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Calculus of axial force in a mechanism using Lagrange equations Thien Van NGUYEN*,1, Roxana Alexandra PETRE1, Ion STROE1 1 *Corresponding author “POLITEHNICA” University of Bucharest, Department of Mechanics, 313 Splaiul Independentei, Bucharest, Romania [email protected]*, [email protected], [email protected] DOI: 10.13111/2066-8201.2016.8.2.8 Received: 14 April 2016 / Accepted: 23 May 2016 Copyright©2016. Published by INCAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 4th International Workshop on Numerical Modelling in Aerospace Sciences, NMAS 2016, 11-12 May 2016, Bucharest, Romania, (held at INCAS, B-dul Iuliu Maniu 220, sector 6) Section 3 – Modelling of structural problems in aerospace airframes Abstract: Lagrange equations are used to study the motion of a system under the action of known external forces. Besides, based on these equations we can determine the internal force in an arbitrary element of the mechanism acting by active force. If an internal force has to be found, a supplementary mobility related to it is considered in the system. The corresponding internal force for new mobility is found for null values of mobility and of its first and second derivatives. Also the determination of the axial force in the connecting rod of the slider-crank mechanism is presented in this paper as an illustration of this method. Key Words: Dynamics, axial force, slider-crank mechanism, constraints. 1. INTRODUCTION Upon to now, several methods have been applied to study and formulate the forward and inverse dynamics such as: Newton-Euler classical procedure, Lagrange equations and multipliers formalism and principle of virtual work, [1], with the goal of giving the kinematic relations and dynamic analysis of the system. Determining the internal forces along with the constraint force analysis are important steps in dynamic analysis, which is the base of the structure design of mechanism. Lu [2] uses the virtual work method to determine the generalized forces of the actuator, in spatial parallel structures and relates them to the real forces that they exert. Geike and Mc Phee [3] proposed a general approach which could determine the inverse dynamics solutions for a planar 3-RRR parallel manipulator and spatial 6-DOF parallel mechanism. Using Newton Euler method and D’Alembert principle Jiang, Li and Wang [9] established the force analysis equations, and also put forward the dynamic analysis model of a parallel mechanism based on the deformation compatibility method. Zhi and Wang [10] improved and applied the solution method of reciprocal screw system to the solution procedure of the constraint forces. Stefan STAICU [5], [7], [8] uses the principle of virtual work for solving the inverse dynamic problem in a 3-DOF parallel mechanism. By using Lagrange equations INCAS BULLETIN, Volume 8, Issue 2/ 2016, pp. 97 – 108 ISSN 2066 – 8201 Thien Van NGUYEN, Roxana Alexandra PETRE, Ion STROE 98 and principle of virtual work, Ion STROE and Stefan STAICU [4] proposed the approach for calculating joint forces in mechanism. The difficulties commonly encountered in dynamic calculus are: complicated kinematical structure possessing of large number of passive degrees of freedom, taking account of inertial forces and constraint forces, problems related to the solution of the inverse dynamics [6]. By using a set of generalized coordinates that are consistent with the constraint relations, we can formulate the equations of motion and calculate the internal forces in the multi-body system without considering the constraint forces, which is the main advantage of Lagrange equations. In fact, the calculus of internal forces for a static system of rigid bodies is quite familiar, but using Lagrange equations for that aim, is an interesting issue considered in recent years. Based on this method, the internal forces in an arbitrary element of the mechanism are simply determined. The slider-crank mechanism is one of the most commonly used machine subsystem in mechanical system design. It is employed as the principal element of internal combustion engines, compressors, fly-ball governors, stamping machines, and many other machines. Therefore, the slider-crank mechanism is considered a simple model for calculating the axial force in connecting rod of the system at an instant moment of time in order to illustrate that method. 