Q5) The translational problem shown below has a linear and a nonlinear spring and is subjected to the applied force fa(t). a) Show that the nonlinear model is 2* + 6x + 3x + x² = fa(t). b) Solve for the operating point x0 when fa(t),= 10 N (assuming that x > 0). AX(s) c) Derive the linearized model and associated transfer function G(s) = A Fa(s) M=2kg fa(t) f:(x)= x² N B=6 N s/m Kj= 3 N/m Q6) The nonlinear resistor in the circuit shown obeys the equation Vr = 2 i3 0.5 H Vin(t) VR = 21 0.5i+3i+2i = v,(1). a) Show that the nonlinear model is b) Solve for the operating point io when vino = 22 volts. ΔΙ(s) c) Derive the linearized model and associated transfer function G(s) AV in(s)

2. Consider the circuit is shown in the figure below. The circuit comprises a voltage supply, v(t), resistor (R), inductor (L) and capacitor (C). Let x1=i and x2= v. be the state variables for this system, where i and v.are the circuit current and capacitor voltage, respectively. For this system, a) Derive the loop equation, involving integro-differential equations, from first principles Write the state differential equations b) c) Write the state-space equation in matrix-vector form d) Sketch the signal-flow graph. Show all your work. R

Analogue 2