On The Chaotic Motion Of Underactuated Planar Five-bar Mechanism With Variable Parameters

Underactuated mechanism is one kind of mechanisms which will existe chaotic motion under certain circumstance. At present, the researches on chaos for underactuated mechanism focus on the test and control of such phenomenon. Actually, the variation of the parameters of a mechanism, both statically and dynamically, will directly affect the state of the mechanism. However, the study on this aspect has seldom been reported.In this thesis, permutation entropy is applied to checking the state of a planar five-bar underactuated mechanism with variable parameters, and finding the edge of chaos. It will make some contribution to the control and anti-control in theory and practice.The main work and results of this thesis are as followings:(1) Modification of calculating permutation entropy is proposed. Homogenization method in calculation of permutation entropy is modified to improve the sensititivoty of permutation entropy when it is used to find the edge of chaos. The modified method is applied to find the edge of chaos for one-dimensional mapping, Logistic system, and two-dimensional mapping, Henon system. The results indicate that the modification is effective. As a simple but sensitive cheching method, permutation entropy is used throughout this thesis.(2) In this thesis, the word, variation of parameter statically, refers to the case when the change of the parameter can be realized only when the motion of the mechanism is stopped. The length of connecting rod and the mass of the slider are taken as parameters varied statically. With SimMechanics toolbox, the motion of the planar five-bar underactuated mechanism is simulated, and the data of displacement of the slider are collected. Futhermore, the states of motion of the mechanism are checked by permutation entropy, and the edge of chaos is found. The resuls show that the variation of the length of connecting rod, or the variation of the mass of the slider, will lead to some changes in the state of motion of the mechanism, either change from periodic motion (quasi-periodic motion) to chaotic motion or change from chaotic motion to periodic motion (quasi-periodic motion). This conclusion is confiremed by the phase portrait, time history of displacement, and Lyaponv exponent derived from the data of the simulation.(3) The word, variation of parameter dynamically, refers to the case when the change of the parameter can be realized at the same time the mechanism is in moving, In this thesis, the mass of the slider and the angular velocity of actuator are taken as parameters varied dynamically. And the studies are carried out by simulation and experiment respectively. The values of the paramenters vary in two fashions, from lower to higher and from higher to lower, and the ranges of variation will include the edge of chaos determined by variation of parameter statically. The results show that the periodic motion of mechanism will be changed into chaotic motion if the masses of the slider vary from the value of periodic motion statically to that of chaotic motion statically, but the chatic motion will be kept if the masses of the slider vary from the value of chaotic motion statically to that of periodic motion statically; and the periodic motion of mechanism will be changed into chaotic motion if the angular velocities of the actuator vary from the value of periodic motion statically to that of chaotic motion statically, and the chatic motion will be changed into periodic motion if the angular velocities of the actuator vary from the value of chaotic motion statically to that of periodic motion statically though the chaotic motion will be kept for a while as the value the periodic motion statically has been attained in some cases. These results might be used to control and anti-control chaos.