Dynamics and control of a four - bar mechanism with relative longitudinal vibration of the coupler link

 Journal of Science & Technology 119 (2017) 006-010 
 Dynamics and Control of a Four-Bar Mechanism with Relative 
 Longitudinal Vibration of the Coupler Link 
 Nguyen Van Khang, Nguyen Sy Nam* 
 1 Hanoi University of Science and Technology – No. 1, Dai Co Viet Str., Hai Ba Trung, Ha Noi, Viet Nam 
 2 National University of Civil Engineering , No. 55, Giai Phong Str., Hai Ba Trung, Ha Noi, Viet Nam 
 Received: October 12, 2016; accepted: June 9, 2017 
 Abstract 
 In the mechanisms and machines operating at high speeds, the elastic vibration of links is inevitable. In this 
 paper the dynamic modeling and controller design for a flexible four-bar mechanism are studied. The fully 
 coupled non-linear equations of motion are obtained by using the Lagrange’s equations with multipliers for 
 constrained multibody systems. The resulting differential-algebraic equations are solved using numerical 
 methods. A simple PD controller is designed to reduce the influence of the elastic link on the desired motion. 
 Keywords: Dynamic analysis, control, Four-bar mechanism, Ritz-Galerkin method, Vibration. 
1. Introduction * A four–bar mechanism OABC is shown in 
 Figure 1. The mechanism consists of the rigid crank 
 Traditionally, dynamic analysis and control of 
 OA of length l , the flexible rod AB of length l and 
mechanisms have been based on the assumption that 1 2
 the rigid rod BC of length l , the distance OC is l , τ 
the links behave as rigid bodies. The demand for high 3 0
 (0)
speed lightweight machinery requires a redesign of is the external torque acting on the crank joint. e1 
the current mechanisms. Unfortunately, reducing the (0)
 and e2 are the unit vectors of the fixed coordinate 
weight of four-bar mechanisms and/or increasing 
 system Ox y . e and e are the unit vectors of the 
their speed may lead to the onset of elastic 0 0 1 2
oscillations, which causes performance degradations reference coordinate system Axy which is rotated 
such as misfeeding in the case of the card feeder with an angle φ2 to the fixed one. Three 
mechanism in Sandor et al [1]. Therefore, the variables φ1, φ2 and φ3 are the angles between the x0-
dynamic analysis and control vibration of flexible axis and crank OA, the x0-axis and flexible link AB, 
mechanisms are required. Although dynamic analysis the x0-axis and output link BC, respectively. 
of flexible mechanisms has been the subject of 
numerous investigations [1-6], the control of such u(l2,t) 
systems has not received much attention [2-4]. Most 
 y l2 B x 
of the work available in the literature which deals u(x,t) 
with vibration control of flexible mechanisms employ x 
an actuator which acts directly on the flexible link. y0 e2
 A φ2 
The effect of the control forces and moments on the e 
overall motion is neglected. An alternative method τ 1
 (0)
would be to control the vibrations through the motion e2 φ3 
 φ1 x0 
of the input link. O C 
 e(0) 
 The current study deals with the control of a 1
 Fig. 1. Diagram of the four-bar mechanism 
four-bar mechanism with a flexible coupler link. An 
actuator is assumed to be placed on the input link 
which applies a control torque. A simple PD control It should be noted that in general the constraint 
is designed which requires measurements of the equations depend on the elastic deformations. In this 
position and angular velocity of the input link only. study, the transverse deformations are neglected. 
