Journal o Mechanical Science and Technology 27 (9) (2013) 2577~2583 www.springerlink.com/content/1738-494 DOI 10.1007/s12206-013-0701-y Stick-slip conditions in the general motion o a planar rigid body Iman Kardan *, Mansour Kabganian, Reza Abiri and Mostaa Bagheri Mechanical Engineering Department, Amirkabir University o Technology, 424 Haez Ave., Tehran, 15875-4413, Iran (Manuscript Received June 11, 2012; Revised March 1, 2013; Accepted April 30, 2013) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract In this paper we study combined translational and rotational (general) motion o planar rigid bodies in the presence o dry coulomb riction contact. Despite the cases where the body has pure translational/ rotational motion or can be assumed as a point mass, during the general motion, distinct points o the rigid body move in dierent directions which cause the riction orce vector at each point to be dierent. Thereore, the direction and the magnitude o the overall riction orce cannot be intuitively deined. Here the concept o instantaneous center o rotation is used as an eective method to determine the resultant riction orce and moment. The main contribution o this paper is to propose novel stick-slip switching conditions or the general in-plane motion o rigid bodies. Simulation results or some combination o eternal orces are provided and some eperimental tests are designed and conducted or practical veriication o the proposed stick-slip conditions. Keywords: Dry coulomb riction; General in-plane motion; Stick-slip phenomenon; riction measurement ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction Due to nonlinearities and other compleities that taking riction orces into account may introduce in complicated systems, it's a common interest to ignore them, supposing that they are negligible comparing to the other orces. Although this ignorance provides us with some ease, it can degrade the desired perormance o the system. Åström [1] showed that dry riction may produce some steady state error in PD-type controllers. On the other hand, limit cycles may occur in PID-type controllers [2]. Recent approaches such as adaptive compensation [3] are also used to suppress the riction eect, which widely depend on the riction ormulation. Hence to improve the perormance and also to get the model more similar to the real system, the riction orces should be somehow ormulated. When modeling dry riction, one o the unavoidable matters to be dealt with is the discontinuity induced by the stick-slip motion [4]. Den Hartog [5] and Hong and Liu [6] have studied the motion o a one-dimensional riction oscillator under harmonic ecitations with zero-stop, two-stops and multi-stops per cycle. Abdo and Abouelsoud [7] used Liapunov second method to estimate the amplitude o the velocity and displacement o the stick-slip motion o a one-dimensional massspring-damper system on a moving belt. Xia [4] proposed a * Corresponding author. Tel.: +98 2164543463 E-mail address: i.kardan@aut.ac.ir Recommended by Associate Editor Cheolung Cheong KSME & Springer 2013 model or investigating the stick-slip motion o a twodimensional riction oscillator under arbitrary ecitations. Duan and Singh [8] proposed a 3-DO rotational model or a torque converter clutch and investigated the stick-slip conditions or that system. They also studied the orced vibration o a reduced 1-DO model o their system assuming that the normal load is varying periodically [9]. Theoretical and eperimental study o the stick-slip motion o a 2-DO system with varying normal load is done by Awrejcewicz and Olejnik [10]. In all the aorementioned cases the bodies in ocus o study have pure translational/rotational motions. When the body has simultaneous rotational and translational motion, which is commonly called general motion, every point o the body moves in a dierent direction. Thus in these cases determining the direction and the magnitude o the resultant riction orce vector is not straight orward. Goyal et al. [11-13] presented a rather comprehensive study on how to determine the magnitude and direction o riction orce/moment acting on a sliding rigid body in the case o isotropic and anisotropic riction laws. They proposed to use the instantaneous center o rotation to determine the overall riction eect which will be ollowed in the present work. They also studied the motion o a reely sliding object which comes to rest under the inluence o dry riction. Assuming a circular contact area, Zhuravlev [14] has ormulated the problem o combined spinning and sliding riction. Kireenkov has used the Zhuravlev theory to study the motion o the bodies with planar dry riction contacts [15]. Kireenkov et al. [16]
