Question

A Mechanical system is shown in Fig. 1.1. a) Obtain a set of simultaneous integro-differential equations, in terms of velocity, to represent the system, where k is spring Constant, m is the mass and b is damper. (Hint: draw two Free Body Diagrams and obtain two equations in terms of x1 and x2 only). X1 - Ft mi m2 M TTTTTTTTYYTTTTTTTTTYYTTTTTTT Fig 1.1 b) What components and energy source do you need to use in order to build an electric circuit who has the same differential equations as you obtained in a)? Can you draw the electric circuit analog of the above mechanical system? Give a try.

Answer 1

a) The integral-differential equation in terms of velocity is derived in the attached image.

b) The series electrical circuit analog is derived in the attached image.The component and Energy source required are mentioned separately in the image.

a) damher Force = b i t bre Coefficient of Damhing i Velocity of end attached to damher Damher Force acts ophasite to the velocity of mass Inertia Force = më m Mass of System i - Acceleration of mass m Inertia Force acts of hosite to the direction of acceleration. biem, blå, ²) mã For writing Samping Force of Damher be we are assuming X2 is greater than %. To stor relatiwe molon between its 2 ends Damhing force of by will att be positive a direction on mi and negative x avection on M2, Writing the Equalion of molion using D'Alembert fremciple, mäi tani + Kx, - be (ág-Åi) - Fe = 0 mär +(+ba) *1 + K2, - baås - Fy0. morget Cb, toba) daxil Kx, – bdas F, = 0 div=v, dx = U, [ Velocity of mass ms] *s=\n de Un [valoky of mass ma]

X2=fuzdt. x = luidt Sulshitule the velocity relation in Eq" O, m, d (day) + (baitba) + k f1h1dt - belle = Ft Anleguo - Differenlich mi dui +(61762) 0,+ kſudt - bolle= F Ed" of mass my at 792 -22 be (2-2, k me k beža tuto Whiling the Equation of Molen using D'Alembert. Principle, mažy +be (hz-*i) + b₂%2 = 0 m₂ % z Hbe tbalů, - bekizo m2 dpm + (b: +65) dez - badacom Substitute the velocity selalion in above Gai?, m() + (by +ba) Us – b, 6,20 galego - wifenilal m, due + Obat b) ve-b90,=0| Equation of mass m2

b) We will need following Element for series analog: > 2 Inductor, Li 8 12, which are Mechanical sylem analog to mass m, & me reshechuvely > 1 Capacitou , es, which is Mechanical Syelem analog of Spring Ki. > 3 Rosi stance, R. , R, and Rg, which are Mechanical syslóm analog to Damher bi, bz, be respectively.... 1 Energy Sowice, élt) which is mechanical System analog of Force FE - L (M) G(A) L2 (m2) -omon HT mo- Energy Sowina L mm Ri(bi) Series Electrical Analoo of the Mechanical Syslom. → The . Favalion for Elecloucal Circuit is : 4 din +(ritre)in +ffüdt = e() Le diz + (Rethz iz - Radi=0.