PHY254H1 F — Homework problem set #2Python

Java Python PHY254H1 F — Homework problem set #2

Due 13 October 2024 at 11:59 p.m.

1 Simple oscillators [30%]

1.1.  Consider a mass m1  attached to an ideal spring (of un-stretched length d) and pulled by constant force F2, with F2  = m2g, as shown in figure 1a. (Note that the definition of x is different than the one we usually use in class.)

1.1.1.  Suppose that the system is in equilibrium when x = L. If L and d are known, what is the spring constant k?

1.1.2.  The system is at rest in the position x = L and the mass m2  is suddenly removed (for example, by cutting the string that connects m1 and m2). We neglect damping. What is

(a) Sketch for Q1.1.

(b) Sketch for Q2.1. Figure 1

the period and amplitude of the oscillations that m1 will start to execute?

1.2.  (Morin 4.27) An overdamped oscillator with natural frequency ω0 and damping coefficient γ starts out at position x0 > 0. What is the maximum initial speed (directed toward the origin) it can have and not cross the origin?

1.3.  (From Morin 4.28) In the absence of damping, a mass on the end of a spring is released from rest at position x0  and over the course of the oscillation, reaches maximum velocity vx . The experiment is repeated, but now with the system immersed in a fluid that causes the motion to be critically damped. This time, the mass reaches vx . Show that the maximum speed of the mass in the first case is e (= exp(1)) times the maximum speed in the second case. That is, show that vx  = e vx .

Note: The fact that the maximum speeds differ by a fixed numerical factor follows from dimen - sional analysis, which tells us that the maximum speed in the first case must be proportional to ω0x0 . And since γ = ω0 in the critical-damping case, the damping doesn’t introduce a new parameter, so the maximum speed has no choice but to again be proportional to ω0x0 .  But showing that the maximum speeds differ by the nice factor of e requires a calculation.

2   Damped and driven harmonic oscillators [10%]

These questions were in last year’s mid-term test and counted for 40% of its total.

2.1.  A torsional harmonic oscillator (cf. figure 1b) is described by the following equation of mo- tion:

where θ is the angle of twist from its equilibrium position in radians, I is the moment of inertia, C is the rotational damping, κ is the torsion spring constant and τ is the drive torque.

2.1.1.  What is its natural angular frequency ω0 ?

2.1.2.  Which range of values of C corresponds to the light damping case?

2.1.3.  For the light damping case, what is the angular frequency of oscillations?

2.1.4.  Suppose a force F, constantin time, is applied at a distance L from the axis so that the torque is τ = FL.  When the oscillatory motion of the transient dies out, what is the resulting angle of twist, θeq?

2.2.  The free oscillations of a mechanical system  PHY254H1 F — Homework problem set #2Python are observed to have a certain angular frequency ω 1 .  The same system, when driven by a force F0cos(ωt) (where F0  is constant, and ω is variable), has a power resonance curve whose angular frequency width, at half-maximum power, is ω1/5.

2.2.1.  At what angular frequency does the maximum power input occur? 2.2.2.  What is the Q of the system?

2.2.3.  The system consists of a mass m on a spring of spring constant k. In terms of m andk, what is the value of the constant bin the resistive term ¡b v

2.2.4.  Sketch the amplitude response curve, marking a few characteristic points on the curve.

3 Vibrating platform [40%]

Figure 2shows a schematic diagram of a system that is used to isolate a platform. from floor vibra- tions.  The mass of the platform. is m, the spring constant of the entire system is k and there is a damping mechanism called a dashpot with damping constant b. The floor is vibrating according to ξ = Afcos(!t) with respect to its equilibrium position.

Figure 2:  Vibration-isolation system showing a platform. mounted on springs with a damping mechanism (called a dashpot) with damping factor b.

3.1.  Show that to find the motion of the platform, you can solve an intermediate ODE that reads

with xi some intermediate position (not that of the platform).  Define all new quantities. Write the expression for Ai, the amplitude of xi .

3.2.  Show that A, the amplitude of x, is characterised by

3.3.  Under which conditions is the device a dampener, and when does it act as an amplifier? How should we tune it in order to dampen vibrations as much as possible?

4 Fourier analysis [20%]

In a lecture, we saw that if a linear damped oscillator is subjected to forcing that is T-periodic (ma = F (t), with F (t) = F (t Å T) = ···), we can decompose this forcing into a Fourier series. This allows us to treat each mode independently, and thanks to the linearity of the system, we can add up each individual response to form. the global response.

In the lecture, we saw how to calculate the Fourier series of the forcing, and we saw what the re- sponse to a monochromatic forcing was, but we never saw what the final superposition looked like! This is the goal of this question.

4.1.  The script. monochromatic-driving.py plots the response of a damped oscillator (natural angular frequency!0 , quality factor Q) to a monochromatic forcing of period T = 2π, angular frequency ! = 2π/T = 1, and amplitude Af = 1. Before you start, make sure you understand every single line, but don’t explain it in your copy.

Adapt this script. to plot the response of the oscillator when the driving is the “even square wave” of lecture 10:

and repeat to make F (t) 2π-periodic. (To keep things simple, have F0 = 1.)

Do not submit your Python script. (you can if you want, but the marker won’t have to look at it). Instead, in your copy, explain the steps you took to adapt the code, the formulas you used (recall that if we derived a formula in a lecture, you can re-use it without justification), the computational choices you made, and why. You don’t have to explain everything, just make sure any randomly picked student in the class would understand how you adapted the code.

4.2.  Explore the parameter regime: vary !0 around! and Q around one, and succinctly explain what you see.

Only do a handful of cases, at least four but it doesn’t have to be much more than that, and only use a handful of lines per case.  This question is only worth 20% of the total, which should tell you that you don’t have to go overboard