solve both questions

Department of Mechanical Engineering Phone: +1.903.566.7003 Fax: +1.903.566.7148 Uttyler.edu/engineering MENG-5314 MEMS Homework Assignment #8 Due on Friday, April 26th, 2024, 11:59 PM Q1. Find the fixed points for the following systems and classify their stability using the graphical approach. a- 𝑥̇ = −𝑥 3 b- 𝑥̇ = 𝑥 3 c- 𝑥̇ = 𝑥 − 𝑥 3 d- 𝑥̇ = 𝑒 𝑥 − cos 𝑥 e- 𝑥̇ = 𝑥1/3 f- 𝑥̇ = 𝑥 2 Q2. Consider the single-sided capacitor of Fig. 1. Assume m=1.784×10−10 kg, k =10.96 N/m, c=0, d = 1.81 μm, and the electrode area A=510×100 μm2. Figure 1 Parallel plate electrostatic capacitor. (a) Plot the potential energy of the system and the corresponding phase portrait for VDC =1, 3, and 4.7 V. (b) Use direct-time integration of the equations of motion to plot the time history of the displacement for VDC = 4 V. Model the voltage load as a step load starting at zero time. Take the system’s initial conditions to be zero velocity and displacement. (c) Repeat (f), but with initial conditions of zero displacement and velocity equal to 0.055 m/s. (d) Plot the system’s potential energy for VDC = 4V and the total energy H for cases (b) and (c) and show the corresponding time trajectories on a phase portrait. (e) Draw conclusions from (c) to (d). Q3. Plot the bifurcation diagrams for the following equations if the control parameter is r. Show the bifurcation points and their types. a) 𝑥̇ = 𝑟 − 𝑐𝑜𝑠ℎ𝑥 1 𝑥 b) 𝑥̇ = 𝑟 + 2 𝑥 − 1+𝑥 c) 𝑥̇ = 𝑥 − 𝑟𝑥(1 − 𝑥)