PHYS 2017/3035 Assignment 2 DUE: Monday 10th August 2015

1) Tuned mass damping Taipei 101 (below) uses tuned mass damping to keep the structure stable. It allowed construction with enough flexibility and strength to withstand earthquakes and typhoons. The stabilising pendulum has a mass m=M/385, where M is the mass of the building, and L=12m. Assume the pendulum hangs from the top floor. The natural period of the building motion is 7s.

Were going to make a very simple model to see how it works.a) Write down equations of motion for this system and include damping. You can use any reasonable simplifications, for example small angle approximations, but you must justify them as you explain your working. b) Use Mathematica to solve these coupled equations and find the damping parameters that give you the most stable building.

2) Stopped clock (a true story) Old mechanical grandfather clocks (left) have a pendulum that is driven by falling weights. In theory, it takes a week for the weight to reach the bottom of its run, and then you have to pull it up again. Someone had a clock that, when wound up on Sunday, would stop sometime in the middle of the night between Thursday and Friday. The stoppage was accompanied by the sound of the weights hitting the side of the case. When this happened, the distance between the centre of mass of the weight and the suspension point was 0.99m. Use coupled modes explain what is happening.

This image shows the internals of a typical grandfather clock.

Model

Were going to make a very simple model to see how it works. a) Write down equations of motion for this system. b) Use small angle assumptions to simplify the equations and add a damping term to the equation of motion of the stabilising pendulum. c) Use Mathematica to solve these coupled equations and find the best damping, include some plots to show what you mean by best damping. Theres a slightly annoying video about the building here: http://tinyurl.com/lk9ylh

4) Stopped clock (a true story) Old mechanical grandfather clocks (left) have a pendulum that is driven by falling weights. In theory, it takes a week for the weight to reach the bottom of its run, and then you have to pull it up again. Someone had a clock that, when wound up on Sunday, would stop sometime in the middle of the night between Thursday and Friday. The stoppage was accompanied by the sound of the weights hitting the side of the case. When this happened, the distance between the centre of mass of the weight and the suspension point was 0.99m. Use coupled modes explain what is happening.

3) Tuned mass damping Taipei 101 (right) uses tuned mass damping to keep the structure stable. It allowed construction with enough flexibility and strength to withstand earthquakes and typhoons. The stabilising pendulum has a mass m=M/385, where M is the mass of the building, and L=12m. Assume the pendulum hangs from the top floor. The natural period of the building motion is 7s. Were going to make a very simple model to see how it works. a) What is the natural period of the pendulum? b) Write an equation for the angle of the building, using the moment of inertia of a uniform bar. Convert this into an equation for x1, the position of the top of the building. Remember to include a term that is the force on the top of the building due to the pendulum. c) Write an equation for the position (x2) of m. You will have to include a damping term, such as c(x1-x2), in this equation. d) Use Mathematica to solve these coupled equations and find the best damping, include some plots to show what you mean by best damping. Theres a slightly annoying video about the building here: http://tinyurl.com/lk9ylh 4) Stopped clock (a true story)

Old mechanical clocks (left) have a pendulum that is driven by falling weights. In theory, it takes a week for the weight to reach the bottom of its run, and then you have to pull it up again. Someone had a clock that, when wound up on Sunday, would stop sometime in the middle of the night between Thursday and Friday. The stoppage was accompanied by the sound of the weights hitting the side of the case. When this happened, the distance between the centre of mass of the weight and the suspension point was 0.99m. Use coupled modes explain what is happening.

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PHYS 2017/3035 Assignment 2 DUE: Monday 10th August 2015

3) Wave reflection

A wave propagates from left to right along a string. The density of the string changes at point p. The diagram below shows the situation after the wave has passed the point p. Figure out what the original wave looked like, (i.e. polarity, height and length) as well as the ratio of the densities to the left and right of p.

4) 3035 Only. Reflection and transmission (Problem 5.5 from Pain)

A point mass M is concentrated at a point on a string of characteristic impedance Z=c. A transverse wave of frequency moves in the positive x direction and is partially reflected and transmitted at the mass. The boundary conditions are that the string displacements just to the left and right of the mass are equal:

(yi, yr and yt are the displacements of the incident, reflected and transmitted waves respectively. ) The difference in the transverse forces just to the left and right of the mass equal the mass times its acceleration. If A1, B1 and A2 are the incident, reflected and transmitted wave amplitudes respectively, show that:

where q=M/2c and i2=-1

Sketch the amplitude and phase of the transmitted wave as a function of q. Comment on the limiting cases as M goes to 0 or infinity.

yi + yr = yt

B1A1

=iq1 + iq

andA2A1

=1

1 + iq