Software Requirements Specification for Double Pendulum

Dong Chen

An outline of all sections included in this SRS is recorded here for easy reference.

Reference Material

This section records information for easy reference.

Table of Units

The unit system used throughout is SI (Système International d’Unités). In addition to the basic units, several derived units are also used. For each unit, the Table of Units lists the symbol, a description, and the SI name.

Symbol	Description	SI Name
${\text{kg}}$	mass	kilogram
${\text{m}}$	length	metre
${\text{N}}$	force	newton
${\text{rad}}$	angle	radian
${\text{s}}$	time	second

Table of Units

Table of Symbols

The symbols used in this document are summarized in the Table of Symbols along with their units. Throughout the document, symbols in bold will represent vectors, and scalars otherwise. The symbols are listed in alphabetical order. For vector quantities, the units shown are for each component of the vector.

Symbol	Description	Units
${a_{\text{x}1}}$	Horizontal acceleration of the first object	$\frac{\text{m}}{\text{s}^{2}}$
${a_{\text{x}2}}$	Horizontal acceleration of the second object	$\frac{\text{m}}{\text{s}^{2}}$
${a_{\text{y}1}}$	Vertical acceleration of the first object	$\frac{\text{m}}{\text{s}^{2}}$
${a_{\text{y}2}}$	Vertical acceleration of the second object	$\frac{\text{m}}{\text{s}^{2}}$
$\boldsymbol{a}\text{(}t\text{)}$	Acceleration	$\frac{\text{m}}{\text{s}^{2}}$
$\boldsymbol{F}$	Force	${\text{N}}$
$g$	Magnitude of gravitational acceleration	$\frac{\text{m}}{\text{s}^{2}}$
$\boldsymbol{g}$	Gravitational acceleration	$\frac{\text{m}}{\text{s}^{2}}$
$\boldsymbol{\hat{i}}$	Unit vector	–
${L_{1}}$	Length of the first rod	${\text{m}}$
${L_{2}}$	Length of the second rod	${\text{m}}$
$m$	Mass	${\text{kg}}$
${m_{1}}$	Mass of the first object	${\text{kg}}$
${m_{2}}$	Mass of the second object	${\text{kg}}$
${p_{\text{x}1}}$	Horizontal position of the first object	${\text{m}}$
${p_{\text{x}2}}$	Horizontal position of the second object	${\text{m}}$
${p_{\text{y}1}}$	Vertical position of the first object	${\text{m}}$
${p_{\text{y}2}}$	Vertical position of the second object	${\text{m}}$
$\boldsymbol{p}\text{(}t\text{)}$	Position	${\text{m}}$
$\boldsymbol{T}$	Tension	${\text{N}}$
${\boldsymbol{T}_{1}}$	Tension of the first object	${\text{N}}$
${\boldsymbol{T}_{2}}$	Tension of the second object	${\text{N}}$
$t$	Time	${\text{s}}$
$\text{theta}$	Dependent variables	${\text{rad}}$
${v_{\text{x}1}}$	Horizontal velocity of the first object	$\frac{\text{m}}{\text{s}}$
${v_{\text{x}2}}$	Horizontal velocity of the second object	$\frac{\text{m}}{\text{s}}$
${v_{\text{y}1}}$	Vertical velocity of the first object	$\frac{\text{m}}{\text{s}}$
${v_{\text{y}2}}$	Vertical velocity of the second object	$\frac{\text{m}}{\text{s}}$
$\boldsymbol{v}\text{(}t\text{)}$	Velocity	$\frac{\text{m}}{\text{s}}$
${w_{1}}$	Angular velocity of the first object	$\frac{\text{rad}}{\text{s}}$
${w_{2}}$	Angular velocity of the second object	$\frac{\text{rad}}{\text{s}}$
${α_{1}}$	Angular acceleration of the first object	$\frac{\text{rad}}{\text{s}^{2}}$
${α_{2}}$	Angular acceleration of the second object	$\frac{\text{rad}}{\text{s}^{2}}$
${θ_{1}}$	Angle of the first rod	${\text{rad}}$
${θ_{2}}$	Angle of the second rod	${\text{rad}}$
$π$	Ratio of circumference to diameter for any circle	–

Table of Symbols

Abbreviations and Acronyms

Abbreviation	Full Form
2D	Two-Dimensional
A	Assumption
DD	Data Definition
DblPend	Double Pendulum
GD	General Definition
GS	Goal Statement
IM	Instance Model
PS	Physical System Description
R	Requirement
RefBy	Referenced by
Refname	Reference Name
SRS	Software Requirements Specification
TM	Theoretical Model
Uncert.	Typical Uncertainty

Abbreviations and Acronyms

Introduction

A pendulum consists of mass attached to the end of a rod and its moving curve is highly sensitive to initial conditions. Therefore, it is useful to have a program to simulate the motion of the pendulum to exhibit its chaotic characteristics. The document describes the program called Double Pendulum , which is based on the original, manually created version of Double Pendulum.

The following section provides an overview of the Software Requirements Specification (SRS) for Double Pendulum. This section explains the purpose of this document, the scope of the requirements, the characteristics of the intended reader, and the organization of the document.

Purpose of Document

The primary purpose of this document is to record the requirements of DblPend. Goals, assumptions, theoretical models, definitions, and other model derivation information are specified, allowing the reader to fully understand and verify the purpose and scientific basis of DblPend. With the exception of system constraints, this SRS will remain abstract, describing what problem is being solved, but not how to solve it.

