Software Requirements Specification for Single Pendulum

Olu Owojaiye

Table of Contents

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
Hz frequency hertz
kg mass kilogram
m length metre
N force newton
rad angle radian
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
ax x-component of acceleration \(\frac{\text{m}}{\text{s}^{2}}\)
ay y-component of acceleration \(\frac{\text{m}}{\text{s}^{2}}\)
a(t) Acceleration \(\frac{\text{m}}{\text{s}^{2}}\)
F Force N
f Frequency Hz
g Gravitational acceleration \(\frac{\text{m}}{\text{s}^{2}}\)
I Moment of inertia kgm2
î Unit vector --
Lrod Length of the rod m
m Mass kg
px x-component of position m
pxi x-component of initial position m
py y-component of position m
pyi y-component of initial position m
p(t) Position m
T Period s
T Tension N
t Time s
vx x-component of velocity \(\frac{\text{m}}{\text{s}}\)
vy y-component of velocity \(\frac{\text{m}}{\text{s}}\)
v(t) Velocity \(\frac{\text{m}}{\text{s}}\)
α Angular acceleration \(\frac{\text{rad}}{\text{s}^{2}}\)
θ Angular displacement rad
θi Initial pendulum angle rad
θp Displacement angle of the pendulum rad
π Ratio of circumference to diameter for any circle --
τ Torque Nm
Ω Angular frequency s
ω Angular velocity \(\frac{\text{rad}}{\text{s}}\)

Table of Symbols

Abbreviations and Acronyms

Abbreviation Full Form
2D Two-Dimensional
A Assumption
DD Data Definition
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
SglPend Single Pendulum
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 Single 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 Single 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 SglPend. 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 SglPend. 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 SglPend 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 (SglPend). 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 Single 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.
  • SglPend 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 SglPend 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 single 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 SglPend, as shown in Fig:sglpend, includes the following elements:

PS1: The rod.

PS2: The mass.

The physical system
Figure: The physical system

Goal Statements

Given the mass and length of the rod, initial angle of the mass and the gravitational constant, the goal statement is:

Motion-of-the-mass: Calculate the motion of the mass.

Solution Characteristics Specification

The instance models that govern SglPend 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 pendulum is attached to the origin.

Theoretical Models

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

Refname TM:acceleration
Label

Acceleration
Equation \[\symbf{a}\text{(}t\text{)}=\frac{\,d\symbf{v}\text{(}t\text{)}}{\,dt}\]
Description
  • a(t) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • t is the time (s)
  • v(t) is the velocity (\(\frac{\text{m}}{\text{s}}\))
Source

accelerationWiki

RefBy

Refname TM:velocity
Label

Velocity
Equation \[\symbf{v}\text{(}t\text{)}=\frac{\,d\symbf{p}\text{(}t\text{)}}{\,dt}\]
Description
  • v(t) is the velocity (\(\frac{\text{m}}{\text{s}}\))
  • t is the time (s)
  • p(t) is the position (m)
Source

velocityWiki
RefBy

Refname TM:NewtonSecLawMot
Label

Newton's second law of motion
Equation \[\symbf{F}=m\,\symbf{a}\text{(}t\text{)}\]
Description
  • F is the force (N)
  • m is the mass (kg)
  • a(t) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Notes

The net force F on a body is proportional to the acceleration a(t) of the body, where m denotes the mass of the body as the constant of proportionality.

Source

--
RefBy

Refname TM:NewtonSecLawRotMot
Label

Newton's second law for rotational motion
Equation \[\symbf{τ}=\symbf{I}\,α\]
Description
  • τ is the torque (Nm)
  • I is the moment of inertia (kgm2)
  • α is the angular acceleration (\(\frac{\text{rad}}{\text{s}^{2}}\))
Notes

The net torque τ on a rigid body is proportional to its angular acceleration α, where I denotes the moment of inertia of the rigid body as the constant of proportionality.

