigid bodies is characterized by a set-valued force law called Signorini’s condition [

Sketch of rigid body contact: (a) normal distance

The classical Coulomb friction law is another typical example that can be considered as a set-valued force law [

Tangential Coulomb’s contact states can be expressed by Sgn function. It also represents a complementarity behavior.

For the model shown in Figure

The relationship between

In order to get the unknown parameters

According to the results of [

There are three equations for the cam-follower impact system. In order to quickly and accurately analyze the complex behavior of cam-follower oblique-impact system, it is better to establish one mathematical equation which can describe separation, impact, and contact motion. Equation measure is introduced here. Equation (

Moreau time-stepping method [

In order to get the expectation LCP condition, we need to separate Sgn function to two Upr functions. For the tangential impact parameter

For the sake of brevity, we define

The full LCP equation of follower can be expressed as

From the solution of (

MATLAB is employed for simulation of the equations. In order to get a proper accuracy, the time step of Moreau midpoint method must be very small. We choose 0.0002 s as the time step, and the values of other parameters can be found in Table

Cam-follower oblique-impact numerical simulation parameters.

Parameter | Value |
---|---|

Length of the follower, |
0.8 m |

Radius of the follower, |
0.01 m |

Mass of the follower, |
1.975 kg |

Eccentricity of the cam, |
0.036 m |

Radius of the cam, |
0.1 m |

Coefficient of restitution in normal direction | 0.42 |

Coefficient of restitution in tangential direction | 0.4 |

Friction coefficient, |
0.2 |

Distance between bear fix point of the follower and rotation center of the cam, |
0.4 m |

Initial angular position of the cam, |
0 rad |

With the increasing of the cam rotational speed, the state of the cam-follower is transformed from the initial permanent contact to the separation and impact. The system performs very complex nonlinear characteristics, such as period, quasiperiod, and chaos response. The bifurcation diagrams are shown in Figure

Bifurcation diagram with the change of cam speed: (a) angular velocity and (b) angular displacement.

When the cam rotational speed is low, the constraining force between cam and follower is less than the follower’s restoring force. Therefore, the cam and follower keep in contact. The angular displacement of time history diagram and phase diagram is shown in Figure

Cam and follower permanent contact when

With the increasing of the cam speed, the cam and follower would be separated. When the cam and follower impact, there are minor collisions after the first impact, and then follower would contact the cam’s surface until the next separation. As shown in Figure

When the cam speed goes higher, as shown in Figures

There also exist

Generally, for vibroimpact system, there are two ways for Poincaré section selection:

Figure

Chaos motion when

Chaos motion when

The cam and follower oblique collision model is established in this paper. In order to simulate the response, the equations of the separation, impact, and contact motion are transformed to a linear complementarity problem. The main conclusions are as follows:

The contact and tangential impact law represent a complementarity behavior. LCP equation method could be used to solve the oblique collision problem.

The system performs very complex nonlinear characteristics, such as period, quasiperiod, and chaos response. For every motion, there is a special collision phenomenon.

Radius of the follower

Bear fix point of the follower

Length of the follower

Angular position of follower with counterclockwise sense of rotation

Angular position of cam with counterclockwise sense of rotation

Eccentricity of the cam

Rotation speed of the cam

Rotation center of the cam

Geometric center of the cam

Initial angular position of the cam

Displacement between cam and follower

Displacement between follower’s fix point and the point which is nearest to the cam.

The authors declare that they have no competing interests.

This work was supported by the National Natural Science Foundation of China (Grants nos. 11272257 and 11672251) and NPU Aoxiang New Star.