2578 I. Kardan et al. / Journal o Mechanical Science and Technology 27 (9) (2013) 2577~2583 also prepared a test bed or eperimental veriication o the 2D models o combined riction. Although the riction modeling or the general motion is provided in these works, they do not study the stick-slip phenomena. In act there seems to be a lack o literatures which have studied the stick-slip switching conditions in simultaneous rotational and translational motion. Thereore, utilizing the method o instantaneous center o rotation to determine the overall riction orce, the stick-slip conditions in the general motion o a planar rigid body are proposed and eperimentally evaluated or the irst time in this paper. It should be noted that only rigid bodies with rectangular cross-sections are considered here. However, the presented ormulations can be easily etended to any other body shapes. This paper is organized as ollows. A simple problem o a planar rigid body with 3 DOs moving under arbitrary eternal orces in the presence o dry riction is deined in section 2. The case o pure translational motion is studied in section 2.1. or the case o general motion, the method o determining the resultant riction orce is presented in section 2.2 and the stickslip switching conditions are proposed in section 2.3. Section 3 provides the numerical simulation results and the eperimental veriication method is described in section 4. inally section 5 concludes the paper. 2. A 3-DO planar rigid body with dry riction contact A rectangular planar rigid body is considered which is moving under arbitrary eternal orces as shown in ig. 1. This body is in dry riction contact with a horizontal surace and has 3 DOs, two o which being in-plane translations along and y aes and the other being a rotation about an ais perpendicular to the plane o motion. Using ewton's second law, one can easily derive the governing equations o the in-plane motion o the body as: m 0 0 ag + 0 m 0 ag y = y + y 0 0 J G α M G + M G where m is the mass o the body and J G is its inertia about an ais perpendicular to the plane o motion passing through the center o mass (CM). a G and a Gy are the linear accelerations o the body along and y aes and α is its angular acceleration. and y are the components o riction orce along the corresponding aes and M G is the riction moment about CM. 2.1 Pure translational motion When the body has pure translational/rotational motion, the riction orce/moment vector can be readily determined knowing the velocity vector. I the body is at rest (stick phase), the (1) ig. 1. A planar rigid body under arbitrary ecitations. riction vector equals the net in-plane ecitation net but in the opposite direction. When the velocity becomes non-zero (slip phase), the riction vector will be in the opposite direction o the velocity vector with a constant magnitude o µ k, in which µ k is the kinetic riction coeicient and is the normal contact orce. These statements are summarized in Eq. (2). net V= 0 = µ k V V 0 V where V is the vector o velocity o the body s center o mass. It should be mentioned that or the sake o simplicity two main assumptions are made here: (1) The body does not take any out o plane rotation. Thus the normal orce is constant and considered to be evenly distributed over the contact surace. (2) The riction coeicients are assumed to be identical allover the contact surace and in all directions. In this simple case the stick-slip conditions can be easily stated: The body remains or enters in stick phase i the velocity is zero and Eq. (3) is satisied, where µ s is the static riction coeicient. µ. (3) net 2.2 General motion s As previously mentioned, during the general motion, at dierent points o the body the riction orce vector has dierent direction which depends on the direction o the velocity vector o those points. So to determine the resultant riction orces it's necessary to determine the direction o motion o every point o the body. To do this, we ollow the method proposed by Goyal [11] and use the instantaneous center o rotation (CR), about which the body has a pure rotational motion at each given moment. (2)
