LAB-Undamped Damped Oscilations PDF

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Summary

In this laboratory we will examine harmonic oscillation. We will model the motion of a mass-spring
system with differential equations....

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Laboratory 3 The Mass-Spring System In this laboratory we will examine harmonic oscillation. We will model the motion of a mass-spring system with differential equations. There are three sections; worth a total of 10 marks. You must submit the following:   

The first section (four questions – pages 5 and 6) deals with the simulation experiment. Section two (two questions) and section three (one question) are on the last page (page 8) of this document. The data you collected, presented in the results table of page 7.

Our objectives are as follows: 1. Determine the effect of parameters on the solutions of differential equations. 2. Determine the behaviour of the mass-spring system from the graph of the solution. 3. Determine the effect of the parameters on the behaviour of the mass-spring. The primary MATLAB commands used are the ode45 function and the masspring.m GUI. The setup is similar to the arrangement shown to the right in order to determine the spring constant k. Various masses m are attached to the spring, and the motion measured.

Linking the period and the mass 1. Set up spring and mass so that vertical oscillations can be measured.

e ke x

k (e+x)

m

1

2. Decide how best to measure the time for one oscillation accurately – this is the period T. 3. Choose a range of masses so that the period varies significantly. 4. Make a number of measurements, taking care not to exceed the elastic limit of the spring, until you have about six pairs of mass-period data. Theory predicts that T  2

m k

for the loaded spring oscillator. 5. Calculate the square of the period T2 for each mass m, and draw an appropriate graph to check whether T2 is proportional to m. 6. Calculate the gradient of your graph and hence find a value of the spring constant k. 7. To check do a simple Hooke’s law experiment with your spring to check this value.

Simulation In this modified laboratory, you will use the simulation of the mass and spring to carry out your tests. The simulation is a simple to use tool, which can enable you to put damping on your spring and weight examples - without making too much mess. ie. Using oil or other viscous fluid, rather than being air damped. The link is here: https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html

2

Mass-Spring System without Damping The motion of a mass suspended to a vertical spring can be described as follows. When the spring is not loaded it has length l0 (situation (a)). When a mass m is attached to its lower end it has length l (situation (b)). From the first principle of mechanics we then obtain

The term g measures the gravitational acceleration (g ’ 9.8m/s 2 ’ 32ft/s 2 ). The quantity k is a spring constant measuring its stiffness. We now pull downwards on the mass by an amount y and let the mass go (situation (c)). We expect the mass to oscillate around the position y = 0. The second principle of mechanics yields

using (L.1). This ODE is second-order.

Equation (L.2) is rewritten

𝑑2𝑦 𝑑𝑡 2

+ 𝜔02 𝑦 = 0

(L.3)

where 𝜔02 = 𝑘/𝑚. Equation (L.3) models simple harmonic motion. Let m = 1 and k = 4. A numerical solution with initial conditions y(0) = 0.1 meter and y’(0) = 0 (i.e., the mass is initially stretched downward 10cms and released, see setting (c) in figure) is obtained by first reducing the ODE to firstorder ODEs. Let v = y’ . Then 𝑣 ′ = 𝑦 ′′ = −𝜔02 = −4𝑦 . Also 𝑣 (0) = 𝑦 ′ (0) = 0 . The following MATLAB program implements the problem with 𝜔0 = 2 .

3

Function Lab01ex1

Note that the parameter 𝜔0 was passed as an argument to ode45 rather than set to its value 𝜔0 = 2 directly in the function f. The advantage is that its value can easily be changed in the driver part of the program rather than in the function, for example when multiple plots with different values of 𝜔0 need to be compared in a single MATLAB figure window.

Fig: Undamped Output.

