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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.
MINISTRY OF EDUCATION AND VIETNAM ACADEMY OF SCIENCE TRAINING 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: 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 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 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 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 and output link can be considered solid, while bars and 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 B A y C x0 O3 O1 D O2 Figure 1.2 Diagram of the six-bar mechanism Figure 1.3 Two degrees of freedom robot Figure 1.5 Diagram of a three degrees of freedom parallel robot 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 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 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, 7377] 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 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 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: B x N w( x, t ) X i ( x)qi (t ) (2.1) y x w i 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]: i X i sin x (2.2) L Similarly, the coordinate system is attached to the two- hinged ends L A Figure 2.1 Two-hinged ends beam y x u x A B Hình 2.2 Two-hinged ends beam bar as shown in Figure 2.2, the axial displacement of the bar in the relative coordinate system is represented as: N u ( x, t ) Yi ( x) pi (t ) (2.3) i 1 2i x Yi ( x) sin (2.4) l 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 A B x have degrees of freedom at L each node include axial q1 q4 displacement, transverse q3 q q6 q5 displacement and the rotate displacement Figure 2.3 Degrees of freedom of element 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]: w( x, t ) X ( x)q2 (t ) X ( x)q3 (t ) X ( x)q5 (t ) X ( x )q6 (t ) (2.5) It is found that [4]: From the boundary conditions we have Hermite’s mode shape functions: x x3 x x X ( x) ; X ( x) x L L L L (2.6) 3 x x x x X ( x ) ; X ( x) L L L L + Longitudinal displacement: u ( x, t ) X ( x)q1 (t ) X ( x)q4 (t ) (2.7) From the boundary conditions we have Hermite’s mode shape functions: X1 x ; L X4 x 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 last node is (i + 1) When deformed, the two-node displacement of element i are q1i , q2 i , q3i at the top node; at the last node are q4 i , q5i , q6i 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, f j ( s1 , s2 , , sm , t ) ( j 1,2, , r ) (2.9) the constraints equations are: The Lagrange’s equations with multipliers for constrained holonomic multibody systems are [5]: d T dt sk r T f Qk i i sk s k i 1 s k ( k 1, 2, , m ) (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 B M mechanism consists of x y x w the rigid crank OA of * M y0 u length l1, the flexible φ2 A τ link AB before deformation of length l2 φ3 x0 φ1 C and the rigid rod BC of O length l3, the distance OC is l0, τ is the external Fig 2.5 Schema of a planar four-bar mechanism with flexible connecting link 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 x0axis and crank OA, the x0-axis and flexible link AB, the x0-axis and output link BC, respectively We have the constraint equations: 11 connecting link only undergoing longitudinal deformations (cutout transverse deformation coordinates) In the case of using the first three modes N1 = 3, N2 = 3, we obtain the nine differential equations of motion with the variables: φ1, φ2, φ3, q1, q2, q3, p1, p2, p3, λ1, λ2 2.3.3 Motion equations of four – bar mechanism when the flexible connecting link is discretized by the finite element method Using an element to discrete the AB bar, substituting boundary conditions into Eq (2.5) and Eq (2.7) we obtained: w( x, t ) X ( x)q3 (t ) X ( x)q6 (t ) (2.20) u ( x, t ) X ( x)q4 (t ) Due to u(l2,t) = q4, constraint equations are as follows: f1 l1 cos 1 l2 q4 cos l3 cos 3 l0 f l1 sin 1 l2 q4 sin l3 sin 3 (2.21) (2.22) Substituting (2.20), (2.21) into (2.12) and (2.13) and then substituting into Eq (2.10) we obtained the equations of motion corresponding to φ1, φ2, φ3 , q3, q4, q6 in short form: M( s )s C( s ,s )s g( s ) τ( t ) ΦTs ( s )λ (2.23) We have differential equations of motion (2.23), these are nonlinear differential equations With two contrained equations (2.22), we have equations with variables of φ1, φ2, φ3, q3, q4, q6 and λ1, λ2 These are generalized equations The individual cases of the differential equations of motion