Symbol	Description	SI Name
kg	mass	kilogram
m	length	metre
N	force	newton
rad	angle	radian
s	time	second

Symbol	Description	Units
a_x1	Horizontal acceleration of the first object	\(\frac{\text{m}}{\text{s}^{2}}\)
a_x2	Horizontal acceleration of the second object	\(\frac{\text{m}}{\text{s}^{2}}\)
a_y1	Vertical acceleration of the first object	\(\frac{\text{m}}{\text{s}^{2}}\)
a_y2	Vertical acceleration of the second object	\(\frac{\text{m}}{\text{s}^{2}}\)
a(t)	Acceleration	\(\frac{\text{m}}{\text{s}^{2}}\)
F	Force	N
g	Magnitude of gravitational acceleration	\(\frac{\text{m}}{\text{s}^{2}}\)
g	Gravitational acceleration	\(\frac{\text{m}}{\text{s}^{2}}\)
î	Unit vector	--
L₁	Length of the first rod	m
L₂	Length of the second rod	m
m	Mass	kg
m₁	Mass of the first object	kg
m₂	Mass of the second object	kg
p_x1	Horizontal position of the first object	m
p_x2	Horizontal position of the second object	m
p_y1	Vertical position of the first object	m
p_y2	Vertical position of the second object	m
p(t)	Position	m
T	Tension	N
T₁	Tension of the first object	N
T₂	Tension of the second object	N
t	Time	s
theta	Dependent variables	rad
v_x1	Horizontal velocity of the first object	\(\frac{\text{m}}{\text{s}}\)
v_x2	Horizontal velocity of the second object	\(\frac{\text{m}}{\text{s}}\)
v_y1	Vertical velocity of the first object	\(\frac{\text{m}}{\text{s}}\)
v_y2	Vertical velocity of the second object	\(\frac{\text{m}}{\text{s}}\)
v(t)	Velocity	\(\frac{\text{m}}{\text{s}}\)
w₁	Angular velocity of the first object	\(\frac{\text{rad}}{\text{s}}\)
w₂	Angular velocity of the second object	\(\frac{\text{rad}}{\text{s}}\)
α₁	Angular acceleration of the first object	\(\frac{\text{rad}}{\text{s}^{2}}\)
α₂	Angular acceleration of the second object	\(\frac{\text{rad}}{\text{s}^{2}}\)
θ₁	Angle of the first rod	rad
θ₂	Angle of the second rod	rad
π	Ratio of circumference to diameter for any circle	--

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

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 L₁).

PS2: The second rod (with length of the second rod L₂).

PS3: The first object.

PS4: The second object.

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.

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

General Definitions

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

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_x1 is the horizontal velocity of the first object (\(\frac{\text{m}}{\text{s}}\)) w₁ is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) L₁ is the length of the first rod (m) θ₁ is the angle of the first rod (rad)
Source	--
RefBy

Detailed derivation of the x-component of velocity:

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

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

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,

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

L₁ 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,

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

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_y1 is the vertical velocity of the first object (\(\frac{\text{m}}{\text{s}}\)) w₁ is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) L₁ is the length of the first rod (m) θ₁ is the angle of the first rod (rad)
Source	--
RefBy

Detailed derivation of the y-component of velocity:

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

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

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,

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

L₁ 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,

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

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_x2 is the horizontal velocity of the second object (\(\frac{\text{m}}{\text{s}}\)) v_x1 is the horizontal velocity of the first object (\(\frac{\text{m}}{\text{s}}\)) w₂ is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) L₂ is the length of the second rod (m) θ₂ is the angle of the second rod (rad)
Source	--
RefBy

Detailed derivation of the x-component of velocity:

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

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

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,

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

L₁ is constant with respect to time, so

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

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_y2 is the vertical velocity of the second object (\(\frac{\text{m}}{\text{s}}\)) v_y1 is the vertical velocity of the first object (\(\frac{\text{m}}{\text{s}}\)) w₂ is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) L₂ is the length of the second rod (m) θ₂ is the angle of the second rod (rad)
Source	--
RefBy

Detailed derivation of the y-component of velocity:

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

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

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,

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

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_x1 is the horizontal acceleration of the first object (\(\frac{\text{m}}{\text{s}^{2}}\)) w₁ is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) L₁ is the length of the first rod (m) θ₁ is the angle of the first rod (rad) α₁ 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:

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

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,

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

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_y1 is the vertical acceleration of the first object (\(\frac{\text{m}}{\text{s}^{2}}\)) w₁ is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) L₁ is the length of the first rod (m) θ₁ is the angle of the first rod (rad) α₁ 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:

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

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,

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

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_x2 is the horizontal acceleration of the second object (\(\frac{\text{m}}{\text{s}^{2}}\)) a_x1 is the horizontal acceleration of the first object (\(\frac{\text{m}}{\text{s}^{2}}\)) w₂ is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) L₂ is the length of the second rod (m) θ₂ is the angle of the second rod (rad) α₂ 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:

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

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

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

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_y2 is the vertical acceleration of the second object (\(\frac{\text{m}}{\text{s}^{2}}\)) a_y1 is the vertical acceleration of the first object (\(\frac{\text{m}}{\text{s}^{2}}\)) w₂ is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) L₂ is the length of the second rod (m) θ₂ is the angle of the second rod (rad) α₂ 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:

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

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

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

Refname	GD:xForce1
Label	Horizontal force on the first object
Units	N
Equation	\[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)+{\symbf{T}_{2}} \sin\left({θ_{2}}\right)\]
Description	F is the force (N) m is the mass (kg) a(t) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) T₁ is the tension of the first object (N) θ₁ is the angle of the first rod (rad) T₂ is the tension of the second object (N) θ₂ is the angle of the second rod (rad)
Source	--
RefBy	IM:calOfAngle2

Detailed derivation of force on the first object:

\[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)+{\symbf{T}_{2}} \sin\left({θ_{2}}\right)\]

Refname	GD:yForce1
Label	Vertical force on the first object
Units	N
Equation	\[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{1}} \symbf{g}\]
Description	F is the force (N) m is the mass (kg) a(t) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) T₁ is the tension of the first object (N) θ₁ is the angle of the first rod (rad) T₂ is the tension of the second object (N) θ₂ is the angle of the second rod (rad) m₁ is the mass of the first object (kg) g is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Source	--
RefBy	IM:calOfAngle2

Detailed derivation of force on the first object:

\[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{1}} \symbf{g}\]

Refname	GD:xForce2
Label	Horizontal force on the second object
Units	N
Equation	\[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}} \sin\left({θ_{2}}\right)\]
Description	F is the force (N) m is the mass (kg) a(t) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) T₂ is the tension of the second object (N) θ₂ is the angle of the second rod (rad)
Source	--
RefBy	IM:calOfAngle2

Detailed derivation of force on the second object:

\[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}} \sin\left({θ_{2}}\right)\]

Refname	GD:yForce2
Label	Vertical force on the second object
Units	N
Equation	\[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{2}} \symbf{g}\]
Description	F is the force (N) m is the mass (kg) a(t) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) T₂ is the tension of the second object (N) θ₂ is the angle of the second rod (rad) m₂ is the mass of the second object (kg) g is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Source	--
RefBy	IM:calOfAngle2

Detailed derivation of force on the second object:

\[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{2}} \symbf{g}\]

Data Definitions

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

Refname	DD:positionGDD
Label	Velocity
Symbol	v(t)
Units	\(\frac{\text{m}}{\text{s}}\)
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	--
RefBy	GD:velocityY2, GD:velocityY1, GD:velocityX2, and GD:velocityX1

Refname	DD:positionXDD1
Label	Horizontal position of the first object
Symbol	p_x1
Units	m
Equation	\[{p_{\text{x}1}}={L_{1}} \sin\left({θ_{1}}\right)\]
Description	p_x1 is the horizontal position of the first object (m) L₁ is the length of the first rod (m) θ₁ is the angle of the first rod (rad)
Notes	p_x1 is the horizontal position p_x1 is shown in Fig:dblpend.
Source	--
RefBy	GD:velocityX1

Refname	DD:positionYDD1
Label	Vertical position of the first object
Symbol	p_y1
Units	m
Equation	\[{p_{\text{y}1}}=-{L_{1}} \cos\left({θ_{1}}\right)\]
Description	p_y1 is the vertical position of the first object (m) L₁ is the length of the first rod (m) θ₁ is the angle of the first rod (rad)
Notes	p_y1 is the vertical position p_y1 is shown in Fig:dblpend.
Source	--
RefBy	GD:velocityY1

Refname	DD:positionXDD2
Label	Horizontal position of the second object
Symbol	p_x2
Units	m
Equation	\[{p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}} \sin\left({θ_{2}}\right)\]
Description	p_x2 is the horizontal position of the second object (m) p_x1 is the horizontal position of the first object (m) L₂ is the length of the second rod (m) θ₂ is the angle of the second rod (rad)
Notes	p_x2 is the horizontal position p_x2 is shown in Fig:dblpend.
Source	--
RefBy	GD:velocityX2

Refname	DD:positionYDD2
Label	Vertical position of the second object
Symbol	p_y2
Units	m
Equation	\[{p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}} \cos\left({θ_{2}}\right)\]
Description	p_y2 is the vertical position of the second object (m) p_y1 is the vertical position of the first object (m) L₂ is the length of the second rod (m) θ₂ is the angle of the second rod (rad)
Notes	p_y2 is the vertical position p_y2 is shown in Fig:dblpend.
Source	--
RefBy	GD:velocityY2

Refname	DD:accelerationGDD
Label	Acceleration
Symbol	a(t)
Units	\(\frac{\text{m}}{\text{s}^{2}}\)
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	--
RefBy

Refname	DD:forceGDD
Label	Force
Symbol	F
Units	N
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 (\(\frac{\text{m}}{\text{s}^{2}}\))
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.