This document will be used as a starting point for subsequent development phases, including writing the design specification and the software verification and validation plan. The design document will show how the requirements are to be realized, including decisions on the numerical algorithms and programming environment. The verification and validation plan will show the steps that will be used to increase confidence in the software documentation and the implementation. Although the SRS fits in a series of documents that follow the so-called waterfall model, the actual development process is not constrained in any way. Even when the waterfall model is not followed, as Parnas and Clements point out parnasClements1986, the most logical way to present the documentation is still to “fake” a rational design process.

Scope of Requirements

The scope of the requirements includes the analysis of a two-dimensional (2D) pendulum motion problem with various initial conditions.

Characteristics of Intended Reader

Reviewers of this documentation should have an understanding of undergraduate level 2 physics, undergraduate level 1 calculus, and ordinary differential equations. The users of DblPend can have a lower level of expertise, as explained in Sec:User Characteristics.

Organization of Document

The organization of this document follows the template for an SRS for scientific computing software proposed by koothoor2013, smithLai2005, smithEtAl2007, and smithKoothoor2016. The presentation follows the standard pattern of presenting goals, theories, definitions, and assumptions. For readers that would like a more bottom up approach, they can start reading the instance models and trace back to find any additional information they require.

The goal statements are refined to the theoretical models and the theoretical models to the instance models.

General System Description

This section provides general information about the system. It identifies the interfaces between the system and its environment, describes the user characteristics, and lists the system constraints.

System Context

Fig:sysCtxDiag shows the system context. A circle represents an entity external to the software, the user in this case. A rectangle represents the software system itself (DblPend). Arrows are used to show the data flow between the system and its environment.

System Context

Figure: System Context

The interaction between the product and the user is through an application programming interface. The responsibilities of the user and the system are as follows:

User Responsibilities
- Provide initial conditions of the physical state of the motion and the input data related to the Double Pendulum, ensuring no errors in the data entry.
- Ensure that consistent units are used for input variables.
- Ensure required software assumptions are appropriate for any particular problem input to the software.
DblPend Responsibilities
- Detect data type mismatch, such as a string of characters input instead of a floating point number.
- Determine if the inputs satisfy the required physical and software constraints.
- Calculate the required outputs.
- Generate the required graphs.

User Characteristics

The end user of DblPend should have an understanding of high school physics, high school calculus and ordinary differential equations.

System Constraints

There are no system constraints.

Specific System Description

This section first presents the problem description, which gives a high-level view of the problem to be solved. This is followed by the solution characteristics specification, which presents the assumptions, theories, and definitions that are used.

Problem Description

A system is needed to predict the motion of a double pendulum.

Terminology and Definitions

This subsection provides a list of terms that are used in the subsequent sections and their meaning, with the purpose of reducing ambiguity and making it easier to correctly understand the requirements.

Gravity: The force that attracts one physical body with mass to another.
Cartesian coordinate system: A coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length (from cartesianWiki).

Physical System Description

The physical system of DblPend, as shown in Fig:dblpend, includes the following elements:

PS1: The first rod (with length of the first rod ${L_{1}}$ ).

PS2: The second rod (with length of the second rod ${L_{2}}$ ).

PS3: The first object.

PS4: The second object.

The physical system

Figure: The physical system

Goal Statements

Given the masses, length of the rods, initial angle of the masses and the gravitational constant, the goal statement is:

motionMass: Calculate the motion of the masses.

Solution Characteristics Specification

The instance models that govern DblPend are presented in the Instance Model Section. The information to understand the meaning of the instance models and their derivation is also presented, so that the instance models can be verified.

Assumptions

This section simplifies the original problem and helps in developing the theoretical models by filling in the missing information for the physical system. The assumptions refine the scope by providing more detail.

twoDMotion: The pendulum motion is two-dimensional (2D).

cartSys: A Cartesian coordinate system is used.

cartSysR: The Cartesian coordinate system is right-handed where positive $x$ -axis and $y$ -axis point right up.

yAxisDir: The direction of the $y$ -axis is directed opposite to gravity.

startOrigin: The first rod is attached to the origin.

firstPend: The first rod has two sides. One side attaches to the origin. Another side attaches to the first object.

secondPend: The second rod has two sides. One side attaches to the first object. Another side attaches to the second object.

Theoretical Models

This section focuses on the general equations and laws that DblPend is based on.

Acceleration

Refname	TM:acceleration
Label	Acceleration
Equation	$\boldsymbol{a}\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}$
Description	$\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) $t$ is the time ( ${\text{s}}$ ) $\boldsymbol{v}\text{(}t\text{)}$ is the velocity ( $\frac{\text{m}}{\text{s}}$ )
Source	accelerationWiki
RefBy

Velocity

Refname	TM:velocity
Label	Velocity
Equation	$\boldsymbol{v}\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}}{\,dt}$
Description	$\boldsymbol{v}\text{(}t\text{)}$ is the velocity ( $\frac{\text{m}}{\text{s}}$ ) $t$ is the time ( ${\text{s}}$ ) $\boldsymbol{p}\text{(}t\text{)}$ is the position ( ${\text{m}}$ )
Source	velocityWiki
RefBy

Newton’s second law of motion

Refname	TM:NewtonSecLawMot
Label	Newton’s second law of motion
Equation	$\boldsymbol{F}=m\,\boldsymbol{a}\text{(}t\text{)}$
Description	$\boldsymbol{F}$ is the force ( ${\text{N}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ )
Notes	The net force $\boldsymbol{F}$ on a body is proportional to the acceleration $\boldsymbol{a}\text{(}t\text{)}$ of the body, where $m$ denotes the mass of the body as the constant of proportionality.
Source	–
RefBy

General Definitions

This section collects the laws and equations that will be used to build the instance models.

