The Motion of a Hanging Chain

Transcription

1 The free end chain with little kick and free falling Altrichter Stacey Qiang Chen Department of Mathematical Sciences University of Delaware December 3, 2007

2 Outline 1 Why this problem? The Mathematical Models 2 3 s and Data 4

3 Outline Why this problem The Mathematical Models 1 Why this problem? The Mathematical Models 2 3 s and Data 4

4 History Role of Dynamics Why this problem The Mathematical Models The hanging chain problem is not of great practical application, but it plays an important role in the history of dynamics. An arbitrary motion of the chain can be expressed as a linear combination of the normal modes. The results have been of inestimatable value in mathematical physics, especially in vibrations and in quantum mechanics. Daniel Bernoulli find the normal modes of the hanging chain in 1732, then Euler discussed it further in And F. W. Bessel investigated the functions that took on his name.

5 Outline Why this problem The Mathematical Models 1 Why this problem? The Mathematical Models 2 3 s and Data 4

6 Why this problem The Mathematical Models The Typical Movement of the Chain

7 PDE and s Why this problem The Mathematical Models Set the coordinates with a vertical x axis and a horizontal y axis, and choose the origin at the equilibrium position of the free end of the chain. Then motion of the chain after kicked is described as following ( T y ) ρ 2 y t 2 = x x where ρ is the linear mass density of the chain, T is the tension in the chain. Note that for the hanging chain, the tension T = ρgx is not constant, g is the gravitational acceleration.

8 PDE and s (Cont d) Why this problem The Mathematical Models Consider that the disturbance moves with the speed which is also given by thus we have the relation dx dt v = T /ρ v = dx dt = T /ρ = gx The condition x = 0 at t = 0 generates x = gt 2 /4

9 Outline 1 Why this problem? The Mathematical Models 2 3 s and Data 4

10 Round-Trip Time By the ODE model, subject to the initial condition x = 0 at t = 0 and the length of the chain L or x = gt 2 /4 t = 2 x/g then the round-trip time is given by t rt = 4 L/g We expect our experiments data of round-trip time matching with this theory

11 Outline 1 Why this problem? The Mathematical Models 2 3 s and Data 4

12 The PDE Equation The PDE model is given by ρ 2 y t 2 = ( T Y x x with the boundary and initial conditions: y(l, t) = 0, y(0, t) is finite, y(x,0)=0, and { y V, 0 x αl, (x, 0) = t 0, αl < x L. where assuming the initial blow imparts a velocity V to the free end for x αl. )

13 Dimension Analysis To solve the partial differential equation with boundary and initial data, we nondimensionlize the variables first. Letting X = x/l, θ = t/ L/g and Y = y/(v L/g), and noting T = ρgx, we have ( 2 X Y ) Y θ 2 = X, X Y (1, θ) = 0, Y (0, θ) finite, Y (X, 0) = 0, and { Y 1, 0 X α, (X, 0) = θ 0, α < X 1.

14 Solving The PDE Using separation of variables, we set Y (X, θ) = w(x)z(θ). From the boundary and initial conditions, we have w n (X) = a n J 0 (λ n X) and zn (θ) = sin(λ n θ), and the solution is Y (X, θ) = a n J 0 (λ n X) sin(λn θ/2) 1 where J 0 is the Bessel function of first kind with parameter 0, λ n are the zeros of J 0.

15 Solving The PDE (Cont d) Then we use the initial condition of Y θ (X, 0) to find out a n, Y θ (X, 0) = 1 a n 2 J 0λ n x = { 1, 0 X α, 0, α < X 1. Then. a n = 2 α 0 J 0(λ n X)dX α λ n 0 J2 0 (λ = 2 α n X)dx λ 2 n J 1 (λ n α) J 2 1 (λ n)

16 The amlitude a n

17 Numerical Solution of The PDE Using the first 100 terms and letting α = 0.025, we solve it numerically using MAPLE and find a graph of the displacement of the chain tip.

18 Outline s and Data 1 Why this problem? The Mathematical Models 2 3 s and Data 4

19 s and Data In this experiment, we first attached a chain to a lab stand so that it hung vertically. The top end of the chain is fixed and the bottom end is free. A small kick is applied to the end of the chain which initiates a traveling wave. This wave will travel up to the fixed end of the chain and bounce back finally reaching the free end. This motion of the wave will continue traveling back and forth, but it will eventually stop due to gravity. The high speed camera with 500 fps was used to capture the evolution of the chain over time and with the help of MATLAB we were able to find the time it takes for the wave to make a round trip.

20 (Cont d) s and Data This is done by finding the frame where to traveling wave begins and the frame where the wave has reached the same spot. For each trial, we found three round trip times and then averaged them. Two different chains used were a heavy metal chain and a lighter metal chain with tiny beads. For each chain, we performed three trials.

21 Outline s and Data 1 Why this problem? The Mathematical Models 2 3 s and Data 4

22 1 s and Data We use large beads chain with the length m.

23 1 (Cont d) s and Data

24 1 (Cont d) s and Data The average of the round-trip time: 1.216s and 1.196s. From the theory, the round-trip time: t rt = 4 L/g = 1.207s, where L = m and g = 9.8m/s 2. The error: /1.207 = and /1.207 =

25 2 s and Data For small beads chain with length m.

26 2 (Cont d) s and Data The average of the round-trip time: 1.024s and 1.077s. From the theory, the round-trip time: t rt = 4 L/g = 1.054s, where L = m and g = 9.8m/s 2. The error: /1.054 = and /1.054 =

27 Outline s and Data 1 Why this problem? The Mathematical Models 2 3 s and Data 4

28 Free falling Chain s and Data We attempte with a short metal chain to see the behavior of free falling with different start positions. By Tomasezewski, Pieranski and Geminard (2006), we have that the acceleration of the chain tip is g/2 instead of g. In here we just do the experiments without analzing the data, because it is very hard for us to use the MATLAB to track the movement of the chain tip.

29 Free Falling Chain 1 s and Data Holding the chain midway between fixed end and taut.

30 Free Falling Chain 2 s and Data Holding the chain close to the fixed end.

31 Some possible sources of error: 1 Error of finding time of complete roud trip. 2 The movement of Chain is not exactly in a plane. 3 The stability of the lab stand as the experiment in progressing. 4 It is diffcult to start kick with a completely still chain.

32 Thank you! Have a Good Holiday!