13e2a46f-0b4d-46d1-a780-5b47e2c22a42 949 4 5 false false false 0 3 decimal

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

References

[1] L. Majkut - Free and Forced Vibrations of Timoshenko Beams Described by Single Difference Equation

[2] 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

[3] K. A. Laith, M.-J. Li , and X.-T. Rui - Transfer Matrix Method for the Determination of the Natural Vibration Characteristics of Realistic Thrusting Launch Vehicle—Part I

[4] J. W. Jaworski, E. H. Dowell - Free vibration of a cantilevered beam with multiple steps: Comparison of several theoretical methods with experiment

[5] left intentionally free

[6] R.D. Blevins - Formulas for Natural Frequencies and Modes Shapes – Ed.1979

time_start 0 time :

General functions

Algorithms/TB_TM algo_original.sm include 33

Algorithms/TB_TM_adim_algo.sm include 72

b h f_Isect b h 3^12 / * :

b h f_Asect b h * :

ρ b h f_mass ρ b * h * :

Min function (root search)

find of occurrence of number in Vector number Vector Match k 1 : i 1 Vector rows range Vector i el number ≡ index k el i : k k 1 + : 21 line if for k 1 ≡ index NO match : index if 31 line :

Numerical Check vs Ref.[4] - aluminium beam with rectangular section

Inputs & Expected results

Figure 1 - from Ref.[1] page

Figure 2 - from Ref.[4] page

Dimensional Inputs

E_test1 60.6 GPa * :

ρ_test1 2664 kg m 3^/ * :

ν 0.33 :

G_test1 E_test1 21 ν + (* / : 2.2782 1010^* Pa *

κ_test1 101 ν + (1211 ν * + / * : 0.8509

b_test1 3.175 mm * :

h_test1 25.4 mm * :

b_test2 h_test1 :

h_test2 b_test1 :

l_test1 463.55 mm * :

mass_test1 ρ_test1 b_test1 h_test1 f_mass : 0.2148 kg m / *

I_test1 b_test1 h_test1 f_Isect : 4.3357 109 -^* m 4^*

I_test2 b_test2 h_test2 f_Isect : 6.7746 1011 -^* m 4^*

A_test1 b_test1 h_test1 f_Asect : 8.0645 105 -^* m 2^*

Characteristic of beams composing the notched beam (from clamping to free end) in-plane bending

l1a 25.4 mm * :

l1b 25.4 mm * :

l1c 50.8 mm * :

l1z 31.75 mm * :

b1a b_test1 :

b1b b_test1 :

b1c b_test1 :

b1z b_test1 :

h1a h_test1 :

h1b 12.7 mm * :

h1c h_test1 :

h1z h_test1 :

I1a I_test1 :

I1b b1b h1b f_Isect :

I1c I_test1 :

I1z I_test1 :

mass1a mass_test1 :

mass1b ρ_test1 b1b h1b f_mass :

mass1c mass_test1 :

mass1z mass_test1 :

A1a A_test1 :

A1b b1b h1b f_Asect :

A1c A_test1 :

A1z A_test1 :

Characteristic of beams composing the notched beam (from clamping to free end) out-of-plane bending

l2a 25.4 mm * :

l2b 25.4 mm * :

l2c 50.8 mm * :

l2z 31.75 mm * :

b2a h_test1 :

b2b 12.7 mm * :

b2c h_test1 :

b2z h_test1 :

h2a b_test1 :

h2b b_test1 :

h2c b_test1 :

h2z b_test1 :

I2a b2a h2a f_Isect :

I2b b2b h2b f_Isect :

I2c b2c h2c f_Isect :

I2z b2z h2z f_Isect :

mass2a ρ_test1 b2a h2a f_mass :

mass2b ρ_test1 b2b h2b f_mass :

mass2c ρ_test1 b2c h2c f_mass :

mass2z ρ_test1 b2z h2z f_mass :

A2a b2a h2a f_Asect :

A2b b2b h2b f_Asect :

A2c b2c h2c f_Asect :

A2z b2z h2z f_Asect :

Non-dimensional Inputs

mult 1 :

EIz01 mult E_test1 * I1a * :

EIz02 mult E_test1 * I2a * :

EIz01 262.7461 m * J *

EIz02 4.1054 m * J *

κGA01 EIz01 l_test1 2^/ :

κGA02 EIz02 l_test1 2^/ :

κGA01 1222.7652 N *

κGA02 19.1057 N *

ρA01 mult ρ_test1 * A1a * :

ρA02 mult ρ_test1 * A2a * :

ρA01 0.2148 kg m / *

ρA02 0.2148 kg m / *

R_ad_1a I1a A1a / sqrt (1 l_test1 / * :