The $x$ -component of velocity of the first object

Refname	GD:velocityX1
Label	The $x$ -component of velocity of the first object
Units	$\frac{\text{m}}{\text{s}}$
Equation	${v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)$
Description	${v_{\text{x}1}}$ is the horizontal velocity of the first object ( $\frac{\text{m}}{\text{s}}$ ) ${w_{1}}$ is the angular velocity of the first object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{1}}$ is the length of the first rod ( ${\text{m}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ )
Source	–
RefBy

Detailed derivation of the $x$ -component of velocity:

At a given point in time, velocity is defined in DD:positionGDD

$\boldsymbol{v}\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}}{\,dt}$

We also know the horizontal position that is defined in DD:positionXDD1

${p_{\text{x}1}}={L_{1}}\,\sin\left({θ_{1}}\right)$

Applying this,

${v_{\text{x}1}}=\frac{\,d{L_{1}}\,\sin\left({θ_{1}}\right)}{\,dt}$

${L_{1}}$ is constant with respect to time, so

${v_{\text{x}1}}={L_{1}}\,\frac{\,d\sin\left({θ_{1}}\right)}{\,dt}$

Therefore, using the chain rule,

${v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)$

The $y$ -component of velocity of the first object

Refname	GD:velocityY1
Label	The $y$ -component of velocity of the first object
Units	$\frac{\text{m}}{\text{s}}$
Equation	${v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)$
Description	${v_{\text{y}1}}$ is the vertical velocity of the first object ( $\frac{\text{m}}{\text{s}}$ ) ${w_{1}}$ is the angular velocity of the first object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{1}}$ is the length of the first rod ( ${\text{m}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ )
Source	–
RefBy

Detailed derivation of the $y$ -component of velocity:

At a given point in time, velocity is defined in DD:positionGDD

$\boldsymbol{v}\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}}{\,dt}$

We also know the vertical position that is defined in DD:positionYDD1

${p_{\text{y}1}}=-{L_{1}}\,\cos\left({θ_{1}}\right)$

Applying this,

${v_{\text{y}1}}=-\left(\frac{\,d{L_{1}}\,\cos\left({θ_{1}}\right)}{\,dt}\right)$

${L_{1}}$ is constant with respect to time, so

${v_{\text{y}1}}=-{L_{1}}\,\frac{\,d\cos\left({θ_{1}}\right)}{\,dt}$

Therefore, using the chain rule,

${v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)$

The $x$ -component of velocity of the second object

Refname	GD:velocityX2
Label	The $x$ -component of velocity of the second object
Units	$\frac{\text{m}}{\text{s}}$
Equation	${v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)$
Description	${v_{\text{x}2}}$ is the horizontal velocity of the second object ( $\frac{\text{m}}{\text{s}}$ ) ${v_{\text{x}1}}$ is the horizontal velocity of the first object ( $\frac{\text{m}}{\text{s}}$ ) ${w_{2}}$ is the angular velocity of the second object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{2}}$ is the length of the second rod ( ${\text{m}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ )
Source	–
RefBy

Detailed derivation of the $x$ -component of velocity:

At a given point in time, velocity is defined in DD:positionGDD

$\boldsymbol{v}\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}}{\,dt}$

We also know the horizontal position that is defined in DD:positionXDD2

${p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)$

Applying this,

${v_{\text{x}2}}=\frac{\,d{p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)}{\,dt}$

${L_{1}}$ is constant with respect to time, so

${v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)$

The $y$ -component of velocity of the second object

Refname	GD:velocityY2
Label	The $y$ -component of velocity of the second object
Units	$\frac{\text{m}}{\text{s}}$
Equation	${v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)$
Description	${v_{\text{y}2}}$ is the vertical velocity of the second object ( $\frac{\text{m}}{\text{s}}$ ) ${v_{\text{y}1}}$ is the vertical velocity of the first object ( $\frac{\text{m}}{\text{s}}$ ) ${w_{2}}$ is the angular velocity of the second object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{2}}$ is the length of the second rod ( ${\text{m}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ )
Source	–
RefBy

Detailed derivation of the $y$ -component of velocity:

At a given point in time, velocity is defined in DD:positionGDD

$\boldsymbol{v}\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}}{\,dt}$

We also know the vertical position that is defined in DD:positionYDD2

${p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)$

Applying this,

${v_{\text{y}2}}=-\left(\frac{\,d{p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)}{\,dt}\right)$

Therefore, using the chain rule,

${v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)$

The $x$ -component of acceleration of the first object

Refname	GD:accelerationX1
Label	The $x$ -component of acceleration of the first object
Units	$\frac{\text{m}}{\text{s}^{2}}$
Equation	${a_{\text{x}1}}=-{w_{1}}^{2}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)$
Description	${a_{\text{x}1}}$ is the horizontal acceleration of the first object ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${w_{1}}$ is the angular velocity of the first object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{1}}$ is the length of the first rod ( ${\text{m}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ ) ${α_{1}}$ is the angular acceleration of the first object ( $\frac{\text{rad}}{\text{s}^{2}}$ )
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of the $x$ -component of acceleration:

Our acceleration is:

$\boldsymbol{a}\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}$

Earlier, we found the horizontal velocity to be

${v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)$

Applying this to our equation for acceleration

${a_{\text{x}1}}=\frac{\,d{w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)}{\,dt}$

By the product and chain rules, we find

${a_{\text{x}1}}=\frac{\,d{w_{1}}}{\,dt}\,{L_{1}}\,\cos\left({θ_{1}}\right)-{w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\,\frac{\,d{θ_{1}}}{\,dt}$

Simplifying,

${a_{\text{x}1}}=-{w_{1}}^{2}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)$

The $y$ -component of acceleration of the first object

Refname	GD:accelerationY1
Label	The $y$ -component of acceleration of the first object
Units	$\frac{\text{m}}{\text{s}^{2}}$
Equation	${a_{\text{y}1}}={w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)$
Description	${a_{\text{y}1}}$ is the vertical acceleration of the first object ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${w_{1}}$ is the angular velocity of the first object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{1}}$ is the length of the first rod ( ${\text{m}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ ) ${α_{1}}$ is the angular acceleration of the first object ( $\frac{\text{rad}}{\text{s}^{2}}$ )
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of the $y$ -component of acceleration:

Our acceleration is:

$\boldsymbol{a}\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}$

Earlier, we found the vertical velocity to be

${v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)$

Applying this to our equation for acceleration

${a_{\text{y}1}}=\frac{\,d{w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)}{\,dt}$

By the product and chain rules, we find

${a_{\text{y}1}}=\frac{\,d{w_{1}}}{\,dt}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\,\frac{\,d{θ_{1}}}{\,dt}$

Simplifying,

${a_{\text{y}1}}={w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)$

The $x$ -component of acceleration of the second object

Refname	GD:accelerationX2
Label	The $x$ -component of acceleration of the second object
Units	$\frac{\text{m}}{\text{s}^{2}}$
Equation	${a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2}\,{L_{2}}\,\sin\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)$
Description	${a_{\text{x}2}}$ is the horizontal acceleration of the second object ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${a_{\text{x}1}}$ is the horizontal acceleration of the first object ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${w_{2}}$ is the angular velocity of the second object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{2}}$ is the length of the second rod ( ${\text{m}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ ) ${α_{2}}$ is the angular acceleration of the second object ( $\frac{\text{rad}}{\text{s}^{2}}$ )
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of the $x$ -component of acceleration:

Our acceleration is:

$\boldsymbol{a}\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}$

Earlier, we found the horizontal velocity to be

${v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)$

Applying this to our equation for acceleration

${a_{\text{x}2}}=\frac{\,d{v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)}{\,dt}$

By the product and chain rules, we find

${a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2}\,{L_{2}}\,\sin\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)$

The $y$ -component of acceleration of the second object

Refname	GD:accelerationY2
Label	The $y$ -component of acceleration of the second object
Units	$\frac{\text{m}}{\text{s}^{2}}$
Equation	${a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2}\,{L_{2}}\,\cos\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)$
Description	${a_{\text{y}2}}$ is the vertical acceleration of the second object ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${a_{\text{y}1}}$ is the vertical acceleration of the first object ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${w_{2}}$ is the angular velocity of the second object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{2}}$ is the length of the second rod ( ${\text{m}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ ) ${α_{2}}$ is the angular acceleration of the second object ( $\frac{\text{rad}}{\text{s}^{2}}$ )
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of the $y$ -component of acceleration:

Our acceleration is:

$\boldsymbol{a}\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}$

Earlier, we found the horizontal velocity to be

${v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)$

Applying this to our equation for acceleration

${a_{\text{y}2}}=\frac{\,d{v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)}{\,dt}$

By the product and chain rules, we find

${a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2}\,{L_{2}}\,\cos\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)$

Horizontal force on the first object

Refname	GD:xForce1
Label	Horizontal force on the first object
Units	${\text{N}}$
Equation	$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)+{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)$
Description	$\boldsymbol{F}$ is the force ( ${\text{N}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${\boldsymbol{T}_{1}}$ is the tension of the first object ( ${\text{N}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ ) ${\boldsymbol{T}_{2}}$ is the tension of the second object ( ${\text{N}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ )
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of force on the first object:

$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)+{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)$

Vertical force on the first object

Refname	GD:yForce1
Label	Vertical force on the first object
Units	${\text{N}}$
Equation	$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}_{1}}\,\cos\left({θ_{1}}\right)-{\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{1}}\,\boldsymbol{g}$
Description	$\boldsymbol{F}$ is the force ( ${\text{N}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${\boldsymbol{T}_{1}}$ is the tension of the first object ( ${\text{N}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ ) ${\boldsymbol{T}_{2}}$ is the tension of the second object ( ${\text{N}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ ) ${m_{1}}$ is the mass of the first object ( ${\text{kg}}$ ) $\boldsymbol{g}$ is the gravitational acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ )
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of force on the first object:

$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}_{1}}\,\cos\left({θ_{1}}\right)-{\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{1}}\,\boldsymbol{g}$