are derived from the general equations 2.4 Establishing the motion equations of six – bar mechanism with two flexible connecting links Considering the motion of a six-bar mechanism shown in Fig 2.6, the flexible links are AB and CD, τ is the external torque acting on the crank joint The angles φ1, φ2, φ3, φ4, φ5 are the positioning angles of the stitches Let u1 and w1 is the relative longitudinal displacement and relative transverse displacement of the point M on AB link u1 and w1 is the relative longitudinal displacement and relative transverse displacement of the point N on CD link 12 D l4 B l2 A y C1 φ1 O1 l3 φ2 τ l1 l0 θ1 C C5 φ4 l5 φ5 C3 O3 φ3 θ2 O2 x Figure 2.7 Schema of a planar four-bar mechanism D y1 B M x1 w1 x1 y2 x2 φ4 φ2 u A N w2 x2 u2 C Figure 2.8 Schema of reference coordinate systems on the elastic links Similar to the four-bar mechanism, we also define the kinetic energy, strain energy, and substituting into the Lagrange’s equation (2.10) We obtained the equations system of six-bar mechanism: M(s)s C(s, s)s g(s) τ(t ) ΦTs (s)λ (2.24) (2.25) f (s ) where M(s) is the n×n inertia matrix, C s, s is the n×n Coriolis/Centripetal matrix, g(s) is the n×1 gravity vector, τ(t) is the n×1 vector of generalized forces, λ = [λ1 λ2 λ3 λ4]T is the r×1 vector of Lagrange multipliers, and Φs(s) denotes the r×n Jacobian matrix, f f1 , f , f , f is vector of constraint equations, s is vector of T generalized coordinates corresponding to each method: + Using Ritz – Galerkin method: s 1 2 3 4 5 q1(1) q2(1) qN(1)1 q1(2) q2(2) qN(2)3 p1(1) p2(1) pN(1)2 p1(2) p2(2) pN(2)4 + Using finite element method (FEM) with an element: s 1 3 5 q3(1) q4(1) q6(1) q3(2) q4(2) q6(2) T T 13 where qi(1) and p (1) j are transverse deformation coordinates and longitudinal deformation coordinates of AB elastic link; qi(2) and p (2) are transverse j deformation coordinates and longitudinal deformation coordinates of CD elastic link Individual cases are also derived from the general equations Conclusion of Chapter 1) It is derived an explicit form of the motion differential equations of the four-bar mechanism owning flexible connecting link and of the sixbar mechanism owning two flexible connecting links 2) Discretization of elastic links is done by using the Ritz-Galerkin method and finite element method 3) The method of setting up any equations of motion presented in this chapter may be used for other mechanisms with different elastic bars CHAPTER 3: FORWARD DYNAMIC ANALYSIS OF PLANAR MECHANISM WITH ELASTIC LINKS Chapter performs the numerical simulation of the dynamics of the mechanism when the external torque is acting on the crank joint, calculates the effect of the deformation on the motion of the mechanism This chapter also performs numerical calculations when the control force is added to minimize the effect of deformation on the motion of the mechanism 3.1 Forward dynamic problem of the constrained multibody systems a) Differential–algebraic equations of motion of constrained multibody systems Differential–algebraic equations of motion of constrained multibody systems can usually be written in the following form [5]: M(s)s C(s, s)s g(s) τ(t ) ΦTs (s)λ f (s ) (3.2) q q T T where s s1 s2 sn , s , z z1 z2 zr , q a , q z e qa q1a q2a qna , qe q1e q2e qne , f na ne , n f r T (3.1) T (3.3) 14 Φs f f f , Φq , Φz , Φ s Φ q Φ z , Φ s r n , Φ q r f , Φ z r r s q z Differentiating Eq (3.2), the system of equations (3.1), (3.2) is given the form: M(s)s ΦTs (s)λ p1 (s,s, t ) (3.4) Φs (s)s p2 (s, s) where p1 (s, s, t ) τ(t ) C(s, s, t )s g(s), p1 (s, s, t ) (3.5) nx1 (3.6) (s) s 2 Φ (s) s 2f (s), p (s, s) rx1 p (s, s) Φ (3.7) s s with α, β are positive constants of the Baumgarte’s stabilization method b) The differential equations of motion in redundant generalized coordinates To eliminate Lagrange multipliers, and to transform the differential– algebraic equation system (3.1), (3.2) to the ordinary differential equation system with the number of equations equal the number of redundant generalized coordinates, we use the rotational matrix R and orthogonal theorem [5] We have the system of equations: RT M(s)s RT p1 (s , s, t ) (3.8) Φs (s)s p2 (s, s, t ) with Ef R (s) 1 , E f fxf , R(s) nxf Φ z Φ q (3.9) (3.10) The system of equations (3.8), (3.9) is the ordinary differential equation of the redundant generalized coordinates s The calculation of the solution of this system is presented in [5] c) The differential equations of