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Research Papers

Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint

[+] Author and Article Information
Vitaliy Fedonyuk

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29631
e-mail: vfedony@g.clemson.edu

Phanindra Tallapragada

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29631
e-mail: ptallap@clemson.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 18, 2015; final manuscript received November 29, 2016; published online January 19, 2017. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 12(3), 031021 (Jan 19, 2017) (8 pages) Paper No: CND-15-1442; doi: 10.1115/1.4035407 History: Received December 18, 2015; Revised November 29, 2016

The Chaplygin sleigh is a canonical problem of mechanical systems with nonholonomic constraints. Such constraints often arise due to the role of a no-slip requirement imposed by friction. In the case of the Chaplygin sleigh, it is well known that its asymptotic motion is that of pure translation along a straight line. Any perturbations in angular velocity decay and result in an increase in asymptotic speed of the sleigh. Such motion of the sleigh is under the assumption that the magnitude of friction is as high as necessary to prevent slipping. We relax this assumption by setting a maximum value to the friction. The Chaplygin sleigh is then under a piecewise-smooth nonholonomic constraint and transitions between “slip” and “stick” modes. We investigate these transitions and the resulting nonsmooth dynamics of the system. We show that the reduced state space of the system can be partitioned into sets of distinct dynamics and that the stick–slip transitions can be explained in terms of transitions of the state of the system between these sets.

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Figures

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Fig. 1

(a) Chaplygin sleigh with a balanced rotor. The center of mass of the arrangement is at a distance b from the rear contact. The picture on the right shows an illustration of a physical cart to realize the Chaplygin sleigh with castors at the front.

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Fig. 2

Phase space plot of the Chaplygin sleigh for parameter values m = 1, b = 1, and I = 1. The energies for the trajectories from the innermost trajectory outward are KE = [0.125, 0.5, 1.125, 2].

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Fig. 3

Cart parameters for initial conditions of ux(0)=5 and ω(0)=2. Required friction for slip is assumed to be infinite.

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Fig. 4

Phase space portrait of the reduced equations of the Chaplygin sleigh. The black curves (the rectangular hyperbolas) denote the critical friction, Fc. The shaded region is the set of initial conditions, which leads to pure nonholonomic motion.

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Fig. 5

S1→S2→S4 transition from stick to slip and then back to stick mode of motion. Initial conditions are ux(0)=0 and ω(0)=2 (starting in S1). (a) The green (in plane trajectory) and red (out of the plane trajectory) represent the stick and slip motion respectively. (b) The dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the blue dashed–dotted curve represents the normal velocity of point P. (c) and (d) show ω(t) and ux(t), respectively. In (e), the black line is the total kinetic energy and the dashed curve is the translational kinetic energy.

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Fig. 6

S1→S2→S1→S2→S4 transition. Initial conditions are (ω(0)=3.3,ux(0)=0)∈S1). (a) The green (in plane trajectory) and red (out of the plane trajectory) represent the stick and slip motion respectively. (b) The dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the blue dashed–dotted curve represents the normal velocity of point P. (c) and (d) show ω(t) and ux(t), respectively.

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Fig. 7

Trajectories of the cart for (ux(0),ω(0))=(0,5.6) (shown purple) and (ux(0),ω(0))=(0,5.7) (shown red). In both cases, the no-slip trajectory confined to the plane uy = 0 is shown in green.

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Fig. 8

Number of slip phases against ω(0) for ux(0)=0

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Fig. 9

Normalized change in kinetic energy for a large range of initial conditions, (ux(0),ω(0))

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Fig. 10

Plot showing the zero contour of the change in kinetic energy surface. Green (shaded) region represents positive change, and white (unshaded) region represents negative change.

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Fig. 11

(a) Values of (ux(0),Δω) for which Δθ is positive (blue squares) and negative (red asterisks), respectively. The ΔKE=0 curve is also pictured in black. (b) The intersection of the ΔKE=0 curve with the Δθ=0 curve is shown. Blue (region 4) marks initial conditions where 10−5<ΔKE<5(10−5). Similarly, cyan (region 5) corresponds to |ΔKE|<10−5 and for initial conditions marked green (region 6) −5(10−5)<ΔKE<−10−5. For red (region 1) initial conditions 5(10−5)<Δθ<5(10−4), for yellow (region 3) |Δθ|<5(10−5), and for purple (region 2) −5(10−4)<Δθ<−5(10−5).

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Fig. 12

Parallel translation for the case where ΔKE=0 and Δθ=0. Initial conditions are ux(0)=1.539646 and Δω=3.020854. In (c), the dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the dashed–dotted curve represents thenormal velocity of point P. In (f), the black line represents the total kinetic energy, and the blue-dashed curve represents the translational kinetic energy.

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Fig. 13

Perturbation is applied periodically such that the cart avoids a rectangular obstacle. In (a), the path of the cart is shown, and ω is shown in (b).

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