Simple Variable Mass 3DOF (Body Axes)

Implement three-degrees-of-freedom equations of motion of simple variable mass with respect to body axes

• Library:

• Aerospace Blockset / Equations of Motion / 3DOF

• 

Description

The Simple Variable Mass 3DOF (Body Axes) block implements three-degrees-of-freedom equations of motion of simple variable mass with respect to body axes. It considers the rotation in the vertical plane of a body-fixed coordinate frame about a flat Earth reference frame. For more information about the rotation and equations of motion, see Algorithms.

Ports

Input

Fx — Applied force along x-axis

scalar

Applied force along the body x-axis, specified as a scalar, in the units selected in Units.

Data Types: double

Fz — Applied force along z-axis

scalar

Applied force along the body z-axis, specified as a scalar.

Data Types: double

M — Applied pitching moment

scalar

Applied pitching moment, specified as a scalar.

Data Types: double

dm/dt — Rate of change of mass

scalar

Rate of change of mass (positive if accreted, negative if ablated), specified as a scalar.

Data Types: double

g — Gravity

scalar

Gravity, specified as a scalar.

Dependencies

To enable this port, set Gravity source to External.

Data Types: double

Vre — Relative velocity

two-element vector

Relative velocity at which mass is accreted to or ablated from the body in body-fixed axes, specified as a two-element vector.

Dependencies

To enable this port, select Include mass flow relative velocity.

Data Types: double

Output

θ — Pitch altitude

scalar

Pitch attitude, within ±pi, returned as a scalar, in radians.

Data Types: double

q — Pitch angular rate

scalar

Pitch angular rate, returned as a scalar, in radians per second.

Data Types: double

dq/dt — Pitch angular acceleration

scalar

Pitch angular acceleration, returned as a scalar, in radians per second squared.

Data Types: double

XeZe — Location of body

two-element vector

Location of the body in the flat Earth reference frame, (Xe, Ze), returned as a two-element vector.

Data Types: double

U w — Velocity of body

two-element vector

Velocity of the body resolved into the body-fixed coordinate frame, (u, w), returned as a two-element vector.

Data Types: double

AxbAzb — Acceleration of body

two-element vector

Acceleration of the body with respect to the body-fixed coordinate frame, (Ax, Az), returned as a two-element vector.

Data Types: double

Fuel — Fuel tank status

scalar

Fuel tank status, returned as:

• 1 — Tank is full.

• 0 — Tank is neither full nor empty.

• -1 — Tank is empty.

Dependencies

To enable this port, set Mass type to Simple Variable.

Data Types: double

AxeAze — Acceleration of body

two-element vector

Accelerations of the body with respect to the inertial (flat Earth) coordinate frame, returned as a two-element vector. You typically connect this signal to the accelerometer.

Dependencies

To enable this port, select the Include inertial acceleration check box.

Data Types: double

Parameters

Main

Units — Input and output units

Metric (MKS) (default) | English (Velocity in ft/s) | English (Velocity in kts)

Input and output units, specified as Metric (MKS), English (Velocity in ft/s), or English (Velocity in kts).

Units	Forces	Moment	Acceleration	Velocity	Position	Mass	Inertia
Metric (MKS)	Newton	Newton-meter	Meters per second squared	Meters per second	Meters	Kilogram	Kilogram meter squared
English (Velocity in ft/s)	Pound	Foot-pound	Feet per second squared	Feet per second	Feet	Slug	Slug foot squared
English (Velocity in kts)	Pound	Foot-pound	Feet per second squared	Knots	Feet	Slug	Slug foot squared

Programmatic Use

Block Parameter: units

Type: character vector

Values: Metric (MKS) | English (Velocity in ft/s) | English (Velocity in kts)

Default: Metric (MKS)

Axes — Body or wind axes

Body (default) | Wind

Body or wind axes, specified as Wind or Body

Programmatic Use

Block Parameter: axes

Type: character vector

Values: Wind | Body

Default: Body

Mass type — Mass type

Simple Variable (default) | Fixed | Custom Variable

Mass type, specified according to the following table.

Mass Type	Description	Default For
Fixed	Mass is constant throughout the simulation.	3DOF (Body Axes) 3DOF (Wind Axes)
Simple Variable	Mass and inertia vary linearly as a function of mass rate.	Simple Variable Mass 3DOF (Body Axes) Simple Variable Mass 3DOF (Wind Axes)
Custom Variable	Mass and inertia variations are customizable.	Custom Variable Mass 3DOF (Body Axes) Custom Variable Mass 3DOF (Wind Axes)

The Simple Variable selection conforms to the equations of motion described in Algorithms.