motion in independent generalized coordinates Using the rotational matrix R, we transform the equation system (3.1), (3.2) to the differential equation system of flexible mechanism of independent generalized coordinates as: (3.11) M q q C q , q q g q τ q where s s(q ), s s(q,q) 15 M q R T s M s R s C q , q R T s M s R s , s C s , s R s g q R T s g s (3.12) 3.2 Forward dynamic control problem of the constrained multibody systems The forward dynamic of the elastic mechanism and its forward dynamic control are as follows: 1) Forward dynamic problem of rigid mechanism: Knowing the τ a τaR (t ) -torque applied to the crank, the parts of the mechanism are considered rigid bodies From the motion equations of the rigid mechanism we can obtain the qaR (t ) - motion of this rigid mechanism (the desired motions/fundamental motion) 2) Forward dynamic problem of flexible mechanism: It is also known that the τ a τaR (t ) -torque is applied to the crank, the mechanism has some elastic links, solving the differential equations of motion of this mechanism q we obtain q a (the motion of the elastic mechanism and the q e deformations) Generally for elastic mechanism: qa qaR (t ), qe (3.13) 3) Forward dynamic control problem: According to the ideas of Karkoub and Yigit [47], we add additional-control torque τC( a) (t ) applying to the crank of elastic mechanism Then, thanks to the additional-control torque, it is possible to make the elastic oscillations of the elastic links minimized and the real motion of the elastic mechanism cling to the fundamental (desired) motions of the solid structure The selected PD additional-control torque has the following form: τC(a) (t ) K P xa K D x a (3.14) where xa qa (t ) qaR (t ) is the difference between the real motion of driver links and the desired fundamental of a constrained multibody system With four-bar and six-bar mechanism that: 16 qa (t ) 1 (t ), qaR (t ) 1R (t ) xa 1 1R (3.15) Rigidmechanism model + Flexible mechanism K + K Figure 3.1 Diagram of PD controller 3.3 Forward dynamics and the ability to control vibrations of four-bar mechanism with flexible connecting link Calculations were performed with three such problems in two cases: the differential equations of motion of mechanism are established using the Ritz-Galerkin method to discrete the elastic rod and using FEM to discrete elastic rod 3.3.1 Establish the differential equation of motion by the Ritz-Galerkin method Numerical calculations will be made in simple to complex cases, including solid structures (for comparison), the connecting link being assumed to be bending only (neglecting the longitudinal deformation) and the full case in which is the structure has a connecting link of both transverse and longitudinal deformation *) Forward dynamics problem: For numerical simulation, the torque applied to the crank is given as: sin(2 t / Tm ) 0 (t ) t Tm t Tm (3.15) where τ0 is the amplitude, Tm is the duration of the torque The results show that: + When the torque amplitude is small, the deformation of the rod is negligible, so that its influence on the motion of the links in the structure is 17 negligible + When the torque amplitude is increased, the deformation is significant, so that the angular displacement distances, the angular velocity of the mechanism are increased Thus, when the elastic deformation is significant, it not only distorts the motion of the elastic link but also distorts the motion of the mechanism In Fig 3.23 to Fig 3.26 is an example of simulation results in the case of contemporaneous bending and compression The torque has τ0 = 0.03 Nm, Tm = 1s Figure 3.23 Crank angle Rigid Flexible Figure 3.24 Output angle … Rigid, Flexible Figure 3.25 Transverse deformation Figure 3.26 Longitudinal deformation of flexible link at x = l2/2 of flexible link *) Forward dynamic control problem Additional-control torque PD has form c k P 1 1R k D 1 1R The results show that: (3.16) 18 + In case deformation is not significant, causing the deviation not too large, this control method has the ability to limit elastic oscillation and control of motion The deviation of the elastic mechanism caused by the deformation is negligible, the motion of the elastic mechanism close to the rigid mechanism + In case deformation is large, causing great deviation in motion, this controller only reduces the deviation in motion without destroying it, this deviation is still significant In Figure 3.29 and Figure 3.30, the control results of the example above The orbital trajectory of the elastic mechanism has followed the rigid mechanism Figure 3.29 Crank angle when controlled … Rigid, Flexible Figure 3.30 Output angle when controlled.