Rolling Wheel - cycloidal path

cycloidal path는 역학계에 많은 의미를 주는 궤도이다.

우선 구르는 바퀴의 한점의 궤도 곡선과 속도를 알알보자.

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The Cycloidal Curve(擺 線).

         In order to simulate the rolling action of two discs, we must consider the path traced by a point on the edge (circumference) of one of the discs.

         First consider the path traced by a point on the edge of a circle that rotates in a stationary position. In this drawing, the point is where the small line meets the edge of the circle. The path is obviously that of a circle.

         Now consider the path traced by this point when the circle is rolling on a horizontal plane. (I have deliberately exaggerated the horizontal movement of the circle in the next drawing to illustrate more clearly the path of the point on the circle.)

        If you draw a line to trace the path of the point as the circle rotates, the result is called a Cycloidal Curve.

        This curve is inaccurate in its shape because of the way I created these drawings, so below is a much more accurate Cycloidal Curve, generated in a chart of the horizontal and vertical coordinates of the path of the point as the circle rotates. The chart is used here to create the graph below (with Rotation in Degrees on the X axis and Displacement in Inches on the Y axis).

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출발점이 어디든 정점에 도달하는 시간이 같다? – ‘등시곡선’

네덜란드의 물리학자 호이겐스는 1673년 『진자시계Horologium Oscilatorium』라는 명저를 통해 진자가 호가 아니라 사이클로이드를 따라 움직일 경우에 진자의 궤도가 등시곡선(tautochrone)이 된다는 것을 증명하고, 이러한 성질을 이용해 진자시계를 만들었다.

등시곡선은 정점에 도달하기 위해서 곡선 상의 어떤 점에서 출발하더라도 도달하는 데 걸리는 시간이 같게 되는 성질을 갖는다. 즉, 그림1에서 보면 A에서 B사이의 곡선은 사이클로이드인데 가장 아래 지점인 C까지 진자가 내려오는 데 걸리는 시간은 이 사이의 어떤 지점에서 출발하더라도 같다. 따라서 등시곡선을 따라 움직이는 사이클로이드 진자는 진폭에 상관없이 일정한 주기를 갖게 되는 것이다.

★ tautochrone problem의 증명

The problem of finding the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. The solution is a cycloid, a fact first discovered and published by Huygens in Horologium oscillatorium (1673). This property was also alluded to in the following passage from Moby Dick: "[The try-pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along a cycloid, my soapstone, for example, will descend from any point in precisely the same time" (Melville 1851).

Huygens also constructed the first pendulum clock with a device to ensure that the pendulum was isochronous by forcing the pendulum to swing in an arc of a cycloid. This is accomplished by placing two evolutes of inverted cycloid arcs on each side of the pendulum's point of suspension against which the pendulum is constrained to move (Wells 1991, p. 47; Gray 1997, p. 123). Unfortunately, friction along the arcs causes a greater error than that corrected by the cycloidal path (Gardner 1984).