R_ad_1b I1b A1b / sqrt (1 l_test1 / * :

R_ad_2a I2a A2a / sqrt (1 l_test1 / * :

R_ad_2b I2b A2b / sqrt (1 l_test1 / * :

R_ad_1c I1c A1c / sqrt (1 l_test1 / * :

R_ad_1z I1z A1z / sqrt (1 l_test1 / * :

R_ad_2c I2c A2c / sqrt (1 l_test1 / * :

R_ad_2z I2z A2z / sqrt (1 l_test1 / * :

k_ad01 0 :

no axial action

k_ad02 0 :

no axial action

α_1a 1 k_ad01 2^κ_test1 G_test1 * A1a * κGA01 / / (+ :

α_1b 1 k_ad01 2^κ_test1 G_test1 * A1b * κGA01 / / (+ :

α_2a 1 k_ad02 2^κ_test1 G_test1 * A2a * κGA02 / / (+ :

α_2b 1 k_ad02 2^κ_test1 G_test1 * A2b * κGA02 / / (+ :

α_1c 1 k_ad01 2^κ_test1 G_test1 * A1c * κGA01 / / (+ :

α_1z 1 k_ad01 2^κ_test1 G_test1 * A1z * κGA01 / / (+ :

α_2c 1 k_ad02 2^κ_test1 G_test1 * A2c * κGA02 / / (+ :

α_2z 1 k_ad02 2^κ_test1 G_test1 * A2z * κGA02 / / (+ :

le aree sono da mettre tutte in Atest1 TBC

ω p_ad2 ρA02 ω2 2^* l_test1 4^* EIz02 / (sqrt :

ω p_ad1 ρA01 ω1 2^* l_test1 4^* EIz01 / (sqrt :

p_ad2 ρA02 l_test1 EIz02 ω2 EIz02 p_ad2 2^* ρA02 l_test1 4^* / (sqrt :

p_ad1 ρA01 l_test1 EIz01 ω1 EIz01 p_ad1 2^* ρA01 l_test1 4^* / (sqrt :

Expected results from Ref.[4]

Figure 3 - from Ref.[4] page

f2b_exp 10.63 66.75 21 mat 2 * π * Hz * :

f2b_Tim 10.745 67.456 21 mat 2 * π * Hz * :

f1c_Tim 54.429 11 mat 2 * π * Hz * :

f1c_exp 49.38 11 mat 2 * π * Hz * :

Dimensional Formulas (clamped - free b.c.)

ω ϕ11 ω κ_test1 l1a I1a A1a mass1a E_test1 G_test1 Φ :

ω ϕ12 ω κ_test1 l1b I1b A1b mass1b E_test1 G_test1 Φ ω κ_test1 l1c I1c A1c mass1c E_test1 G_test1 Φ * ω κ_test1 l1b I1b A1b mass1b E_test1 G_test1 Φ ω κ_test1 l1c I1c A1c mass1c E_test1 G_test1 Φ * * :

ω ϕ13 ω κ_test1 l1b I1b A1b mass1b E_test1 G_test1 Φ ω κ_test1 l1c I1c A1c mass1c E_test1 G_test1 Φ * ω κ_test1 l1b I1b A1b mass1b E_test1 G_test1 Φ ω κ_test1 l1z I1z A1z mass1z E_test1 G_test1 Φ * * :

ω φ1 ω ϕ11 ω ϕ12 * ω ϕ12 * ω ϕ13 * :

600 Hz * φ1 2.6077 0.6107 m * 0.0002 s 2^kg / * 0.0015 s 2^kg m * / * 14.1959 1 m / * 2.5946 0.0015 s 2^kg m * / * 0.0076 s 2^kg m 2^* / * 39788.7842 kg s 2^/ * 7721.0535 kg m * s 2^/ * 2.666 13.7935 1 m / * 7706.9156 kg m * s 2^/ * 1118.9223 kg m 2^* s 2^/ * 0.6182 m * 2.5303 44 mat

600 Hz * φ1 33 el 2.666

ω Zφ1 ω φ1 33 el ω φ1 44 el * ω φ1 34 el ω φ1 43 el * - 11 line :

600 Hz * Zφ1 1.7807 -

600 Hz * Zφ1 1 / abs log10 0.2506

ω ϕ21 ω κ_test1 l2a I2a A2a mass2a E_test1 G_test1 Φ :

ω ϕ22 ω κ_test1 l2b I2b A2b mass2b E_test1 G_test1 Φ ω κ_test1 l2c I2c A2c mass2c E_test1 G_test1 Φ * ω κ_test1 l2b I2b A2b mass2b E_test1 G_test1 Φ ω κ_test1 l2c I2c A2c mass2c E_test1 G_test1 Φ * * :

