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ɇȺɍɑɇɂɌɊɍȾɈȼȿɇȺɊɍɋȿɇɋɄɂəɍɇɂȼȿɊɋɂɌȿɌ-201ɬɨɦ4ɫɟɪɢɹ2 -- Ⱦɢɧɚɦɢɤɚɧɚɜɢɛɪɨɛɨɬ ɂɜɚɧɅɭɤɚɧɨɜȼɟɧɤɨȼɢɬɥɢɟɦɨɜɂɜɟɥɢɧɂɜɚɧɨɜ Dynamics of a VibroBot: The mathematical model of a unidirectionally moving prototype of a mobile mechanical system with vibration propulsion is formulated. The results obtained from the numerical experiment with the robot dynamic model having linear elastic and dissipative characteristics are attached. The effects of variation of the robot’s parameters upon its dynamics were investigated and analyzed. Key words: VibroBot, Vibrational drive, Mathematical Model, Dynamics, Numerical experiment. ɍȼɈȾ Ɇɨɛɢɥɧɢɬɟɪɨɛɨɬɢɫɜɢɛɪɚɰɢɨɧɧɨɡɚɞɜɢɠɜɚɧɟɜɢɛɪɨɛɨɬɢɫɚɨɛɟɤɬɧɚɡɚɫɢɥɟɧ ɢɧɬɟɪɟɫ ɦɨɬɢɜɢɪɚɧɨɬɧɹɤɨɢɬɟɯɧɢɩɪɟɞɢɦɫɬɜɚɩɪɢɞɜɢɠɟɧɢɟɜɧɟɬɪɚɞɢɰɢɨɧɧɢɭɫɥɨ- ɜɢɹɫɴɱɟɬɚɧɢɫɴɫɡɚɜɢɲɟɧɢɢɡɢɫɤɜɚɧɢɹɡɚɦɢɧɢɚɬɸɪɧɨɫɬɟɧɟɪɝɨɟɮɟɤɬɢɜɧɨɫɬɢɟɤɨ- ɥɨɝɨɫɴɨɛɚɡɧɨɫɬ ȼɢɛɪɨɛɨɬɢɬɟ ɢɦɚɬ ɩɪɨɫɬɚ ɤɨɧɫɬɪɭɤɰɢɹ ɡɚɳɨɬɨ ɧɟ ɢɡɢɫɤɜɚɬ ɫɩɟ- ɰɢɚɥɧɢ ɞɜɢɠɟɳɢ ɦɟɯɚɧɢɡɦɢ Ⱦɜɢɠɟɧɢɟɬɨ ɢɦ ɫɟ ɨɫɴɳɟɫɬɜɹɜɚ ɨɬ ɢɧɟɪɰɢɨɧɧɢ ɫɢɥɢ ɜɴɡɧɢɤɜɚɳɢ ɨɬ ɩɟɪɢɨɞɢɱɧɨ ɪɟɥɚɬɢɜɧɨ ɞɜɢɠɟɧɢɟ ɧɚ ɧɟɭɪɚɜɧɨɜɟɫɟɧɢ ɦɚɫɢ ȼɢɛɪɨ- ɡɚɞɜɢɠɜɚɳɢɹɬ ɦɟɯɚɧɢɡɴɦ ɮɭɧɤɰɢɨɧɢɪɚ ɜ ɨɤɨɥɧɨɫɬ ɧɚ ɨɫɧɨɜɧɢɹ ɪɟɡɨɧɚɧɫ Ɍɨɜɚ ɨɛɭɫɥɚɜɹ ɩɨɹɜɚɬɚ ɧɚ ɧɟɥɢɧɟɣɧɢ ɞɢɧɚɦɢɱɧɢ ɩɪɨɰɟɫɢ ɤɨɢɬɨ ɡɚɬɪɭɞɧɹɜɚɬ ɢɡɫɥɟɞ- ɜɚɧɟɬɨ ɧɚ ɬɨɡɢ ɜɢɞ ɦɟɯɚɧɢɱɧɢ ɫɢɫɬɟɦɢ ɉɪɟɨɛɥɚɞɚɜɚɳɚ ɱɚɫɬ ɨɬ ɩɪɨɜɟɞɟɧɢɬɟ ɢɡɫɥɟɞɜɚɧɢɹ ɜɴɪɯɭ ɜɢɛɪɨɛɨɬɢ ɫɚ ɫɜɴɪɡɚɧɢ ɫ ɚɧɚɥɢɬɢɱɧɨ ɦɨɞɟɥɢɪɚɧɟ ɟɤɫɩɟɪɢɦɟɧ- ɬɚɥɧɨɩɪɨɬɨɬɢɩɧɨɩɪɨɭɱɜɚɧɟ ɢɤɨɦɩɸɬɴɪɧɨɫɢɦɭɥɢɪɚɧɟɧɚɞɢɧɚɦɢɱɧɨɬɨɢɦɩɨɜɟɞɟ- ɧɢɟ >@ >@ >@ >@ ȼ ɧɚɫɬɨɹɳɚɬɚ ɪɚɡɪɚɛɨɬɤɚ ɟ ɮɨɪɦɭɥɢɪɚɧ ɦɟɯɚɧɨ-ɦɚɬɟɦɚɬɢɱɟɧ ɦɨɞɟɥ ɧɚ ɤɨɥɟɫɟɧ ɜɢɛɪɨɛɨɬ ɫ ɟɞɧɨɩɨɫɨɱɧɨ ɞɜɢɠɟɧɢɟ ɨɫɧɨɜɚɧ ɧɚ ɟɤɫɩɟɪɢɦɟɧɬɚɥɧɢ ɞɚɧɧɢ ɨɬ >@ ɡɚ ɨɩɢɬɟɧ ɨɛɪɚɡɟɰ ɢɡɫɥɟɞɜɚɧ ɜ >7@ [@ ɋ ɩɨɦɨɳɬɚ ɧɚ ɱɢɫɥɟɧ ɟɤɫɩɟɪɢ- ɦɟɧɬɟɩɪɨɜɟɞɟɧɞɢɧɚɦɢɱɟɧɚɧɚɥɢɡɧɚɫɴɡɞɚɞɟɧɢɹɦɨɞɟɥ ɆȿɏȺɇɈ-ɆȺɌȿɆȺɌɂɑȿɇɆɈȾȿɅ ɇɚ ɮɢɝ ɟ ɩɨɤɚɡɚɧ ɞɢɧɚɦɢɱɟɧ ɦɨɞɟɥ ɧɚ ɪɟɚɥɟɧ ɨɩɢɬɟɧ ɨɛɪɚɡɟɰ ɧɚ ɜɢɛɪɨɛɨɬ [7@ ȼ ɧɟɝɨ ɟ ɩɪɢɟɬɨ ɱɟ ɜɫɢɱɤɢ ɟɤɜɢɜɚɥɟɧɬɧɢ ɦɚɫɢ ɫɚ ɪɚɡɩɪɟɞɟɥɟɧɢ ɜ ɪɚɜɧɢɧɧɢ ɮɢɝɭɪɢɤɨɢɬɨɫɟɞɜɢɠɚɬɜɴɜɜɟɪɬɢɤɚɥɧɚɪɚɜɧɢɧɚɤɨɹɬɨɫɴɜɩɚɞɚ ɫɪɚɜɧɢɧɚɬɚ XOZ ɧɚ ɚɛɫɨɥɸɬɧɚɬɚɤɨɨɪɞɢɧɚɬɧɚɫɢɫɬɟɦɚOXYZ Ɋɚɡɝɥɟɠɞɚɧɚɬɚ ɦɟɯɚɧɢɱɧɚ ɫɢɫɬɟɦɚ ɩɪɢɬɟɠɚɜɚ ɞɢɧɚɦɢɱɧɚ ɫɢɦɟɬɪɢɹ ɩɨ ɧɚɩɪɚɜ- ɥɟɧɢɟ ɧɚ ɨɫɬɚ OY Ɍɨɜɚ ɞɨɩɭɫɤɚɧɟ ɟ ɜɹɪɧɨ ɡɚ ɜɴɡɛɭɠɞɚɳɨɬɨ ɜɴɡɞɟɣɫɬɜɢɟ ɧɚ ɞɜɟɬɟ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɜɴɪɬɹɳɢ ɫɟ ɧɟɭɪɚɜɧɨɜɟɫɟɧɢ ɦɚɫɢ m ɢ ɡɚ ɫɢɦɟɬɪɢɱɧɨ ɩɨɡɢ- ɰɢɨɧɢɪɚɧɢɬɟ ɟɥɚɫɬɢɱɧɢ ɢ ɞɢɫɢɩɚɬɢɜɧɢ ɜɪɴɡɤɢ ɧɨ ɟ ɩɪɢɛɥɢɡɢɬɟɥɧɨ ɨɬɧɨɫɧɨ ɪɚɡɩɨ- ɥɨɠɟɧɢɟɬɨɧɚɦɚɫɨɜɢɬɟɰɟɧɬɪɨɜɟɧɚɫɴɫɬɚɜɧɢɬɟɬɟɥɚ Ɍɟɥɚɬɚ ɢɫɬɟɝɥɚ G ɢ G 2 ɫɟɞɜɢɠɚɬɬɪɚɧɫɥɚɰɢɨɧɧɨ-ɩɪɚɜɨɥɢɧɟɣɧɨȼɴɡɛɭɠ- ɞɚɳɨɬɨɞɟɣɫɬɜɢɟ ɧɚɧɟɭɪɚɜɧɨɜɟɫɟɧɢɬɟ ɦɚɫɢm ɟɧɚɫɨɱɟɧɨɩɨɨɫɬɚɈXɌɨɦɨɠɟɞɚ ɫɟ ɫɢɦɭɥɢɪɚ ɱɪɟɡ ɩɟɪɢɨɞɢɱɧɨ ɞɜɢɠɟɧɢɟ ɨɬɧɨɫɧɨ ɬɹɥɨ ɧɚ ɟɤɜɢɜɚɥɟɧɬɧɨ ɬɹɥɨ ɦɨɞɟɥɢɪɚɧɨ ɤɚɬɨɦɚɬɟɪɢɚɥɧɚɬɨɱɤɚɫɬɟɝɥɨG = m gɤɨɟɬɨ ɫɟɞɜɢɠɢɩɨɡɚɞɚɞɟɧɯɚɪ- ɦɨɧɢɱɟɧɡɚɤɨɧ K t) ȼɴɜɟɠɞɚɦɟ ɬɪɢ ɨɛɨɛɳɟɧɢ ɤɨɨɪɞɢɧɚɬɢ x , x 2 , x , ɤɨɢɬɨ ɨɩɪɟɞɟɥɹɬ ɩɨɥɨɠɟɧɢɟɬɨ ɧɚ ɬɟɥɚɬɚ ɢ ɫɩɪɹɦɨ ɤɨɨɪɞɢɧɚɬɧɚɬɚ ɫɢɫɬɟɦɚ XOZ Ʉɨɨɪɞɢɧɚɬɢɬɟ x , x 2 ɢ x ɫɚ ɜɡɚɢɦɨɫɜɴɪɡɚɧɢɱɪɟɡɡɚɜɢɫɢɦɨɫɬɢɬɟ x 2 t) = x t) + l + [ t), x t) = x 2 t) + l 2 + K t), ɤɴɞɟ- ɬɨ K t) = U VLQ Z t + I ); ɪɚɡɫɬɨɹɧɢɹɬɚ l ɧɚɱɚɥɧɚɬɚ ɞɴɥɠɢɧɚ ɧɚ ɟɤɜɢɜɚɥɟɧɬɧɚɬɚ ɩɪɭ- ɠɢɧɚ, l 2 ɢ U, ɧɚɱɚɥɧɚɬɚ ɮɚɡɚ I ɢ ɴɝɥɨɜɚɬɚ ɫɤɨɪɨɫɬ Z ɫɚ ɡɚɞɚɞɟɧɢ ɩɨɫɬɨɹɧɧɢ ɜɟɥɢ- ɱɢɧɢ [ ɟɦɹɪɤɚɬɚɧɚɢɡɦɟɧɟɧɢɟɬɨ ɧɚɞɴɥɠɢɧɚɬɚɧɚɟɤɜɢɜɚɥɟɧɬɧɚɬɚɩɪɭɠɢɧɚɫɩɪɹɦɨ ɧɟɣɧɨɬɨɧɚɱɚɥɧɨɫɬɚɬɢɱɧɨɫɴɫɬɨɹɧɢɟ Ʉɨɥɟɥɚɬɚɧɚɤɨɥɢɱɤɚɬɚɦɨɞɟɥɢɪɚɧɢɤɚɬɨɬɜɴɪɞɢɟɞɧɚɤɜɢɩɨɞɜɨɣɤɢɬɟɥɚɢ ɢɡɜɴɪɲɜɚɬ ɪɚɜɧɢɧɧɨ ɞɜɢɠɟɧɢɟ Ɇɚɫɨɜɢɬɟ ɢɦ ɰɟɧɬɪɨɜɟ ɢɦɚɬ ɫɤɨɪɨɫɬɢ ɟɞɧɚɤɜɢ ɫɴɫ

