Vibration analysis of the planar flexible mechanism using the redundant generalized coordinates

MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ...*** NGUYEN SY NAM VIBRATION ANALYSIS OF THE PLANAR FLEXIBLE MECHANISM USING THE REDUNDANT GENERALIZED COORDINATES Major: Engineering Mechanics Code: 9 52 01 01 SUMMARY OF THE DOCTORAL THESIS Hanoi – 2018 The thesis has been completed at Graduate University of Science and Technology, Vietnam Academy of Science and Technology Supervisor 1: Prof. Dr. Sc. Nguyen Van Khang Supervisor 2: Assoc. Prof. Dr. Le Ngoc Chan Reviewer 1: Reviewer 2: Reviewer 3: The thesis is defended to the thesis committee for the Doctoral Degree, at Graduate University of Science and Technology - Vietnam Academy of Science and Technology, on Date Month Year 2018 Hardcopy of the thesis can be found at: - Library of Graduate University of Science and Technology - National Library of Vietnam 1 PREFACE Rationale of the study In order to save the amount of needed materials, to reduce the inertia and to speed up the work, the bars of a machine structure can be slimmer and more compact. However, vibrations usually occur when the machines work, especially at high speeds, or when speeding up or down due to the decreased stiffness of the slender sections which are not large enough. These vibrations reduce the precision of the high-precision requirements, which delays the successive operations of the machine because of the existing vibration for a certain period of time. Moreover, it also makes the substantial reaction force on joints. Therefore, the elasticity of the bars should be considered when studying the mechanics of the machine. Objectives of the thesis The thesis will focus on studying the dynamic behavior of a planar mechanism which has one or more elastic bars, such as calculating the elastic deformation of the links, and assessing the effect of the deformation back on the movement of the structure during the work. The ultimate goal is to help minimize the negative impact of the elastic vibrations as well as limiting the elastic vibrations. Object and scope of the study The thesis will focus on studying planar elastic mechanisms, performing numerical simulations and surveying the responds to a number of specific planar structural models such as the four - bar mechanism, six – bar mechanism. Methodologies of the study Using analytic methods to construct differential equations of motion, linearization of differential equations of the motion, and numerical simulation on software such as Matlab and Maple to calculate and to stimulate the dynamic process of the system. Main research content of the thesis + Derivation of the equations of motion of flexible mechanisms. + Dynamic analysis of the elastic mechanism when there is no 2 control force and when there is additional control force. + Linearization of dynamic equations and vibration analysis of flexible mechanisms in steady-state. Determination of the research problems The thesis consists of four chapters + Chapter 1: Overview of elastic mechanisms and elastic robots. + Chapter 2: Representation of the set of differential equations of motion of some mechanisms with one or more elastic links. + Chapter 3: This chapter also investigates the control this systems problem by adding a control force on the input links to limit the effect of elastic deformation on the motions of system. Numerical calculations and numerical simulation of dynamic problems of flexible mechanisms. + Chapter 4: Proposed approach for linearization of the equations of constrained multibody systems. It then uses the Newmark method to calculate steady-state periodic vibrations of the parametric vibration of constrained dynamical models. CHAPTER 1. OVERVIEW OF RESEARCH PROBLEMS 1.1. Mechanisms have flexible body Depending on the size, the characteristics of the bearing force, as well as the technical requirements, each part of the mechanisms can be considered as rigid body or flexible body. According, the systems may be considered owing zero, one, two or more elastic body. For example, in Fig 1.2, the 6-bar mechanism diagram, driving 1, plate 3 and output link 5 can be considered solid, while bars 2 and 4 are generally longer and thinner so they can be considered as elastic body. Thus, this mechanism is considered to have two elastic segments that are suitable. In case of a two degrees of freedom robot as shown in Fig 1.3, the accuracy of the location of the end point of impact is important, therefore the links are considered elastic links. Also, another example is a three degrees of freedom parallel robot in Fig. 1.5. In this case the legs of the robot are usually slender but require very high precision, so the consideration of the robot legs as the elastic part is necessary. 