I. Kardan et al. / Journal o Mechanical Science and Technology 27 (9) (2013) 2577~2583 2579 become: pg G CR = RCR q G yg y CR (6) where p and q indicate the components o the transormed coordinate system and R CR is the rotation matri. inally the resultant riction components can be computed by integration: ig. 2. Inertial and transormed coordinate systems. Knowing linear and angular velocity o the body's center o mass (G), the location o CR is determined as: y ( ω V ) = CR G G y = ( ω y + V )/ ω CR G G / ω (4) M p q = d sin( β ) = µ ksign( ω) A = d cos( β ) = µ ksign( ω) A CR = R d B CR p p + q p p + q dpdq dpdq = µ ksign( ω) p + q dpdq A (7) (8) (9) here ( G,y G ) and ( CR,y CR ) deine the location o center o mass and instantaneous center o rotation respectively, ω is the angular velocity o the body and V G and V Gy are its linear velocities along and y aes. ow considering an element with dimensions o d and dy in the location o point B as shown in ig. 2, the velocity o this point o the body and the riction orce acting on the element can be determined as: V = ω R B d B CR V = µ k A V B B ddy where R B-CR is the position o point P relative to CR. It's assumed that the normal orce is distributed evenly on the contact surace, so that the normal orce on each element can be considered to be proportional to its area. Knowing the riction orce at the location o each element, the resultant riction orce and moment can be computed by integrating over the entire body. In order to acilitate the integration process, we propose a coordinate transormation. The coordinate system is translated to the location o CR and rotated as much as θ which is the body's in-plane rotation. This transormation simpliies the integration limits. The new coordinates o center o mass will (5) where p and q are the components o riction orce along p and q aes, M CR is the resultant riction moment about CR, and L and W are the dimensions o the body. ow the components should be transormed to inertial coordinates and the riction moment should be taken about the center o mass: T p = RCR y q M = M + y ( y ) ( ). G CR G CR y G CR (10) inally the accelerations are computed using Eq. (1) and by integrating them, velocity and position o the body in the net time step are deined. According to Eq. (4) as ω goes to zero, position o CR shits to ininity, implicating that the body has a pure translational motion. The method to deal with this case is discussed in section 2.1. 2.3 Stick-slip switching conditions The conditions o switching between stick and slip phases in a general motion o a planar rigid body are derived in this section. By deinition, using CR to describe the motion o a rigid body combines the rotational and translational motions
2580 I. Kardan et al. / Journal o Mechanical Science and Technology 27 (9) (2013) 2577~2583 into a pure rotation about CR. Thereore while every one o linear or angular velocities is non-zero, the body remains in slip phase. In order to enter or remain in stick phase, all the velocities should be zero and the eternal orces should be below a maimum. To determine this upper limit the concept o instantaneous center o zero acceleration is employed, which is a point with zero acceleration at each instant and is abbreviated hereater as CA. It is obvious that during the stick phase all the accelerations are zero and thereore, using the CA seems not to be applicable here. But we use the CA to determine the stick-slip switching conditions by making an assumption such that: In the stick phase and under the inluence o eternal orces, the body tends to move in the direction that it would move in the absence o riction. This assumption could be justiied regarding the act that riction does not cause any motion by itsel and just tend to counteract the relative motion o bodies in contact. Assuming that there is no riction, the linear and angular accelerations o the body can be calculated as: 1 a i m 0 0 ayi = 0 m 0 y α 0 0 J i G M G (11) where the subscript i indicates that the quantities are computed assuming a rictionless contact. The location o CA is then determined using Eq. (12). y CA CA α a = α i G yi α y + a = α i i G i i. (12) By setting the origin o transormed coordinate system to CA, the maimum moment that static riction can eert about CA could be deined as: ( M ) CA ma = R d B CA Gi 2 Gi 2 = µ ssign( αi ) p + q dpdq A pgi = qgi y y G CA R CA G CA Gi 2 Gi 2 (13) (14) where R CA is the rotation matri between inertial coordinate system and the transormed one located at CA. inally the stick condition can be stated as: The body will remain or enter in stick phase, i all the velocities o the body are zero and the resultant moment o eternal orces about CA is less than the maimum moment which static riction can ig. 3. o-slip responses. ig. 4. Zero-stop responses. apply, or in the other words: M ( M ). (15) CA CA ma 3. Simulation results 3.1 Pure translational motion Xia [4] states that when the ecitations along and y directions are sinusoidal with a dierence o π/2 in their phases but identical amplitudes and