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... The proof given in [6, p. 480] uses polar coordinates and is based on purely geometric arguments (see also [8], [10]). Our purpose is to outline an analytical proof of Theorem 1 (see [2]) and to give the integral representation of the roulette equations. ...
... We briefly sketch his considerations. The method of S. S. Bjushgens includes the approach of Hall and Wagon (see [8]), who considered the special case when the roulette is the x-axis. ...
The purpose of this note is to bring to light a nearly forgotten theory of roulettes that was developed more than 100 years ago. It leads to a differential equation which contains the roulette lemma as a special case and uses complex-valued functions to describe curves in a plane.
... Another mechanics study related the exact meshing relationship of square and irregularly shaped wheels on matching unevenly shaped roads (similar to a rack and pinion) [13]. Although this study discusses the rolling kinematics of the irregular shapes to its predefined uneven road, the dynamics forces acting on such a system were not taken into account. ...
... To confirm our derived dynamic KS equations of motion ( (12), (13), and (14)), we slightly modify them to match our physical experimental setup. This modification serves as a base for verification, while introducing an extension for practical applications of these dynamic KS equations. ...
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A kinetic shape (KS) is a smooth two- or three-dimensional shape that is defined by its predicted ground reaction forces as it is pressed onto a flat surface. A KS can be applied in any mechanical situation where position-dependent force redirection is required. Although previous work on KSs can predict static force reaction behavior, it does not describe the kinematic behavior of these shapes. In this article, we derive the equations of motion for a rolling two-dimensional KS (or any other smooth curve) and validate the model with physical experiments. The results of the physical experiments showed good agreement with the predicted dynamic KS model. In addition, we have modified these equations of motion to develop and verify the theory of a novel transportation device, the kinetic board, that is powered by an individual shifting their weight on top of a set of KSs.
... Smooth surfaces, however, are associated with substantial slip with inefficiencies that set the upper limit on translation velocity. Here, we identify surface topographies that register with mwheel structure, taking inspiration from the mathematics of roads and wheels where it can be shown that, for any given wheel shape, there is a complementary road for optimal translation (21,22). For example, smooth-riding bicycles can be made with square-shaped wheels on roads constructed from a series of truncated catenaries (Fig. 1). ...
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Microbot locomotion is challenging because of the reversible nature of microscale fluid flow, a limitation that can be overcome by breaking flowfield symmetry with a nearby surface. We have used this strategy with rotating wheel-shaped microbots, microwheels (μwheels), that roll on surfaces leading to enhanced propulsion and fast translation speeds. Despite this, studies to date on flat surfaces show that μwheels roll inefficiently with substantial slip. Taking inspiration from the mathematics of roads and wheels, we demonstrate that μwheel velocities can be significantly enhanced by changing microroad topography. Here, we observe that periodic bumps in the road can be used to enhance the traction between μwheels and nearby walls. Whereas continuous μwheel rotation with slip is observed on flat surfaces, a combination of rotation with slip and nonslip flip occurs when μwheels roll on surfaces with periodic features, resulting in up to fourfold enhancement in translation velocity. The unexpectedly fast rolling speed of μwheels on bumpy roads can be attributed to the hydrodynamic coupling between μwheels and road surface features, allowing nonslip rotation of entire wheels along one of their stationary edges. This road-wheel coupling can also be used to enhance μwheel sorting and separation where the gravitational potential energy barrier induced by topographic surfaces can lead to motion in only one direction and to different rolling speeds between isomeric wheels, allowing one to separate them not based on size but on symmetry.
... Another approach of calculating an area of cardioid and other shapes of closed curves is presented using the surveyor's method [5]. A road-wheel relationship by rolling a cardioid wheel on an inverted cycloid is discussed in [11]. Cardioid finds various applications in fractals, complex analysis, plant physiology and engineering. ...
This article revisits an integral of radical trigonometric functions. It presents several methods of integration where the integrand takes the form $\sqrt{1 \pm \sin x}$ or $\sqrt{1 \pm \cos x}$. The integral has applications in Calculus where it appears as the length of a cardioid represented in polar coordinates.
Our objective is to design innovative robot wheels capable of rolling on staircases, without sliding and without bouncing. This is the first step to reach the ultimate goal of building wheelchairs capable to overcome the obstacles imposed by staircases on people with limited mobility. We show that, given a staircase with equal steps, there is an infinite number of wheels that roll over it, with the constraints of no-sliding and no-bouncing. Some of these wheels appear to be very interesting for real applications. We also present an algorithm for the construction of a wheel, which depends only on the measures of the tread (the part of the staircase that is stepped on) and the riser (the vertical portion between each tread) of the step.
