Where is the rigid-body mass placed and how are the nodal forces calculated in the "Reduced-order Flexible Solid"?

I'd like to model a flexible beam under distributed loads using the reduced-order flexible solid in Simscape. The beam is discretized into N elements and the global mass and stiffness matrices are calculated from external codes. No modal coordinates are involved and I defined N+1 interface frames to help the block locate the nodes. I have a couple of questions regarding the rigid-body mass partition and the load application:

• When determining the rigid body dynamics, Is the mass of the beam taken as a point mass or the element-based distributed masses? I read from a report (which might be an old-version of the reduced-order flexible solid implementation) that the inertia can be partitioned with several different options (section 3.3.2). It's not clear to me how the mass/inertia is partitioned in the new reduced-order flexible solid block.
• I was wondering how the block deals with an external force that is applied in between two nodes. How the nodal forces are calculated without the definition of a shape function?

I do appreciate it if anyone could help provide some relevant reports or papers of the theory and modelling details behind the block.

Answer 1

[Kartik Saxena]

Hi,

The Simscape Reduced-Order Flexible Solid block allows you to model the dynamics of a flexible body by providing the mass and stiffness matrices, which you have calculated using external codes. The block can use these matrices to compute the deformation of the solid under loads and the resulting effect on the overall system dynamics.

Regarding your questions:

1. Rigid-Body Mass Partition: The mass of the beam is not taken as a point mass; rather, the mass distribution is determined by the provided mass matrix. The Reduced-Order Flexible Solid block uses the mass matrix to calculate the rigid-body dynamics and the flexible dynamics separately. The rigid-body inertia properties are derived from the mass matrix, taking into account the distribution of mass. The block automatically partitions the mass matrix into rigid-body and flexible components. The inertia properties (center of mass, moments of inertia, etc.) are calculated based on the mass distribution described by the mass matrix.

2. External Force Application: When an external force is applied between two nodes, the block needs to distribute this force to the nodes in some manner. In a finite element model, this would typically be handled using shape functions. However, the Reduced-Order Flexible Solid block in Simscape does not explicitly use shape functions. Instead, it uses the provided mass and stiffness matrices to determine how forces are distributed and how the solid deforms. If the force is applied at a location that does not correspond to a node, the block will internally determine an equivalent nodal force distribution based on the provided system matrices.

Refer to the following MathWorks documentation for more information:

https://www.mathworks.com/help/physmod/sm/ref/reducedorderflexiblesolid.html

I hope this resolves your issue.

[Jiayao]

Hi Kartik,

Many thanks for your answers, they make perfect sense to me!

It would be interesting to know how the block extracts those inertia properties from the given reduced-order mass matrix. According to my observation, Simulink threw an error when I gave a reduced-order mass matrix calculated from, say, a beam model with center of mass offsets between neighboring elements. I suspected that the error was because the block has a specific structure of mass matrix so that it can extract inertial properties and wouldn't support some extra terms in the matrix introduced by the CoM offset transformation.

Best wishes and Happy New Year,

Jiayao