requencies, the response will have a circular orbit. In act in this case the resultant eternal orce vector will have a constant magnitude. I the magnitude is greater than riction limit orce, the body will always slip (zero-stop). Otherwise, no motion will occur (no-slip). Thereore in this case, stick-slip motion won't happen. To assess these statements, the values o parameters are chosen as: µ s = µ k = 0.4, L = 0.03 [m], W = 0.01 [m], m = 0.0234 [kg], J G = 1.95e-6 [kg.m 2 ], and two dierent sets o eternal orces are applied to the body. The irst set o ecitations is chosen to have an amplitude o 0.08 [] with M G = 0, = 0.08cos(5t), and y = 0.08cos(5t+π/2). As can be seen rom ig. 3, in this case the magnitude o the applied orce is less than the riction limit and no slip occurs. The second set o ecitations has an amplitude o 0.1 [] with M G = 0, = 0.1cos(5t), and y = 0.1cos(5t+π/2). In this case the magnitude o the applied orce always remains greater than the riction limit and the body never enters the stick phase (ig. 4). As shown in ig. 5, a case o multi-stop per cycle is also simulated using the same value o parameters but a dierent set o ecitations with M G = 0, = 0.08cos(5t), and y = 0.073sin(3t).
I. Kardan et al. / Journal o Mechanical Science and Technology 27 (9) (2013) 2577~2583 2581 ig. 5. our-stop responses. ig. 7. Schematic overview o the setup. ig. 6. The general in-plane responses. It is notable that the initial position and orientation o the body is chosen in such a way that makes the responses symmetric about the origin. 3.2 General motion To impose a general motion to the body, the ecitations M G = 0.0007cos(4t+π/4), = 0.07cos(5t), and y = 0.06sin(3t) are applied with the same values o parameters as beore. Responses are illustrated in ig. 6. Simulations results indicate that although or the pure translational motions the magnitude o the resultant riction orce equals the value o µ k during the slip phase, this no longer holds or the case o general motion. However this magnitude always remains within the circle o riction limit as epected. The results also reairm that stick phase happens simultaneously or all the degrees-o-reedom. 4. Eperimental validation In order to practically validate the presented riction modeling, some deinite orces and torques should be applied to the body in a way that a general motion is imposed. Then the body's displacements should be measured and compared to the simulation results. This procedure is almost impossible in practice because o the diiculties related to applying a desired orce to a moving body. So the whole presented riction ormulation could not be veriied eperimentally. But evaluation o the stick-slip switching condition, which is the main contribution o this work, is possible by conducting a test as ollows. It is known that a point orce which its line o action does not pass through the center o mass can be replaced by the same orce at the center o mass and a moment. The magnitude o the moment is a unction o the eccentricity and the magnitude o the applied orce. So, i a body at rest is subjected to an increasing eccentric point orce, its initial motion will be a general motion. By measuring the size and knowing the location o the applied orce, the required orce to impose an initial general motion to the rest body can be measured practically. The proposed stick-slip switching condition can be veriied by comparing the measured orce to the theoretically computed one. It is evident that in this test only the magnitude o the applied orce at the beginning o the motion is measured and thereore just the condition o switching rom stick to slip phase is veriied. However according to Eq. (15) it is clear that the validity o condition o entering the stick phase (switching rom slip to stick phase) can be directly concluded rom the validity o the condition o switching rom stick to slip phase. Thereore it can be declared that by running this test we can veriy the whole presented stick-slip switching conditions. To perorm the described test an eperimental setup is prepared (igs. 7 and 8). The setup includes a load cell SS2 by Sherborne Sensors mounted on a 2-DO motorized translation stage by Standa. The load cell signal is transerred to a PC through a DAQ board PCI 1716 by Advantech. As illustrated in ig. 9, two L-shaped links are prepared to transer orce between the body and the load cell. The one with a lat end is used in measuring the riction coeicients and the other one is used to apply eccentric point orces to the body. The body is a 30 10 10 [mm 3 ] aluminum rectangular cube which is placed on a ground steel surace.
