System Dynamics

For each of the following ODEs determine if the eigenvalues are (a) real and distinct, (b) repeated, (c) complex conjugate pairs:

3.) Derive the relationship between the height h, and the time for the hydraulic system shown in Fig 17.25, pp 437. Neglect inertance. [20 pts]

5.) Derive the differential equation for a motor driving a load through a gear system as shown in Figure 17.27 on page 438, which relates the angular displacement of the load with time.

2. Determine the total response of the 2nd-order linear differential equation with initial conditions, y(0) = 1 and ÿ(0) = 1 \ddot{y}+5 \dot{y}+4 y=8 Determine the characteristic equation and root, the homogeneous solution y#(t), the particular solution yp(t) and the total solution y(t). Plot the homogeneous, particular and total solution with MATLAB.

Consider the closed-loop control system shown below: whcre K1 and K2 arc the positive constants. Derive the closed-loop sensitivity function: S(s) = E(s)/R(e). (2) (2.5 points) Determine K, and K2 such that wn = 4 rad/sec, and t, = 1 sec. Note:uhere u and t are the natural freguency and damning ratio respectivel: t_{s}=\frac{4}{\omega_{n}} \text {, where } \omega_{n} \text { and } \zeta \text { are the natural frequency and damping ratio, respectively. }

6. Do Problem 9.20 from the textbook. [Statement: A smooth, flat plate of length I= 6m and width b = 4 m is placed in water with an upstream velocity of U = 0.5 m/s. Determine the boundary layer thickness and the wall shear stress at the center and the trailing edge of the plate. Assume a laminar boundary layer.]

. A system with controller gain K has the following transfer function: \frac{s+2}{s^{4}+12 s^{3}+4 s^{2}+3 s+K} Showing your working, use the Routh method to determine the range of the values of K which will achieve a stable response.

: Consider the following spring-mass-damper system: Draw free body diagrams for each mass. i) Write the equations of motion for each mass as differential equations in the time domain. iii) Convert the equations of motion for each mass into algebraic equations using the Laplacetransform. (Assume zero initial conditions.) (3 points) iv) Solve for the transfer function G(s) = X2(s)/F(s). (You do not need to simplify your answer orconvert the transfer function back to the time domain.) (3 points) v) Find a state-space representation of the equations of motion.

1. The machine in the picture is a turbine working at 50H7 To mitigate the vibrations transmitted to the ground, the turbine is mounted on an isolator. The total mass of the turbine is 100Kg. We need to design the isolator such that the transmissibility is 0.5, with a natural frequency of vibration equal to the half of the working frequency of 50H7.Assuming that the isolator can be schematized by a spring and a damper in parallel, findthe values of k and h that satisfy the above described requirements

Determine expression of y(t) based on taking the FFT of the data. Data is tab delimited in 2 columns t and y(t). Verify your expression for y(t) by plotting your expression over top of the data from the file. Be sure to clearly label your plots and submit your code. (30 pts)