3 1.2. Review of research in the world Dynamics of flexible multibody systems is the field of science that attracts the attention of many scientists in the world. To study the problems of flexible multibody systems, a common approach is to create those dynamic models. These models will be a basis for numerical simulations, investigating the response of the systems, control design and the optimal design problem. Study on creation of dynamic models. The most widely used three methods for setting up mathematical models [86] is: a) Floating frame of reference formulation: In this formulation, two sets of coordinates are used to describe the configuration of the deformable bodies; one set describes the location and orientation of a selected body coordinate system, while the second set describes the deformation of the body with respect to its coordinate system. Using the principle of virtual work in dynamics or Lagrange’s equation we can systematically develop the dynamic equations of motion of the deformable bodies that undergo large reference displacements. In the floating frame of reference Figure 1.3. Two degrees of freedom robot Figure 1.5. Diagram of a three degrees of freedom parallel robot O1 A B y x0 O2 C D O3 Figure 1.2. Diagram of the six-bar mechanism 1 2 3 4 5 0 0 0 4 formulation, the equations of motion are expressed in terms of a coupled set of reference and elastic coordinates. The reference coordinates define the location and the orientation of a selected body coordinate system, while the elastic coordinates define the deformation of the body with respect to its reference. The elastic coordinates can be introduced using component mode methods, the finite element method or experimental identification techniques. When the deformations equal zeros, equations of motion of rigid mechanisms can be obtained. This formulation is currently the most widely used high precision method. b) Finite segment method: In this approach, the deformable body is assumed to consist of a set of rigid bodies which are connected by springs and/or dampers. c) Linear theory of elastodynamics: The solution strategy which was used in the past is to consider the multibody system first a collection of rigid bodies. General purpose multibody computer methodologies and programs can then be used to solve for the inertia and reaction forces. These inertia and reaction forces obtained from the rigid body analysis are then introduced to a linear elasticity problem in order to solve for the deformation of the flexible components in the system. The total motion of the deformable bodies is then obtained by superimposing the small elastic deformation on the gross rigid body motion. Amongst the above methods, due to many advantages the floating frame of reference formulation will be used the thesis will to establish the differential equations of motion of mechanisms. In addition, while previous studies have often established this motion differential equations as implicit matrix, in this thesis we will establish equations in the explicit analytics form. Some studies on stability and control. If the deformations affect the motion of the system, the problem now is to control these systems so that the impact of deformation on the motion of mechanism is minimal or to reduce that elastics vibration. In the literature, the solutions to this problem mainly focused on robots or manipulator arms, and the mechanisms is less interested. About control of mechanisms, although dynamic analysis of flexible mechanisms has been the subject of numerous investigations, the 5 control of such systems has not received much attention. Most of the work available in the literature which deals with vibration control of flexible mechanisms employ an actuator which acts directly on the flexible link. However, The effect of the control forces and moments on the overall motion is neglected. In addition, the implementation of such controllers may require sophisticated and expensive design. In the study of Karkoub and Yigit [47], an alternative method would be to control the vibrations through the motion of the input link. An actuator is assumed to be placed on the input link which applies a control torque. This study deals with control of a four-bar mechanism with a transverse defomation coupler link. A control torque placed on the input link to limit the effect of elastic