We propose a simple approach to determine all possible wheels that can roll smoothly without slipping on a periodic roadbed, while maintaining the center of mass at a fixed height. We also address the inverse problem that of obtaining the roadbed profile compatible with a specific wheel and all other related “quantized wheels.” The role of symmetry is highlighted, which might preclude the center of mass from remaining at a fixed height. A straightforward consequence of such geometric quantization is that the gravitational potential energy and the moment of inertia are discrete, suggesting a parallelism between macroscopic wheels and nano-systems, such as carbon nanotubes.
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A circular shape placed on an incline will roll; similarly, an irregularly shaped object, such as the Archimedean spiral, will roll on a flat surface when a force is applied to its axle. This rolling is dependent on the specific shape and the applied force (magnitude and location). In this paper, we derive formulas that define the behavior of irregular 2D and 3D shapes on a flat plane when a weight is applied to the shape's axle. These kinetic shape (KS) formulas also define and predict shapes that exert given ground reaction forces when a known weight is applied at the axle rotation point. Three 2D KS design examples are physically verified statically with good correlation to predicted values. Motion simulations of unrestrained 2D KS yielded expected results in shape dynamics and self-stabilization. We also put forth practical application ideas and research for 2D and 3D KS such as in robotics and gait rehabilitation.
A theorem of Apostol and Mnatsakanian states that as a circle rolls on a line, the area of the cycloidal sector traced by a point on the circle is always three times the area of the corresponding segment cut from the rolling circle. We generalize this result by showing that sinusoidal and logarithmic spirals rolling on lines have similar area ratio properties. We then extend our ideas to include one curve rolling on another. Such pairs of curves are a natural generalization of road-wheel pairs.
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This paper began with experiments using the computer algebra system Mathe? matica to draw trochoids, the kinds of curves produced by the Spirograph? drawing sets. If two tangent circles have their centers on the same side of the common tangent line, and one circle remains fixed while the other is rolled around it without slipping, a hypotrochoid is traced by any point on a diameter or extended diameter of the rolling circle. If two tangent circles have their centers on opposite sides of the common tangent line, and one circle remains fixed while the other is rolled around it without slipping, an epitrochoid is traced by any point on a diameter or extended diameter of the rolling circle. A hypocycloid is a hypotro? choid for which the tracing point is on the circumference of the rolling circle, and an epicycloid is an epitrochoid for which the tracing point is on the circumference of the rolling circle. The term trochoid is used to refer to either a hypotrochoid or an epitrochoid. Either radius, but not both, can be infinite, so that cycloids and trochoids obtained by rolling a circle along a straight line, and also certain spirals and involutes are covered by the nomenclature, but in this paper we shall assume both radii are finite. The Spirograph? produces graphs of trochoids using toothed disks and rings to prevent slipping, but because none of the holes for the pen reach the circumference of the disks, the Spirograph? cannot be used to draw true hypocycloids or epicycloids. Since the graph of a trochoid depends on four parameters that are fixed and one that is variable, all these quantities will be part of the notation. The hypotrochoid denoted by hy[t;n,m,r,a] is generated by a rolling (moving) circle of radius m and a fixed (rconmoving) circle of radius n, with rm the distance from the center of the rolling circle to the tracing point. Assume the center of the fixed circle is at the origin and denote the initial position of the tracing point (and the center of the
Handbook of Mathematical Functions
  • M Abramowitz
  • I Stegun
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1965.
Rockers and rollers, this MAGAZINE
  • G B Robison
G. B. Robison, Rockers and rollers, this MAGAZINE 33 (1960), 139-144.
If you ask mathematicians what they do, you always get the same answer They think. They think about difficult and unusual problems. They do not think about ordinary problems: they just write down the answers
  • D G Wilson
  • Problem
  • E
D. G. Wilson, Problem E1668, Amer. Math. Monthly 72 (1965), 82-83. If you ask mathematicians what they do, you always get the same answer.' They think. They think about difficult and unusual problems. They do not think about ordinary problems: they just write down the answers. M. Egrafov (translated from Russian), contributed by the late R. P. Boas, Jr. This content downloaded from on Mon, 02 Mar 2015 15:04:31 UTC All use subject to JSTOR Terms and Conditions