c4_legi_principale_cu_grafice.doc
TRANSCRIPT
3
3.2. Legea LiniarMicarea tachetului dup legea liniar este o micare uniform. Ecuaiile generale sub form adimensional sunt:
unde ; .
Pentru determinarea coeficienilor A i B se pun condiiile la capetele intervalului:
B = 0;
.
Prin nlocuirea coeficienilor A i B cu valorile lor, se obin ecuaiile legii Liniare sub forma adimensional:
Graficele sunt date n figura 3.2.
EMBED Excel.Chart.8 \s
EMBED Excel.Chart.8 \s
Fig.3.2
3.3. Legea parabolicLegea parabolic este definit pe dou subintervale, care pot fi egale sau inegale. Considerm cazul cnd acestea sunt egale (figura 3.3a). Pe primul subinterval, deplasarea tachetului variaz dup o ramur de parabol cu vrful n punctul [0;0] care este originea axelor, iar pe al doilea subinterval, deplasarea tachetului variaz dup o ramur de parabol cu vrful n punctul de coordonate [1;1].
Pe subintervalul micarea este accelerat, iar pe subintervalul micarea este ncetinit. Pentru ca micarea s fie simetric, se pun condiiile:
pentru x =1/2, deplasarea tachetului s fie y = 1/2;
cele dou parabole s admit derivat unic n punctul de coordonate [1/2 ;1/2] .
Fig.3.3a
a). Subintervalul ;
Ecuaiile generale sub form adimensional sunt:
Pentru determinarea coeficienilor A i B se pun condiiile la capetele intervalului:
C = 0;
.
y =1/2 A = 2.
Prin nlocuirea coeficienilor A i B cu valorile lor, se obin ecuaiile legii parabolice, sub forma:
b). Subintervalul ;
Ecuaiile generale sub form adimensional sunt:
Pentru determinarea coeficienilor A i B se pun condiiile la capetele intervalului:
C = 1;
;
y =1/2 A = - 2.
Prin nlocuirea coeficienilor A i B cu valorile lor, se obin ecuaiile legii parabolice, sub forma:
Graficele de variaie sunt date n figura
EMBED Excel.Chart.8 \s
EMBED Excel.Chart.8 \s
Fig.3.3b
3.4. Legea sinusoidalEcuaiile generale sub form adimensional sunt:
unde ; . Legea sinusoidal corespunde unei micri variate. In prima parte a intervalului, micarea este accelerat, iar n a doua parte a intervalului, micarea este ncetinit. Pentru a asigura simetria micrii, se pune condiia ca funcia de transmitere a acceleraiilor () s se anuleze la jumtatea intervalului.
Pentru x = 1/2. i = 0 , rezult:
EMBED Equation.3
Pentru determinarea coeficienilor A , C i D se pun condiiile la capetele intervalului:
D = 0;
C = 1;
EMBED Equation.3 .
Prin nlocuirea coeficienilor A i B cu valorile lor, se obin ecuaiile legii Sinusoidale sub forma:
Graficele sunt date n figura 3.4.
EMBED Excel.Chart.8 \s
EMBED Excel.Chart.8 \s
Fig.3.4
3.5. Legea CosinusoidalEcuaiile generale sub form adimensional sunt:
unde ; .
Legea cosinusoidal corespunde unei micri variate. In prima parte a intervalului, micarea este accelerat, iar n a doua parte a intervalului, micarea este ncetinit. Pentru a asigura simetria micrii, se pune condiia ca funcia de transmitere a acceleraiilor () s se anuleze la jumtatea intervalului.
Pentru x = 1/2 i = 0 , rezult:
EMBED Equation.3
Pentru determinarea coeficienilor A , C i D se pun condiiile la capetele intervalului:
D + A = 0;
-A + C + D = 1;
C = 0. Rezult: ; ; C = 0.
Prin nlocuirea coeficienilor A, B i C cu valorile lor, se obin ecuaiile legii Cosinusoidale sub forma: ; ; .
Graficele sunt date n figura 3.5.
EMBED Excel.Chart.8 \s
EMBED Excel.Chart.8 \s
Fig.3.5
3.6. Legi Polinomiale de grad superior
Ecuaia general a deplasrii sub form adimensional este dat de un polinom de gradul r :
unde ; .
S-a considerat un polinom cu patru termeni pentru care: , iar toi termenii necunoscutei de grad mai mic dect m sunt zero.
Pentru determinarea coeficienilor se pun condiiile la capetele intervalului.
Din condiia x = 0 ; y = 0 se determin termenul liber. Aceast condiie este satisfcut de ecuaia deplasrii.
Ceilali coeficieni ai polinomului sunt determinai din condiia la captul intervalului, x = 1 ; y = 1 i condia ca toate derivatele de ordinul (r 1) s se anuleze: , , ... ,,
In cazul polinomului considerat, pentru a determina cei patru coeficieni , , , sunt necesare patru ecuaii. Acestea sunt date de ecuaia deplasrii i primele trei derivate:
Substituind: x = 1 ; y = 1 ; ; ; , se obine:
Sistemul poate fi pus sub forma:
Determinatul principal este de tip Vandermonde, iar coeficienii se determin cu relaiile:
; ;
; .
3.6.1. Legea polinomial 4 6 8 - 10
Se d : ; ; ; .
Se determin coeficienii:
;
; .
Ecuaiile legii polinomiale 4-6-8-10, sub form adimensional sunt:
;
;
.
Graficele sunt date n figura 3.6a.
EMBED Excel.Chart.8 \s
EMBED Excel.Chart.8 \s
Fig.3.6a
3.6.2. Legea polinomial 2 3 Se d : ; .
Se determin coeficienii:
; .
Ecuaiile legii polinomiale 2-3, sub form adimensional sunt:
;
;
.
Graficele sunt date n figura 3.6b.
EMBED Excel.Chart.8 \s
EMBED Excel.Chart.8 \s
Fig.3.6b
Aplicaia 1. S se determine funciile de transmitere reale : ; ; pentru legile de micare :polinomial 2-3, polinomial 4-6-8-10, liniar, parabolic, sinusoidal; cosinusoidal;, cunoscnd ecuaiile sub form adimensional. Exemplu: legea polinomial 2 3 are ecuaiile adimensionale: ; ; .
Cunoscnd c
; ; , i c
rezult:
; ;
.Lucrarea nr.1 Funciile de transmitere cu mrimi adimensionale i cu mrimi realeNr
crtLegea
de micareFunctiile de transmitere cu mrimi adimensionaleFunctiile de transmitere cu mrimi reale
1Polinomial2-3
2Polinomial4-6-8-10
3Liniar
4Parabolic
Parabolic
5Sinusoidal
6Cosinusoidal
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