Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 3, pp. 665-676, Warsaw 2014 DYNAMICAL ANALYSIS OF A CONSTRAINED FLEXIBLE EXTENSIBLE LINK WITH RIGID SUPPORT AND CLEARANCE Mihai Dupac School of Design, Engineering and Computing Bournemouth University, Dorset, United Kingdom e-mail: mdupac@bournemouth.ac.uk Dynamic response of robotic systems is affected bydeformation of their flexible components, velocity andmass of the systems, as well as by the presence of clearance or impact between the components. Since accurate simulations of such robotic systems are increasingly impor- tant, themodelling and dynamical behaviour of an extensiblemechanismwith a rigid crank and a flexible link is investigated in this paper. The equations of motion of the extensible flexible link, constrained to a circular, Cartesian, elliptical, Cassinian, Lame or pear-shaped quartic path, are presented. A dynamical analysis is carried out in order to compare the dynamical response of the flexible link vs. a rigid link under the combined effect of different parameters such as flexibility and clearance. The simulation result shows clear trajectories divergence due to the impact effect of the flexible link on the rigid crank. Keywords: multibody dynamics, impact analysis, contact forces 1. Introduction Themodelling, simulation and control of mechanical and robotic systems have been frequently studied for several years. For the study of their dynamical properties, mechanical models conta- ining rigid links connected by joints are considered. Their dynamical response is usually affected by deformation of the links, by the presence of clearance in the joints as well as by the impact of the components/links. All the mentioned factors affect system performance, which results in a low ability to perform high-precision manipulation, increased vibration and high levels of noise. Moreover, the dynamical stress caused by the motion and the impact of the links affects vibration characteristics of themechanical system, and canmake the linkage fail. However, small amount of information to date about constrained extensible flexible systems can be found in the literature, so any relevant contribution to the topics in particular and knowledge in general may be considered important. Some classical examples of the modelling and simulation of flexible systems, their forward, inverse and impulsive dynamics, and their stability were discussed by Garcia and Bayo (1994), Beale et al. (1998), Kvecses and Cleghorn (2004). The kinematics and equilibrium of an exten- sible rod under axial load was presented in Filipich and Rosales (2000). Dynamic analysis of planar mechanisms with rotating slider joint and clearance received attention in Stoenescu and Marghitu (2003), and the simulation of the non-smooth translational joints with clearance was considered in the work of Zhuang andWang (2012). In Rubinstein (1999), Dupac andMarghitu (2006), a dynamical analysis of some systems with lumpedmasses was considered. The effect of clearance and cracks on some lumped mass systems and the impact equations using Newton’s law were discussed by Brach (1989), Dupac and Beale (2010). Themodelling and simulation of impacting systems as well as the study of some contact force models were considered in Lanka- rani and Nikravesh (1990), Dupac and Beale (2010), Flores et al. (2010, 2011), Machado et al. (2012). 666 M. Dupac The control (by suppressing oscillations) of systems undergoing rotations and impacts, and periodic impacts between elastic beams and rigid beams was studied in Boghiu et al. (1996), Marghitu et al. (1999). A study concerning the free vibration of an extensible rotating beam was presented in the work of Leea and Sheub (2007). Combined vibration control of a flexible linkage mechanism for suppressing its vibrations and impact effects, as well as impact control in the case of a flexible link were discussed by Izumi and Hitaka (1993), Jin et al. (2004). The impact dynamics and control of flexible-joint and dual-arm robots were simulated in Zhang and Angeles (2005), Liu et al. (2007), and some systems with extensible members were studied by Fritzkowski and Kaminski (2009), Dupac (2012, 2013). In this paper, the modelling and simulation of an extensible mechanism with a rigid crank and a flexible link constrained to follow a specific pathwas considered. Accurate simulations for some constrained trajectories, i.e., circular, Cartesian, elliptical, Cassinian, Lameor pear-shaped quartic, explored mechanism behaviour under the combined effect of different parameters such as flexibility, impact due to clearance, and trajectory constraints. A dynamical analysis was carried out in order to compare the dynamical behaviour of the extensible mechanism with a flexible link vs. rigid link behaviour. 