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Page 1: b[jh[hl - uni-ruse.bgconf.uni-ruse.bg/bg/docs/cp15/2/2-4.pdf · >bgZfbdZgZ\b[jh[hl B\ZgEmdZgh\

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Dynamics of a VibroBot: The mathematical model of a unidirectionally moving prototype of a mobile mechanical system with vibration propulsion is formulated. The results obtained from the numerical experiment with the robot dynamic model having linear elastic and dissipative characteristics are attached. The effects of variation of the robot’s parameters upon its dynamics were investigated and analyzed.

Key words: VibroBot, Vibrational drive, Mathematical Model, Dynamics, Numerical experiment.

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x , x2, x , XOZ x , x2 x

x2 t) = x t) + l + t), x t) = x2 t) + l2 + t), - t) = t + ); l -, l2 , -

Page 2: b[jh[hl - uni-ruse.bgconf.uni-ruse.bg/bg/docs/cp15/2/2-4.pdf · >bgZfbdZgZ\b[jh[hl B\ZgEmdZgh\

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V x t -

a 2x t2 V t, a2 2x2 t2 V2 t, a

2x t2 = a2 + 2 t 2 = a2 – 2 , = a r, -

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[ , -Gi, i = 2, …, ; Nj, Tj, j = 2; -Rv, v = 2, , Fk, Fb; l, l = 2, …, ; M , = Gi = mi g Fk = – k ,

= x2 – x + l Fk Fk = k s – l ), s x2 Fb = – b|V2 – V | –

Fb, V2 – V ) – l = – ml al M = – m r/ 2)a –

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R , R2, R , P , P2, P , P , N , T , N2, T2 a , a2 -

l

a = Fa m + m + m a2 = m 2 – Fa) / m2 + m ),

a = a2 – 2 , R = – m a , R2 = m g, R = m2 + m )g,

l = l + {m [g l2 – l ) – a h2 – Fa h – h )} m2 + m )g, P = m a ,

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P2 = {– Fah + m2 + m d – d ) + m d – d2 g – m a h d , P = m a , P = {Fah + m2 + m )d + m d2 g + m a h d , N = m g + {– Fah + m2 + m d – d ) + m d – d2 g – m a h d , T = m a , T2 = m a , N2 = m g + {Fah + [ m2 + m )d + m d2 g + m a h d ,

Fa = k x2 – x – l ) + b|V2 – V |

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Page 4: b[jh[hl - uni-ruse.bgconf.uni-ruse.bg/bg/docs/cp15/2/2-4.pdf · >bgZfbdZgZ\b[jh[hl B\ZgEmdZgh\

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x t = V , x = , 2) V t = [k x2 – x – l ) + b|V2 – V | / [m + m + m V =

x2 t = V2, x2 = s ,

V2 t = m 2 t + ) – [k x2 – x – l ) + b|V2 – V | / m2 + m ), V2 =

) V = V t), V t) > ; V = V t)

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– 2

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e) )

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Page 8: b[jh[hl - uni-ruse.bgconf.uni-ruse.bg/bg/docs/cp15/2/2-4.pdf · >bgZfbdZgZ\b[jh[hl B\ZgEmdZgh\

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i) )

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