Simcenter Madymo - How to deal with mass moments and products of inertia in Multi-Body modelling?

When dealing with dynamics of Multi-Body systems (i.e. excluding prescribed motions), not only mass of rigid bodies but also mass moments and products of inertia become relevant when the motion also involves rotation. In Simcenter Madymo, all the inertia properties related to a rigid body are defined within the BODY.RIGID element by the following attributes:

 

Fig. 1 - Description of the element BODY.RIGID (keyword help)

 

• MASS: mass of the rigid body, expressed in [kg] - by default, value 1.0 is used
• CENTER_OF_GRAVITY: XYZ coordinates of the center of gravity in the body's local coordinate system expressed in [m] - by default, value 0.0 0.0 0.0 is used
• INERTIA: moments and products of inertia of the rigid body IXX IYY IZZ IXY IYZ IZX expressed in [Kg.m²] - by default, value 1.0 1.0 1.0 0.0 0.0 0.0 is used
• ORIENT_INERTIA*: orientation of the inertia's coordinate system - default colinear to the body's local coordinate system, with the origin of the inertia's coordinate system at the center of gravity

 

*this option might be useful when the calculated moments and products of inertia of the rigid body are not expressed in the same coordinate system of the rigid body. To check the correct position of the center of gravity and the orientation of the inertia's coordinate system w.r.t to the body's local coordinate system, both can be visualized in the XMADgic Viewer using the Display Settings option as shown below; the appearance of the icons can be customized (color and size).

 

Fig. 2 - Representation of body's coordinate systems and center of gravity

 

Hints

1) Unless dealing with pure translational motion, always define moments and products of inertia. The default values being relatively high, especially for light mass bodies, the resulting kinematic might be incorrect. As a good practice, it is recommended to use for intermediate rigid bodies a low mass (for example 0.001Kg) associated to a low inertia such as 1.0E-06Kg.m².

2) For more complex shapes, the user can look on the internet, for example (https://en.wikipedia.org/wiki/List_of_moments_of_inertia)

3) Remember that inertia moments are additive when expressed at the same point. This means that a complex geometry can be divided into several basic geometries; once the inertia moments of each subpart of the assembly are calculated at a point P, they can be added to get the total inertia of the assembly. The same principle can also be applied to determine the inertia moments of hollow shapes by subtracting the inertia of the removed volume. 

 

Based on the available data of the object to be modelled, the user has the following options.

 

Option 1 - CAD geometry is available and inertia properties are known

The user can define the mass, location of the center of gravity and the moments of inertia of the object directly in the BODY.RIGID. If the object is likely to interact with the environment, the user can additionally:

• either attach a SURFACE.* element to the rigid body or
• attach a facet (fully supported nodes of the FE mesh on the rigid body) representation of the envelop of the object with the option UPDATE_BODY_INERTIA=OFF and using MATERIAL.NULL in combination with DENSITY_NULL=0 for the element material properties.

Then the user will have to define a compliance characteristic (CHARACTERISTIC.CONTACT). This is the methodology applied for instance to the sub-components of the Madymo ATD's (MB or facet model) with an example below, showing the head model for both the MB and the facet dummies.

Fig. 3 - ATD's head models

 

Option 2 - CAD geometry available but inertia properties unknown

The user can mesh the geometry, define the correct material attributes (density, thickness) and finally support the entire mesh on the rigid body; however, the rigid body's mass and inertia properties must be defined very low since the mesh's mass and inertial properties will be transferred to the rigid body using the attribute UPDATE_BODY_INERTIA=ON. This work with structures modelled with 2D or 3D elements. Below an example with a wooden body block with the origin of the rigid body chosen arbitrarily.

 

Fig. 4 - Wooden body block model

 

After having performed a null run, the mass and inertia properties of the FE.MODEL can be checked in the reprint (.rep) file (values expressed in the global coordinate system and not in the local coordinate system of the FE.MODEL).

 

 

Option 3 - No CAD and inertia properties available

This use case can be encountered for example in very early project phases when no data is available yet but a first estimation of the kinematic is needed. In that case, inertia properties can be approximated by considering the complex structure or object as a basic geometry, for instance considering a complete car wheel (tire + rim) as a simple cylinder with uniform weight distribution. In that case, the moments of inertia can be calculated using the attached Excel spreadsheet and the values directly input into the Madymo model.

Below an example of calculation is shown for a complete wheel of 20Kg, having a diameter of 600mm and a width of 200mm. Enter as input these parameters in the Excel sheet related to the tab Thick cylinder.

Note: enter r1=0 for solid cylinders

 

Fig. 5 - Extract from the calculator for cylinders

 

The calculated moments of inertia are: "5.167E-01    5.167E-01    9.000E-01    0.000E+00    0.000E+00    0.000E+00". Copy the complete string and paste it in INERTIA attribute of the rigid body.

 

Fig. 6 - Moments of inertia copied to the INERTIA attribute of the rigid body

 

Note: the calculation can be also automated by using DEFINE variables and expressions in the value of the INERTIA attribute (useful when running DoE).

 

General remark

Always check the orientation of the body's local coordinate system in Madymo and the orientation of the inertia's coordinate system as depicted in the Excel sheet (in which the calculated moments of inertia are expressed). In case that the coordinate system in which the inertia properties were calculated do not coincide with the body's local coordinate system, the inertia's coordinate system must be oriented accordingly. Alternatively, the scalars can be manually swapped to align the inertia moments and products to the actual body's local coordinate system.

The example below shows a car wheel simplified model with the wheel's local coordinate system defined with the X-axis as the longitudinal axis of the wheel. However in the calculator, the longitudinal axis is defined as Z and not X, therefore the inertia's coordinate system must be aligned as depicted below (rotated by 90° around Y-axis of the body's coordinate system so that the Z-axis of the inertia's coordinate system corresponds to the longitudinal axis of the wheel).

 

Fig. 7 - Alignment of inertia's coordinate system to calculated inertia properties

 

Please contact your local Simcenter Madymo support organization for further questions/details on this topic.