Horizontal force on the second object

Refname	GD:xForce2
Label	Horizontal force on the second object
Units	${\text{N}}$
Equation	$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)$
Description	$\boldsymbol{F}$ is the force ( ${\text{N}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${\boldsymbol{T}_{2}}$ is the tension of the second object ( ${\text{N}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ )
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of force on the second object:

$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)$

Vertical force on the second object

Refname	GD:yForce2
Label	Vertical force on the second object
Units	${\text{N}}$
Equation	$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{2}}\,\boldsymbol{g}$
Description	$\boldsymbol{F}$ is the force ( ${\text{N}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${\boldsymbol{T}_{2}}$ is the tension of the second object ( ${\text{N}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ ) ${m_{2}}$ is the mass of the second object ( ${\text{kg}}$ ) $\boldsymbol{g}$ is the gravitational acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ )
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of force on the second object:

$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{2}}\,\boldsymbol{g}$

Data Definitions

This section collects and defines all the data needed to build the instance models.

Velocity

Refname	DD:positionGDD
Label	Velocity
Symbol	$\boldsymbol{v}\text{(}t\text{)}$
Units	$\frac{\text{m}}{\text{s}}$
Equation	$\boldsymbol{v}\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}}{\,dt}$
Description	$\boldsymbol{v}\text{(}t\text{)}$ is the velocity ( $\frac{\text{m}}{\text{s}}$ ) $t$ is the time ( ${\text{s}}$ ) $\boldsymbol{p}\text{(}t\text{)}$ is the position ( ${\text{m}}$ )
Source	–
RefBy	GD:velocityY2, GD:velocityY1, GD:velocityX2, and GD:velocityX1

Horizontal position of the first object

Refname	DD:positionXDD1
Label	Horizontal position of the first object
Symbol	${p_{\text{x}1}}$
Units	${\text{m}}$
Equation	${p_{\text{x}1}}={L_{1}}\,\sin\left({θ_{1}}\right)$
Description	${p_{\text{x}1}}$ is the horizontal position of the first object ( ${\text{m}}$ ) ${L_{1}}$ is the length of the first rod ( ${\text{m}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ )
Notes	${p_{\text{x}1}}$ is the horizontal position ${p_{\text{x}1}}$ is shown in Fig:dblpend.
Source	–
RefBy	GD:velocityX1

Vertical position of the first object

Refname	DD:positionYDD1
Label	Vertical position of the first object
Symbol	${p_{\text{y}1}}$
Units	${\text{m}}$
Equation	${p_{\text{y}1}}=-{L_{1}}\,\cos\left({θ_{1}}\right)$
Description	${p_{\text{y}1}}$ is the vertical position of the first object ( ${\text{m}}$ ) ${L_{1}}$ is the length of the first rod ( ${\text{m}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ )
Notes	${p_{\text{y}1}}$ is the vertical position ${p_{\text{y}1}}$ is shown in Fig:dblpend.
Source	–
RefBy	GD:velocityY1

Horizontal position of the second object

Refname	DD:positionXDD2
Label	Horizontal position of the second object
Symbol	${p_{\text{x}2}}$
Units	${\text{m}}$
Equation	${p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)$
Description	${p_{\text{x}2}}$ is the horizontal position of the second object ( ${\text{m}}$ ) ${p_{\text{x}1}}$ is the horizontal position of the first object ( ${\text{m}}$ ) ${L_{2}}$ is the length of the second rod ( ${\text{m}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ )
Notes	${p_{\text{x}2}}$ is the horizontal position ${p_{\text{x}2}}$ is shown in Fig:dblpend.
Source	–
RefBy	GD:velocityX2

Vertical position of the second object

Refname	DD:positionYDD2
Label	Vertical position of the second object
Symbol	${p_{\text{y}2}}$
Units	${\text{m}}$
Equation	${p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)$
Description	${p_{\text{y}2}}$ is the vertical position of the second object ( ${\text{m}}$ ) ${p_{\text{y}1}}$ is the vertical position of the first object ( ${\text{m}}$ ) ${L_{2}}$ is the length of the second rod ( ${\text{m}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ )
Notes	${p_{\text{y}2}}$ is the vertical position ${p_{\text{y}2}}$ is shown in Fig:dblpend.
Source	–
RefBy	GD:velocityY2

Acceleration

Refname	DD:accelerationGDD
Label	Acceleration
Symbol	$\boldsymbol{a}\text{(}t\text{)}$
Units	$\frac{\text{m}}{\text{s}^{2}}$
Equation	$\boldsymbol{a}\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}$
Description	$\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) $t$ is the time ( ${\text{s}}$ ) $\boldsymbol{v}\text{(}t\text{)}$ is the velocity ( $\frac{\text{m}}{\text{s}}$ )
Source	–
RefBy

Force

Refname	DD:forceGDD
Label	Force
Symbol	$\boldsymbol{F}$
Units	${\text{N}}$
Equation	$\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}$
Description	$\boldsymbol{F}$ is the force ( ${\text{N}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ )
Source	–
RefBy

Instance Models

This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.

