Transfer matrix method for determination of the natural vibration characteristics of elastically coupled launch vehicle boosters

Abstract

The analysis of natural vibration characteristics has become one of important steps of the manufacture and dynamic design in the aerospace industry. This paper presents a new scenario called virtual cutting in the context of the transfer matrix method of linear multibody systems closed-loop topology for computing the free vibration characteristics of elastically coupled flexible launch vehicle boosters. In this approach, the coupled system is idealized as a triple-beam system-like structure coupled by linear translational springs, where a non-uniform free-free Euler–Bernoulli beam is used. A large thrust-to-weight ratio leads to large axial accelerations that result in an axial inertia load distribution from nose to tail. Consequently, it causes the development of significant compressive forces along the length of the launch vehicle. Therefore, it is important to take into account this effect in the transverse vibration model. This scenario does not need the global dynamics equations of a system, and it has high computational efficiency and low memory requirements. The validity of the presented scenario is achieved through comparison to other approaches published in the literature.

Appendix

Transfer matrix and the geometric matrix of the dummy body (rigid body) with two input ends (one is beam and one is spring) and single output end (beam) are

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$$\begin{aligned}&\underbrace{{{\mathbf {H}}}_{j,I_1}}_{1\times 4}\underbrace{{{\mathbf {Z}}}_{j,I_1}}_{4\times 1}=\underbrace{{{\mathbf {H}}}_{j,I_2}}_{1\times 2}\underbrace{{{\mathbf {Z}}}_{j,I_2}}_{2\times 1}\;\hbox {where}\,{{\mathbf {H}}}_{j,I_1} =\left[ {{\begin{array}{cccc} 1&{} 0&{} 0&{} 0 \\ \end{array}}} \right] \nonumber \\&\quad \hbox {and}\,{{\mathbf {H}}}_{j,I_2} =\left[ {{\begin{array}{cc} 1&{} 0 \\ \end{array}}} \right] . \end{aligned}$$

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Transfer matrix and the geometric matrix of the dummy body (rigid body) with two input ends (one is beam and one is also beam) and single output end (spring) are

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Transfer matrix and the geometric matrix of the dummy body (rigid body) with three input ends (one is a beam, one is a spring and one is also a spring) and the single output end (beam) are

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Supplementary equation of the dummy body (rigid body) with two input ends (one is a beam and one is also a beam) and the single output end (spring) is

$$\begin{aligned}&\left. {{{\mathbf {E}}}_{j,I_1}} \right| _{1\times 4} \left. {{{\mathbf {Z}}}_{j,I_1}} \right| _{4\times 1} =\left. {{{\mathbf {E}}}_{j,I_2}} \right| _{1\times 4} \left. {{{\mathbf {Z}}}_{j,I_2}} \right| _{4\times 1}, \nonumber \\&\hbox {where}\,{{\mathbf {E}}}_{j,I_1} =\left[ {{\begin{array}{cccc} 0&{} 0&{} 1&{} 0 \\ \end{array}}} \right] \;,\;{{\mathbf {E}}}_{j,I_2} =\left[ {{\begin{array}{cccc} 0&{} 0&{} {-1}&{} 0 \\ \end{array}}} \right] . \end{aligned}$$

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Transformation matrix from $\bar{{x}}\bar{{y}}\rightarrow xy$ system coordinates for linear translational spring, beam, and dummy body components

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