2. KINEMATICS OF MULTIBODY SYSTEM Let two bodies (i) and (j) be with motions constrained by a coupling mechanism, which is defined by points Oi, Oj (Fig. 1). The motion of the body (i) with respect to the inertial reference frame O0x0y0z0 is determined by the position vector of the mass center O0Ci and by matrix Ai0 which gives the attitude of moving reference frame Cixiyizi, attached with (i) body, with respect to inertial reference frame O0x0y0z0. In the similar way, the position vector O0C j and matrix A j 0 are defined for the body (j). Figure 1 - System of rigid bodies Each body, (i) or (j), has 6 degrees of freedom when it has no constraint with others. The number of degrees of freedom is reduced by the number of constraints which are imposed by the coupling mechanism. If the general motions of bodies (i) and (j) with respect to the inertial reference frame O0x0y0z0 are known, then the relative motion of the body (i) with respect to the body (j) can be determined by the position vector INCAS BULLETIN, Volume 8, Issue 2/ 2016 99 Calculus of axial force in a mechanism using Lagrange equations O j Oi (O0Ci Ci Oi ) (O0C j C j O j ) (1) and by matrix Aij , which gives the attitude of the body (i) with respect to the body (j) Aij Ai 0 Aj 0 T (2) The matrix Ai 0 allows expressing the unit vectors of the moving reference frame Cixiyizi with respect to unit vectors of inertial reference frame O0x0y0z0 ii i0 ji Ai 0 j0 ki k0 (3) For the unit vectors of the moving reference frame Cjxjyjzj, the relation can be written as following: i j i0 j j A j0 j0 k j k0 (4) The attitude of the body (i) with respect to the body (j) is given by matrix Aij using the relation bellow: i j ii j A i ij j j k j ki (5) In contrast, the attitude of the body (j) with respect to the body (i) is given by matrix A ji with the following relation: i j ii j j A ji ji ki k j (6) By comparing relations (5) and (6), it follows: Aij A ji Ai 0 T for T (7) T relation (3), and multiplying left with A j 0 for relation (4), then adding them, we obtain: Multiplying left with INCAS BULLETIN, Volume 8, Issue 2/ 2016 Thien Van NGUYEN, Roxana Alexandra PETRE, Ion STROE Ai 0 T ii T ji A j 0 ki i j jj k j 100 (8) By multiplying left with Ai 0 for relation (8), the below relation will be obtained: i j (9) jj k j From relations (5) and (9), relation (2) is obtained, which is used to compute the matrix Aij , if the matrices Ai 0 and A j 0 are known. The terms of matrices Ai 0 and A j 0 depend on the attitude angles of the bodies (i) and (j) with respect to the inertial ii T ji Ai 0 A j 0 ki reference frame. If the orientation of the body (i) with respect to the inertial reference frame is defined by 1i , 2i , 3i angles, which correspond to the sequence of rotations 1-2-3 with respect to a reference frame parallels with the inertial reference frame O0x0y0z0, then the matrix below c2i c3i s1i s2i c3i s3i c1i c1i s2i c3i s1i s3i Ai 0 c2i s3i s1i s2i s3i c1i c3i c1i s2i s3i s1i c3i s1i c2i c1i c2i s2i (10) and the angular velocity c2i c3i s3i 0 1i ωi 0 c2i s3i c3i 0 2i s2i 0 1 3i are obtained. In the above relations, notations of the following form were used: (11) s1i sin 1i , c1i cos 1i s2i sin 2i , c2i cos 2i (12) s3i sin 3i , c3i cos 3i Coupling mechanism between the bodies (i) and (j) imposes restrictions on the relative motion of (i) with respect to (j). 