Calculation of angle of first rod

Refname	IM:calOfAngle1
Label	Calculation of angle of first rod
Input	${L_{1}}$ , ${L_{2}}$ , ${m_{1}}$ , ${m_{2}}$ , ${θ_{1}}$ , ${θ_{2}}$
Output	${θ_{1}}$
Input Constraints	${L_{1}}\gt{}0$ ${L_{2}}\gt{}0$ ${m_{1}}\gt{}0$ ${m_{2}}\gt{}0$
Output Constraints
Equation	${α_{1}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{-g\,\left(2\,{m_{1}}+{m_{2}}\right)\,\sin\left({θ_{1}}\right)-{m_{2}}\,g\,\sin\left({θ_{1}}-2\,{θ_{2}}\right)-2\,\sin\left({θ_{1}}-{θ_{2}}\right)\,{m_{2}}\,\left({w_{2}}^{2}\,{L_{2}}+{w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{1}}\,\left(2\,{m_{1}}+{m_{2}}-{m_{2}}\,\cos\left(2\,{θ_{1}}-2\,{θ_{2}}\right)\right)}$
Description	${α_{1}}$ is the angular acceleration of the first object ( $\frac{\text{rad}}{\text{s}^{2}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ ) ${w_{1}}$ is the angular velocity of the first object ( $\frac{\text{rad}}{\text{s}}$ ) ${w_{2}}$ is the angular velocity of the second object ( $\frac{\text{rad}}{\text{s}}$ ) $g$ is the magnitude of gravitational acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${m_{1}}$ is the mass of the first object ( ${\text{kg}}$ ) ${m_{2}}$ is the mass of the second object ( ${\text{kg}}$ ) ${L_{2}}$ is the length of the second rod ( ${\text{m}}$ ) ${L_{1}}$ is the length of the first rod ( ${\text{m}}$ )
Notes	${θ_{1}}$ is calculated by solving the ODE here together with the initial conditions and IM:calOfAngle2.
Source	–
RefBy	IM:calOfAngle2, FR:Output-Values, and FR:Calculate-Angle-Of-Rod

Calculation of angle of second rod

Refname	IM:calOfAngle2
Label	Calculation of angle of second rod
Input	${L_{1}}$ , ${L_{2}}$ , ${m_{1}}$ , ${m_{2}}$ , ${θ_{1}}$ , ${θ_{2}}$
Output	${θ_{2}}$
Input Constraints	${L_{1}}\gt{}0$ ${L_{2}}\gt{}0$ ${m_{1}}\gt{}0$ ${m_{2}}\gt{}0$
Output Constraints
Equation	${α_{2}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{2\,\sin\left({θ_{1}}-{θ_{2}}\right)\,\left({w_{1}}^{2}\,{L_{1}}\,\left({m_{1}}+{m_{2}}\right)+g\,\left({m_{1}}+{m_{2}}\right)\,\cos\left({θ_{1}}\right)+{w_{2}}^{2}\,{L_{2}}\,{m_{2}}\,\cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{2}}\,\left(2\,{m_{1}}+{m_{2}}-{m_{2}}\,\cos\left(2\,{θ_{1}}-2\,{θ_{2}}\right)\right)}$
Description	${α_{2}}$ is the angular acceleration of the second object ( $\frac{\text{rad}}{\text{s}^{2}}$ ) ${θ_{1}}$ is the angle of the first rod ( ${\text{rad}}$ ) ${θ_{2}}$ is the angle of the second rod ( ${\text{rad}}$ ) ${w_{1}}$ is the angular velocity of the first object ( $\frac{\text{rad}}{\text{s}}$ ) ${w_{2}}$ is the angular velocity of the second object ( $\frac{\text{rad}}{\text{s}}$ ) ${L_{1}}$ is the length of the first rod ( ${\text{m}}$ ) ${m_{1}}$ is the mass of the first object ( ${\text{kg}}$ ) ${m_{2}}$ is the mass of the second object ( ${\text{kg}}$ ) $g$ is the magnitude of gravitational acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ ) ${L_{2}}$ is the length of the second rod ( ${\text{m}}$ )
Notes	${θ_{2}}$ is calculated by solving the ODE here together with the initial conditions and IM:calOfAngle1.
Source	–
RefBy	IM:calOfAngle2, IM:calOfAngle1, FR:Output-Values, and FR:Calculate-Angle-Of-Rod

Detailed derivation of angle of the second rod:

By solving equations GD:xForce2 and GD:yForce2 for ${\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)$ and ${\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)$ and then substituting into equation GD:xForce1 and GD:yForce1 , we can get equations 1 and 2:

${m_{1}}\,{a_{\text{x}1}}=-{\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)-{m_{2}}\,{a_{\text{x}2}}$

${m_{1}}\,{a_{\text{y}1}}={\boldsymbol{T}_{1}}\,\cos\left({θ_{1}}\right)-{m_{2}}\,{a_{\text{y}2}}-{m_{2}}\,g-{m_{1}}\,g$

Multiply the equation 1 by $\cos\left({θ_{1}}\right)$ and the equation 2 by $\sin\left({θ_{1}}\right)$ and rearrange to get:

${\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)\,\cos\left({θ_{1}}\right)=-\cos\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{x}1}}+{m_{2}}\,{a_{\text{x}2}}\right)$

${\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)\,\cos\left({θ_{1}}\right)=\sin\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{y}1}}+{m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g+{m_{1}}\,g\right)$

This leads to the equation 3

$\sin\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{y}1}}+{m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g+{m_{1}}\,g\right)=-\cos\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{x}1}}+{m_{2}}\,{a_{\text{x}2}}\right)$