deformation. Simulation results demonstrate that the proposed controllers are effective in suppressing the vibrations as well as in accurate positioning of the mechanism. This idea has made the control of mechanisms easier. However, more comprehensive research on this issue is needed. Some studies on linearization of the differential equations of motion: The differential equations of flexible multibody systems usually are complex non-linear equations. An effective solution to solve those equations is using the numerical methods [5, 23], however, it is quite complex and time consuming. Therefore, for simpler calculation, the differential equations are linearized. However, The linearization of motion equations of constrained multibody systems is also a complex problem. Previously linearized methods were quite difficult to apply for elastic mechanisms. In The thesis, we propose a simple and convenient linearization method when applying numerical calculations. 1.3. Researches in our country In the study of dynamics of the elastic mechanism, there are very few studies in Vietnam. A number of studies on dynamics of elastic mechanics have been done by Prof. Nguyen Van Khang et al. [7,8,10, 73- 77] at the Hanoi University of Science and Technology. 1.4. Determination of the research problems Problem one: Applying the general method to set the dynamic differential equation of motion for planar elastic mechanisms in which the elastic link 6 is discretized by a number of methods such as the Ritz-Galerkin method, finite element method (FEM). Problem two: Dynamic calculation, elastic-deformation calculation, assessment of the elastic bars’ impact on the motions of mechanism. Using the control method to minimize the effect, as well as eliminating the elastic oscillations Problem three: Machine mechanisms usually work in steady-state mode, where defomations will cause small oscillations around that stabilizing motion. The thesis will study and propose the method of linearisation of the motion of the mechanism around the stabilization motion, apply the Newmark method to calculate circular oscillations in the stable mode, from which the dynamic analysis in some cases. CHAPTER 2. ESTABLISHING THE MOTION EQUATIONS OF FLEXIBLE MULTIBODY SYSTEMS 2.1. Discretized Lagrange coordinates The elasticity in the structure is a continuous system characterized by an infinite number of degrees of freedom. These elastic rods are often discretized into finite degrees of freedom by methods, most commonly the Ritz-Galerkin method and the Finite Element Method (FEM). 2.1.1. Discretized Lagrange coordinates by Ritz-Galerkin method In the case of two- hinged ends beam, the transverse displacement w(x, t) in the Axy coordinate system attached to the beam, with Ax axes along AB will be expressed as: 1 ( , ) ( ) ( ) N i i i w x t X x q t   (2.1) with Xi (x) are dependent on boundary conditions; qi(t) are elastic coordinates. According to the Ritz-Galerkin method, in this case are of the form [4]: sini iX x L      (2.2) Similarly, the coordinate system is attached to the two- hinged ends x L w x y A B Figure 2.1. Two-hinged ends beam x y x u A B Hình 2.2. Two-hinged ends beam 7 bar as shown in Figure 2.2, the axial displacement of the bar in the relative coordinate system is represented as: 1 ( , ) ( ) ( ) N i i i u x t Y x p t   (2.3) It is found that [4]: 2 1( ) sin 2i i xY x l      (2.4) 2.1.2. Discretized Lagrange coordinates by finite element method (FEM) In this method, the elastic link is divided into finite numbers. The element ith in the plane will have 3 degrees of freedom at each node include axial displacement, transverse displacement and the rotate displacement. a) In case using an element to discrete. Considering the AB bar with the assumptions that it is straight, homogeneous, and the cross sectional area remains constant, AB is considered a Euler - Bernoulli beam. + Transverse displacement of the bar [50]: 2 2 3 3 5 5 6 6( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )w x t X x q t X x q t X x q t X x q t    (2.5) From the boundary conditions we have Hermite’s mode shape functions: 2 3 2 3 2 3 2 2 3 2 3 5 62 3 2 ( ) 1 3 2 ; ( ) 2 ( ) 3 2 ; ( ) x x x xX x X x x L L L L x x x xX x X x LL L L                     (2.6) + Longitudinal displacement: 1 1 4 4( , ) ( ) ( ) ( ) ( )u x t X x q t X x q t  (2.7) From the boundary conditions we have Hermite’s mode shape functions: 1 41 ; x xX X L L    (2.8) b) In case using more elements to discrete. By Spliting the elastic link AB into N equal elements, the length of each element is l = L / N. Consider element i-th, whose first node is i, the A x B L q1 q2 q3 q5 Figure 2.3. Degrees of freedom of element q4 q6 8 last node is (i + 1). When deformed, the two-node displacement of element i are 1 2 3, ,i i iq q q at the top node; at the last node are 4 5 6, ,i i iq q q . Thus the total number of co-ordinates determines the deformation of the beam AB when dividing the beam into N elements of 3(N + 1). 