3. The Duffing equation describes a nonlinear oscillator with a nonlinear spring (energy storage component) and possibly a damping component: d²x dx +c +kx+Ex³ = f(t), where c, k and & are constants. 2 dt dt ; Chow that the Duffing equation fails to meet the requirements for a linear operator equation. ;. • Develop the general solution for the case where & =0 and f(t) = 0, and the initial conditions are x(0) = xo, dx/dt(0) = vo; and show why the approach will not work for the case when & #0. For the case where c = 0 and f(t) = 0, the homogeneous DE falls into one of the categories of second order ODE's that can be integrated using a transformation of variables. Using that transformation, carry the solution process as far as you can.

answer completely and neatly

For point a, suggest a different form but the same equation without removing u

T C We consider a rotating shaft. A torque T is applied, resuting in a rotation of the shaft. The bearing system and the output connections yield a viscous damping torque D. T: input torque 2: output angular velocity Dan: viscous damping = J: moment of inertia Governing equation (obtained using Euler equation for rotating solids): JN+an = T αΩ