<content> <p> Golden ratio min/max </p> </content>

a182c594-dd1e-4061-b7a8-fbc97fbb606b 28 4 5 false false 0 0 decimal

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

References:

[1] Timoshenko - Vibration Problems in Engineering pag 342

[2]

[3]

t0 1 time :

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

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 * + (* :

D1 0.40 m * :

D2 0.20 m * :

D3 0.14 m * :

d1 0.16 m * :

d2 0.10 m * :

d3 0.04 m * :

L1 2 m * :

L2 4 m * :

L3 2 m * :

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^- (* * :

L L1 :

mass ρ D1 d1 f_mass : Isect D1 d1 f_Isect : Asect D1 d1 f_Asect : 31 mat 844.4601 0.0012 0.1056 31 mat

ω Φ1 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 :

L L2 :

mass ρ D2 d2 f_mass :

Isect D2 d2 f_Isect :

Asect D2 d2 f_Asect :

ω Φ2 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 :

L L3 :

mass ρ D3 d3 f_mass :

Isect D3 d3 f_Isect :

Asect D3 d3 f_Asect :

ω Φ3 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 :

ω Φ ω Φ1 ω Φ2 * ω Φ3 * eval :

1 minte ω detZ ω Φ 33 el ω Φ 44 el * ω Φ 34 el ω Φ 43 el * - eval :

local definition ω Z ω detZ 108^/ 11 line :

x λ 1 2.5 - x ≤ (x 2.5 ≤ (& cases :

slow plot
10 min

0.01 x Z * x λ * x Z x Z

x Z x 5 - 0.25 - solve 4.4157 - 2.3954 - 2.2842 - 0.7904 - 41 mat

4.4157 - 2.3954 - 2.2842 - 0.7904 - 41 mat

x y axes x y / - :

x Z axes 2 1 sys

1 time t0 - min 8

<content> <p> Golden ratio min/max </p> </content>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 71 line :

a b + 2 / iter 21 mat

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f x Z eval :

ε 1012 -^:

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 41 line :

X f 0.75 - 0.625 - y1 y2 ≥ ε Extrema :

X 0.715159821081 -

on failure , increase TOL TOL 109 -^:

r 0.85 - 0.75 - TOL D : 0.790367969 -

X 0.715159821081 -

1 time t0 - min 13.7

Export/inspect the project

ω 4.415 - 4.415 4.4 - range :

4.4157 - 2.3954 - 2.2842 - 0.7904 - 41 mat

20 minutes data ω ω Z vectorize augment :

The ranged project data data eval :

inspect data 150 row 2.18 - 2.6497 105^* - 12 mat

1 time t0 - min 33.5

data rows 589

data 1 row data data rows row 21 sys 4.415 - 6.6891 108^* - 12 mat 4.405 9.8793 109^* - 12 mat 21 sys

sub data 130460 range 12 range el :

sub 0.79 - 0 o 10 red 1 5 mat 0.79 0 o 10 red 1 5 mat 2.284 - 0 . 14 red 1 5 mat 2.284 0 . 14 red 1 5 mat 2.3954 - 0 . 14 black 1 5 mat 2.3954 0 . 14 black 1 5 mat 7 1 sys sub 0.79 - 0 . 14 red 1 5 mat 0.79 0 . 14 red 1 5 mat 2.284 - 0 . 14 red 1 5 mat 2.284 0 . 14 red 1 5 mat 2.3954 - 0 . 14 black 1 5 mat 2.3954 0 . 14 black 1 5 mat 7 1 sys