Time-dependent mass and inertia parameters

First Generation/Sensors & Actuators

This content is specific to Simscape™ Multibody™ First Generation software. First-generation features are slated to be deprecated and should be avoided.

The Variable Mass & Inertia Actuator block allows you to
vary the **mass** *m* and/or **inertia tensor** ** I** of
the Body to which it is connected. The general form of Newton's second
law for linear or angular motion is

(*mass* or *inertia*)
* *acceleration* = external *force* or *torque*

This block externally varies the leftmost parameter in this
law of motion with a Simulink^{®} signal.

The Variable Mass & Inertia Actuator does *not* apply
any thrust forces or torques associated with the Body's mass loss
or gain. Such thrust effects would occur on the left-hand side of
the force or torque law as terms proportional to the time derivatives
of the mass or inertia tensor, *dm/dt* or *d**I**/dt*, multiplied by the
related thrust velocities. You must separately apply such thrust forces
or torques to the Body with Body Actuators.

You connect the Variable Mass & Inertia Actuator block to
the original Body at a **Body coordinate system** (CS).
You can connect multiple Variable Mass & Inertia Actuators to
a single Body, each Actuator at a separate Body CS port. If Body CS
ports are lacking, open the Body dialog and create them as needed.

At each Body CS so connected, the Variable Mass & Inertia
Actuator creates an invisible body. The attachment is equivalent to
connecting another Body with a Weld, except that the other body's
mass properties vary with time. This invisible body has a time-varying
mass and/or symmetric inertia tensor supplied by the external Simulink signal.
The **center of gravity** coordinate
system (CG CS) of the invisible body is identical to the attached
Body CS. The inertia tensor of the invisible body is evaluated at
this CS, in this coordinate system's axes.

Once started, a Simscape
Multibody simulation creates a combined
or *composite* body, made of the invisible, time-varying
body created by the Actuator and the original Body. The total mass
of the composite body is the sum of the visible Body and the invisible
body's masses. The CG of this composite body is recomputed at each
time step. The inertia tensor of the composite body is formed at each
time step by combining the inertia tensors of the visible Body and
the invisible body. The combined inertia tensor is then evaluated
at the composite body's new CG.

The time-varying mass and inertia tensor of the invisible body must satisfy these requirements:

The mass and

**principal inertial moments**can be positive, negative, or zero.The only restriction is that the total mass and the principal inertial moments of the composite body be nonnegative.

The time-varying inertia tensor of the invisible body must be symmetric.

You can mix variable mass and/or variable inertia tensor actuation.

Actuation | Effect on Connected Body |
---|---|

Variable mass alone | Adds a time-varying point mass at the attached Body CS |

Variable inertia tensor alone | Adds time-varying inertia tensor at the attached Body CS without changing the composite body's total mass |

Variable mass and inertia tensor combined | Adds invisible body with time-varying mass and inertia tensor at the attached Body CS |

While the invisible, attached body and the invisible composite body have time-varying mass properties, you do not see any visible changes in the original Body that you are actuating. The mass properties in its dialog do not change.

If you are visualizing the varying-mass/inertia actuated Body as an equivalent ellipsoid, the ellipsoid is displayed using the static data in the Body dialog itself. The displayed ellipsoid ignores the effect of any Variable Mass & Inertia Actuators attached to the Body. See About Body Color and Geometry.

The dialog has one active area, **Actuation**.

You can apply a variable mass, a variable inertia tensor, or both, to a body.

If you apply both, you need to bundle the variable mass and inertia tensor into a 10-component signal, in the order shown in the dialog.

**Mass**Select the check box to apply an external time-varying mass from the input Simulink signal. In the pull-down menu to the right, select units for this time-varying mass. The default is

`kg`

(kilograms).**Inertia tensor**Select the check box to apply an external time-varying inertia tensor from the input Simulink signal. In the pull-down menu to the right, select units for this time-varying inertia tensor. The default is

`kg-m`

^{2}(kilogram-meters^{2}).

The Simulink input signal has the following components. For variable mass or inertia tensor actuation alone, omit the missing components.

Time-varying mass (scalar) | Time-varying inertia tensor (9-vector):(I_{11} ,
I_{21} , I_{31} ,
I_{12} , ... ) |

[1] Corbin, H. C., and P. Stehle, *Classical
Mechanics*, Second Edition, New York, Dover Publications,
1994 (original edition, 1960), chapters 2, 5, and 9.

[2] Goldstein, H., *Classical Mechanics*,
Second Edition, Reading, Massachusetts, Addison-Wesley, 1980, chapters
4 and 5.

[3] Piscane, V. L., and R. C. Moore, eds., *Fundamentals
of Space Systems*, Johns Hopkins University/Applied Physics
Laboratory Series, New York, Oxford University Press, 1994, chapters
3, 4, and 5.

See Varying a Body's Mass and Inertia Tensor for more on varying the mass and inertia tensor of a body.

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