Source

--
RefBy

IM:calOfAngularDisplacement and GD:angFrequencyGD

General Definitions

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

Refname GD:velocityIX
Label

The x-component of velocity of the pendulum

Units

\(\frac{\text{m}}{\text{s}}\)
Equation \[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
Description
  • vx is the x-component of velocity (\(\frac{\text{m}}{\text{s}}\))
  • ω is the angular velocity (\(\frac{\text{rad}}{\text{s}}\))
  • Lrod is the length of the rod (m)
  • θp is the displacement angle of the pendulum (rad)
Source

--
RefBy

Detailed derivation of the x-component of velocity:

At a given point in time, velocity may be defined as

\[\symbf{v}\text{(}t\text{)}=\frac{\,d\symbf{p}\text{(}t\text{)}}{\,dt}\]

We also know the horizontal position

\[{p_{\text{x}}}={L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]

Applying this,

\[{v_{\text{x}}}=\frac{\,d{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)}{\,dt}\]

Lrod is constant with respect to time, so

\[{v_{\text{x}}}={L_{\text{rod}}}\,\frac{\,d\sin\left({θ_{p}}\right)}{\,dt}\]

Therefore, using the chain rule,

\[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
Refname GD:velocityIY
Label

The y-component of velocity of the pendulum

Units

\(\frac{\text{m}}{\text{s}}\)
Equation \[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
Description
  • vy is the y-component of velocity (\(\frac{\text{m}}{\text{s}}\))
  • ω is the angular velocity (\(\frac{\text{rad}}{\text{s}}\))
  • Lrod is the length of the rod (m)
  • θp is the displacement angle of the pendulum (rad)
Source

--
RefBy

Detailed derivation of the y-component of velocity:

At a given point in time, velocity may be defined as

\[\symbf{v}\text{(}t\text{)}=\frac{\,d\symbf{p}\text{(}t\text{)}}{\,dt}\]

We also know the vertical position

\[{p_{\text{y}}}=-{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]

Applying this,

\[{v_{\text{y}}}=-\left(\frac{\,d{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)}{\,dt}\right)\]

Lrod is constant with respect to time, so

\[{v_{\text{y}}}=-{L_{\text{rod}}}\,\frac{\,d\cos\left({θ_{p}}\right)}{\,dt}\]

Therefore, using the chain rule,

\[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
Refname GD:accelerationIX
Label

The x-component of acceleration of the pendulum

Units

\(\frac{\text{m}}{\text{s}^{2}}\)
Equation \[{a_{\text{x}}}=-ω^{2}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
Description
  • ax is the x-component of acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • ω is the angular velocity (\(\frac{\text{rad}}{\text{s}}\))
  • Lrod is the length of the rod (m)
  • θp is the displacement angle of the pendulum (rad)
  • α is the angular acceleration (\(\frac{\text{rad}}{\text{s}^{2}}\))
Source

--
RefBy

Detailed derivation of the x-component of acceleration:

Our acceleration is:

\[\symbf{a}\text{(}t\text{)}=\frac{\,d\symbf{v}\text{(}t\text{)}}{\,dt}\]

Earlier, we found the horizontal velocity to be

\[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]

Applying this to our equation for acceleration

\[{a_{\text{x}}}=\frac{\,dω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)}{\,dt}\]

By the product and chain rules, we find

\[{a_{\text{x}}}=\frac{\,dω}{\,dt}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)-ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\,\frac{\,d{θ_{p}}}{\,dt}\]

Simplifying,

\[{a_{\text{x}}}=-ω^{2}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
Refname GD:accelerationIY
Label

The y-component of acceleration of the pendulum

Units

\(\frac{\text{m}}{\text{s}^{2}}\)
Equation \[{a_{\text{y}}}=ω^{2}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
Description
  • ay is the y-component of acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • ω is the angular velocity (\(\frac{\text{rad}}{\text{s}}\))
  • Lrod is the length of the rod (m)
  • θp is the displacement angle of the pendulum (rad)
  • α is the angular acceleration (\(\frac{\text{rad}}{\text{s}^{2}}\))
Source

--
RefBy

Detailed derivation of the y-component of acceleration:

Our acceleration is:

\[\symbf{a}\text{(}t\text{)}=\frac{\,d\symbf{v}\text{(}t\text{)}}{\,dt}\]

Earlier, we found the vertical velocity to be

\[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]

Applying this to our equation for acceleration

\[{a_{\text{y}}}=\frac{\,dω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)}{\,dt}\]

