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

Table 1

Frequency parameter 𝑝² for a free-free Euler-Bernoulli square beam under axial load for 𝐸𝐼_𝑧=200×10⁹ N/m²; 𝐺=1×10⁹⁰ N/m²; 𝜌=7800 kg/m³; 𝑏=ℎ=0.01 m; 𝐿_𝑚=1 m.

Axial load 𝛾	Mode	𝑝^2 [24]	𝑝^2 [TMM]
−0.999	1	0.84680644	0.84680659
	2	7.22386030	7.223860352
	3	10.6308452	10.630845189
0.0	1	0.00000000	0.000000000
	2	4.73004074	4.7300407450
	3	7.85320462	7.853204624
	4	10.9956078	10.995607838
1.0	1	3.26787488	3.267874878
	2	5.59376731	5.593767306
	3	8.35398454	8.353984538
	4	11.3264204	11.326420425