Transfer Matrix Method for the Determination of the Natural Vibration Characteristics of Realistic Thrusting Launch Vehicle—Part I

Table 4

First three frequencies (rad/sec) of high and low aspect ratio for a multistage launch vehicle: comparison between [5] and TMM.

Mode	Timoshenko beam						Euler-Bernoulli beam
	With rotary inertia and shear deformation		With rotary inertia only		With shear deformation only		(without rotary inertia and shear deformation)
	Reference [5]	TMM	Reference [5]	TMM	Reference [5]	TMM	Reference [5]	TMM
Whole multistage launch vehicle (𝐿/𝐷≈25)
1	16.017	16.005	16.054	16.042	16.083	16.069	16.120	16.107
2	46.521	46.492	47.195	47.172	46.770	46.735	47.457	47.428
3	93.943	93.710	97.566	97.393	94.727	94.471	98.686	98.322
Upper stage of the launch vehicle (𝐿/𝐷≈5)
1	334.776	335.019	400.968	401.468	348.945	349.146	438.557	439.003
2	760.445	760.596	1104.970	1106.305	803.663	803.364	1367.317	1369.986
3	960.819	959.593	1655.463	1659.978	1049.897	1048.378	2292.381	2290.001