By the product and chain rules, we find

\[{a_{\text{y}}}=\frac{\,dω}{\,dt}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\,\frac{\,d{θ_{p}}}{\,dt}\]

Simplifying,

\[{a_{\text{y}}}=ω^{2}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
Refname GD:hForceOnPendulum
Label

Horizontal force on the pendulum
Units

N
Equation \[\symbf{F}=m\,{a_{\text{x}}}=-\symbf{T}\,\sin\left({θ_{p}}\right)\]
Description
  • F is the force (N)
  • m is the mass (kg)
  • ax is the x-component of acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • T is the tension (N)
  • θp is the displacement angle of the pendulum (rad)
Source

--
RefBy

Detailed derivation of force on the pendulum:

\[\symbf{F}=m\,{a_{\text{x}}}=-\symbf{T}\,\sin\left({θ_{p}}\right)\]
Refname GD:vForceOnPendulum
Label

Vertical force on the pendulum
Units

N
Equation \[\symbf{F}=m\,{a_{\text{y}}}=\symbf{T}\,\cos\left({θ_{p}}\right)-m\,\symbf{g}\]
Description
  • F is the force (N)
  • m is the mass (kg)
  • ay is the y-component of acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • T is the tension (N)
  • θp is the displacement angle of the pendulum (rad)
  • g is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Source

--
RefBy

Detailed derivation of force on the pendulum:

\[\symbf{F}=m\,{a_{\text{y}}}=\symbf{T}\,\cos\left({θ_{p}}\right)-m\,\symbf{g}\]
Refname GD:angFrequencyGD
Label

The angular frequency of the pendulum
Units

s
Equation \[Ω=\sqrt{\frac{\symbf{g}}{{L_{\text{rod}}}}}\]
Description
  • Ω is the angular frequency (s)
  • g is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • Lrod is the length of the rod (m)
Notes

The torque is defined in TM:NewtonSecLawRotMot and frequency is f is defined in DD:frequencyDD.

Source

--
RefBy

GD:periodPend and IM:calOfAngularDisplacement

Detailed derivation of the angular frequency of the pendulum:

Consider the torque on a pendulum defined in TM:NewtonSecLawRotMot. The force providing the restoring torque is the component of weight of the pendulum bob that acts along the arc length. The torque is the length of the string Lrod multiplied by the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement:

\[\symbf{τ}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right)\]

So then

\[\symbf{I}\,α=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right)\]

Therefore,

\[\symbf{I}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right)\]

Substituting for I

\[m\,{L_{\text{rod}}}^{2}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right)\]

Crossing out m and Lrod we have

\[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right)\,\sin\left({θ_{p}}\right)\]

For small angles, we approximate sin θp to θp

\[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right)\,{θ_{p}}\]

Because this equation, has the same form as the equation for simple harmonic motion the solution is easy to find. The angular frequency

\[Ω=\sqrt{\frac{\symbf{g}}{{L_{\text{rod}}}}}\]
Refname GD:periodPend
Label

The period of the pendulum
Units

s
Equation \[T=2\,π\,\sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}}\]
Description
  • T is the period (s)
  • π is the ratio of circumference to diameter for any circle (Unitless)
  • Lrod is the length of the rod (m)
  • g is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Notes

The frequency and period are defined in the data definitions for frequency and period respectively

Source

--
RefBy

Detailed derivation of the period of the pendulum:

The period of the pendulum can be defined from the general definition for the equation of angular frequency

\[Ω=\sqrt{\frac{\symbf{g}}{{L_{\text{rod}}}}}\]

Therefore from the data definition of the equation for angular frequency, we have

\[T=2\,π\,\sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}}\]

Data Definitions

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

Refname DD:positionIX
Label

x-component of initial position
Symbol

pxi
Units

m
Equation \[{{p_{\text{x}}}^{\text{i}}}={L_{\text{rod}}}\,\sin\left({θ_{i}}\right)\]
Description
  • pxi is the x-component of initial position (m)
  • Lrod is the length of the rod (m)
  • θi is the initial pendulum angle (rad)
Notes

pxi is the horizontal position

pxi is shown in Fig:sglpend.