… Rigid, _ Flexible 3.3.2 Establish the differential equation of motion by the finite element method The calculations for the cases are the same as in section 3.3.1 The computational results show that the dynamic behavior in this case and the case where the differential equation of motion is established using the RitzGalerkin method (Section 3.3.1) is similar, the difference is negligible 3.4 Forward dynamics and the ability to control vibrations of six-bar mechanism with two flexible connecting links Numerical calculations are performed in the following cases: rigid mechanism, elastic mechanism owning two flexible connecting links with only axial deformation (neglecting the transverse deformation) and elastic mechanism owning two flexible conecting links with only transverse 19 deformation (neglecting the longitudinal deformation) With the torque as (3.15), the calculated result shows that: + When the torque is small, the deformations of the links are negligible, so that its influence on the motion of the part in the mechanism is small + As the torque increase increases, the deformation is increased significantly, so the angular deviation and angular velocity of the parts increase significantly + When adding PD control torque, the control result of the trajectory of the elastic mechanism is similar to the trajectory of the rigid mechanism and the elastic oscillation is minimized Conclusion of Chapter In chapter 3, numerical calculation of dynamic problems and control of the four-bar elastic mechanism with an elastic connecting link and 6-bar mechanism with two elastic conecting links have been carried out The results of the numerical simulation show that by adding extra control forces to the cranks, we can control the oscillation generated by the elastic coupling when the velocity was sufficiently small Numerical simulation also shows that in cases when the velocity was not small, this controlling way can not eliminate the vibration generated by the elastic bonding Therefore, in such cases, more appropriate controlling methods will be needed Through those simulation examples, the use of the Ritz-Galerkin method is equivalent to using the finite element method CHAPTER LINEARIZATION AND PERIODIC VIBRATION ANALYSIS OF PLANAR FLEXIBLE MECHANISMS This thesis proposes a linearization approach for the motion equations of constrained multibody systems around the fundamental motion of the mechanism In that fundamental motion of this mechanism is the motion of the rigid mechanism which angular velocity of the crank (??) The idea of this approach is to transform differential - algebraic equations to ordinary differential equations by eliminating Lagrange multipliers, then it will linearize the ordinary differential equation by Taylor series expansion of these equations around the fundamental motion 20 4.1 A new approach of linearization of the motion equations of constrained multibody systems After introducing the differential-algebraic equation system of of constrained multibody system (3.1), (3.2) on the ordinary differential equations (3.8), (3.9), Next, we will linearize these equations around the fundamental motion of the mechanism Let sR (t ) be the desired fundamental motion and x the difference between the real motion and the desired fundamental of a constrained multibody system By introducing s sR x, s sR x, s sR x (3.1) and: f1 (s, s) R (s)M(s)s , k1 (s, s, t ) R p1 (s, s, t ) (3.2) f (s, s) Φ s (s)s , k (s, s, t ) p (s, s, t ) (3.3) T T Conducting Taylor series expansion of functions f1 (s, s) , k (s, s, t ) , f ( s , s ) , k (s, s, t ) s R , s R , s R , Substituting (3.8), (3.9), and neglecting nonlinear terms, we obtain the linearized differential equations: M R (t ) x C R (t ) x K R (t ) x h R (t ) f1 k1 where M R (t ) s R , C R (t ) s f k s s R k (s , s , t ) f1 (s R , s R ) h R (t ) R R k (s R , s R , t ) f (s R , s R ) f1 s R , K R (t ) f2 R s k s R k s R (3.4) R R (3.5) (3.6) In this method, the fundamental motions must be determined In the problem of elastic mechanism, by solving the equation (4.4), we obtain the solution x, x , x as the deviation of the real motions of the elastic mechanism versus the fundamental motion of the rigid mechanism and the deformation components