80107b9c-20df-430b-9a4a-172f68387ef9 20 4 5 false false false 0 3 decimal

&[DATE] &[TIME] - &[FILENAME]

References:

[1] T. A. El-Sayed, S. H. Farghaly - A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with
Experimental and FE(3D) Methods

[2] left intentionally free

[3] left intentionally free

[4] J. Y. Shen, E. G. Abu-Saba, W. M. McGinley, L. Sharpe, Jr, - A Piecewise Continuous Timoshenko Beam Model for the Dynamic Analysis of Tapered Beam-like Structures

[5] L. K. Abbas, X. Rui - Free Vibration Characteristic of Multilevel Beam Based on Transfer Matrix Method of Linear Multibody Systems- Free Vibration Characteristic of Multilevel Beam Based on Transfer Matrix Method of Linear Multibody Systems

[6] F. Brusa - Modal Analysis Of Multistepped Beam-Like Structures_Part_I

As suggested by Prof KRASKA all the regions are set to "optimisation / numerical"....strangely some were to numerical other on symbolic, even in the same region.....

t0 0 time :

Inputs & Expected results

Case S3-80/200

Figure 1 - from Ref.[1] page 12

Figure 3 - from Ref.[1] page 15

Calculations

Common inputs

E 2.068 1011^* Pa * :

ρ_steel 7850 kg m 3^/ * :

ν 0.298 :

Ltot 500 mm * :

TMMT algo_under_crit.sm include 33

Common functions

G E 21 ν + (* / : 7.9661 1010^* Pa *

k_circ_full 61 ν + (76 ν * + / * : 0.8862

D d f_Isect π 64 / D (4^d (4^- (* :

D d f_Asect π D 2 / (2^d 2 / (2^- (* :

ρ D d f_mass ρ π D 2 / (2^d 2 / (2^- (* * :

ε k_circ_full :

Beam 1

D1 20 mm * :

d1 0 mm * :

L1 225 mm * :

mass1 ρ_steel D1 d1 f_mass : 2.4662 kg m / *

Isect1 D1 d1 f_Isect : 7.854 109 -^* m 4^*

Asect1 D1 d1 f_Asect : 0.0003 m 2^*

ω α ω ε L1 Isect1 Asect1 mass1 E G Φ :

Beam 2

D2 80 mm * :

d2 0 mm * :

L2 200 mm * :

mass2 ρ_steel D2 d2 f_mass : 39.4584 kg m / *

Isect2 D2 d2 f_Isect : 2.0106 106 -^* m 4^*

Asect2 D2 d2 f_Asect : 0.005 m 2^*

ω β ω ε L2 Isect2 Asect2 mass2 E G Φ :

Beam 3

D3 D1 :

d3 d1 :

L3 Ltot L1 - L2 - :

mass3 ρ_steel D3 d3 f_mass : 2.4662 kg m / *

Isect3 D3 d3 f_Isect : 7.854 109 -^* m 4^*

Asect3 D3 d3 f_Asect : 0.0003 m 2^*

ω δ ω ε L3 Isect3 Asect3 mass3 E G Φ :

ω Ω ω δ ω β * ω α * :

ω Z ω Ω 31 el ω Ω 42 el * ω Ω 32 el ω Ω 41 el * - 11 line :

x 0 > ( x 150 ≤ ( & x kHz * Z 1 kg * GW * kHz * / abs log10 if 1 1 sys 19 ( x 19.25 ≤ ( & x kHz * Z 1 kg * GW * kHz * / abs log10 if 1 1 sys x 35 > ( x 37.5 ≤ ( & x kHz * Z 1 kg * GW * kHz * / abs log10 if 1 1 sys x 35 > ( x 37.5 ≤ ( & x kHz * Z 1 kg * GW * kHz * / abs log10 if 1 1 sys x 45 > ( x 47.5 ≤ ( & x kHz * Z 1 kg * GW * kHz * / abs log10 if 1 1 sys x 97.5 > ( x 102.5 ≤ ( & x kHz * Z 1 kg * GW * kHz * / abs log10 if 1 1 sys x 127.5 > ( x 130 ≤ ( & x kHz * Z 1 kg * GW * kHz * / abs log10 if 1 1 sys

ω Ω1 ω α ω β * ω δ * :

ω Z1 ω Ω1 31 el ω Ω1 42 el * ω Ω1 32 el ω Ω1 41 el * - 11 line :

x 0 > ( x 150 ≤ ( & x kHz * Z1 1 kg * GW * kHz * / abs log10 if 1 1 sys

t1 0 time :

t1 t0 - 1402.062 s *