ω ϕ23 ω κ_test1 l2b I2b A2b mass2b E_test1 G_test1 Φ ω κ_test1 l2c I2c A2c mass2c E_test1 G_test1 Φ * ω κ_test1 l2b I2b A2b mass2b E_test1 G_test1 Φ ω κ_test1 l2z I2z A2z mass2z E_test1 G_test1 Φ * * :

ω φ2 ω ϕ21 ω ϕ22 * ω ϕ22 * ω ϕ23 * :

600 Hz * φ2 62.9722 5.1067 m * 0.012 s 2^kg / * 0.1364 s 2^kg m * / * 737.1721 1 m / * 60.6267 0.1388 s 2^kg m * / * 1.6011 s 2^kg m 2^* / * 3.3777 105^* kg s 2^/ * 27672.015 kg m * s 2^/ * 65.1386 749.1481 1 m / * 27990.5745 kg m * s 2^/ * 2324.7057 kg m 2^* s 2^/ * 5.3505 m * 62.3193 44 mat

600 Hz * φ2 33 el 65.1386

ω Zφ2 ω φ2 33 el ω φ2 44 el * ω φ2 34 el ω φ2 43 el * - 11 line :

600 Hz * Zφ2 51.1089

600 Hz * Zφ2 1 / abs log10 1.7085

Adimensional Formulas (clamped - free b.c.)

p_ad2 ϕad21 E_test1 I2a * EIz02 / p_ad2 R_ad_2a ρ_test1 A2a * ρA02 / κ_test1 G_test1 * A2a * κGA02 / k_ad02 α_2a l2a l_test1 / Trad :

50 ϕad21 1.0009 0.0548 0.0015 - 2.675 105 -^* - 0.0686 1.0009 0.0548 - 0.0015 - 3.7532 - 0.068 - 1.0009 0.0548 137.0099 - 3.7532 - 0.0686 1.0009 44 mat

p_ad2 ϕad22 E_test1 I2b * EIz02 / p_ad2 R_ad_2b ρ_test1 A2b * ρA02 / κ_test1 G_test1 * A2b * κGA02 / k_ad02 α_2b l2b l_test1 / Trad E_test1 I2c * EIz02 / p_ad2 R_ad_2c ρ_test1 A2c * ρA02 / κ_test1 G_test1 * A2c * κGA02 / k_ad02 α_2c l2c l_test1 / Trad * E_test1 I2b * EIz02 / p_ad2 R_ad_2b ρ_test1 A2b * ρA02 / κ_test1 G_test1 * A2b * κGA02 / k_ad02 α_2b l2b l_test1 / Trad E_test1 I2c * EIz02 / p_ad2 R_ad_2c ρ_test1 A2c * ρA02 / κ_test1 G_test1 * A2c * κGA02 / k_ad02 α_2c l2c l_test1 / Trad * * :

50 ϕad22 2.549 0.4227 0.0714 - 0.0081 - 21.3792 2.6711 0.5618 - 0.086 - 131.9848 - 12.9875 - 2.1754 0.4148 862.3481 - 113.6398 - 13.3237 2.1479 44 mat

p_ad2 ϕad23 E_test1 I2b * EIz02 / p_ad2 R_ad_2b ρ_test1 A2b * ρA02 / κ_test1 G_test1 * A2b * κGA02 / k_ad02 α_2b l2b l_test1 / Trad E_test1 I2c * EIz02 / p_ad2 R_ad_2c ρ_test1 A2c * ρA02 / κ_test1 G_test1 * A2c * κGA02 / k_ad02 α_2c l2c l_test1 / Trad * E_test1 I2b * EIz02 / p_ad2 R_ad_2b ρ_test1 A2b * ρA02 / κ_test1 G_test1 * A2b * κGA02 / k_ad02 α_2b l2b l_test1 / Trad E_test1 I2z * EIz02 / p_ad2 R_ad_2z ρ_test1 A2z * ρA02 / κ_test1 G_test1 * A2z * κGA02 / k_ad02 α_2z l2z l_test1 / Trad * * :

50 ϕad23 1.8589 0.3331 0.056 - 0.0055 - 13.6557 1.9588 0.4678 - 0.065 - 93.6232 - 8.3805 - 1.7418 0.335 666.7364 - 82.3865 - 9.3233 1.6873 44 mat

p_ad2 φad2 p_ad2 ϕad21 p_ad2 ϕad22 * p_ad2 ϕad22 * p_ad2 ϕad23 * :

