3d trajectory of system of wheels

In summary, the conversation discusses the development of equations of motion for a system of wheels with varying radii and angular velocities. The equations are provided for systems with two and three wheels, but the development of equations for a system of four wheels is still in progress. The equations involve trigonometric functions and allow for determining the trajectory of the system. The speaker also expresses curiosity about whether the third wheel will scratch the first.
  • #1
snay
1
0
Hello!

I tried to develop equations of motion of system of wheels, where the first one with radius of r is rotated with some angular velocity v, the center of the second wheel is connected with the first on its perimeter and its plane of rotation is perpendicular to the plane of rotation of the first wheel. The second wheel has a radius of r/2 and angular velocity of 2v. The third wheel is connected with the second in the same way and its plane of rotation is perpendicular to the plane of rotation of the second wheel. The radius of the third wheel is r/4 and the velocity - 4v. I tried to develop general equations, which allow to determine trajectory for the system of arbitrary number of wheels. I have developed equations for the system of two and three wheels, but stuck in developing the equations for the system of four wheels. Here I post my solutions:

- for the system of two wheels
x1=(cos(phi)+1/q*cos(phi)*cos(q*phi))*r;
y1=(sin(phi)+1/q*sin(phi)*cos(q*phi))*r;
z1=(1/q*sin(q*phi))*r;

- for the system of three wheels
x2=(cos(phi)+1/q*cos(phi)*cos(q*phi)+1/q^2*(cos(q^2*phi)*cos(phi-pi/2)*cos(phi+pi/2)+cos(q*phi)*cos(phi)*cos(phi)))*r;
y2=(sin(phi)+1/q*sin(phi)*cos(q*phi)+1/q^2*(sin(q^2*phi)*cos(phi-pi/2)*cos(phi+pi/2)-sin(q*phi)*cos(phi)*cos(phi)))*r;
z2=(-1/q*sin(q*phi)-1/q^2*sin(q^2*phi)*cos(phi)+1/q*sin(phi)*cos(phi)*cos(phi))*r;

I will be very thankful for any comments.
 
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  • #2
Just as a curiosity, I wonder if the third wheel will scratch the first.
 

1. What is a 3D trajectory of a system of wheels?

The 3D trajectory of a system of wheels refers to the path or motion that the system takes in three-dimensional space. It takes into account the movement of all the wheels in the system and how they interact with each other to determine the overall trajectory.

2. How is the 3D trajectory of a system of wheels calculated?

The 3D trajectory of a system of wheels is calculated using mathematical equations and principles of physics, such as Newton's laws of motion and kinematics. It takes into account factors such as the size and shape of the wheels, their rotation speed, and the forces acting on them.

3. Why is it important to study the 3D trajectory of a system of wheels?

Understanding the 3D trajectory of a system of wheels is important in various fields, such as robotics, automotive engineering, and aerospace. It allows for the prediction and optimization of the system's movement, which can improve its efficiency, stability, and safety.

4. How does the 3D trajectory of a system of wheels differ from a 2D trajectory?

A 2D trajectory only considers the movement of an object in two-dimensional space, while a 3D trajectory takes into account the object's movement in three dimensions. This means that a 3D trajectory is more complex and can provide a more accurate representation of the object's motion.

5. Can the 3D trajectory of a system of wheels be affected by external factors?

Yes, the 3D trajectory of a system of wheels can be affected by external factors such as friction, air resistance, and external forces. These factors can alter the system's movement and trajectory, and must be considered in its calculation and analysis.

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