2582 I. Kardan et al. / Journal o Mechanical Science and Technology 27 (9) (2013) 2577~2583 ig. 10. A sample o the measured orce or pure translation motion. ig. 8. The prepared eperimental setup. ig. 11. A sample o the measured eccentric orce or h = 10 mm. ig. 9. L-shaped connecting links. ig. 12. A sample o the measured eccentric orce or h = 5 mm. As the irst step o the validation procedure the riction parameters should be determined practically. The motorized stage moves the load cell with a constant speed and by using the lat-end link a pure translational motion is imposed to the body. As the link touches the body, because o the elastic property o the load cell tip, the applied orce begins to increase. This increase continues till the eerted orce overcomes the static riction and the body starts to move. At this moment the measured orce decreases rapidly and or the rest o the motion gains an almost ied value with small luctuations. Beore starting to move and while moving with a constant speed the body has zero acceleration, so it can be claimed that the highest value o the measured orce equals the maimum value o the static riction µ s and its inal ied value equals the kinetic riction µ k. A sample o the applied orce is shown in ig. 10. Knowing the weight o the body, the riction coeicients are calculated to be µ s = 0.55 and µ k = 0.52. To evaluate the proposed stick-slip switching conditions or the general motion, the sharp-end link is used to apply an eccentric orce to the body. As shown in ig. 7 the orce is eerted perpendicular to the body's side with a distance o h millimeter rom the center o mass. Similar to the previous case the load cell is moved with a constant speed and when the link touches the body, the measured orce increases rapidly till the motion is started. The measured orce or two dierent values o h = 10 [mm] and h = 5 [mm] is depicted in igs. 11 and 12, respectively. A comparison o eperimental and theoretical orces which are required or initiating a general motion is provided in Table 1, in which the theoretical orces are calculated as ollows: The Inertial coordinate system is deined such that θ = 0 and the eternal orce lies along direction. Thereore a yi = 0 and R CA = I where I is the identity matri. Then knowing mass and moment o inertia o the body, the accelerations a i and α i are computed using Eq. (11) and the location o CA is deined through Eq. (12). inally (M CA ) ma is determined using Eq. (13) and the eternal orce which can produce such moment about CA is considered as the theoretical orce. The good agreement between theoretical and eperimental results indicates that the proposed model predicts the switching conditions very well. The results also show that by increasing the eccentricity, as can be inerred intuitively, the
I. Kardan et al. / Journal o Mechanical Science and Technology 27 (9) (2013) 2577~2583 2583 Table 1. Comparison o eperimental and theoretical orce or initiating a general motion. Eccentricity [mm] body starts to move with less eort. It should be also noted that because o the compleity o the imposed motion no analysis can be done about the changes in the orce measured ater the motion is started. 5. Conclusions General motion o a planar rigid body in the presence o dry riction is ormulated and stick-slip switching conditions are proposed. Although the eternal orces can be o any orm, simulation results or some combinations o harmonic ecitations are provided. The results show that the presented ormulation eectively models the cases o zero-stop, one-stop and multi-stops per cycle in all the 1, 2 and 3-DO in-plane motions. Some tests are designed and conducted or practical validation o the proposed stick-slip conditions. An eperimental setup is prepared or applying eccentric orces to the body and measuring the orce which initiates a general motion. The