Next, multiply equation GD:xForce2 by $\cos\left({θ_{2}}\right)$ and equation GD:yForce2 by $\sin\left({θ_{2}}\right)$ and rearrange to get:

${\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)\,\cos\left({θ_{2}}\right)=-\cos\left({θ_{2}}\right)\,{m_{2}}\,{a_{\text{x}2}}$

${\boldsymbol{T}_{1}}\,\sin\left({θ_{2}}\right)\,\cos\left({θ_{2}}\right)=\sin\left({θ_{2}}\right)\,\left({m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g\right)$

which leads to equation 4

$\sin\left({θ_{2}}\right)\,\left({m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g\right)=-\cos\left({θ_{2}}\right)\,{m_{2}}\,{a_{\text{x}2}}$

By giving equations GD:accelerationX1 and GD:accelerationX2 and GD:accelerationY1 and GD:accelerationY2 plus additional two equations, 3 and 4, we can get IM:calOfAngle1 and IM:calOfAngle2 via a computer algebra program:

Data Constraints

The Data Constraints Table shows the data constraints on the input variables. The column for physical constraints gives the physical limitations on the range of values that can be taken by the variable. The uncertainty column provides an estimate of the confidence with which the physical quantities can be measured. This information would be part of the input if one were performing an uncertainty quantification exercise. The constraints are conservative to give the user of the model the flexibility to experiment with unusual situations. The column of typical values is intended to provide a feel for a common scenario.

Var	Physical Constraints	Typical Value	Uncert.
${L_{1}}$	${L_{1}}\gt{}0$	$1.0$ ${\text{m}}$	10 $\%$
${L_{2}}$	${L_{2}}\gt{}0$	$1.0$ ${\text{m}}$	10 $\%$
${m_{1}}$	${m_{1}}\gt{}0$	$0.5$ ${\text{kg}}$	10 $\%$
${m_{2}}$	${m_{2}}\gt{}0$	$0.5$ ${\text{kg}}$	10 $\%$

Input Data Constraints

Properties of a Correct Solution

The Data Constraints Table shows the data constraints on the output variables. The column for physical constraints gives the physical limitations on the range of values that can be taken by the variable.

Var	Physical Constraints
${θ_{1}}$	${θ_{1}}\gt{}0$
${θ_{2}}$	${θ_{2}}\gt{}0$

Output Data Constraints

Requirements

This section provides the functional requirements, the tasks and behaviours that the software is expected to complete, and the non-functional requirements, the qualities that the software is expected to exhibit.

Functional Requirements

This section provides the functional requirements, the tasks and behaviours that the software is expected to complete.

Input-Values: Input the values from Tab:ReqInputs.

Verify-Input-Values: Check the entered input values to ensure that they do not exceed the data constraints. If any of the input values are out of bounds, an error message is displayed and the calculations stop.

Calculate-Angle-Of-Rod: Calculate the following values: ${θ_{1}}$ and ${θ_{2}}$ (from IM:calOfAngle1 and IM:calOfAngle2).

Output-Values: Output ${θ_{1}}$ and ${θ_{2}}$ (from IM:calOfAngle1 and IM:calOfAngle2).

Symbol	Description	Units
${L_{1}}$	Length of the first rod	${\text{m}}$
${L_{2}}$	Length of the second rod	${\text{m}}$
${m_{1}}$	Mass of the first object	${\text{kg}}$
${m_{2}}$	Mass of the second object	${\text{kg}}$

Required Inputs

Non-Functional Requirements

This section provides the non-functional requirements, the qualities that the software is expected to exhibit.

Correctness: The outputs of the code have the properties of a correct solution.

Portability: The code shall be portable to multiple environments, particularly Windows, Mac OSX, and Linux.

Traceability Matrices and Graphs

The purpose of the traceability matrices is to provide easy references on what has to be additionally modified if a certain component is changed. Every time a component is changed, the items in the column of that component that are marked with an “X” should be modified as well. Tab:TraceMatAvsA shows the dependencies of the assumptions on each other. Tab:TraceMatAvsAll shows the dependencies of the data definitions, theoretical models, general definitions, instance models, requirements, likely changes, and unlikely changes on the assumptions. Tab:TraceMatRefvsRef shows the dependencies of the data definitions, theoretical models, general definitions, and instance models on each other. Tab:TraceMatAllvsR shows the dependencies of the requirements and goal statements on the data definitions, theoretical models, general definitions, and instance models.

	A:twoDMotion	A:cartSys	A:cartSysR	A:yAxisDir	A:startOrigin	A:firstPend	A:secondPend
A:twoDMotion
A:cartSys
A:cartSysR
A:yAxisDir
A:startOrigin
A:firstPend
A:secondPend

Traceability Matrix Showing the Connections Between Assumptions and Other Assumptions

	A:twoDMotion	A:cartSys	A:cartSysR	A:yAxisDir	A:startOrigin	A:firstPend	A:secondPend
DD:positionGDD
DD:positionXDD1
DD:positionYDD1
DD:positionXDD2
DD:positionYDD2
DD:accelerationGDD
DD:forceGDD
TM:acceleration
TM:velocity
TM:NewtonSecLawMot
GD:velocityX1
GD:velocityY1
GD:velocityX2
GD:velocityY2
GD:accelerationX1
GD:accelerationY1
GD:accelerationX2
GD:accelerationY2
GD:xForce1
GD:yForce1
GD:xForce2
GD:yForce2
IM:calOfAngle1
IM:calOfAngle2
FR:Input-Values
FR:Verify-Input-Values
FR:Calculate-Angle-Of-Rod
FR:Output-Values
NFR:Correctness
NFR:Portability

