Computer lab assignment 2022

Assessment No: 3
Assessment Type: Individual Computer Lab Assignment
Due Date: Friday, 11th October 2019
Student ID:
Name:

Introduction, Background and Project Overview
A dynamic problem can be split in two steps: obtaining the equation of motion and solving it. To obtain
the equation of motion we use common dynamics theory, such as Newton’s second law. The equation of
motion can then be solved through analytical or numerical techniques. So far, you have mostly looked at
analytical solutions to problems, however, even for simple dynamics problems these can rapidly become
complex and unwieldy. In actuality, the presence of a true analytical solution is rare in many real-world
problems and rely on simplifications and assumptions (e.g. neglecting drag). Numerical solutions on the
other hand can be readily used to solve simple and complex dynamics problems. For numerical solutions
the main challenges become computational resources and the elegance of the model.
Elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of
mass is connected by a spring. Compared with the ideal pendulum, not only angle, but also the length of
string (spring) is changing in the process, which makes the problem nonlinear, and extremely difficult to
solve analytically. An example in reality for elastic pendulum is the bungee jump. In this assignment, we
will first try to get the analytical solution for elastic pendulum with multiple approximations. Then finite
difference method will be adopted to get the numerical solution. In the end, we will try to build up a
simple numerical model for the whole bungee jump process.
Figure 1. The schematic graph for an elastic
pendulum. The grey curve is the trajectory of the
mass.
Table 1. Physical quantities in the system.
Property Symbol Quantity Unit
Item
Total mass m 0.1 kg
Initial angle θ0 0.05 rad
Initial angular
velocity ̇θ0 0.01 rad/s
Initial radius
velocity ̇r0 0.1 m/s
Spring
Free length l0 20 m
Initial elongation dr0 2 m
Elastic coefficient k 20 N/m
For the following questions:
 This elastic pendulum is swinging in one plane, so two dimensional coordinate system is
adequate to describe the system.
 The equation of motion for a harmonic oscillator is in the form of :
̈θ ( t )+ a ⋅(θ ( t )−C)=0( a>0)
a and C are constants, which can vary in different equation of motion, and need to be found by
rearranging. Such form of equation has the analytical solution like:
θ ( t )= A cos ( ωn ⋅t−φ ) +C
Where
Angular frequency ωn= √ a
Amplitude A= √(θ0 −C)2+ ¿ ¿

Phase φ =atan ¿
 Small angle theorem: when θ<5o , sinθ ≈ θ, cosθ ≈ 1.

Equation of Motion
1. Draw the free body diagram for the elastic pendulum model in the Fig 1. Write down the
equations of motion in polar coordinate systems, and Cartesian coordinate system.
Answer within this box. Change the size of the box if needed.