- Consider A Flexible Rope Or Chain That Hangs Along X Direction From The Ceiling Under The Sole Action Of Gravity If 1 (244.63 KiB) Viewed 25 times
Consider a flexible rope (or chain) that hangs (along x-direction) from the ceiling under the sole action of gravity. If we pull the rope to one side and let it go in lateral direction, then vibration of a hanging rope will occur. From Newton's second law of motion one can show that the lateral displacement Y(x) of the rope is governed by the differential equation: Y(x) (pg[L-x]Y') + pw²Y = 0 ; 0<x<L where is the mass per unit length, 8 is the acceleration gravity, is the temporal frequency and Lis the length of the rope. 1. Derive the general solution of the lateral displacement Y(x) of the rope. 2. One of the constants in the general solution should be zero, why? 3. Use the second condition: Y(0)-0, to find the exact solution. 4. Find the first ten natural frequencies of the rope. 5. What do you think if the mass per unit length is a function of x, is the above solution valid, discuss? x