Question

Equations of Motion: Lagrange's Method Use Lagrange's Method to find the Equations of Motion for the following systems. Define a datum point at the static equilibrium point, solve for the initial spring forces, and substitute them in to get simplified answers. M M

Answer 1

Solve for equations of motion,

lagrange's method. 315 fon Initially, in static equilibrium Condition, x=0 3K2 2 Kinetic energy of the system, KE = – Mr2 - 1. Potential energy of the system; 2 x2 + Mg x - , x² + 1 kan? е 5% big Pit - Rix 2 Calculate L. L= K.E-PE (= -1 k, 22 2 - 1 2 K 2 2 ² Mg x + 12 Mic²

t. 1 Me? - x2 / 2 x 2 mg r 0 1 1 0 - 2 k. (2x) - 2 kg (24) - Mg ey hinta-mg] pe Mio Now, equation of motion of the system, 27 d [mi] - [ch, x-kit-ng] =0 Mäi + 48, +42)x+mg = 0 - 0 Lequn of motion. Initial spring force; X(t = 0) = Est [ (ik, tk) &st = Mg] - 2 Substitute egun 2 in ① Mät (k, tkz) x + (k, + k ₂) Ist = 0 Mx + ( k, + k₂Xxx & st) = 0

Substitute X = xt6st x = x Simplify above equation, MX (k, tk) x=0 find equation of motion. TtFct) C Kinetic energy, T= 1 M ² Potential energy. U= 1 K x ² +mgn Dissipative energy, R = 1 / ² Q = f(t). fint, aT - Mi au = kx. tmg an OT 0. de = of-ci di

Equation of motion off ( 3 ) = Mix Ang - Mä -o + i + Kx=Q +mg - Mätcxt k ne f(t). - (Müt cx+ Kx + Mg = f(t) o Initially, equilibrium condition, too, - K&st + Mg +0= F(0) [ Mg = F(o)tk Ist - Substitute 2 in Mät in tkxt f(0)+ K8 st = F(t) =) Mx tatk(x++) = f(t)-f(o). » Mi 4 ( 4 k(x + 6+ - (+)- F(0)

let X = x + f + f(0) x= i =ä simplify the above equation; Mä t ä+ K (x + fs + + f(o)) = f(t) - Mrtcut KX=f(t) - MY + cÝ + KX=f(t) final equation of motion. aflt) ay Kinetic energy & 7. disipatins energy. - T Į Mih RTL-00) Potential ergy- V-LR6xy)? 4,4 %(vy) O=f(t) tmgr U = klx - y) = mga

> 2 kcery) + klx - y) ting ob = c (x - y) Abe lagsange's equn :- (25) -3+ - Mi - 0 + ¿cr-y) + 2k(ory) + Mg = f(t) » (Mž + ccm-y) + 2+ (x+y)+ My = Ft Initial equilibrium condition, (Mg - 2kSSHFO) - substitute equn & in equal Mätclx- y) + 2k(nry)+2k8 st f(0) = f(t)

Mix tchery) + 2 kln-y) +2k Ist + F(0)=f(+) Më tele- g) + 2k (x -y f S + + f(o)) = f(t)

simplify using x= x + 8 st 2 (x-7)= x-y + Set + F1 / Y = g - flo) Ý į

simplify for substituting in abone equn. MY+ c(8-4) +2k (x-7)= F(t) +Mx +2k (X-Y) + C(X-Y)=f(t)) final equation of mohon. Kinetic energy; T = Iw2 1L hI OM TE LCH127003 Potential energy, U= 1 60² + Mgh here, h= L(1- coso) (U= 160² + Mg LC1-coso) find; 2T . ME I . MEö also, 27. M17(0) = 0

30 6e + mg (0+ sin®) au = GO + Mg sino use lagranges equ" :- 00: Substitute all the values, ML²ö -0+0+ GO + Mg(sin O=0 - M120 +GO+Mg sino=0 - for small O, sindro IME²0 + (6 +mg L) 0 =0 final agen of motion.