Then we determine the real motion based on (4.1) In the case the fundamental motison is steady-state mode (crank velocity is constant), the coefficient – Matrices in Eq (4.4) are timeperiodic elements, the method (4.4) will be a the following the computer features the following computer The best solution for this case is the Newmark integration method for the early detection of linear differential 21 equation [72] the MAPLE software can be used To determine the MR, CR, KR, hR matrices, where input parameters are M, p1, f and fundamental motions s R , s R , s R of the rigid mechanism It is Simple and convenient The above matrices will be converted to the code of MATLAB software for numerical calculation 4.2 Calculating periodic solutions of linear dynamic with time-priodic coefficients based on Newmark integration method Using Newmark integration method, Prof Nguyen Van Khang et al have proposed algorithm for finding the solutions of the periodic differential equation [67,72] 4.3 Periodic vibration analysis of four-bar mechanism with flexible connecting link In this section, we will calculate the periodic vibration of the fourbar mechanism with flexible connecting link when crank velocity is constant with the case: the conecting link only undergos axial deformation (neglecting the transverse deformation) and only undergos transverse deformation (neglecting the longitudinal deformation) 4.3.1 The elastic mechanism has flexible connecting links only undergoing transverse deformation given the fundamental motions that are the motion of the rigid mechnism with constant crank velocity: 1R t 1R (0) t , 1R , 1R (3.7) Calculated results are shown in Table 4.2 The results computed by this method are compared with the linearization method used in the literature [10, 74] as shown in Table 4.2 and Fig 4.6 Table 4.2 Calculation results: Bending amplitude (w) in the middle of the elastic Angular velocity link x = l2/2(mm) (rpm) Well-known approach Proposed approach [10,74] 600 0.2605 0.2835 900 0.6167 0.6242 22 1200 1.1330 1.119 Conclusion: From numerical results it can be shown that the bending deformation of the flexible rod increases as the speed increases The results calculated by the method proposed that differed from the results calculated by the previous method are negligible This contributes to the reliability of the proposed linearization method 1.5 w [mm] 0.5 -0.5 -1 -1.5 10 12 14 t [rad] Figure 4.6 Bending amplitude in the middle of the elastic link x = l2/2, n = 1200 rpm Proposed approach, … Well-known approach [10, 74] 4.4 Periodic vibration analysis of six-bar mechanism with two flexible connecting links undergoing longitudinal deformation The simulation results are calculated in the case of a rotating crank with a constant angular velocity of 210 rpm Calculated time is Figure 4.23 Longitudinal deformation of the link done in a duration (a AB, - - - Experimental [66], Theoretical round) Fig 4.23 and Fig 4.24 show the longitudinal deformation curve of the link AB and the link CD using the proposed linearization Figure 4.24 Longitudinal deformation of the link method (solid line) CD, - - - Experimental [66], Theoretical 23 and the experimental results [66] (dashed line) On the deformation curves of the linearization method Goc khau dan [do] there are deviation from Figure 4.25 Angle erroneous of output the experimental curves link O3D, ε5 [rad] (The maximum deviation of the link AB is about 0.3mm, the link CD is about 0.2mm) However, the experimental curves and theoretical curves have the same shape This is acceptable it can be explained that due to the fact that the theoretical calculations have not been fully considered In addition, we have linearized the equation of motion to approximate it Fig 4.25 is the angle erroneous graph of output link O3D of the elastic mechanism versus the rigid mechanism ( 5 5R ) due to elastic deformation, amplitude 0.2 [rad] 0.1 -0.1 -0.2 90 180 270 vibration is about 0.15 rad (~ 8.6o) Conclusion of Chapter The main results achieved in this chapter are: Developing a new method of linearization of the differentialalgebraic equations of constrained flexible multibody system Generalized linearization algorithm and a quite simple calculation diagram obviously are possible to apply a software such as MAPLE, MATLAB into the linearization process Applying the Newmark method to find the periodicity condition of the linear differential equation of the periodic