Dynamic Analysis and Experimental Validation of Periodically Wrapped Cable-Harnessed Plate Structures

Abstract

Background

The impact of power and signal cable harnesses on the dynamic behavior of lightweight space structures has come into the spotlight in the recent past. The previous analytical modelling efforts in this area have primarily focused on either periodically wrapped beam structures or plate structures of parallel cable wrapping patterns.

Objective

The presented paper aims at experimental validation for an analytical model developed by the authors for cable harnessed plate structures with periodic patterns such as zigzag and diagonal. The work includes an extensive analysis of the vibration behavior of cable-harnessed plates in comparison to their host plates with no cables attached.

Methods

An energy equivalence homogenization technique is developed to model the vibrations behavior of the cable harnessed plates of periodic patterns. Experimental modal analysis is performed on cable-harnessed plates using impact hammer excitation and laser doppler vibrometry. The natural frequencies, mode shapes, and frequency response functions are obtained for comparisons with the model predictions.

Results

The impacts of the harnessing cables on plate dynamics are modeled and compared to the experimental frequency response functions for periodically wrapped cabled plates in four different test structures. The impact of the cable wrapping directions along the length and width on the modes that experience the largest stiffening effects are clearly discussed and validated with the test results for the two major wrapping patterns, diagonal and zigzag. It is shown that for a given number of fundamental elements and cable wrapping rows along the length, the parallel pattern experiences the largest stiffening along the length followed by the diagonal and zigzag patterns respectively. Also, the twist mode was stiffened the most for the zigzag pattern and the least in the parallel cabled plate. Finally, the wrapping angle for which the torsional frequency is maximum for each of the diagonal and zigzag patterns is found to be the same. A computational study is also performed to further analyze the system dynamics by varying the host plate dimensions.

Conclusions

The model and test results are shown to be in very good agreement. The analytical model is found to well predict the cables' mass and stiffening effects for all the wrapping patterns studied in this research.

Appendices

Appendix

Using the energy-equivalent homogenization technique, the total strain energy of the cable-harnessed plate system with periodic cable wrapping pattern was obtained as [34]:

$$\begin{aligned}\left({U}_{sys}\right)=&\frac{1}{2}{\int }_{0}^{a}{\int }_{0}^{b}\Big\{{H}_{1}+{H}_{2}w{,}_{xx}^{2}+{H}_{3}w{,}_{yy}^{2}+{H}_{4}w{,}_{xy}^{2}\\&+{H}_{5}w{,}_{x}^{2}+{H}_{6}w{,}_{y}^{2}-{H}_{7}w{,}_{xx}+{H}_{8}w{,}_{xy}\\&-{H}_{9}w{,}_{yy}+{H}_{10}w{,}_{xx}w{,}_{xy}+{H}_{11}w{,}_{yy}w{,}_{xy}\\&+{H}_{12}w{,}_{x}w{,}_{y}+{H}_{13}w{,}_{xx}w{,}_{yy}\Big\}dxdy\end{aligned}$$

(4)

And, the kinetic energy was obtained as [34]:

$${T}_{sys}=\frac{1}{2}{\int }_{0}^{b}{\int }_{0}^{a}\left\{{K}_{1}{\left(\dot{w}\right)}^{2}\right\}dxdy$$

(5)

where the coefficients ${H}_{i}\ (i=1\ to\ 13)$ and ${K}_{1}$ are mentioned for both the patterns as follows:

Zigzag Pattern

$${H}_{1}=\frac{{T}^{2}}{{E}_{c}{A}_{c}{L}_{2}\mathrm{cos}\theta }+\frac{1}{h{E}_{p}}\left({N}_{x}^{2}+{N}_{y}^{2}-2\nu {N}_{x}{N}_{y}+2\left(1+\nu \right){N}_{xy}^{2}\right)$$

$${H}_{2}=D+\frac{{E}_{c}{A}_{c}{z}_{c}^{2}{\mathrm{cos}}^{3}\theta }{{L}_{2}}+\frac{T{z}_{c}^{2}\mathrm{cos}\theta }{{L}_{2}}+\frac{{N}_{x}{h}^{2}}{24}$$

$${H}_{3}=D+\frac{{E}_{c}{A}_{c}{z}_{c}^{2}{\mathrm{sin}}^{4}\theta }{{L}_{2}\mathrm{cos}\theta }+\frac{T{z}_{c}^{2}{\mathrm{sin}}^{2}\theta }{{L}_{2}\mathrm{cos}\theta }+\frac{{N}_{y}{h}^{2}}{24}$$

$${H}_{4}=2D\left(1-\nu \right)+\frac{4{E}_{c}{A}_{c}{z}_{c}^{2}\mathrm{cos}\theta {\mathrm{sin}}^{2}\theta }{{L}_{2}}+\frac{T{z}_{c}^{2}}{{L}_{2}\mathrm{cos}\theta }+\frac{{N}_{x}{h}^{2}}{24}+\frac{{N}_{y}{h}^{2}}{24}$$

$${H}_{5}=\frac{T\mathrm{cos}\theta }{{L}_{2}}+{N}_{x}=0$$

$${H}_{6}=\frac{T{\mathrm{sin}}^{2}\theta }{{L}_{2}\mathrm{cos}\theta }+{N}_{y}=0$$

$${H}_{7}=0$$

$${H}_{8}=0$$

$${H}_{9}=0$$

$${H}_{10}=\frac{{N}_{xy}{h}^{2}}{6}=0$$

$${H}_{11}=\frac{{N}_{xy}{h}^{2}}{6}=0$$

$${H}_{12}=2{N}_{xy}=0$$

$${H}_{13}=2\nu D+\frac{2{E}_{c}{A}_{c}{z}_{c}^{2}\mathrm{cos}\theta {\mathrm{sin}}^{2}\theta }{{L}_{2}}$$

$${K}_{1}={\rho }_{p}h+\frac{{\rho }_{c}{A}_{c}}{{L}_{2}\mathrm{cos}\theta }+\frac{2{\rho }_{c}{A}_{c}h}{{L}_{1}{L}_{2}}$$

In these above mentioned coefficients for the zigzag pattern, the variables ${N}_{x}$, ${N}_{y}$, and ${N}_{xy}$ are written as [34]:

$${N}_{x}=-\frac{nT\mathrm{cos}\theta }{b}$$

$${N}_{y}=-\frac{2mT\mathrm{sin}\theta }{a}$$

$${N}_{xy}=0$$

where $m$ is the number of fundamental elements in each row and $n$ is the number of rows of repeating fundamental elements in the zigzag pattern. Additionally, the wrapping angle $\theta$ can be calculated for zigzag pattern as

$$\mathrm{tan}\theta =\frac{2{L}_{2}}{{L}_{1}}=\frac{2bm}{an}$$

Diagonal Pattern

$${H}_{1}=\frac{{T}^{2}}{{E}_{c}{A}_{c}\left({L}_{1}+{L}_{2}\mathrm{cos}\theta \right)}+\frac{1}{h{E}_{p}}\left({N}_{x}^{2}+{N}_{y}^{2}-2\nu {N}_{x}{N}_{y}+2\left(1+\nu \right){N}_{xy}^{2}\right)$$

$${H}_{2}=D+\frac{{E}_{c}{A}_{c}{z}_{c}^{2}{\mathrm{cos}}^{3}\theta }{{L}_{2}}+\frac{T{z}_{c}^{2}\mathrm{cos}\theta }{{L}_{2}}+\frac{{N}_{x}{h}^{2}}{24}$$