2.2. Establishing the motion equations of constrained multibody systems by Lagrange’s equations with multipliers Consider constrained holonomic multibody systems, with m redundant generalized coordinates. Systems have r holonomic constraints, the constraints equations are: 1 2( , ,..., , ) ( 1,2,..., )j mf s s s t j r (2.9) The Lagrange’s equations with multipliers for constrained holonomic multibody systems are [5]: 1 ( 1, 2,..., ) r i k i ik k k k fd T T Q k m dt s s s s                   (2.10) 2.3. Establishing the motion equations of four – bar mechanism with flexible connecting link. Considering the motion of a four-bar mechanism OABC, which is shown in Fig. 2.5, The mechanism consists of the rigid crank OA of length l1, the flexible link AB before deformation of length l2 and the rigid rod BC of length l3, the distance OC is l0, τ is the external torque acting on the crank joint. 2.3.1. The kinetic energy, strain energy and constraints equations a) Coordinate systems and constraints equations. The fixed coordinate system Ox0y0, the reference coordinate system Axy which is rotated with an angle φ2 to the point A. The angles φ1, φ2, φ3 are the angles between the x0- axis and crank OA, the x0-axis and flexible link AB, the x0-axis and output link BC, respectively. We have the constraint equations: O A B φ1 y0 x0 φ2 φ3 C x y w x Fig. 2.5. Schema of a planar four-bar mechanism with flexible connecting link M* M u τ 9     1 1 1 2 2 3 3 0 2 1 1 2 2 3 3 cos cos cos 0 sin sin sin 0 B B f l l u l l f l l u l                  (2.11) b) The kinetic energy of mechanism:                2 2 2 22 2 2 2 2 2 1 3 1 1 2 0 1 1 1 2 1 1 2 1 2 1 1 1 2 1 1 2 1 2 2 2 1 1 1 2 2 2 2 sin 2 cos 2 cos 2 sin 2 2 l O C u wT I I l w x u t t u wl l x u l t t u wl w w x u dx t t                                                                           c)The strain energy of mechanism 2 2 22 2 2 0 0 1 1 2 2 l lu wEA dx EI dx x x               (2.13) where E, I, A, μ are modulus of elasticity, area moment of inertia of the coupler link, cross sectional area, mass per unit length of the coupler link, respectively. 2.3.2. Motion equations of four – bar mechanism when the flexible connecting link is discretized by the Ritz-Galerkin method According to the Ritz-Galerkin method, transverse and longitudinal vibrations are of the form: 1 1 ( , ) ( ) ( ) N i i i w x t X x q t   2 1 ( , ) ( ) ( ) N k k k u x t Y x p t   (2.14) By substituting Eqs. (2.14) into Eq. (2.12), (2.13) and then substituting into Eq. (2.10) we obtained the equations of motion of the system as: *) The equation for φ1 coordinate:                 2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 1 2 1 1 2 1 1 1 2 2 21 2 2 1 2 1 2 1 2 1 2 1 2 1 cos( ) cos 2 sin sin cos sin 2 cos sin 2 N O k k k N N N i i k k i i i k i N k k k k l lI l l l H p l C q l H p l C q l l l H p l H                                                                        2 1 1 1 2 2 1 2 1 2 1 2 1 1 1 1 1 2 1 1 2 sin cos sin cos N k k N N i i i i i i p l C q l C q l l                           (2.15) (2.12) 10 *) The equation for φ2 coordinate:       2 1 1 1 2 2 2 1 2 1 1 2 2 2 1 2 1 2 1 2 1 1 1 1 3 2 2 1 1 1 1 1 1 1 1 1 1 cos cos sin 2 2 3 2 N N k k i i k i N N N N N N N N ij i j k k kl k l ik i k i i i j k k l i k i N N ik i k i k l H p C q l l m q q F p b p p n q p D q n q p                                                                          1 1 2 2 2 2 1 2 2 1 1 1 1 1 2 2 2 21 2 1 1 2 1 1 1 2 1 1 1 2 1 1 2 2 1 2 2 2 2 sin sin cos 2 sin . cos . N N N N N ij i j k k kl k l i j k k l N N k k i i k i B B m q q F p b p p l l l H p l C q l u l u                                                      (2.16) *) The equation for φ3 coordinate:    3 3 3 1 3 3 2sin cos 0CI l l       (2.17) *) The equations for qi coordinates (i = 1,2,..., N1):     2 1 2 1 1 1 1 2 2 1 1 2 2 1 1 1 2 2 2 1 1 1 cos sin 2 0 N N i i ik k ij j k j N N N i ik k ij j ij j k j j l C D n p m q l C n p m q EI k q                                             (2.18) *) The equations for pk coordinates (k = 1,2,...N2):       1 2 1 2 2 2 1 1 1 2 2 1 1 1 2 1 1 2 2 2 1 2 2 2 1 1 1 sin cos 2 cos sin 0 N N k ik i kl l k i l N N N ik i k kl l kl l k i l l