coefficient to calculate the periodic vibration of the four-bar mechanism and the six- bar mechanism with elastic connecting links The results, calculated according to the method proposed in the thesis, are consistent with the experimental results and the results calculated by using other methods 360 24 CONCLUSIONS AND RECOMMENDATIONS Main results 1) Applying some methods of flexible multibody system dynamics has provided an explicit set of motion equations for the mechanism owing elastic links 2) Forward dynamic analysis of planar mechanism owing elastic links was conducted when the driving torque was applied to the crank Then, it is possible to calculate the deformation of the elastic links, to evaluate the effect of deformation on the movement of the parts in the mechanism 3) In order to limit the effect of deformation and eliminate the elastic oscillation, the controlling scheme for oscillation through the additional control torque applied to the crank has been applied The simulation results show that the controller performs very well when the conduction runs at a not too high velocity When the speed of conduction is high, the proposed controlling method is not appropriate, other control methods need to be studied 4) It has proposed the linearization method of the differentialalgebraic equations of constrained flexible multibody system around the fundamental motion to solve the equation This method is general, simple, convenient and can be automated, thank to some softwares such as MAPLE, MATLAB, This method to solving Periodic vibration problems was applied when carrying out this thesis These applications are quite convenient and accounting time is greatly reduced Some issues and further research directions 1) A more complete consideration of the factors influencing the model dynamics of the system such as the internal inhibition, extanal inhibition, 2) Studying the multibody systems owing the elastic bars driven by electric motors 3) Applying to some objects such as elastic robots, mechanism in space 4) Applying the modern controlling methods such as sliding control, neural network control, LIST OF PUBLISHED WORKS Nguyen Van Khang, Nguyen Sy Nam, Nguyen Van Quyen (2018), Symbolic linearization and vibration analysis of constrained multibody systems, Archive of Applied Mechanics 88(8), pp 1369 – 1384 Nguyen Van Khang, Nguyen Phong Dien, Nguyen Sy Nam (2016), An efficient numerical procedure for calculating periodic vibrations of elastic mechanisms, Vietnam Journal of Mechanics, VAST, Vol 38, No (2016), pp 15 – 25 Nguyen Van Khang, Nguyen Sy Nam (2017), Dynamics and control of a four-bar mechanism with relative longitudinal vibration of the coupler link, Journal of Science & Technology (Technical Universities), 119, pp 006-010 Nguyen Van Khang, Nguyen Sy Nam, Nguyen Phong Dien (2017), Modelling and model-based control of a four-bar mechanism with a flexible coupler link Proceedings of the 5th IFToMM International Symposium on Robotics and Mechatronics (ISRM2017), Sydney (accepted) Nguyễn Văn Khang, Nguyễn Sỹ Nam (2015), Tính toán dao động đàn hồi cấu sáu phương pháp Newmark, Tuyển tập cơng trình hội nghị học kỹ thuật toàn quốc, NXB Đà Nẵng 2015, tr 189 – 199 Nguyễn Văn Khang, Nguyễn Sỹ Nam (2017), Động lực học điều khiển cấu bốn khâu lề với khâu nối đàn hồi, Tuyển tập công trình Hội nghị khoa học tồn quốc lần thứ Cơ kỹ thuật tự động hóa, Nhà xuất Bách Khoa - Hà Nội, tr 40 – 47 Nguyen Sy Nam, Le Ngoc Phuong, Pham Hong Anh (2016), Dynamics and control of a four-bar mechanism with relative transverse vibration of the coupler link, Proceedings of the International Conference on Sustainable Development in Civil Engineering 2016, Construction Publishing House, pp 275 – 283 Nguyễn Văn Khang, Nguyễn Sỹ Nam (2018), Tính tốn dao động tuần hồn cấu sáu khâu có hai khâu nối đàn hồi, Tuyển tập cơng trình Hội nghị Cơ học tồn quốc lần thứ X, Hà Nội, 8-9/12/2017, Tập Động lực học điều khiển, Cơ học máy NXB Khoa học tự nhiên Công nghệ, tr 403 – 412 ... 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... systems around the fundamental motion of the mechanism In that fundamental motion of this mechanism is the motion of the rigid mechanism which angular velocity of the crank (??) The idea of this approach... DYNAMIC ANALYSIS OF PLANAR MECHANISM WITH ELASTIC LINKS Chapter performs the numerical simulation of the dynamics of the mechanism when the external torque is acting on the crank joint, calculates the