$$\begin{aligned}{H}_{3}=D+{E}_{c}{A}_{c}&\left\{\frac{1}{{L}_{2}\mathrm{cos}\theta }\left(\frac{T{z}_{c}^{2}\mathrm{sin}\theta }{{E}_{c}{A}_{c}}+{z}_{c}^{2}{\mathrm{sin}}^{4}\theta \right)\right.\\&+\left.\frac{1}{{L}_{1}}\left(\frac{T{z}_{c}^{2}}{{E}_{c}{A}_{c}}+{z}_{c}^{2}\right)\right\}+\frac{{N}_{y}{h}^{2}}{24}\end{aligned}$$

$${H}_{4}=2D\left(1-\nu \right)+\frac{4{E}_{c}{A}_{c}{z}_{c}^{2}\mathrm{cos}\theta {\mathrm{sin}}^{2}\theta }{{L}_{2}}+\frac{T{z}_{c}^{2}}{{L}_{2}\mathrm{cos}\theta }+\frac{{N}_{x}{h}^{2}}{24}+\frac{{N}_{y}{h}^{2}}{24}$$

$${H}_{5}=\frac{T\mathrm{cos}\theta }{{L}_{2}}+{N}_{x}=0$$

$${H}_{6}=\frac{T{\mathrm{sin}}^{2}\theta }{{L}_{2}\mathrm{cos}\theta }+\frac{T}{{L}_{1}}+{N}_{y}=0$$

$${H}_{7}=\frac{2T{z}_{c}\mathrm{cos}\theta }{{L}_{2}}$$

$${H}_{8}=\frac{4T{z}_{c}\mathrm{sin}\theta }{{L}_{2}}$$

$${H}_{9}=-\frac{2T{z}_{c}{\mathrm{sin}}^{2}\theta }{{L}_{2}\mathrm{cos}\theta }+\frac{2T{z}_{c}}{{L}_{1}}$$

$${H}_{10}=\frac{{N}_{xy}{h}^{2}}{6}+\frac{2T{z}_{c}^{2}\mathrm{sin}\theta }{{L}_{2}}+\frac{4{E}_{c}{A}_{c}{z}_{c}^{2}{\mathrm{cos}}^{2}\theta \mathrm{sin}\theta }{{L}_{2}}$$

$${H}_{11}=\frac{{N}_{xy}{h}^{2}}{6}+\frac{2T{z}_{c}^{2}\mathrm{sin}\theta }{{L}_{2}}+\frac{4{E}_{c}{A}_{c}{z}_{c}^{2}{\mathrm{sin}}^{3}\theta }{{L}_{2}}$$

$${H}_{12}=\frac{2T\mathrm{sin}\theta }{{L}_{2}}+2{N}_{xy}=0$$

$${H}_{13}=2\nu D+\frac{2{E}_{c}{A}_{c}{z}_{c}^{2}\mathrm{cos}\theta {\mathrm{sin}}^{2}\theta }{{L}_{2}}$$

$${K}_{1}={\rho }_{p}h+\frac{{\rho }_{c}{A}_{c}}{{L}_{2}\mathrm{cos}\theta }+\frac{{\rho }_{c}{A}_{c}}{{L}_{1}}+\frac{2{\rho }_{c}{A}_{c}h}{{L}_{1}{L}_{2}}$$

In the above mentioned coefficients for the diagonal pattern, the variables ${N}_{x}$, ${N}_{y}$, and ${N}_{xy}$ are written as [34]:

$${N}_{x}=-\frac{nT\mathrm{cos}\theta }{b}$$

$${N}_{y}=-\frac{m\left(T+T\mathrm{sin}\theta \right)}{a}$$

$${N}_{xy}=-\frac{nT\mathrm{sin}\theta }{b}$$

where $m$ is the number of fundamental elements in each row and $n$ is the number of rows of repeating fundamental elements in the diagonal pattern. Additionally, the wrapping angle $\theta$ can be calculated for diagonal pattern as

$$\mathrm{tan}\theta =\frac{{L}_{2}}{{L}_{1}}=\frac{bm}{an}$$