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Section 1: Questions From The Simulation (3 marks) Questions Undamped 1 & 2 1. From the graph in Fig: Undamped Output, answer the following questions. (1 Mark) a. What is the period of the motion? Answer this question first graphically (by reading the period from the graph) and then analytically (by finding the period using 𝜔0 ). b. When will the mass-spring system come to rest? Why? c. What is the maximum velocity (in magnitude) attained by the mass, and when is it attained? d. How does the size of the mass m and the stiffness k of the spring affect the motion? Your answer should be supported graphically (by modifying the file LAB08ex1.m by changing the values of m and k) and analytically (by a theoretical analysis on how 𝜔0 – and therefore the period of the oscillation – is related to m and k). 2. Plotting (½ Marks) a. Plot the quantity 𝐸 =

1 2

1

𝑚𝑣 2 + 2 𝑘𝑦 2 as a function of time. What do you observe?

b. Plot v vs y (phase plot). Does the curve ever get close to the origin? Why or why not? What does that mean for the mass-spring system?

Mass-Spring System with Damping When the movement of the mass is damped due to viscous effects (e.g., the mass moves in a cylinder containing oil, situation (d)), an additional term proportional to the velocity must be added. The resulting equation becomes

𝑚

𝑑2𝑦 𝑑𝑡 2

+𝑐

by setting 𝑝 =

𝑑𝑦 𝑑𝑡 𝑐 2𝑚

+ 𝑘𝑦 = 0 or

𝑑𝑡 2

+ 2𝑝

𝑑𝑦 𝑑𝑡

+ 𝜔 02 𝑦 = 0

. The program LAB01ex1 is updated by modifying the function f:

Function Lab01ex2

5

𝑑2𝑦

(L.4)

Questions Damped 3 & 4 3. Damped Harmonic Motion (1 Marks) Fill in LAB01ex2.m to reproduce Fig. Damped Output, and then answer the following questions. a. For what minimal time t 1 will the mass-spring system satisfy |𝑦(𝑡)| < 0.01 for all 𝑡 > 𝑡1 . You can answer the question either by magnifying the MATLAB figure using the magnify button, or use the following MATLAB commands (explain):

Fig: Damped Output

b. (b) What is the maximum (in magnitude) velocity attained by the mass, and when is it attained? c. How does the size of c affect the motion? Run the file LAB08ex1.m for different values of c to support your answer. 4. Plotting ( ½ Marks) 1

1

a. Plot the quantity 𝐸 = 2 𝑚𝑣 2 + 2 𝑘𝑦 2 as a function of time. What do you observe? b. Plot v vs y (phase plot). Comment on the behaviour of the curve in the context of the motion of the spring. 6

Laboratory 3 Results

Trial

Damped / Undamped

1

Undamped

2

Undamped

3

Undamped

4

Undamped

5

Damped

6

Damped

7

Damped

8

Damped

9

Simulation Undamped

10

Simulation Undamped

11

Simulation Damped

12

Simulation Damped

Mass

Extension

Maximum Amplitude

Time to decay

Name___________________________________ Student ID ________________________________

Lecturers Name ___________________________________

Session ________________________

Lecturer’s Signature ________________________________

Date __________________________

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Section 2: Solve the following Ordinary Differential Equations (4 marks) A) Solve this ordinary differential equation using Laplace transforms. 𝑑 𝑦(𝑡) = 3𝑒 −5𝑡 𝑑𝑡

Having initial conditions 𝑦(0) = 5 𝑦 ′ (0) = 0

B) Solve this ordinary differential equation using Laplace transforms. 𝑑2𝑦 + 16𝑥 = 1 𝑑𝑥 2 having initial conditions 𝑦 (0) = 20 𝑑𝑦 =0 𝑑𝑥

Section 3: Engineering Application (3 marks) The following differential equation is the equation of motion for an ideal spring-mass system with damping and an external force 𝐹(𝑡).

𝑚. 𝑥 ′′ (𝑡) + 𝑐. 𝑥 ′ (𝑡) + 𝑘. 𝑥(𝑡) = 𝐹(𝑡)

If 𝑚 = 2𝑘𝑔 , 𝑐 = 10𝑁. 𝑠/𝑚, 𝑘 = 8 𝑁/𝑚, 𝐹(𝑡) = 1𝑁

And 𝑥 ′ (0) = 𝑥(0) = 0

Find the solution of this differential equation using Laplace transforms.

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