(Imagine a setup almost like a cassette tape arrangement). The description below is sufficient to draw a diagram of the mechanical arrangement and vectors.
Consider 2 spindles s1 and s2 that can rotate on their axes.
Their centers are located along the x-axis:
s1 at the origin (0,0) and
s2 at a distance d, (d, 0).
Call the constant spindle radius of s1, r1.
Call the constant spindle radius of s2, r2.
An unknown length of tape, L, and constant thickness, e, is tightly wound on s1.
The radius of the WOUND spindle s1 with the tape is R1, measured from the origin.
The end of this tape is attached to s2. R1>=r1.
The radius of the WINDING spindle s2 with the tape is R2, measured from the axis of s2. R2>=r2
The tape leaves s1 at point A, and winds into s2 at point B. Both A and B are located on the 1st quadrant. Assume negligible sagging of the tape from A to B. Therefore the tape leaves A tangent to R1 and enters B tangent to R2.
Let s2 be the driver spindle rotating at constant clockwise angular velocity omega2.
Let s1 be the driven spindle with unknown angular velocity omega1, unknown angular acceleration alpha1
Find:
What is the linear velocity V of the tape at any given time, t?
What is the angle of AB relative to the x-axis, call this angle, gamma, at any given time, t?
What is the angular velocity omega1?
What is the angular acceleration alpha1?
What is the rate of change of R1 at any time, t? (dR1/dt)
What are the radii R1 and R2 at any given time t?
What is the total time, T, to unwind the entire tape?
How many clockwise REVOLUTIONS (or if you prefer, radians), N2, will it take s2 to completely unwind s1? Call the angle subtended by s2 in any given time t, in radians, theta2.
Call the angle subtended by s1 in any given time T, in radians, theta1.
State or label other necessary assumptions or conventions.
It can be solved without going into vectors but if you are going to use vector notation, assume a xyz-coordinate system:
x-axis is positive going to the right of the page,
y-axis is positive going to the top of the page,
z-axis is positive going out of the page
using the right-hand rule of rotation, therefore, the k vector of omega2 is negative since it is clockwise
and show the vector differential equations of motion and derivation.
This is a lot harder than it seems and I haven't had the brains to solve it.
Thanks!
Consider 2 spindles s1 and s2 that can rotate on their axes.
Their centers are located along the x-axis:
s1 at the origin (0,0) and
s2 at a distance d, (d, 0).
Call the constant spindle radius of s1, r1.
Call the constant spindle radius of s2, r2.
An unknown length of tape, L, and constant thickness, e, is tightly wound on s1.
The radius of the WOUND spindle s1 with the tape is R1, measured from the origin.
The end of this tape is attached to s2. R1>=r1.
The radius of the WINDING spindle s2 with the tape is R2, measured from the axis of s2. R2>=r2
The tape leaves s1 at point A, and winds into s2 at point B. Both A and B are located on the 1st quadrant. Assume negligible sagging of the tape from A to B. Therefore the tape leaves A tangent to R1 and enters B tangent to R2.
Let s2 be the driver spindle rotating at constant clockwise angular velocity omega2.
Let s1 be the driven spindle with unknown angular velocity omega1, unknown angular acceleration alpha1
Find:
What is the linear velocity V of the tape at any given time, t?
What is the angle of AB relative to the x-axis, call this angle, gamma, at any given time, t?
What is the angular velocity omega1?
What is the angular acceleration alpha1?
What is the rate of change of R1 at any time, t? (dR1/dt)
What are the radii R1 and R2 at any given time t?
What is the total time, T, to unwind the entire tape?
How many clockwise REVOLUTIONS (or if you prefer, radians), N2, will it take s2 to completely unwind s1? Call the angle subtended by s2 in any given time t, in radians, theta2.
Call the angle subtended by s1 in any given time T, in radians, theta1.
State or label other necessary assumptions or conventions.
It can be solved without going into vectors but if you are going to use vector notation, assume a xyz-coordinate system:
x-axis is positive going to the right of the page,
y-axis is positive going to the top of the page,
z-axis is positive going out of the page
using the right-hand rule of rotation, therefore, the k vector of omega2 is negative since it is clockwise
and show the vector differential equations of motion and derivation.
This is a lot harder than it seems and I haven't had the brains to solve it.
Thanks!
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