 Therefore, the constraint equations here depend on 
2. Equations of motion of the four-bar mechanism 
 the longitudinal deformations: 
 f1 l1 cos 1 l2 u( l2 ,t ) cos 2 l3 cos 3 l0 0
 f l sin l u( l ,t ) sin l sin 0
 * Corresponding author: Tel.: (+84) 977.244.292 2 1 1 2 2 2 3 3
Email: nam.nguyensy@gmail.com (1) 
 6 
 Journal of Science & Technology 119 (2017) 006-010 
 Ritz-Galerkin method is applied to the coupler The strain energy according to Reddy [9] is given by 
link. The axial deformation of the coupler link is 
 l 2
written as [7, 8] 1 2 u 1 N N
  EA dx EA k q q (9) 
 2 x 2  ij i j
 N 0 i 1 j 1
 u( x,t ) X ( x ).q ( t ) (2) 
  i i l
 2
 i 1
 where k X X dx 
 ij i j
where qi(t) are the modal coordinates and Xi(x) are 0
the mode shapes of the rod. The boundary conditions 
 The Lagrange's equations for constrained 
are as follows: 
 holonomic systems are [8]: 
 u(l ,t)
 u(0,t) = 0; EA 2 0 (3) d T T  2 f 
 x  k Q (10) 
   k j
 dt  j  j  j k 1  j 
The mode shapes are given as [7]: 
 where ηj are the generalized coordinates which 
 2i 1 x 
 X ( x ) B sin (4) include rigid body coordinates φ1, φ2, φ3 as 
 i i well as elastic modal coordinates q ; f are the 
 2 l2 j k
 constraint equations, λ1 and λ2 are Lagrange 
To simplify the equation (4) take Bi = 1. multipliers; and Qj are the generalized forces. By 
 substituting equations (1, 8, 9) into equation (10), we 
 The kinetic energy of the four-bar mechanism 
 obtained the equations of motion of the system as: 
shown in Figure 1 is given by 
 2
 l 2 l1l2
 2 I l l   cos( )
 1 2 1 2 1 2 O 1 2 1 2 2 1 2
 T TOA TBC TAB IO 1 IC 3 rM dx (5) 
 2 2 2 N N
 0  
 l1 2 cos 1 2 Ci qi l1 sin 1 2 Ci qi
 i 1 i 1
where IO and IC are the mass moments of inertia of 
 l l 2 N
the input and output links with respect to the joint 1 2  2  2
 2 sin( 1 2 ) l1 2 sin 1 2 Ci qi (11)
axes, respectively, μ is mass per unit length of the 2 i 1
 N
coupler link, and rM is the position vector of a point  
 2l1 2 cos 1 2 Ci qi l1 sin 1.1 l1 cos 1.2 
M on the coupler link given as i 1
 l l 2 N l 3
 r r x u e (6) 1 2   2 
 M A 1 1 cos 1 2 l1 1 cos 1 2 Ci qi 2
 2 i 1 3
The coordinates of the point M in the fixed 
 N N N l l 2
coordinate system are given as:   2 D q m q q 1 2  2 sin 
 2  i i  ij i j 1 1 2 
 i 1 i 1 j 1 2
 x l cos ( x u )cos 
 M 1 1 2 N N N N 
 (7) l  2 sin C q 2  D q m q q 
 yM l1 sin 1 ( x u )sin 2 1 1 1 2  i i 2  i i  ij i j 
 i 1 i 1 i 1 j 1 
 By differentiating the equations (7) combined l2 uB sin 2.1 l2 uB cos 2.2
with equation (4), and then substituting into the (12) 
equation (5), the kinetic energy is obtained as 
 IC 3 l3 sin 3 1 l3 cos 3 2 0 (13) 
 1 1 l l 2
 T I l 2l  2 I  2 1 2   cos 
 2 O 1 2 1 2 C 3 2 1 2 1 2 N
 l1 1 sin 1 2 Ci  mij q j
 3 N N N 
 l j 1
 2 2  2 
 2 2 2 Di qi mij qi q j
   N
 6 2 i 1 i 1 j 1 
 l  2 cos C   2 D m q (14) 
  N N N 1 1 1 2 i 2 i  ij j 
     j 1 
 mij qi q j l1 1 2 cos 1 2 Ci qi
 2 i 1 j 1 i 1 N
 N EA kij q j 1 cos 2 2 sin 2 i
   j 1
 l1 1 sin 1 2 Ci qi (8)
 i 1
 l2 l2 l2 where i = 1,2,..,N; 
where C X dx , D xX dx ; m X X dx 
 i i i i ij i j 1 when i 2k 1, k 1,2,...
 0 0 0
 i 
 1 when i 2k, k 1,2,...
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 Journal of Science & Technology 119 (2017) 006-010 
3. Dynamic analysis clearer, since the larger torque causes larger elastic 
 deformations. 
 A set of (3+N) differential equations (11-14) and 
two algebraic constraint equations (1) are the motion Table 1. Parameters of four-bar mechanism 
equations of mechanism with an elastic link. 
