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Definition of cycloid

Cycloid is one of many fascinating curves in mathematics. It is defined as follows: a circle rolls slowly along a straight line, and the trajectory of a fixed point on the circle is called a cycloid.

Cycloidal nicknames and their causes

When a circle rolls on a fixed straight line, the trajectory of a fixed point on the circumference. Also known as wheel line. The initial position of the fixed point on the circle is the coordinate origin, and the straight line is the X axis. When the circle rolls by J angle, the fixed point on the circle reaches the position of P from the position of O point. When the circle rolls once, that is, J changes from O to 2π, the first arch of the cycloid is drawn on a fixed point on the moving circle. Scroll forward for another week, draw a second arch on the moving circle, and continue to scroll to get the third arch and the fourth arch ... All these arches are exactly the same in shape, with an arch height of 2a (the diameter of the circle) and an arch width of 2πa (the circumference of the circle). Cycloid has an important property, that is, when an object slides from point A to point B which is not directly below it by gravity, it takes the shortest time to slide along the cycloid between A and B, so cycloid is also called the steepest descent curve.

Properties of cycloid

By the17th century, it was found that the cycloid had the following properties:

1. Its length is equal to four times the diameter of the circle of revolution. It is particularly interesting that its length is a rational number independent of π.

2. The area under the arc is three times that of the circle of revolution.

3. The points on the circle describing the cycloid have different speeds-in fact, it is even at rest somewhere.

When the marbles are released from different points of the cycloidal container, they will reach the bottom at the same time.

The appearance and controversy of cycloid

The first appearance of cycloid can be found in a book published by C. Powell in 150 1 But in the17th century, a large number of outstanding mathematicians (such as Galileo, Pascal, Torricelli, Descartes, Fermat, Wu Ren, Varis, Huygens, johann bernoulli, Leibniz, Newton, etc. ) I am keen to study the properties of this curve. 17th century is an era when people are interested in mathematical mechanics and mathematical kinematics, which can explain why people are interested in cycloids. During this period, along with many discoveries, there have also been many disputes about the right of discovery, accusations of plagiarism, and neglect of others' work. Therefore, the cycloid is called the controversial "golden apple" and.

Related stories of cycloid

Clock and cycloid

The clock has become one of the indispensable tools for modern people. Without a clock, people will lose track of time and miss many important appointments. When you look at your watch, I wonder if you ever want to know what is hidden in the clock. A grain of sand is a world, and many things we take for granted are accumulated bit by bit by the blood and sweat of our ancestors.

Hiding in the clock to write these theories is a thick book! Looking back at the previous medieval navigation times, mastering time is a major event related to the safety of the whole ship. Time is an indispensable factor in fighting against the sea. In ancient times, the hourglass water clock was used for timing, but these timing tools were quite inaccurate. In order to increase the chances of the crew's survival, it became a top priority for the scientific community at that time to invent an accurate timer.

At that time, Galileo, a young Italian scientist, happened to find an interesting phenomenon on the leaning tower of Pisa. When the church chandelier swings back and forth, no matter whether it is big or small, the time spent swinging each time is equal. At that time, he calculated the time according to his heartbeat pulse. From then on, Galileo forgot to eat and sleep, and devoted himself to studying physics and mathematics. He did the simple pendulum experiment again with his own drops. The results show that the swing time of a simple pendulum has nothing to do with the swing amplitude, but only with the length of a single cycloid. This phenomenon made Galileo think that it was possible to make an accurate clock with a simple pendulum, but he never put this ideal into practice.

Galileo's discovery inspired the scientific community, but it was soon discovered that the swinging periods of a simple pendulum were not exactly equal. Galileo's observations and experiments were not accurate enough. In fact, the greater the swing amplitude of the pendulum, the longer the swing period, but the change of this period is very small. Therefore, if this pendulum is used as a clock, the amplitude of the pendulum will become smaller and smaller because of friction and air resistance, and the clock will go faster and faster.

Soon after, Dutch scientists decided to make an accurate clock. Galileo's simple pendulum swings on an arc, so we also call it a circular pendulum. Dutch scientists want to find a curve so that when the pendulum swings along such a curve, the swing period is completely independent of the swing amplitude. This group of scientists gave up physical experiments and studied purely on mathematical curves. After many failures, I finally found such a curve, which is called "cycloid" in mathematics.

If you cut a circle out of cardboard and fix a pencil on the edge of the circle, the pencil will draw a cycloid when the circle rolls along a straight line. I believe many people have seen and played with this kind of toy. I used to see street vendors selling this kind of cycloidal toy again. Many people praise the beauty of cycloid, but they don't know the correlation between cycloid and clock. Pendulums in watch shops are all made of cycloidal characteristics.

Basic principle of planetary cycloidal transmission mechanism

In cycloidal pin-wheel planetary transmission, the tooth profile curve of cycloidal wheel adopts short epicycloid generated by internal meshing circle. The generating principle of this cycloidal curve is shown in the drawing.

There is a generating circle (rolling circle) with a radius of rp' and a base circle with a radius of rc'. When the generated circle makes pure rolling around the base circle, and its center Op is at the positions of Op 1, Op2, Op3, Op4, Op5, Op6, etc. , points m merged on the plane of generating circle pass through M 65438+ respectively.

If the geometric relationship generated by the cycloid still maintains the inscribed rolling relationship, and the base circle and the cycloid are regarded as rigid bodies in relative motion, then the cycloid pattern moves relative to the center of the circle Op in a planetary way, which is the basic principle of the planetary cycloid transmission mechanism.

PS: cycloidal equation

x=a(t-sint)

Y = a( 1- cost)

Cycloid is the fastest curve. Only when you have studied advanced mathematics can you understand it better and accept it.