60 φad2 752.3635 90.6806 14.697 - 1.942 - 5809.3318 700.0884 113.6322 - 15.0148 - 34801.6251 - 4188.0132 - 679.6704 89.9373 2.8452 105^* - 34246.3628 - 5549.7273 734.3042 44 mat

p_ad2 Zφad2 p_ad2 φad2 33 el p_ad2 φad2 44 el * p_ad2 φad2 34 el p_ad2 φad2 43 el * - 11 line :

60 Zφad2 42.6901 -

60 Zφad2 1 / abs log10 1.6303

Plot : adimensional formulas vs dimensional formulas (clamped - free b.c.) out-of-plane bending

x 0 > ( x 40 ≤ ( & 100 x Zφad2 1 / abs log10 * if 1 1 sys x 0 > ( x 40 ≤ ( & 100 EIz02 x 2 ^ * ρA02 l_test1 4 ^ * / ( sqrt Zφ2 1 / abs log10 * if 1 1 sys 2 1 sys ρA02 EIz02 / f2b_exp 1 el ( 2 ^ * l_test1 4 ^ * ( 0.5 ^ 800 - ρA02 EIz02 / f2b_exp 1 el ( 2 ^ * l_test1 4 ^ * ( 0.5 ^ 800 2 2 mat ρA02 EIz02 / f2b_Tim 1 el ( 2 ^ * l_test1 4 ^ * ( 0.5 ^ 800 - ρA02 EIz02 / f2b_Tim 1 el ( 2 ^ * l_test1 4 ^ * ( 0.5 ^ 800 2 2 mat 2 1 sys ρA02 EIz02 / f2b_exp 2 el ( 2 ^ * l_test1 4 ^ * ( 0.5 ^ 800 - ρA02 EIz02 / f2b_exp 2 el ( 2 ^ * l_test1 4 ^ * ( 0.5 ^ 800 2 2 mat ρA02 EIz02 / f2b_Tim 2 el ( 2 ^ * l_test1 4 ^ * ( 0.5 ^ 800 - ρA02 EIz02 / f2b_Tim 2 el ( 2 ^ * l_test1 4 ^ * ( 0.5 ^ 800 2 2 mat 2 1 sys 2 1 sys 2 1 sys 1 1 sys 1 1 sys 1 1 sys 1 1 sys 1 1 sys

ρA02 EIz02 / f2b_exp 1 el (2^* l_test1 4^* (0.5^3.2831

ρA02 EIz02 / f2b_exp 2 el (2^* l_test1 4^* (0.5^21.073

ρA02 EIz02 / f2b_Tim 1 el (2^* l_test1 4^* (11.0132

ρA02 EIz02 / f2b_Tim 2 el (2^* l_test1 4^* (0.5^20.8339

Comparison prediction vs experimental out-of-plane bending

Prediction from above plot

f1b_TMM_ad 3.358 :

estimate on above plot

f2b_TMM_ad 30.6 :

estimate on above plot

ρA02 EIz02 / f2b_exp 1 el (2^* l_test1 4^* (0.5^3.2831

ρA02 EIz02 / f2b_exp 2 el (2^* l_test1 4^* (0.5^20.6159

ρA02 EIz02 / f2b_Tim 1 el (2^* l_test1 4^* (0.5^3.3186

ρA02 EIz02 / f2b_Tim 2 el (2^* l_test1 4^* (0.5^20.8339

Err%_1st_freq_out_of_plane_bend f1b_TMM_ad ρA02 EIz02 / f2b_exp 1 el (2^* l_test1 4^* (0.5^- (ρA02 EIz02 / f2b_exp 1 el (2^* l_test1 4^* (0.5^/ 100 * :

Err%_1st_freq_out_of_plane_bend 47.1171

Err%_1st_freq_out_of_plane_bend1 f1b_TMM_ad ρA02 EIz02 / f2b_Tim 1 el (2^* l_test1 4^* (0.5^- (ρA02 EIz02 / f2b_Tim 1 el (2^* l_test1 4^* (0.5^/ 100 * :

Err%_1st_freq_out_of_plane_bend1 45.5426

Err%_2nd_freq_out_of_plane_bend f2b_TMM_ad ρA02 EIz02 / f2b_exp 2 el (2^* l_test1 4^* (0.5^- (ρA02 EIz02 / f2b_exp 2 el (2^* l_test1 4^* (0.5^/ 100 * :

Err%_2nd_freq_out_of_plane_bend 48.4292

Err%_2nd_freq_out_of_plane_bend1 f2b_TMM_ad ρA02 EIz02 / f2b_Tim 2 el (2^* l_test1 4^* (0.5^- (ρA02 EIz02 / f2b_Tim 2 el (2^* l_test1 4^* (0.5^/ 100 * :

Err%_2nd_freq_out_of_plane_bend1 46.8758

time_end 0 time :

Δtime time_end time_start - :

Δtime 21704.044 s *