Source

--
RefBy

Refname DD:positionIY
Label

y-component of initial position
Symbol

pyi
Units

m
Equation \[{{p_{\text{y}}}^{\text{i}}}=-{L_{\text{rod}}}\,\cos\left({θ_{i}}\right)\]
Description
  • pyi is the y-component of initial position (m)
  • Lrod is the length of the rod (m)
  • θi is the initial pendulum angle (rad)
Notes

pyi is the vertical position

pyi is shown in Fig:sglpend.

Source

--
RefBy

Refname DD:frequencyDD
Label

Frequency
Symbol

f
Units

Hz
Equation \[f=\frac{1}{T}\]
Description
  • f is the frequency (Hz)
  • T is the period (s)
Notes

f is the number of back and forth swings in one second

Source

--
RefBy

GD:periodPend, DD:periodSHMDD, and GD:angFrequencyGD

Refname DD:angFrequencyDD
Label

Angular frequency
Symbol

Ω
Units

s
Equation \[Ω=\frac{2\,π}{T}\]
Description
  • Ω is the angular frequency (s)
  • π is the ratio of circumference to diameter for any circle (Unitless)
  • T is the period (s)
Notes

T is from DD:periodSHMDD

Source

--
RefBy

GD:periodPend
Refname DD:periodSHMDD
Label

Period
Symbol

T
Units

s
Equation \[T=\frac{1}{f}\]
Description
  • T is the period (s)
  • f is the frequency (Hz)
Notes

T is from DD:frequencyDD

Source

--
RefBy

GD:periodPend and DD:angFrequencyDD

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.

Refname IM:calOfAngularDisplacement
Label

Calculation of angular displacement
Input

Lrod, θi, g

Output

θp
Input Constraints \[{L_{\text{rod}}}\gt{}0\] \[{θ_{i}}\gt{}0\] \[\symbf{g}\gt{}0\]
Output Constraints \[{θ_{p}}\gt{}0\]
Equation \[{θ_{p}}\left(t\right)={θ_{i}}\,\cos\left(Ω\,t\right)\]
Description
  • θp is the displacement angle of the pendulum (rad)
  • t is the time (s)
  • θi is the initial pendulum angle (rad)
  • Ω is the angular frequency (s)
Notes

The constraint θi > 0 is required. The angular frequency is defined in GD:angFrequencyGD.

Source

--
RefBy

FR:Output-Values and FR:Calculate-Angular-Position-Of-Mass

Detailed derivation of angular displacement:

When the pendulum is displaced to an initial angle and released, the pendulum swings back and forth with periodic motion. By applying Newton's second law for rotational motion, the equation of motion for the pendulum may be obtained:

\[\symbf{τ}=\symbf{I}\,α\]

Where τ denotes the torque, I denotes the moment of inertia and α denotes the angular acceleration. This implies:

\[-m\,\symbf{g}\,\sin\left({θ_{p}}\right)\,{L_{\text{rod}}}=m\,{L_{\text{rod}}}^{2}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}\]

And rearranged as:

\[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}}\,\sin\left({θ_{p}}\right)=0\]

If the amplitude of angular displacement is small enough, we can approximate sin(θp) = θp for the purpose of a simple pendulum at very small angles. Then the equation of motion reduces to the equation of simple harmonic motion:

\[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}}\,{θ_{p}}=0\]

Thus the simple harmonic motion is:

\[{θ_{p}}\left(t\right)={θ_{i}}\,\cos\left(Ω\,t\right)\]

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.
Lrod Lrod > 0 44.2 m 10%
θi θi > 0 2.1 rad 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
α α > 0
θp θp > 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-Angular-Position-Of-Mass: Calculate the following values: θ and θp (from IM:calOfAngularDisplacement).

Output-Values: Output Lrod (from IM:calOfAngularDisplacement).

Symbol Description Units
Lrod Length of the rod m
m Mass kg
α Angular acceleration \(\frac{\text{rad}}{\text{s}^{2}}\)
θi Initial pendulum angle rad
θp Displacement angle of the pendulum rad

Required Inputs following FR:Input-Values

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:twoDMotion
A:cartSys
A:cartSysR
A:yAxisDir
A:startOrigin

Traceability Matrix Showing the Connections Between Assumptions and Other Assumptions

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