Programmatic Use

Block Parameter: mtype

Type: character vector

Values: Fixed | Simple Variable | Custom Variable

Default: 'Simple Variable'

Initial velocity — Initial velocity of body

100 (default) | scalar

Initial velocity of the body, (V₀), specified as a scalar.

Programmatic Use

Block Parameter: v_ini

Type: character vector

Values: '100' | scalar

Default: '100'

Initial body attitude — Initial pitch altitude

0 (default) | scalar

Initial pitch attitude of the body, (θ₀), specified as a scalar.

Programmatic Use

Block Parameter: theta_ini

Type: character vector

Values: '0' | scalar

Default: '0'

Initial body rotation rate — Initial pitch rotation rate

0 (default) | scalar

Initial pitch rotation rate, (q₀), specified as a scalar.

Programmatic Use

Block Parameter: q_ini

Type: character vector

Values: '0' | scalar

Default: '0'

Initial incidence — Initial angle

0 (default) | scalar

Initial angle between the velocity vector and the body, (α₀), specified as a scalar.

Programmatic Use

Block Parameter: alpha_ini

Type: character vector

Values: '0' | scalar

Default: '0'

Initial position (x,z) — Initial location

[0 0] (default) | two-element vector

Initial location of the body in the flat Earth reference frame, specified as a two-element vector.

Programmatic Use

Block Parameter: pos_ini

Type: character vector

Values: '[0 0]' | two-element vector

Default: '[0 0]'

Initial mass — Initial mass

1.0 (default) | scalar

Initial mass of the rigid body, specified as a scalar.

Programmatic Use

Block Parameter: mass

Type: character vector

Values: '1.0' | scalar

Default: '1.0'

Empty mass — Mass of body when fuel tank is empty

0.5 (default) | scalar

Mass of body when fuel tank is empty, specified as a scalar.

Programmatic Use

Block Parameter: mass_e

Type: character vector

Values: '0.5' | scalar

Default: '0.5'

Full mass — Mass of body when fuel tank is full

3.0 (default) | scalar

Mass of body when fuel tank is full, specified as a scalar.

Programmatic Use

Block Parameter: mass_f

Type: character vector

Values: '3.0' | scalar

Default: '3.0'

Empty inertia — Body inertia when fuel tank is full

0.5 (default) | scalar

Body inertia when the fuel tank is full, specified as a double scalar.

Programmatic Use

Block Parameter: Iyy_e

Type: character vector

Values: '0.5' | scalar

Default: '0.5'

Full inertia — Full inertia

3.0 (default) | scalar

Full inertia of the body, specified as a scalar.

Programmatic Use

Block Parameter: Iyy_f

Type: character vector

Values: '3.0' | scalar

Default: '3.0'

Gravity Source — Gravity source

Internal (default) | External

Gravity source, specified as:

External	Variable gravity input to block
Internal	Constant gravity specified in mask

Programmatic Use

Block Parameter: g_in

Type: character vector

Values: 'Internal' | 'External'

Default: 'Internal'

Acceleration due to gravity — Gravity source

9.81 (default) | scalar

Acceleration due to gravity, specified as a double scalar and used if internal gravity source is selected. If gravity is to be neglected in the simulation, this value can be set to 0.

Dependencies

• To enable this parameter, set Gravity Source to Internal.

Programmatic Use

Block Parameter: g

Type: character vector

Values: '9.81' | scalar

Default: '9.81'

Include mass flow relative velocity — Mass flow relative velocity port

off (default) | on

Select this check box to add a mass flow relative velocity port. This is the relative velocity at which the mass is accreted or ablated.

Programmatic Use

Block Parameter: vre_flag

Type: character vector

Values: off | on

Default: 'off'

Limit mass flow when mass is empty or full — Limit mass flow

on (default) | off

Select this check box to limit the input mass flow rate when one of these is true:

• Fuel tank is full and input mass flow rate is positive.

• Fuel tank is empty and input mass flow rate is negative.

When the input mass flow rate might cause the mass to exceed its limits, the block uses a zero mass flow rate value in the equations of motion. For more information, see Algorithms.

If you do not want to limit the input mass flow rate, clear this check box.