2. System model In this section, aflexible extensible linkwhich rotates about the“fixed end” C is considered.The extensible link is composed of a rigid guide of length lCQ (having its “left” end denoted by C and its “right” end denoted by Q), and a moving part (flexible link) which in an un-deformed state has the length lAS. Amotor torque M1 acts on the rigid guide of themechanical system. The flexible link slides inside the guide CQ and has its “left” end denoted by A and its “right” end denoted by S as shown in Fig. 1. The distance between the left end C of the guide and the left end A of the slider (moving part) is denoted by l and represent the constrained trajectory of the end A of the flexible link. Fig. 1. Constrained flexible extensible link with rigid support model 2.1. Flexible link model For the study of the flexible extensible mechanism with a rigid crank, a mechanical model similar to the one described byDupac andMarghitu (2006), Dupac andBeale (2010) and shown in Fig. 2 is considered. The flexible link is modelled as a mechanical system with rigid rods connected by torsional springs. Dynamical analysis of a constrained flexible extensible link... 667 Fig. 2. Mechanical model of the flexible link modelled using n successive equal rigid rods QiQi+1, i=1,n, the absolute angles φi and relative angles θi between two successive links The flexible link shown in Fig. 2 ismodelled using n=6 successive equal rigid rods denoted by QiQi+1, i = 1,n, connected with torsional springs. Each one of the rods QiQi+1 has the mass mi = m and moment of inertia J. The spring constant used to model link flexibility is computed using Rubinstein (1999), Mitiguy and Banerjee (2000), as ks =EJ/lAS, where E is the Young modulus and lAS = ∑n i=1 lQiQi+1 is the length of the flexible link AS when the link is not deformed. The rod Q1Q2 is linked to the constrained trajectory with a slot-joint at Q1. Each one of the rods QiQi+1, i = 1,n, has the length lQiQi+1 = lAS/n. The angles θi, i= 1,n−1, represent the relative angle between the links QiQi+1 and Qi+1Qi+2. The angles between the link QiQi+1 and the horizontal direction denoted by φi are named the absolute an- gles. The system of two successive rods QiQi+1 and Qi+1Qi+2, and the correspondingmoments Mi,Mi+1 and Mi+2 at the nodes Qi,Qi+1 and Qi+2, are presented in Fig. 3. Fig. 3. System of two successive links QiQi+1 and Qi+1Qi+2 and their associatedmoments Mi,Mi+1, Mi+2 at the corresponding nodes Qi,Qi+1,Qi+2 2.2. Constrained trajectories of the flexible link The end A of the flexible link (moving part) is constrained to follow one of the constrained trajectories shown in Fig. 4, that is, a circular, Cassinian, Cartesian (called also Descartes), elliptic, Lame (also known as a super ellipse) or a pear-shaped quartic trajectory respectively. One can see that the left end C of the guide shown in Fig. 4 is not the same as the center O of the constrained trajectories to which the end A is constrained. The extensible link shown 668 M. Dupac Fig. 4. Constrained flexible extensible link about: (a) Cartesian (called also Descartes), (b) Cassinian, (c) elliptic, (d) circular, (e) Lame, (f) pear-shaped quartic trajectory in Fig. 4, composed by the guide CQ and flexible link AS has the end C of the guide located on the Ox axis, the end A of the flexible link follows the constrained trajectory, distance dCA between the end C and the left end A equal to l, i.e., dCA = l, and the length between the end C and point A′ denoted by dCA′ equal to d. Since it is important to describe themotion so that the point C represents the origin of the motion, a geometrical method developed in Dupac (2012, 2013), has been used to derive the associated equations of motion. The general Cartesian and polar equations of motion for all the circular, Cassinian, Cartesian, elliptical, Lame and pear-shaped quartic trajectories are shown in Table 1. The parameters for the Cartesian equations in Table 1 are defined as: a, b and c are real numbers, n is a natural number, and r is the distance between the center O of the Cartesian reference frame and the “left end” A of the flexible link. The parameters for the polar equations inTable 1 are defined as: a, b, c, d and m are real numbers, n1,n2 and n3 are natural numbers, r is the distance between the center O of the Cartesian reference frame and the “left end” A of the flexible link, α is the angle between Ox axis and the radius, and θ is the angle between the Ox axis and the rigid crank. 3. Dynamic model of the flexible link Theposition vector of the center ofmass Wi of each rigid rod QiQi+1, i=1,n of the constrained flexible extensible link is given by rWi =xWii+yWij, where i and j are the unit vectors of the associated Cartesian reference frame Oxy. The horizontal and the vertical coordinates of the center of mass for each rod i=1,n can be expressed as Dynamical analysis of a constrained flexible extensible link... 669 Table 1.Cartesian equations of motion for six possible constrained trajectories Constrained Cartesian equation Polar equation trajectory Circular (x−a)2+(y− b)2 = r2 l2=r2+(r−d)2−2r(r−d) lcosθ+(r−d) r Elliptical (x a )2 + (y b )2 =1 l= b2 a−cos(π−θ) √ a2− b2 Lame (x a )n + (y b )n =1 l= ( ∣ ∣ ∣ 1 a cos mθ 4 ∣ ∣ ∣ n2 + ∣ ∣ ∣ 1 b sin mθ 4 ∣ ∣ ∣ n3)− 1 n1 Cartesian √ x2+y2+ b √ (x−d)2+y2 = a r2−2(a+ bcosθ)r+ c2 =0 Cassinian (x2+y2)2−2a2(x2−y2)+ c4= a4 r4+a4+2r2a2cos(2θ)= c4 Pear-shaped b2y2 =x3(a−x)2 – xWi = dCQ1 cosθ+ i ∑ j=1 lQjQj+1 cosφj − 1 2 lQiQi+1 cosφi yWi = dCQ1 sinθ+ i ∑ j=1 lQjQj+1 sin(χφj)− 1 2 lQiQi+1 sinφi (3.1) where dCQ1 = l can be computed based on Table 1 for each constrained trajectory, and the angles Θi (named absolute angles) are shown in Fig. 2. It can be observed that if all the connecting rods QjQj+1, j = 1,k are constrained by the driver CQ shown in Fig. 1, i.e., dCQ1 + ∑k j=1 lQjQj+1 ¬ lCQ1, then all the angles φj, j = 1,k equal to the angle θ, φ1 = φ2 = . . . = φk = θ. The velocity vector of the center of mass Wi of each rigid rod QiQi+1 of the constrained flexible extensible link is the derivative with respect to time of the position vector rWi given by vWi = ṙWi = ẋWii+ ẏWij where vxWi = ẋWi = ḋCQ1 cosθ−dCQ1θ̇ sinθ− i ∑ j=1 lQjQj+1φ̇j sinφj + 1 2 lQiQi+1φ̇i sinφi vyWi = ẏWi = ḋCQ1 sinθ+dCQ1θ̇cosθ+ i ∑ j=1 lQjQj+1φ̇j cosφj − 1 2 lQiQi+1φ̇icosφi (3.2) 3.1. Dynamic model without clearance For themodel without clearance, the flexible link translates parallel to its support (the rigid guide) and no clearance between the rigid guide and the flexible link is possible. The dynamics of the flexible extensible link can be expressedusing theLagrange differential equation ofmotion d dt (∂T ∂q̇i ) − ∂T ∂qi =Ri (3.3) where Ri are the generalized forces and qi = θi are the generalized coordinates, the subscript i represents the number of the generalized forces/coordinates, andwhere the generalized forces Ri acting on each link can be written as in Dupac and Marghitu (2006). The total kinetic energy of the system in Eq. (3.3) can be expressed as 670 M. Dupac T = n ∑ i=1 Ti = 1 2 n ∑ i=1 (miv 2 Wi + IWiω 2 i) (3.4) where Ti is the kinetic energy of each ith component, IWi = m(l 2 QiQi+1 +h2)/12 is the mass moment of inertia with respect to the center of mass Wi of each link and ωi = ωik is the angular velocity and h is the height of the rod i. Equation (3.3) can be expressed in a matrix formby replacing/computing the derivatives inside the equation. A straightforward example for expressing the differential equations of motion in their non-dimensional form can be followed in Awrejcewicz et al. (2004). 3.2. Dynamic model with clearance For themodelwith clearance, theflexible linkof the extensible crank can translate and rotate about its support (rigid guide) as shown in Fig. 5. Due to the clearance model, the flexible link may exhibit impacts on the rigid crank as shown in Fig. 5. To illustrate the impacts of the flexible link with the guide, Fig. 5 was made with a very large backlash in order to make the clearance clearly visible. The possible impact model for the flexible link is shown in Fig. 5, that is (a) no con- tact/impact between themoving link and the guide, (b) impact ona single point, and, (c) impact on two points. In the dynamics of the flexible linkwith impact, the conditions for switching from one impact model to another depends on the initial conditions, on the dynamic response of the system, as well as on thematerial colliding properties. Sincemultiple impacts at the same time instant can be statistically excluded, simultaneous impacts on two points are not considered in this study but only presented. Fig. 5. Constrained flexible extensible link with rigid support and the clearancemodel, (a) no impact, (b) impact on one point, (c) impact on two points Different contact forcemodels such as the one discussed in Lankarani andNikravesh (1990), Flores et al. (2010, 2011), may be considered to model the dynamical response of the system. The system dynamics is modelled by introducing the contact forcemodels into the equations of motion as external generalized forces. The equations of motion can be expressed as inMarghitu et al. (1999), Dupac and Beale (2010), Flores et al. (2011). Considering the contact force com- ponents to be elastic and dissipative components, the expression of the contact (impact) force can be expressed as in Lankarani and Nikravesh (1990) by Fn =κδ n+Dδ̇ (3.5) where κδn represents the elastic force, Dδ̇ represents the dissipation of energy during impact, κδ is a positive real number which specifies the stiffness of the boundary, κ is the stiffness coefficient at the contact interface, δ is the penetration, n is an exponential coefficient, D is the hysteresis as a function of penetration, and δ̇ is the relative impact velocity. Based on the Dynamical analysis of a constrained flexible extensible link... 671 impact force model with a compliant force formulation (Ryan, 1990) as well as Lankarani and Nikravesh (1990), Stoenescu andMarghitu (2003), Sharf and Zhang (2006), Yang et al. (2009), Machado et al. (2012), the impact force can be evaluated using Eq. (3.5) by replacing D with the viscous damping coefficient c(δ) (Ryan, 1990) as Fn = { κδn+ c(δ)δ̇ δ > 0 0 δ¬ 0 (3.6) where c(δ) =          0 δ¬ 0 cmax δ2 dmax− δ ( 3− 2δ dmax− δ ) 0<δ