3. LAGRANGE EQUATIONS 3.1. Equations of Motion of the System A significant advantage of the Lagrange approach for developing equations of motion for complex systems comes as we leave the Cartesian xi coordinate system and move into a general coordinate system. We introduce a general notation for the relationship between h Cartesian variables of position xi and their description in generalized coordinates (for some systems, the number of generalized coordinates is lager then the number of degrees of INCAS BULLETIN, Volume 8, Issue 2/ 2016 101 Calculus of axial force in a mechanism using Lagrange equations freedom and this is accounted for by introducing constraints on the system). In general case, each xi could be dependent upon every qk. xi xi q1 , q2 ...qk ..., qh (13) When constraints are expressed by functions of coordinates, the motion of the systems can be studied with Lagrange equations for holonomic systems with dependent variables, whereas other conditions of constraint are expressed by velocities, the motion is described with Lagrange equations for non-holonomic systems. For a non-holonomic system, the Lagrange equations corresponding to a system of h generalized coordinates d E dt qk E Qk qk p a i ik , k 1, 2,..., h (14) i 1 are completed with the constraints h a ik qk i 1, 2,..., p , bi 0, (15) k 1 where E is the kinetic energy expressed with respect to an inertial reference frame, conventionally considered as fixed, and Wk Qk (16) qk are the generalized forces, while Wk is the virtual work produced by the forces acting upon the system, corresponding to the virtual displacement qk . By solving system (14), of h equation, and (15), of p equations, coordinates qk and Lagrange multipliers i will be found. From (14), the equations for the holonomic system can be obtained by replacing functions aik . In the case of a holonomic system, constraints are of the form: i q1 ,..., qh , t 0, i 1,2,..., p (17) From the above formula, the differential form is obtained: h i q k 1 qk bi 0, i 1, 2,..., p (18) k By comparing relations (18) and (15), it follows: i aik qk (19) Then equations (14) become: d E E Qk dt qk qk qk Let define the analytical function U p , k 1, 2,..., h i i (20) i 1 p i i (21) i 1 then equations (20) can be written in the form: INCAS BULLETIN, Volume 8, Issue 2/ 2016 Thien Van NGUYEN, Roxana Alexandra PETRE, Ion STROE d E dt qk 102 E U Qk , k 1, 2,..., h qk qk (22) Starting from these h differential equations and using p relations of constraint, we determine just the generalized coordinates qk and the Lagrange multipliers. 3.2. Calculus of Internal Forces For a mechanical system with h degrees of freedom represented by independent generalized coordinates qk (k=1,2…,h), the Lagrange equations are expressed as following: d E dt qk E U Qk* , (k 1,..., h) qk qk (23) An internal force Qh 1 , as new generalized force, can be found if a new fictitious mobility according to internal force is considered. Then the mechanical system is considered as the one with h+1 degrees of freedom. Equation for the new mobility is d E E U Qh 1 . dt qh 1 qh 1 qh 1 (24) Considering again the mechanism, the internal force h1 is easily obtained from (24) in the following form d E E U h 1 dt qh 1 qh 1 qh 1 qqhh 11 00 (25) qh 1 0 4. EXAMPLE: SLIDER-CRANK MECHANISM As an example, the one degree of freedom system of slider-crank mechanism is considered (Fig.2). The slider-crank mechanism consisting of a crank (1) characterized of the length OA= r, mass m1; connecting rod (2) characterized by the length AB= l, mass m2 and a slider 3 characterized by mass m3. The crank OA rotates with the motion law o t o 2 t 2 by the active torque Mo. A 1 2 C2 C1 Mo B 3 o Figure 2 - Slider-crank mechanism The Lagrange equations of motion of a system are written in the form: d E E U Qk , k 1, 2...h (26) d t qk qk qk We first apply Lagrange equations to derive the equations of motion of the mechanism. This is one degree of freedom system, so we choose the generalized coordinates as q ( ) . INCAS BULLETIN, Volume 8, Issue 2/ 2016 103 Calculus of axial force in a mechanism using Lagrange equations It is convenient to describe the positions of the mechanism mass centers with Cartesian coordinates, and then to express the kinetic energy E and the force function U using the polar angle. Moments of inertia of the crank 1 and the connecting rod 2 are written: Jo m1.r 2 , 3 and J C 2 (27) m2 .l 2 . 