Dependencies

To enable this parameter, set Mass type to Simple Variable.

Programmatic Use

Block Parameter: mdot_flag

Type: character vector

Values: 'off' | 'on'

Default: 'on'

Data Types: double

Include inertial acceleration — Include inertial acceleration port

off (default) | on

Select this check box to add an inertial acceleration in flat Earth frame output port. You typically connect this signal to the accelerometer.

Dependencies

To enable the A_xeA_ze port, select this parameter.

Programmatic Use

Block Parameter: abi_flag

Type: character vector

Values: 'off' | 'on'

Default: 'off'

State Attributes

Assign a unique name to each state. You can use state names instead of block paths during linearization.

• The number of names must match the number of states, as shown for each item, or be empty. Set all or none of the block states.

• To assign names to single-variable states, enter unique names between quotes, for example, 'q' or "q".

• To assign names to two-variable states, enter a comma-separated list surrounded by braces, for example, {'Xe','Ze'}.

• If a state parameter is empty (' '), no name is assigned.

• To assign state names with a variable in the MATLAB^® workspace, enter the variable without quotes. A variable can be a character vector, cell array of character vectors, or string.

Velocity: e.g., {'u, 'w'} — Velocity state name

'' (default) | comma-separated list surrounded by braces

Velocity state names, specified as a comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: vel_statename

Type: character vector

Values: '' | comma-separated list surrounded by braces

Default: ''

Pitch attitude: e.g., 'theta' — Pitch attitude state name

'' (default)

Pitch attitude state name, specified as a character vector or string.

Programmatic Use

Block Parameter: theta_statename

Type: character vector | string

Values: ''

Default: ''

Position: e.g., {'Xe', 'Ze'} — Position state name

'' (default) | comma-separated list surrounded by braces

Position state names, specified as a comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: pos_statename

Type: character vector

Values: '' | comma-separated list surrounded by braces

Default: ''

Pitch angular rate e.g., 'q' — Pitch angular rate state name

'' (default)

Pitch angular rate state name, specified as a character vector or string.

Programmatic Use

Block Parameter: q_statename

Type: character vector | string

Values: '' | scalar

Default: ''

Mass: e.g., 'mass' — Mass state name

'' (default) | scalar

Mass state name, specified as a character vector or string.

Programmatic Use

Block Parameter: mass_statename

Type: character vector | string

Values: '' | scalar

Default: ''

Algorithms

It considers the rotation in the vertical plane of a body-fixed coordinate frame about a flat Earth reference frame.

The equations of motion are

$$\begin{array}{l}{A}_{xb}=\dot{u}=A_{xe}-qw\\ A_{zb}=\dot{w}=A_{ze}+qu\\ A_{xe}=\frac{\left(F_x-\dot{m}{u}_{re}\right)}{m}-g\sin \theta \\ A_{ze}=\frac{\left(F_z-\dot{m}{w}_{re}\right)}{m}+g\cos \theta \\ {\dot{X}}_e=u\cos \theta +w\sin \theta \\ {\dot{Z}}_e=-u\sin \theta +w\cos \theta \\ \dot{q}=\frac{M_y-{\dot{I}}_{yy}q}{I_{yy}}\\ \dot{\theta }=q\\ {\dot{I}}_{yy}=\frac{I_{yy\_full}-I_{yy\_empty}}{m_{full}-m_{empty}}\dot{m}\\ I_{yy}=I_{yy\_empty}+\left(I_{yy\_full-}{I}_{yy\_empty}\right)\frac{m-m_{empty}}{m_{full}-m_{empty}}\end{array}$$

where the applied forces are assumed to act at the center of gravity of the body. Input variables are F_x, F_z, M_y, $$\dot{m}$$. u_re, w_re, and g are optional input variables. Mass m is limited to between m_empty and m_full. Whenever mass is saturated at empty or full, for consistency, limit $$\dot{m}$$ within the equations of motion.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2006a

Simple Variable Mass 3DOF (Body Axes) Block Changes

The 3DOF equations of motion have been updated. Existing models created prior to R2021b that contain 3DOF equations of motion blocks continue to run. If you replace a pre-R2021b version of a 3DOF equation of motion block with an R2021b or later version, your updated model might have a higher tendency for algebraic loops. For an example of how to remove algebraic loops using unit delays, see Remove Algebraic Loops. For further information about algebraic loops, see Identify Algebraic Loops in Your Model.