In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the centre of mass of a body.
Figure (1) shows the force applying and velocity of a particle moving in circular path at several angular positions whereas figure (2) shows the position of a particle at angular displacement θ.
Applications of circular motion
Examples of circular motion include: an artificial satellite orbiting the Earth at a constant height, a stone which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.
For this tutorial we will look at the example of an artificial satellite orbiting the earth in a circular orbit.
For an artificial satellite if the launch speed was too small, it would eventually fall to earth. But if launched with sufficient speed, the projectile would fall towards the earth at the same rate that the earth curves. This would cause the projectile to stay the same height above the earth and to orbit in a circular path. And at even greater launch speeds, a cannonball would once more orbit the earth, but now in an elliptical path. At every point along its trajectory, a satellite is falling toward the earth. Yet because the earth curves, it never reaches the earth.
So what launch speed does a satellite need in order to orbit the earth? The answer emerges from a basic fact about the curvature of the earth. For every 8000 meters measured along the horizon of the earth, the earth’s surface curves downward by approximately 5 meters. So if you were to look out horizontally along the horizon of the Earth for 8000 meters, you would observe that the Earth curves downwards below this straight-line path a distance of 5 meters. For a projectile to orbit the earth, it must travel horizontally a distance of 8000 meters for every 5 meters of vertical fall. It so happens that the vertical distance that a horizontally launched projectile would fall in its first second is approximately 5 meters (0.5*g*t2).
For this reason, a projectile launched horizontally with a speed of about 8000 m/s will be capable of orbiting the earth in a circular path. This assumes that it is launched above the surface of the earth and encounters negligible atmospheric drag. As the projectile travels tangentially a distance of 8000 meters in 1 second, it will drop approximately 5 meters towards the earth. Yet, the projectile will remain the same distance above the earth due to the fact that the earth curves at the same rate that the projectile falls. If shot with a speed greater than 8000 m/s, it would orbit the earth in an elliptical path.
Now if a satellite is moving in a circular path at a constant height above the surface of earth, its dynamic equations can be described as of a particle moving in a circular path:
Where v is the tangential velocity and is the angular velocity of the satellite.
Simulation of satellite moving in a circular orbit using Simulink
To simulate the motion of a satellite moving in circular orbit we use of dynamic equations of movement of satellite. Which are:
Examining the equations we know that there are special blocks needed for sin and cos. If we look at the Sources and Math operator libraries, we find that three blocks are available for trigonometric functions: the Sine wave block from Sources library, Sine wave function and Trigonometric Function blocks from the Math operation library. The documentation for these blocks shows that the Sine wave Function block is used in certain special cases, while the Trigonometric Function block is more suitable for a term like θ than sin term that we have, and Trigonometric Function block would need the external input requiring additional blocks in the model. So the best block to use here is the Sine Wave block from the Sources library.
The Sine Wave block from Sources Library
The Sine Wave block provides the basic function a sin (+α) for a model. The parameters window, shown in figure below, allows to choose all the parameters in the function above to customize the function for our needs. The documentation indicates that the sine type parameter can be time-base with the simulation time as the time (t) parameter. Since our model supplies the amplitude of the sine wave separately, we choose an amplitude of 1. The sine wave is centred at 0, so the bias is set to 0.
Note that choosing a phase of π/2 will cause the function to be equivalent of cos ωt, so that we can provide either function using same block. Finally, Sample time parameter is chosen to be 0, causing the sin wave function to be continuous output block.
The Gain Block from the Math Operation Library
To produce a gain of ratio of v and , we use the Gain block from Math Operation Library. This block provides a multiplicative constant and we can enter -1 as its value in parameters window. The value of fraction is input to the gain block and reverses the sign.
We construct the simulation model as shown in figure below with two scope blocks for output x(t) and y(t) and constant blocks for input parameters and initial conditions.
We note that while we could have set the amplitude of sine wave block to account for input parameters v/ we have chosen to make these inputs an explicitly visible block rather than embed them in sine wave block parameters. That makes our model easier to understand and to change for experimenting with values.
Above used blocks can be found in the library browser at these locations:
Sine Wave Block: Simulink/sources
Gain Block: Simulink/commonly used blocks
Constant: Simulink/commonly used blocks
Product: Simulink/commonly used blocks
Scope: Simulink/commonly used blocks
XY Graph: Simulink/Sinks
When we run the simulation, we choose a time range of 10.0. We get the results shown in figure.