Question

3. Consider the equation of motion of a single-degree-of-freedom system: mi + ci + kx = F(1) Derive the condition that leads to divergent oscillations in each of the following cases: (a) when the forcing function is proportional to the displacement, F(t) = F,x(t); when the forcing function is proportional to the velocity, F(t) = Fox(t); and (c) when the forcing function is proportional to the acceleration, F(t) = F,x(t).

Answer 1

Solution:

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solution (12 (11) given, moč + cod + Ko=F matcátkol = Fox mätcod +(K-Fo) x = 0 Laplace forums Form, LCX(+)] = X(5) L [oi (A)] = sX(3) L [6i (4)] = 5 x (5) Therefore, m 3*XC9) + C5 X(5) + CK-FO) *(3) = 0 [ms + CS +(K-Fo]*(5) = 0 s? + ES + CK-F) = 0 (2) S2 = =p-ig Roots: si optiq with P and a are real numbers. polynomial Equation: (5-51) 65-52)=3-(31+52) 5 + 5152 ² - (51752) 5 + 5,52 = 8 + C s +(K-FO) - (S1+52) = -(Cotia +(P-11)) = -2P herre, (70 There Fore, (K-FO) = 5,52 (K-FO) - (P+i 9) (P-197 = p2+92 here, K-Foz 0 = K3 Fo The dumping co-efficient must be positive, c>,0 The stifness must be eque or greater than the amplitude OF Force, k> Fo

(6) mätcat kx =F mät coc + Kx = Food mic + ((-Fo) oi+ ka=0 L (16+)] = X(5) L Coc (t)) = $ XC5) 1 [ši (t) ] - ${xC5) There Fore, mŠ x (s) + ((-fo) s X45) + KX65) = 0 [ms+ CC-FO)S+K] x(3) = 0 $+ CC-fo) s + =0 Roots: Si=pria 52 = pria 1 polynomial Equation: (5-5)) (5-52)= 5- (51752)5 +5,52 S-651752)5 + 5152 = $+ CC-fo) st 20 m (c-Fo) = -(51752) (C-FO) -- (Pria +P-19) =-2P m here, c-Fo 7,0 = (> Fo k = 5,2 k = (Prig) (P-19) = p²79² m K3,0 The damping coefficient must be Equal or greater them the amplitude of Force. C> Fo The stifness must be positive, kao

(C) mä +catka =F (m-Fo) si + cåt kx=0 Laplace transform, L["(t)] = X(5) 1 Cocct)] = 3 XC5) [së ct)] = $x(5) There Fore, (m-f)x(s) $ + esxc5)+ FX CS) = 0 [cm-Fo) sº + C5+*] (5) = 0 Št. stk (m-Fo) m-Fo Roots: pria = si p-ig = S2 / polynomial Equation: (5-51) (5-52)=8-(51+52) 5 + 5,52 5 - (5, +52) 5 + 5152 = 8 + CC-FO).S+ E = 0 m Analyze co-efficient s. c =-(51752) m-Fo с m-fo .(priat p-iq) =-2P a »0 = c>m-Fo m-fo pe+9² = 51 52 =) k= (M-F) (Pria) (P-14) ) K = m-fo m-fo K 30 K (m-Fo) m-fo The damping coefficient, c> (m-Fo) and The stifness must be, k (m-Fo) Thank you.