Question

Modelling of mechanical dynamic systems Exercice 1 An disc of an inertia J and radius r is attached to a fixed axis of rotation A as shown below. The disc is in contact with a mass M attached via a spring of stiffness K to a fixed wall. The inertia-mass contact is subject to viscous friction of coefficientſ. The motion of the mass with respect to the horizontal floor is subject to the same viscous friction coefficient f.. The system input is a horizontal force f(1) on the mass M and the output is the rotation Oft) of the disc. 1 Establish the Free-body diagram of the mass M and apply the 2nd Newton's low for linear motion x(t). 2. Establish the Free-body diagram of the inertia J (disc) and apply the 2nd Newton's low for angular motion O(t). 3. Write the equation of motion involving the angular motion 00 (suppose initial conditions Zero) and state if the system is linear or nonlinear 4. Find the transfer function of the system if f(b) is the input and (t) is the output

Answer 1

Let the angobbar displacement of the mass M bex, and angular dis placement of rotational body be o Then linear displacement of a point on the circumference of rotating body = +2=r0 free body diagram of rotating body i. f (x -x) Equation of motion :- JO + f (x2-x; ) 2 = 0 Jöt fro- fuxir = 0 Taking Laplace Transform. T²0(s) + fyrs Ols) - fyrs X, (s) = 0 free body diagram for mars Mi- for (x2 - 4) amxi Kime src) f(t). til Kilic f(x) Mxi - f. (x2-x) + fuxi + Kx = f (4) Taking Laplace Transform: - Msx,(s) + 2f,s x,(s) – fus X2Cs) + kXy66) - FC) FI MS*x,(s) + 28,5 xils) – Fosro(s) + kx,(s) = FC) Writing in matix form Ju+furs - furs Tols) 7 l-fors Me²+ 2F vs + K L

Solving by Cramer's Rule:- e ) 0 F(s) -fins M3 + 2fyrstk Isht furs - fors T-fors As + 28, stk) fors FG) (Tst for?) (M5 + 2F 3 +K) - (fors?? - F(s) (Testfr) (M5²4 283 tk) (fyrs) 2 which is the required transfer function.