Example 2 Fabric dynamics worn by a rigid link pendulum
See the code for this section here.
Dynamics of a Fabric Sleeve
Pendulum with Fabric Sleeve Setup
A fabric sleeve, made of Jersey fabric, was cut and clamped on a rigid-link pendulum, cut to shape out of ABS clear plastic (Fig. below).
This shape is modeled on a standard sized human arm, and used to simulate the effect of clothing movement given wearer motion. The pendulum was fixed with a 1 DOF joint at the top and passively swings. The model was intended to capture the fabric dynamics due to the pendulum free motion. Three magnetic trackers were used to measure the link COM motion, and deformation of two points on the fabric ("s_1,s_2").
The results from such research can be useful for research on wearable sensors. Capturing the dynamics of soft fabrics can provide many benefits to textile-embedded human motion analysis systems, such as those used for computer animation or rehabilitation feedback . The fabric parameters are not known and identified to present the best correlation with the experiments. Table below presents the fabric and setup dimensions and the simulation parameters.
Modeling Assumptions & Program Input
The fabric can be modeled as a membrane which is a 2D tension-only continuum geometry that does not withstand bending or compression. This can be done by assuming the fabric as a net of equally distributed masses with connecting linear springs. Setting "dir=1" in the package express that the springs are tension-only elements. We have used a similar method to model a spider web with "TMTDyn" recently. A drawback of such assumptions is that the model does not capture the crumbling of the fabric between the mesh nodes. Also, the membrane assumption may not be accurate for thick fabrics, such as the Jersey fabric used in these experiments. This is more important when the fabric takes shapes such as hollow columns, e.g. a fabric sleeve, that is more resistive against buckling and bending.
The fabric behaves like a shell, a 2D continuum geometry that withstands bending and compression too, in these scenarios. We used EB beam elements to resolve this issue without the need to increase the number of nodes or introducing diagonal connections, e.g. forming a tetrahedron mesh. This is a simplifying assumption that may be accurate enough to capture the underlying physics of a system with a thin membrane or shell geometries.
To model the system, we focus on the fabric model and import the link motion in the form of a constraint that follows an already recorded path ("chi_c=rho_com") based on the experimental recording of "rho_com". We use a constrained EOM to model the proposed constrained system. One geometric constraint is enough to fully define the link 1 DOF motion. Due to the constrained motion of the link, the values for "m_1" & "I_{m_1}" are not important for our analysis. Index 1 is used to define the link parameters and index 2 for the fabric ones. The fabric deforms when clamped on the link. The overall geometry of the clamped fabric is modeled with FreeCAD software as a wireframe sketch with a "3 x 5" grid of "n_d=15" nodes and 22 edges as in previous figures. The CAD model is stored in iges format to be imported into the model later. Here, nodes 12 & 13 are equivalent to "s_1" & "s_2". The fabric is clamped at nodes 14 & 16 to the link at "[+-l_{c_x},0,l_{c_z}]". Two sets of six constraints are defined to fully fix each of these two nodes to the link.
The fabric mesh is modeled with lumped masses at the CAD-file wireframe nodes that are interconnected with EB beams. The system states are described with absolute states of the masses. This is similar to the EBA model for continuum rods that was discussed in the previous section. Nodes are rigid lumped masses with an equally distributed mass of "m_2/n_d". The relation for thin plates is used to derive their second moment of inertia as
Links are considered as EB ribbons with linear elasticity "K_epsilon=diag(a_c[G,G,E])" and "K_alpha=diag([E,E,G]).diag(J)" as in the case of continuum rods in previous section. Here, "I = [ l_{m2_y}^3 l_{m2_b}, l_{m2_y} l_{m2_b}^3, l_{m2_y}^3 l_{m2_b} + l_{m2_y} l_{m2_b}^3 ]" is a "1 x 3" vector consisting of the cross-section second moments of areas, where "l_{m2_b}=(l_{m2_x}+l_{m2_z})/2" is the mean width of the ribbons in the "x,z"-axis directions.
The only parameter that needs updating during simulation is "u_c", the desired acceleration of the pendulum COM position, either in the "x-" or "z-"axis direction. To map the nodes motion to the beams deformation map, one way is to define the "xaxis" vector that defines beam specific frames. This results in better accuracy but cumbersome calculations to derive and simulate the system EOM. An alternative assumption is to have the beams in the local body frames, but assuming that they are initially deformed to reach the second connecting body. This is possible by setting "init = nan" which adopts the beam initial geometry ("epsilon_0,alpha_0") to the system initial condition. We use the second method where 2D continuum geometries are modeled. This results in an almost 10 times decrease in the size of the file storing the system EOM. The inputs for the "TMTDyn" package to model this setup is as in Tables below.
