Topics in Modal Analysis II, Volume 6 pp 565-579 | Cite as

# A Dynamic Model Tailored to Flexible Launch Vehicle Umbilical Analysis

## Abstract

Dynamic analysis of flexible launch vehicle umbilicals, such as cables and hoses, has presented a significant challenge to engineers and scientists in the aerospace industry. When high fidelity analysis of such components is required, it is often performed using complex finite element models, which are costly and time consuming to construct and solve using most commercially available finite element packages. The highly coupled, non-linear nature of the equations of motion demands analysis tools and techniques specifically tailored to optimizing solution time while producing accurate results. A three-dimensional flexible-body dynamic model was developed specifically for analysis of cables and hoses. The governing equations of motion were derived using both the Euler-Lagrange equation and the Raleigh dissipation function.

### Keywords

Torque### Nomenclature

- \( {{A}_0} \)
Material cross-sectional area of the umbilical \( \left( {{{m}^2}} \right) \)

- \( c \)
Translation damping coefficient of each element \( \left( {\frac{{N \cdot s}}{m}} \right) \)

- \( {{c}_r} \)
Rotational damping coefficient of each element \( \left( {\frac{{N \cdot m \cdot s}}{{rad}}} \right) \)

- \( d(s) \)
Horizontal location between supports at a given point along a catenary’s length \( (m) \)

- \( E \)
Material modulus of elasticity \( \left( {Pa} \right) \)

- \( {{{\overset{\rightharpoonup} {F}}}_{{ext}}} \)
External force vector \( (N) \)

- \( {\overset{\rightharpoonup} {g}} \)
Gravity vector \( \left( {\frac{m}{{{{s}^2}}}} \right) \)

- \( h \)
Vertical distance between a catenary’s supports \( (m) \)

- \( H \)
Horizontal component of catenary tension \( (N) \)

- \( {\overset{\rightharpoonup} {j}} \)
Unit vector normal at an attachment surface \( (m) \)

- \( k \)
Translation stiffness of each element \( \left( {\frac{N}{m}} \right) \)

- \( {{k}_r} \)
Rotational (bending) stiffness of each element \( \left( {\frac{{N \cdot m}}{{rad}}} \right) \)

- \( l \)
Horizontal distance between a catenary’s supports \( (m) \)

- \( {{L}_0} \)
Unstretched length of the umbilical \( (m) \)

- \( M \)
Mass of each element used to represent the umbilical \( \left( {kg} \right) \)

- \( R \)
Total energy dissipation \( (J) \)

- \( s \)
Distance along the length of a catenary \( (m) \)

- \( t \)
Time \( (s) \)

- \( T \)
Total kinetic energy \( (J) \)

- \( V \)
Total potential energy \( (J) \)

- \( {{V}_t} \)
Vertical component of catenary tension \( (N) \)

- \( {{W}_g} \)
Total weight of the umbilical \( (N) \)

- \( z(s) \)
Depth of a catenary profile at a given point along its length \( (m) \)

### References

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