Supplementary Document: Analytical Study of Coupling Effects for Vibrations of Cable-Harnessed Beam Structures

This paper presents a distributed parameter model to study the effects of the harnessing cables on the dynamics of a host structure motivated by space structures applications. The structure is modeled using both Euler–Bernoulli and Timoshenko beam theories (TBT). The presented model studies the effects of coupling between various coordinates of vibrations due to the addition of the cable. The effects of the cable's offset position, pretension, and radius are studied on the natural frequencies of the system. Strain and kinetic energy expressions using linear displacement field assumptions and Green–Lagrange strain tensor are developed. The governing coupled partial differential equations for the cable-harnessed beam that includes the effects of the cable pretension are found using Hamilton's principle. The natural frequencies from the coupled Euler, Bernoulli, Timoshenko and decoupled analytical models are found and compared to the results of the finite element analysis (FEA).


S1. Supplementary to Mathematical Modeling
In the section 2, mathematical modeling of the paper, Equations. (13a)-(13f) and (17a) -(17d) are coupled through stiffness terms. All the coordinates of motion are coupled because of the pre-tension in the cable, Young's modulus and radius of the cable. In mathematical terms, the first derivative of displacement represents the slope, second derivative represents moment, third derivative represents shear and the fourth derivative represents the intensity of load. Mathematically, Equations. (17b) and (17c) corresponding to the in-plane and out-of-plane bending coordinates. The axial and torsion coordinates are coupled to these modes because of equivalent shear terms (third derivative of displacement and second derivative of angle). The torsion mode Equation. (17d) is coupled to the in-plane and out-of-bending modes because of equivalent moment terms. The axial mode Equation. (17a) is coupled to the bending coordinates because of equivalent shear terms. Equations (17b) and (17c) show that the coupling term related to the in-plane and out of plane bending is fourth derivative which physically corresponds to load. In Timoshenko model, Equations (13a)-(13f), the coupling coefficients in addition to depending on the cable parameters like position coordinates along y and z axis, cable radius and cable pre tension, also depends on the geometry of the host structure.
In a Timoshenko beam, apart from the cable coupling, the rotation of cross section are geometrically coupled to the bending coordinates. In Equation (13a), the axial mode is coupled to the rotations of crosssections through the cable parameters. In Equation (13b), the in-plane bending mode is coupled to the torsion mode through the cable parameters and to the rotation of cross-section about z axis because of geometry of the beam ( 11 ). Similarly, in Equation (13c) the out of plane bending mode is coupled to the 2 torsion mode through cable parameters and to the rotation of cross-section through the geometric term. In Equation (13d), the torsion mode is coupled to the bending terms through the cable parameters. Similarly, in Equations (13e) and (13f), the rotations of cross-section about z and y-axis are coupled to other coordinates through the cable parameters and beam geometry terms. In Timoshenko beam, we can also observe that unlike Euler-Bernoulli, we do not see presence of in-plane bending terms in the out of plane bending mode equation (Equation (13c)) and vice-versa (Equation (13b)). The two bending terms here are coupled through the rotations of cross-section related terms (Equations (13e) and (13f)). For the mode shape analysis, the mass normalized mode shapes obtained from the coupled Euler

S2. Supplementary to Results and Discussions section.
Bernoulli model are presented. The results in Figure. (s1) for fixed-fixed boundary condition indicate that for the 4 th mode, the out-of-plane bending is the dominant mode. The 3 rd mode is predominantly an in-plane bending mode, and the 5 th mode is the torsional mode. The first predominantly axial mode is also shown in this figure, which corresponds to the 22 nd mode. The mode shape results in Figure. (s2) pertain to the cantilever boundary conditions. For this boundary condition, it is shown that the out-of-plane bending is dominant in the third and the fourth modes; the torsional mode is dominant at the fifth frequency, and the sixteenth mode shown corresponds to the first axial mode. For the simply supported boundary condition, Figure.
The mode shape constant can be found out by using the following mass normalization criterion.
The coupled mode shapes corresponding to the lower and higher natural frequency roots of Equation. (32) are plotted in Figures. s4 (a) and s4 (b) respectively. In Figure s4   bending and torsional modes with respect to the cable pre-tension for the system parameters shown in Table   (1) of the paper. From this figure, it can be understood that the pre-tension has negligible effect on the system's natural frequencies. This is because of the relatively large bending stiffness that makes it less susceptible to the effects of tension. To further, study the impact of tension on the natural frequencies, an I-beam cross-section shown in Figure. (s6) (Front View) with the numerical parameters presented in Table (s1) is also considered. The position coordinates of the center of the cable in this case are ( , ) = (0.0048,0.0052) . This geometry was chosen due to its smaller torsional stiffness. As shown in Figure. (s7a), the fundamental mode for the fixed-fixed boundary condition corresponds to the torsional dominant mode. In Figures. (s7b) and (s7c), for cantilever and simply supported boundary conditions, the fundamental mode corresponds to the in-plane bending dominant mode.

Fig. s6: I-beam cross section and dimensions
As expected for the I cross-section, the in-plane bending has much smaller critical loading compared to the out-of plane bending due to the smaller moment of inertia in that direction. Therefore, the in-plane bending is shown to be more prone to buckling in Figures. (s7b) and (s7c). Also, the critical loading for the simply supported is shown to be larger than the cantilever beam as expected. For fixed-fixed boundary condition, because the torsion mode is the fundamental one, the I section beam experiences torsional buckling. For cantilver and simply supported boundary conditions, the system experiences buckling in the in-plane direction.