12 (28) respectively. Thus, the kinetic energy is 1 1 1 1 E 1T .J o .1 rCT2 .m2 .r .C 2 2T .J C 2 .2 rCT3 .m3 .r .C 3 2 2 2 2 E sin 2 m1.r 2 . 2 l 2 .cos 2 r.sin 2 .cos m2 . 2 2 6 6(l r 2 .sin 2 ) 2. l 2 r 2 .sin 2 1 r 2 .cos 2 r.cos m3 . 2 2 2(l r 2 .sin 2 ) 2 l r 2 .sin 2 2 2 .r . (29) 2 2 2 .r . .sin The fore function U has the expression (m1 m2 ) g.r.sin . 2 The external generalized force acting on the mechanism is written L M . Q M U (30) (31) Applying (26), the differential equation governing the motion is obtained m1 m2 .l 2 cos 2 m3 .r 2 .sin 2 2 m2 m3 .sin 2 2 2 3 3 l r .sin 2 (m2 2m3 ).r.sin .cos l 2 r 2 .sin 2 2 r 3m2 r.sin 3 m2 m .sin .cos m . 3 3 2 l 2 r 2 .sin 2 4 2 2 2 2 m2 2m3 . r.sin .cos m2 m3 r .sin .cos .(cos sin ) l 2 r 2 .sin 2 l 2 r 2 .sin 2 4 r 2 2 m2 .l 2 .sin .cos r 3 .sin 3 .cos 2 m2 2 2 2 2 m3 . 2 l r 2 .sin 2 l 2 r 2 .sin 2 12(l r .sin ) 2 2 3 m2 .r .l .sin .cos 12(l 2 r 2 .sin 2 ) 2 m m2 .g.r.cos M 1 2 (32) INCAS BULLETIN, Volume 8, Issue 2/ 2016 Thien Van NGUYEN, Roxana Alexandra PETRE, Ion STROE 104 - The axial force in the connecting rod AB at an instant moment of time can be determined based on the Lagrange equations of second kind. At that time, the generalized coordinates of the slider-crank mechanism are represented by: q1 ; q2 u. And u is considered as a supplemental mobility. A 1 C1 C2' 2' 2'' N C2'' Mo B 3 o Figure 3 - Slider-crank mechanism with virtual supplemental displacement on the connecting rod d E E U N h 1 u 0 d t u u u u 0 (33) u 0 Suppose that the connecting rod AB is divided into two parts (it has the mass: m2’ and m2’’, respectively) at the position (0≤ ≤ l). By geometrical relation shown in Fig. 3, we have: r.sin (l u)sin (34) sin r.sin l u (l u )2 r 2 .sin 2 l u By carrying out the total differential equation (36), the following is obtained cos r. .cos u.sin r.(l u ). .cos r.u.sin (l u ).cos (l u ). (l u ) 2 r 2 .sin 2 (35) (36) (37) The kinetic energy will be expressed as following: 1 1 1 1 E 1T .J o .1 rCT2' .m2' .rC 2' 2'T .J C 2' .2' rCT2'' .m2'' .rC 2'' 2 2 2 2 (38) 1 T 1 T 2'' .J C 2'' .2'' rC 3 .m3 .rC 3 2 2 where 2' , 2'' , which are the angular velocity vectors of the two parts m2’ and m2’’, respectively, are expressed as 0 , ω2' ω2'' α 0 (39) r.(l u ). .cos r.u.sin (l u ). (l u ) 2 r 2 .sin 2 and JC2’, JC2’’, which are the tensor of inertia of the crank 1, the two parts m2’, m2’’ with respect to C2’, C2’’, are written in the forms: INCAS BULLETIN, Volume 8, Issue 2/ 2016 105 Calculus of axial force in a mechanism using Lagrange equations m2 . 3 , 12.l (40) m2 .(l )3 , 12.l (41) J C 2' J C 2'' and rC 2' , rC 2'' , rC 3 , which are the position vectors of mass centers of the two parts m2’, m2’’ and slider m3, respectively, are expressed as following: 2 2 2 r.cos 2(l u ) . (l u ) r .sin r. r.sin .sin , 2(l u ) 0 (42) (l 2u ) 2 2 2 r.cos 2(l u ) . (l u ) r .sin r.(l 2u ) r.sin .sin , 2(l u ) 0 (43) r.cos (l u ) 2 r 2 .sin 2 0 . 0 (44) rC 2' rC 2'' rC 3 By derivation rC 2' , rC 2'' , rC 3 in respect to time, rC 2' r 2 . . .sin .cos r 2 . .u.sin 2 r. .sin 2(l u ). (l u ) 2 r 2 .sin 2 2(l u ) 2 . (l u ) 2 r 2 .sin 2 r. . .cos r. .u.sin r. .cos 2(l u ) 2(l u ) 2 0 , (45) rC 2'' r 2 .(l 2u ). .sin .cos 2(l u )3 .u r 2 .(l ).u.sin 2 r. .sin 2 2 2 2( l u ) ( l u ) r .sin 2(l u ) 2 . (l u ) 2 r 2 .sin 2 r.(l 2u ). .cos r.(l ).u.sin , r. .cos 2(l u ) 2(l u ) 2 0 (46) INCAS BULLETIN, Volume 8, Issue 2/ 2016 Thien Van NGUYEN, Roxana Alexandra PETRE, Ion STROE rC 3 r 2 . .sin .cos (l u ).u r. .sin 2 2 2 (l u ) r .sin (l u )2 r 2 .sin 2 0 0 106 . (47) The force function U has the expression m .g.r.l.sin m2 .g.(l ).r.u.sin m U 1 m2 .g.r.sin 2 2(l u ) l.(l u ) 2 (48) Applying (33), we obtain the axial force N in the connecting rod AB 6(m2 m3 ).l 5 6.m2 .l 4 . 3m2 .r 4 .(2 l ).sin 4 3(3m2 2m3 ).r 2 .l 3 .sin 2 12m2 .r 2 .l 2 . .sin 2 (5m2 6m3 ).r.l 3 .cos . l 2 r 2 .sin 2 3m .r 3 .(2 l ).sin 2 .cos . l 2 r 2 .sin 2 6m .r.l 2 . .cos . l 2 r 2 .sin 2 2 2 N 2 2 2 2 2 2 6.l .