comparison o theoretical and eperimental results reveals the eectiveness o the proposed model in predicting the conditions o switching between stick and slip phases. Reerences Theoretical orce [m] Measured orce [m] Relative rror % h = 5 34.7 36.5 4.93 h = 10 26.7.6 5.07 [1] K. J. Åström, Control o systems with riction, proceeding o the 4th international conerence on motion and vibration control, Zurich, Switzerland (1998) 25-32. [2] H. Olsson and K. J. Åström, riction generated limit cycles, IEEE Transactions on Control and Systems Technology, 9 (4) (2001) 629-636. [3] V. Eranian and M. Kabganian, Adaptive trajectory control and dynamic riction compensation or a leible-link robot, Journal o Mechanics, 26 (2010) 205-217. [4]. Xia, Modelling o a two-dimensional coulomb riction oscillator, Journal o Sound and Vibration, 265 (2003) 1063-1074. [5] J. P. Den Hartog, orced vibrations with combined coulomb and viscous riction, Transactions o the American Society o Mechanical Engineers, 53 (1931) 107-115. [6] H. K. Hong and C. S. Liu, Coulomb riction oscillator: modelling and responses to harmonic loads and base ecitations, Journal o Sound and Vibration, 229 (2000) 1171-1192. [7] J. Abdo and A. A. Abouelsoud, Analytical approach to estimate amplitude o stick-slip oscillations, Journal o Theoretical and Applied Mechanics, 49 (4) (2011) 971-986. [8] C. Duan and R. Singh, Stick-slip behavior in torque converter clutch, SAE Transactions, Journal o Passenger Car: Mechanical Systems, 116 (2005) 2785-2795. [9] C. Duan and R. Singh, orced vibrations o a torsional oscillator with coulomb riction under a periodically varying normal load, Journal o Sound and Vibration, 325 (2009) 499-506. [10] J. Awrejcewicz and P. Olejnik, Occurrence o stick-slip phenomenon, Journal o Theoretical and Applied Mechanics, 45 (1) (2007) 33-40. [11] S. Goyal, Planar sliding o a rigid body with dry riction: limit suraces and moment unction, Ph.D. Thesis, Cornell University, Ithaca, Y, USA, (1989). [12] S. Goyal, A. Ruina and J. Papadopoulos, Planar sliding with dry riction. Part1. limit surace and moment unction, Wear, 143 (1991) 307-330. [13] S. Goyal, A. Ruina and J. Papadopoulos, Planar sliding with dry riction. Part2. dynamics o motion, Wear, 143 (1991) 331-352. [14] V. G. Zhuravlev, The model o dry riction in the problem o the rolling o rigid bodies, J. Appl. Math. Mech, 62 (5) (1998) 705-710. [15] A. A. Kireenkov, Combined model o sliding and rolling riction in dynamics o bodies on a rough plane, Mechanics o Solids, 43 (3) (2008) 412-425. [16] A. A. Kireenkov, S. V. Semendyaev and V.. ilatov, Eperimental study o coupled two-dimensional models o sliding and spinning riction, Mechanics o Solids, 45 (6) (2010) 921-930. Iman Kardan received his B.Sc. and M.Sc. degree in Mechanical Engineering rom Amirkabir University o Technology (Tehran Polytechnic), Tehran, Iran in 2008 and 2011, respectively. Since 2008, he has been a research assistant in the System Dynamics and Control Research Laboratory, mechanical engineering department, Amirkabir University o Technology. His research interest is in the area o robotics, microrobotics, control and smart materials. Mansour Kabganian received his B.Sc. rom Amirkabir University o Technology (Tehran Polytechnic), Tehran, Iran, in 1975, M.Sc. rom University o Tarbiat Modarres, Tehran, in 1988, and Ph.D. rom the University o Ottawa, Ottawa, Canada, in 1995, all in mechanical engineering. He joined the Department o Mechanical Engineering at Amirkabir University o Technology in 1988, where he is a Proessor and the Head o the System Dynamics and Control Research Laboratory. He is also the Head o the Space Dynamics and Control Research Center, which he established in 2009. His main research interests are control, stability, and dynamical systems such as spacecrats, vehicles, and microrobotics.