# Documentation

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# Variable Mass & Inertia Actuator

Time-dependent mass and inertia parameters

## Library

First Generation/Sensors & Actuators

## Description

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.

 Caution   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 dI/dt, multiplied by the related thrust velocities. You must separately apply such thrust forces or torques to the Body with Body Actuators.

### How the Actuator Varies a Body's Mass and Inertia Tensor

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.

### The Composite Body

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.

### What The Invisible Body Requires

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.

ActuationEffect on Connected Body
Variable mass aloneAdds a time-varying point mass at the attached Body CS
Variable inertia tensor aloneAdds time-varying inertia tensor at the attached Body CS without changing the composite body's total mass
Variable mass and inertia tensor combinedAdds invisible body with time-varying mass and inertia tensor at the attached Body CS

### What Does Not Vary in the Original Body

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.

## Dialog Box and Parameters

The dialog has one active area, Actuation.

## 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-meters2).

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):(I11 , I21 , I31 , I12 , ... )

## References

[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.