fb200cb2-2945-4542-ba13-fd5031213d92 313 Alvaro Diaz Falconi 4 5 false false 0 10 decimal

References:

[1] Timoshenko - Vibration Problems in Engineering pag 342

[2]

[3]

ρ 8000 kg m 3^/ * :

ν 0.298 :

E 198000 MPa * :

G E 21 ν + (* / :

κ_hollow 21 ν + (43 ν * + / * : 0.5304

Hollow cylinder cross section

mass 0.98 kg m / * :

Isect 2.331 108 -^* m 4^* :

Asect 1.225 104 -^* m 2^* :

L 0.32 m * :

Isect Asect r Isect Asect / (sqrt :

ε κ_hollow :

Isect mass a E Isect mass / * (sqrt :

ω Isect mass β ω 2^Isect mass a 2^/ (4 nthroot :

L Isect Asect α 1 ε / Isect Asect r L / (2^* E G / * :

L Isect Asect γ Isect Asect r L / (2^:

ω Isect Asect mass L A ω Isect mass β L * (4^L Isect Asect α L Isect Asect γ + (* :

ω Isect Asect mass L B ω Isect mass β L * (8^L Isect Asect α (L Isect Asect γ (- (2^* 4 ω Isect mass β L * (4^* (+ (sqrt :

ω Isect Asect mass L η 2 sqrt 2 L * / ω Isect Asect mass L A - ω Isect Asect mass L B + (sqrt * :

ω Isect Asect mass L θ 2 sqrt 2 L * / ω Isect Asect mass L A ω Isect Asect mass L B + (sqrt * :

ω Isect Asect mass L σ1 1 L / ω Isect Asect mass L θ L * L Isect Asect α ω Isect Asect mass L θ L * / ω Isect mass β L * (4^* - (* :

ω Isect Asect mass L σ2 1 L / ω Isect Asect mass L η L * L Isect Asect α ω Isect Asect mass L η L * / ω Isect mass β L * (4^* + (* :

Isect k E Isect * :

ω Isect Asect mass L σ11 ω Isect Asect mass L θ 3^L Isect Asect α L 2^* ω Isect mass β 4^* ω Isect Asect mass L θ * - L Isect Asect γ L 2^* ω Isect mass β 4^* ω Isect Asect mass L σ1 * - (:

ω Isect Asect mass L σ21 ω Isect Asect mass L η 3^L Isect Asect α L 2^* ω Isect mass β (4^* ω Isect Asect mass L η * + L Isect Asect γ L 2^* ω Isect mass β 4^* ω Isect Asect mass L σ1 * + (:

ω Isect Asect mass L σ12 ω Isect Asect mass L θ (2^L Isect Asect α L 2^* ω Isect mass β (4^* - (:

ω Isect Asect mass L σ21 ω Isect Asect mass L η (3^L Isect Asect α L 2^* ω Isect mass β (4^* ω Isect Asect mass L η * + L Isect Asect γ L 2^* ω Isect mass β (4^* ω Isect Asect mass L σ2 * + (:

ω Isect Asect mass L σ22 ω Isect Asect mass L η (2^L Isect Asect α L 2^* ω Isect mass β (4^* + (:

ω Isect Asect mass L φ11 1 ω Isect Asect mass L σ12 ω Isect Asect mass L σ22 + / ω Isect Asect mass L σ22 ω Isect Asect mass L θ L * cos * ω Isect Asect mass L σ12 ω Isect Asect mass L η L * cosh * + (* :

ω Isect Asect mass L φ12 1 ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * + / ω Isect Asect mass L σ21 ω Isect Asect mass L θ L * (sin * ω Isect Asect mass L σ11 ω Isect Asect mass L η L * (sinh * + (* :

ω Isect Asect mass L φ13 1 Isect k (ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * + (* / ω Isect Asect mass L σ2 - (ω Isect Asect mass L θ L * (sin * ω Isect Asect mass L σ1 ω Isect Asect mass L η L * (sinh * + (* :

ω Isect Asect mass L φ14 1 Isect k (ω Isect Asect mass L σ12 ω Isect Asect mass L σ22 + (* / ω Isect Asect mass L θ L * cos - ω Isect Asect mass L η L * (cosh + (* :

ω Isect Asect mass L φ21 1 ω Isect Asect mass L σ12 ω Isect Asect mass L σ22 + / ω Isect Asect mass L σ1 ω Isect Asect mass L σ22 ω Isect Asect mass L θ L * (sin * * ω Isect Asect mass L σ2 ω Isect Asect mass L σ12 * ω Isect Asect mass L η L * (sinh * - (* :

ω Isect Asect mass L φ22 1 ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * + / ω Isect Asect mass L σ2 ω Isect Asect mass L σ21 ω Isect Asect mass L θ L * ((cos * * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * ω Isect Asect mass L η L * ((cosh * + (* :

ω Isect Asect mass L φ23 ω Isect Asect mass L σ1 ω Isect Asect mass L σ2 * Isect k (ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * + (* / ω Isect Asect mass L θ L * cos - ω Isect Asect mass L η L * (cosh + (* :

ω Isect Asect mass L φ24 1 Isect k (ω Isect Asect mass L σ12 ω Isect Asect mass L σ22 + (* / ω Isect Asect mass L σ1 ω Isect Asect mass L θ L * (sin * ω Isect Asect mass L σ2 ω Isect Asect mass L η L * (sinh * + (* :