Refname	IM:calOfAngle1
Label	Calculation of angle of first rod
Input	L₁, L₂, m₁, m₂, θ₁, θ₂
Output	θ₁
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	α₁ is the angular acceleration of the first object (\(\frac{\text{rad}}{\text{s}^{2}}\)) θ₁ is the angle of the first rod (rad) θ₂ is the angle of the second rod (rad) w₁ is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) w₂ is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) g is the magnitude of gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) m₁ is the mass of the first object (kg) m₂ is the mass of the second object (kg) L₂ is the length of the second rod (m) L₁ is the length of the first rod (m)
Notes	θ₁ is calculated by solving the ODE here together with the initial conditions and IM:calOfAngle2.
Source	--
RefBy	FR:Output-Values, FR:Calculate-Angle-Of-Rod, and IM:calOfAngle2

Refname	IM:calOfAngle2
Label	Calculation of angle of second rod
Input	L₁, L₂, m₁, m₂, θ₁, θ₂
Output	θ₂
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	α₂ is the angular acceleration of the second object (\(\frac{\text{rad}}{\text{s}^{2}}\)) θ₁ is the angle of the first rod (rad) θ₂ is the angle of the second rod (rad) w₁ is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) w₂ is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) L₁ is the length of the first rod (m) m₁ is the mass of the first object (kg) m₂ is the mass of the second object (kg) g is the magnitude of gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) L₂ is the length of the second rod (m)
Notes	θ₂ is calculated by solving the ODE here together with the initial conditions and IM:calOfAngle1.
Source	--
RefBy	FR:Output-Values, FR:Calculate-Angle-Of-Rod, IM:calOfAngle2, and IM:calOfAngle1

Detailed derivation of angle of the second rod:

By solving equations GD:xForce2 and GD:yForce2 for T₂ sin(θ₂) and T₂ cos(θ₂) and then substituting into equation GD:xForce1 and GD:yForce1 , We can get equations 1 and 2:

\[{m_{1}} {a_{\text{x}1}}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)-{m_{2}} {a_{\text{x}2}}\]

\[{m_{1}} {a_{\text{y}1}}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{m_{2}} {a_{\text{y}2}}-{m_{2}} g-{m_{1}} g\]

Multiply the equation 1 by cos(θ₁) and the equation 2 by sin(θ₁) and rearrange to get:

\[{\symbf{T}_{1}} \sin\left({θ_{1}}\right) \cos\left({θ_{1}}\right)=-\cos\left({θ_{1}}\right) \left({m_{1}} {a_{\text{x}1}}+{m_{2}} {a_{\text{x}2}}\right)\]

\[{\symbf{T}_{1}} \sin\left({θ_{1}}\right) \cos\left({θ_{1}}\right)=\sin\left({θ_{1}}\right) \left({m_{1}} {a_{\text{y}1}}+{m_{2}} {a_{\text{y}2}}+{m_{2}} g+{m_{1}} g\right)\]

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(θ₂) and equation GD:yForce2 by sin(θ₂) and rearrange to get:

\[{\symbf{T}_{2}} \sin\left({θ_{2}}\right) \cos\left({θ_{2}}\right)=-\cos\left({θ_{2}}\right) {m_{2}} {a_{\text{x}2}}\]

\[{\symbf{T}_{1}} \sin\left({θ_{2}}\right) \cos\left({θ_{2}}\right)=\sin\left({θ_{2}}\right) \left({m_{2}} {a_{\text{y}2}}+{m_{2}} g\right)\]

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.
L₁	L₁ > 0	1.0 m	10%
L₂	L₂ > 0	1.0 m	10%
m₁	m₁ > 0	0.5 kg	10%
m₂	m₂ > 0	0.5 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
θ₁	θ₁ > 0
θ₂	θ₂ > 0

Output Data Constraints

	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

	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:Correct
NFR:Portable

	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:Correct
NFR:Portable

Software Requirements Specification for Double Pendulum

Dong Chen

Table of Contents

Reference Material

Table of Units

Table of Symbols

Abbreviations and Acronyms

Introduction

Purpose of Document

Scope of Requirements

Characteristics of Intended Reader

Organization of Document

General System Description

System Context

User Characteristics

System Constraints

Specific System Description

Problem Description

Terminology and Definitions

Physical System Description

Goal Statements

Solution Characteristics Specification

Assumptions

Theoretical Models

General Definitions

Detailed derivation of the x-component of velocity:

Detailed derivation of the y-component of velocity:

Detailed derivation of the x-component of velocity:

Detailed derivation of the y-component of velocity:

Detailed derivation of the x-component of acceleration:

Detailed derivation of the y-component of acceleration:

Detailed derivation of the x-component of acceleration:

Detailed derivation of the y-component of acceleration:

Detailed derivation of force on the first object:

Detailed derivation of force on the first object:

Detailed derivation of force on the second object:

Detailed derivation of force on the second object:

Data Definitions

Instance Models

Detailed derivation of angle of the second rod:

Data Constraints

Properties of a Correct Solution

Requirements

Functional Requirements

Non-Functional Requirements

Traceability Matrices and Graphs

Values of Auxiliary Constants

References