Traceability Matrix Showing the Connections Between Assumptions and Other Items

	DD:positionGDD	DD:positionXDD1	DD:positionYDD1	DD:positionXDD2	DD:positionYDD2	GD:accelerationX1	GD:accelerationY1	GD:accelerationX2	GD:accelerationY2	GD:xForce1	GD:yForce1	GD:xForce2	GD:yForce2	IM:calOfAngle1	IM:calOfAngle2
DD:positionGDD
DD:positionXDD1
DD:positionYDD1
DD:positionXDD2
DD:positionYDD2
DD:accelerationGDD
DD:forceGDD
TM:acceleration
TM:velocity
TM:NewtonSecLawMot
GD:velocityX1	X	X
GD:velocityY1	X		X
GD:velocityX2	X			X
GD:velocityY2	X				X
GD:accelerationX1
GD:accelerationY1
GD:accelerationX2
GD:accelerationY2
GD:xForce1
GD:yForce1
GD:xForce2
GD:yForce2
IM:calOfAngle1															X
IM:calOfAngle2						X	X	X	X	X	X	X	X	X	X

Traceability Matrix Showing the Connections Between Items and Other Sections

	IM:calOfAngle1	IM:calOfAngle2
GS:motionMass
FR:Input-Values
FR:Verify-Input-Values
FR:Calculate-Angle-Of-Rod	X	X
FR:Output-Values	X	X
NFR:Correctness
NFR:Portability

Traceability Matrix Showing the Connections Between Requirements, Goal Statements and Other Items

The purpose of the traceability graphs is also to provide easy references on what has to be additionally modified if a certain component is changed. The arrows in the graphs represent dependencies. The component at the tail of an arrow is depended on by the component at the head of that arrow. Therefore, if a component is changed, the components that it points to should also be changed. Fig:TraceGraphAvsA shows the dependencies of assumptions on each other. Fig:TraceGraphAvsAll shows the dependencies of data definitions, theoretical models, general definitions, instance models, requirements, likely changes, and unlikely changes on the assumptions. Fig:TraceGraphRefvsRef shows the dependencies of data definitions, theoretical models, general definitions, and instance models on each other. Fig:TraceGraphAllvsR shows the dependencies of requirements and goal statements on the data definitions, theoretical models, general definitions, and instance models. Fig:TraceGraphAllvsAll shows the dependencies of dependencies of assumptions, models, definitions, requirements, goals, and changes with each other.

TraceGraphAvsA

Figure: TraceGraphAvsA

TraceGraphAvsAll

Figure: TraceGraphAvsAll

TraceGraphRefvsRef

Figure: TraceGraphRefvsRef

TraceGraphAllvsR

Figure: TraceGraphAllvsR

TraceGraphAllvsAll

Figure: TraceGraphAllvsAll

For convenience, the following graphs can be found at the links below:

Values of Auxiliary Constants

There are no auxiliary constants.

References

[1]: Hibbeler, R. C. Engineering Mechanics: Dynamics. Pearson Prentice Hall, 2004. Print.

[2]: Koothoor, Nirmitha. A Document Driven Approach to Certifying Scientific Computing Software. McMaster University, Hamilton, ON, Canada: 2013. Print.

[3]: Parnas, David L. and Clements, P. C. “A rational design process: How and why to fake it.” IEEE Transactions on Software Engineering, vol. 12, no. 2, Washington, USA: February, 1986. pp. 251–257. Print.

[4]: Smith, W. Spencer and Koothoor, Nirmitha. “A Document-Driven Method for Certifying Scientific Computing Software for Use in Nuclear Safety Analysis.” Nuclear Engineering and Technology, vol. 48, no. 2, April, 2016. http://www.sciencedirect.com/science/article/pii/S1738573315002582. pp. 404–418.

[5]: Smith, W. Spencer and Lai, Lei. “A new requirements template for scientific computing.” Proceedings of the First International Workshop on Situational Requirements Engineering Processes - Methods, Techniques and Tools to Support Situation-Specific Requirements Engineering Processes, SREP’05. Edited by PJ Agerfalk, N. Kraiem, and J. Ralyte, Paris, France: 2005. pp. 107–121. In conjunction with 13th IEEE International Requirements Engineering Conference,

[6]: Smith, W. Spencer, Lai, Lei, and Khedri, Ridha. “Requirements Analysis for Engineering Computation: A Systematic Approach for Improving Software Reliability.” Reliable Computing, Special Issue on Reliable Engineering Computation, vol. 13, no. 1, February, 2007. https://doi.org/10.1007/s11155-006-9020-7. pp. 83–107.

[7]: Wikipedia Contributors. Acceleration. June, 2019. https://en.wikipedia.org/wiki/Acceleration.

[8]: Wikipedia Contributors. Cartesian coordinate system. June, 2019. https://en.wikipedia.org/wiki/Cartesian_coordinate_system.

[9]: Wikipedia Contributors. Velocity. June, 2019. https://en.wikipedia.org/wiki/Velocity.