 Constant Description Value 
Therefore, we have (5+N) differential–algebraic 
 [unit] 
equations with (5+N) variables given as 
 l0[m] Length of the ground link 0.4064 
 M( s,t)s ΦT (s,t )λ p (s,s,t ) (15) 
 s 1 l1[m] Length of the input link 0.0635 
 f(s,t) = 0 (16) l2[m] Length of the coupler link 0.3048 
 T l3[m] Length of the output link 0.3048 
where s = [φ1 φ2 φ3 q1 qN λ1 λ2] ; λ is vector of 2
 IO[kgm ] Moment of inertia of the 7.466 
Lagrange multipliers; f is the vector of constraint -6
equations. In this paper, the Lagrange multipliers input link x10 
 2
partition method is used to solve the system of IC[kgm ] Moment of inertia of the 2.002 
 -3
differential-algebraic equations (15) and (16) output link x10 
numerically. Some other algorithms can be found in μ[kg/m] Mass per unit length 0.2237 
Khang [8]. E[N/m2] Modulus of elasticity 2.06x1011 
 For comparison purposes, another set of 
 A[m2] Cross-sectional area of 8.19 
simulations is carried out by assuming that the 
 the coupler link x10-6 
coupler link behaves as a rigid body (i.e., neglecting 
 m [kg] Mass of the input link 0.0142 
the longitudinal deformations). The values of the 1
parameters used in the simulations are given in Table m2[kg] Mass of the coupler link 0.0682 
1. The torque on the input link is given by: 
 m3[kg] Mass of the output link 0.0682 
 0 sin(2 t / Tm ) t Tm
 (t ) (17) 
 0 t Tm
where τ0 is the peak torque and Tm is the duration of 
the torque. 
 The initial conditions are selected as follows: 
angle of input φ10 = π/2, angular velocity of input link 
 10 0 , elastic deformations qi0 = 0 and elastic 
deformations velocity qi0 0 . By using the Newton - 
Raphson method for solving constraint equations (1a, 
b) with the initial conditions as above, we obtain the 
initial positions of the other links as: 
 0.6752rad , 2.1564 rad ,  0,  0 . Fig. 2. Crank angle, τ0 = 0.03 Nm. 
 20 30 20 30 rigid, flexible. 
 Figures 2 and 3 compare responses of rigid and 
flexible mechanism with a peak torque magnitude of 
τ0 = 0.03 Nm and Tm = 1s. As mentioned in section 1, 
because of the fully coupled nature of the equations, 
the rigid body coordinates (e.g., input and output link 
angular displacements) are affected by the elastic 
deformation of the coupler link. However, this effect 
is negligible when the peak of torque is small. 
 Another set of simulations is carried out with a 
peak torque magnitude of τ0 = 0.1 Nm, Tm = 1s and 
the simulations are performed during the period from 
0 to 3s. The responses for the flexible and rigid 
models of the four-bar mechanism are shown in 
Figures 4 through 6. The effect of flexibility is now Fig. 3. Output angle, τ0 = 0.03 Nm. 
 rigid, flexible. 
 8 
 Journal of Science & Technology 119 (2017) 006-010 
 4. Control flexible four - bar mechanism 
 The governed equations are highly non-linear 
 differential equations. Most control designs require a 
 linear unconstrained model. In this case a 
 straightforward linearization is not possible since it 
 may be difficult to find an operating point. However, 
 the equations can be linearized around a rigid body 
 trajectory. Since the main goal of the controller is to 
 suppress vibrations, a rigid body trajectory can be 
 used as the nominal trajectory and the deviations 
 from this can be assumed small. The real trajectory of 
 flexible mechanism can be obtained as follows 
 1 1d y1
 2 2d y2
 Fig. 4. Crank angle, τ0 = 0.1 Nm (18) 
 rigid, flexible. 3 3d y3
 y q
 3 i i
 where φ1d, φ2d, φ3d represent the rigid mechanism 
 trajectory; y1, y2, y3 are deviation of flexible trajectory 
 versus rigid trajectory. It is assumed that y1, y2, y3 , 
 y3+i are small. 