(l r .sin ). l r .sin 2 6(m2 m3 ).r.l 7 .cos 3m2 .r 7 .(2 l ).sin 6 .cos 6m2 .r.l 6 . .cos 3(5m2 4m3 ).r 3 .l 5 .sin 2 .cos 6(2m2 m3 ).r 5 .l 3 .sin 4 .cos 3 2 2 2 2 2 18m2 .r .l . .(l r .sin ).sin .cos 2 2 2 2 2 2 2 2 6l .(l r .sin ) . l r .sin 6 6 2 5 2 2 5 2 3m2 .(2 l ).r .sin 3( m2 2m3 ).r .l .cos (5m2 6m3 ).r .l .sin 2(4m 3m ).r 4 .l 3 .sin 4 6m .r 2 .l 2 . .(l 2 2r 2 .sin 2 ).sin 2 2 3 2 m .r 4 .l 3 .sin 2 .cos 2 3m .r 2 .l. 2 .(l 2 r 2 .sin 2 ).cos 2 2 2 6l 2 .(l 2 r 2 .sin 2 ) 2 (m m ).g .r.sin (m1.l 2m2 . ).g .r.sin 1 2 2l 2l 2 2 (49) In the inverse dynamics, we suppose that the history of rotational motion of the crank is known by the following function: o t o t 2 , where o 25 (rad / s), o (rad / s 2 ) 2 100 For the simulative purpose let’s consider the slider-crank mechanism, which has the following characteristics: m1=0.1(kg), r=0.1 (m), m2=0.1 (kg), l=0.2 (m), m3=0.1 (kg). Using MATLAB software, a program was developed to solve the inverse dynamics of the slider-crank mechanism. The variations of the active torque, rotational angle versus time are shown in Fig. 4 and Fig. 5, the variation of the axial force with respect to ratio /l is shown in Fig.6, and the variation of the axial forces at three specified positions versus time is shown in Fig. 7. INCAS BULLETIN, Volume 8, Issue 2/ 2016 107 Calculus of axial force in a mechanism using Lagrange equations Figure 4 - The active torque of the crank M. Figure 5 - The rotational angle . Figure 6 - The axial force with respect to “l” at the instant time t=0.03(s). Figure 7 - The axial forces at the three specified positions versus time. 5. CONCLUSIONS With the Lagrange equations, the position, velocity and acceleration of each element of mechanism in real-time revolution can be released from the differential equations of motion in the kinetic analysis. In addition, the active torque consistent with the given motional law has been determined taking account the masses and forces of inertia introduced by links of the mechanism in the model dynamics. A new method to compute axial forces is presented in this paper, but the method can be extended to all internal forces. REFERENCES [1] L.-W. Tsai, Robot analysis: The Mechanics of Serial and Parallel Manipulators, Wiley, New York, ISBN: 978-0-471-32593-2, 1999. [2] Y. Lu, Using virtual work theory and CAD functionalities for solving active force and passive force of spatial parallel manipulators, Mechanism and Machine Theory, Elsevier, 42, pp. 839-858, 2007. [3] T. Geike, J. Mc Phee, Inverse dynamic analysis of parallel manipulators with full mobility, Mechanism and Machine Theory, Elsevier, 38, pp. 549-562, 2003. [4] I. Stroe, S. Staicu, Calculus of joint forces using Lagrange equations and principle of virtual work, Proceedings of the Romanian Academy, Series A, Volume 11, no. 3, pp. 253-260, 2010. INCAS BULLETIN, Volume 8, Issue 2/ 2016 Thien Van NGUYEN, Roxana Alexandra PETRE, Ion STROE 108 [5] S. Staicu, I. Stroe, Comparative analysis in dynamics of the 3-RRR planar parallel robot, Proceedings of the Romanian Academy, Series A, Volume 11, no. 4, pp. 347-354, 2010. [6] R. Voinea, I. Stroe, A general method for kinematics pairs synthesis, Mechanism and Machine Theory, Elsevier, 30, 3, pp. 461-470,1995. [7] S. Staicu, D. Zhang, A novel dynamic modeling approach for parallel mechanism analysis, Robotics and Computer-Integrated Manufacturing, Elsevier, 24, 1, pp. 167-172, 2008. [8] S. Staicu, Inverse dynamics of the spatial 3-RPS parallel robot, Proceedings of the Romanian Academy, Series A, Volume 13, pp. 62-70, 2012. [9] Y. Jiang, T. M. Li, L. P. Wang, Research on the dynamic model of an over-constrained parallel mechanism, Journal of Mechanical Engineering, vol. 49, no. 17, pp. 123–129, 2013. [10] Ch. Zhi, S. Wang, Y. Sun, B. Li, A novel analytical solution method for constraint forces of the kinematic pair and its applications, Mathematical problems in engineering, Hindawi Publishing Corporation, Volume 2015, 8 pages, http://dx.doi.org/10.1155/2015/371342, 2015. INCAS BULLETIN, Volume 8, Issue 2/ 2016