ω Isect Asect mass L φ31 Isect k (ω Isect Asect mass L σ12 ω Isect Asect mass L σ22 + (/ ω Isect Asect mass L σ11 ω Isect Asect mass L σ22 * ω Isect Asect mass L θ L * (sin * ω Isect Asect mass L σ12 ω Isect Asect mass L σ21 * ω Isect Asect mass L η L * sinh * + (* :

ω Isect Asect mass L φ32 Isect k ω Isect Asect mass L σ11 ω Isect Asect mass L σ21 * * - (ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * + (/ ω Isect Asect mass L θ L * cos ω Isect Asect mass L η L * (cosh - (* :

ω Isect Asect mass L φ33 1 ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * + (/ ω Isect Asect mass L σ2 ω Isect Asect mass L σ21 * ω Isect Asect mass L θ L * cos * ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L η L * ((cosh * + (* :

ω Isect Asect mass L φ34 1 ω Isect Asect mass L σ12 ω Isect Asect mass L σ22 + / ω Isect Asect mass L σ11 - (ω Isect Asect mass L θ L * sin * ω Isect Asect mass L σ21 ω Isect Asect mass L η L * (sinh * + (* :

ω Isect Asect mass L φ41 Isect k (ω Isect Asect mass L σ12 * ω Isect Asect mass L σ22 * - ω Isect Asect mass L σ12 ω Isect Asect mass L σ22 + (/ ω Isect Asect mass L θ L * cos ω Isect Asect mass L η L * (cosh - (* :

ω Isect Asect mass L φ42 Isect k (- ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * + (/ ω Isect Asect mass L σ12 ω Isect Asect mass L σ21 ω Isect Asect mass L θ L * (sin * * ω Isect Asect mass L σ11 ω Isect Asect mass L σ22 * ω Isect Asect mass L η L * (sinh * - (* :

ω Isect Asect mass L φ43 1 ω Isect Asect mass L σ1 ω Isect Asect mass L σ21 * ω Isect Asect mass L σ2 ω Isect Asect mass L σ11 * + (/ ω Isect Asect mass L σ2 ω Isect Asect mass L σ12 ω Isect Asect mass L θ L * (sin * * ω Isect Asect mass L σ1 ω Isect Asect mass L σ22 * ω Isect Asect mass L η L * (sinh * + (* :

ω Isect Asect mass L φ44 1 ω Isect Asect mass L σ12 ω Isect Asect mass L σ22 + (/ ω Isect Asect mass L σ12 ω Isect Asect mass L θ L * ((cos * ω Isect Asect mass L σ22 ω Isect Asect mass L η L * ((cosh * + (* :

added eval here

ω Φ ω Isect Asect mass L φ11 ω Isect Asect mass L φ12 ω Isect Asect mass L φ13 ω Isect Asect mass L φ14 ω Isect Asect mass L φ21 ω Isect Asect mass L φ22 ω Isect Asect mass L φ23 ω Isect Asect mass L φ24 ω Isect Asect mass L φ31 ω Isect Asect mass L φ32 ω Isect Asect mass L φ33 ω Isect Asect mass L φ34 ω Isect Asect mass L φ41 ω Isect Asect mass L φ42 ω Isect Asect mass L φ43 ω Isect Asect mass L φ44 44 mat eval :

detZ ω Isect Asect mass L φ31 ω Isect Asect mass L φ42 * ω Isect Asect mass L φ32 ω Isect Asect mass L φ41 * - :

detZ num2str strlen 25257

very very long

detZ detZ ratsimp Maxima :

detZ num2str strlen 8158

very long

ω detZ detZ :

detZ' xx detZ xx Diff ratsimp Maxima :

can't use ω here, looks like a bug in maxima for the greek letter.

ω detZ detZ eval :

xx detZ' detZ' eval :

1 F x NR MI h εx 13 mat 25105 -^x UnitsOf * 105 -^x UnitsOf * 13 mat : iter 1 MI range Fx x F : x' x h Fx * x h + F Fx - / - eval : x x' - abs εx > x x' : break if 31 line for x 31 line :

n 18 range Ω n el x detZ 20 n * kHz * NR : for

n 17 range Ω' n el x detZ' 1020 n 1 - (* + (kHz * NR : for

ΔZ Ω detZ MN 2 -^* vectorize : normi 3.4 106 -^*

ΔZ' Ω' detZ' MN 2 -^* kHz * vectorize : normi 1.6 108 -^*

Ω kHz 13.7131 32.8356 55.3997 78.9188 102.4893 125.3542 146.9698 162.0383 81 mat

Ω' kHz 10.9222 27.4924 47.5194 68.9282 90.8209 112.5643 133.5257 71 mat

Brute force imag part detection

ωi Ω 8 el :

Δωi 1 kHz * :

n 16 range ωi detZ MN 2 -^* Im 0 ≡ 0 try ωi ωi Δωi + : while ωi ωi Δωi - : Δωi 0.1 Δωi * : 12 mat 21 line for

ωi kHz 163.0244

ωi Ω 8 el - 986.04 Hz *

'last' real value:

ωi detZ MN 2^29.9022 -

The units for detZ are Mega Newtons.

P ω 0 ωi kHz / ltlt ω kHz * detZ MN 2 -^* eval if ω' 0 ωi kHz / ltlt ω' kHz * detZ' MN 2 -^* kHz * eval if Ω kHz / ΔZ o augment Ω' kHz / ΔZ' o augment ωi kHz / ωi detZ MN 2 -^* x augment 51 sys :

P P

The issue in the plot is related with the problem that there are not a true NaN in SMath. For example, try to change the value for else in P with zero instead "".