 In order to investigate whether it is possible to 
 suppress vibrations of the flexible link by a control 
 torque applied to the input link, a simple control 
 strategy, in particular PD controller is used. The 
 target of the controller is that y1, y2, y3 , y3+i approach 
 zero when time approaches infinity (or large 
 enough). The control torque applied to the input link 
 is given by: 
  c KP y1 Kd y1 (19) 
 where Kp, Kd are the PD controller gains, 
 Fig. 5. Output angle, τ0 = 0.1 Nm. y1 1 1d ; y1 1 1d . Thus, the total torque 
 rigid, flexible. acting on the input link is τ + τC. 
Fig. 6. Longitudinal deformation of flexible coupler Fig. 7. Crank angle with PD controller, τ0 = 0.1 Nm 
link, τ0 = 0.1 Nm. rigid, flexible. 
 9 
 Journal of Science & Technology 119 (2017) 006-010 
 numerically to simulate the system behavior. A 
 simple PD controller was designed which does not 
 require measurement of the elastic deformations. The 
 controller has been shown to be efficient in 
 suppressing the vibrations of the flexible link as well 
 as controlling the rigid body motion. 
 References 
 [1] Sandor, G. N. and Erdman, A. G. (1984) Advanced 
 Mechanism Design: Analysis and Synthesis. 
 Englewood Cliffs, NJ: Prentice-Hall. 
 [2] Sung, C. K. and Chen, Y. C (1991) Vibration control 
 of the elastodynamic response of high-speed flexible 
 linkage mechanisms. ASME Journal of Vibration and 
 Acoustics 113, 14-21. 
Fig. 8. Output angle with PD controller. τ0 = 0.1 Nm. 
 rigid, flexible. [3] Beale, D. G. and Lee, S. W. (1995) The applicability 
 of fuzzy control for flexible mechanisms. ASME 
 Design Engineering Technical Conferences DE-Vol. 
 84-1, 203-209. 
 [4] Liao, W. H. Chou, J. H. and Horng, I. R (1997). 
 Robust vibration control of flexible linkage 
 mechanisms using piezoelectric films. Smart Materials 
 and Structures 6, 457-463. 
 [5] Schwab, A. L. and Meijard, J. P. (1997) Small 
 vibrations superim - Posedon non-linear rigid body 
 motion. Proceedings of the 1997 ASME Design 
 Engineering Technical Conferences, Sacramento, 
 CA,1-7. 
 [6] Yigit, A. Scott, R.A. and Ulsoy, A.G (1988), “Flexural 
 motion of aradially rotating beam attached to a rigid 
 body”. Journal of Sound and Vibration 121, 201 - 210. 
 [7] Khang, N.V (2005), Engineering vibrations (in 
Fig. 9. Longitudinal deformation of flexible coupler Vietnamese), Science and Technics Publishing House 
link with PD controller, τ = 0.1 Nm. 
 0 [8] Khang, N.V (2017), Dynamic of multibody system (in 
 rigid, flexible. 
 Vietnamese, 2. Edition) Science and Technics 
 Now, we will suppress vibrations in case the Publishing House. 
peak torque magnitude of τ0 = 0.1 Nm and Tm = 1s [9] Reddy, J.N (2002), Energy Principles and Variational 
(e.g., Figures 4 – 6). The parameters used in the Methods in Applied Mechanics, John Wiley and Son, 
simulations are given in Table 1. The controller gains ISBN: 978-0-471-17985-6. 
are chosen as K = 0.5, K = 0.2. The calculating 
 p d [10] Bajodah, A. H. Hodges, D. H. and Chen, Y. H (2003) 
results are shown in Figs 7 – 9. These “New form of Kane’s equations for constrained 
results show that controller is able to suppress the systems”. Journal of Guidance, Control, and Dynamics 
vibrations and control the link angular motions. In 26:79–88. 
Figure 9 the longitudinal deformation is suppressed 
within 0.85 s. 
5. Conclusions 
 In this study, dynamic modeling and control of a 
four-bar mechanism with flexible coupler link has 
been investigated and it was assumed that there is 
only axial deformation. The non-linear equations of 
motion are obtained through a constrained 
Lagrangian approach and described by a set of 
differential-algebraic equations. The governed 
differential-algebraic equations are solved 
 10