# System Dynamics

1. The equations of motion of this system are +3y + 4y -32-4Z = 0 2 +52 +62-5ý-6y = f(t)

1. Find the transfer function.

For each of the ODEs in the 1st column, indicate whether it is: 1. linear time-invariant (LTI), linear time-varying (LTV), or nonlinear 2. 1st order, 2nd order, or higher order 3. homogeneous or inhomogeneous by marking the appropriate column. The unknown function x(t) represents the state of some mechanical system and t represents time. Hint: An ODE is considered nonlinear only if the nonlinearity involves the unknown function x(t).

Consider the following IVP: with initial condition x(to) = -10 and to = 0. x*+2x=0 1. What is the particular solution, x(t)? 2. What is the value of x as time t → ∞? A. x → →∞0 B. x → -10 C. x→0 D. x → +10 E. x → +∞

Consider the ODE x* +2x = e-2t with initial condition x(to) = 10 and to = 0. What is the particular solution, x(t)?

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

For each of the following linear, time-invariant, 2nd order homogeneous ODEs solve for the particular solution that satisfies the initial values given.

Match each one of the responses shown below (Cases 1-4) with one of the following IVPS

1. What is maximum force in the tow cable and when does it occur? Does this maximum force occur when the submarine goes through its maximum acceleration? Explain your anwer. 2. What is the elongation in the tow cable due to the drag of the submarine at steady state? 3. What is the maximum velocity of the submarine and when does it occur? You recommend that the senior officer should select a cable length that would minimize the value of the peak force in the cable, while keeping the cable from being unnecessarily long. Therefore, you suggest that the system should have a damping ratio = 0.707., 1, 1.3 1. Find the length of the cable for damping ratio = 0.707., 1, 1.3 2. Adjust the gain values of your block diagram to agree with this cable length. 3. Simulate and determine the maximum velocity and maximum cable force. Compare the results with the 300 ft. cable. 4. What is the physical significance of = 0.707., 1, 1.3 in a dynamical system in relation to the shape of the response? Why is it the advantage of the cable with = 0.707. over 1, 1.3?

Analyze the problem statement to prepare a free body diagram of the submarine (shown in Figure 1). Use the prepared free body diagram to find the equation of motion for the submarine. Prepare diagrams of the model under various conditions in Simulink and run each simulation for 30 seconds. Use the given damping coefficient values to determine the stiffness values. Run each simulation using the stiffness values calculated in step 4. Determine the maximum velocity of the submarine and the maximum force and elongation in the towing cable for the system using each calculated stiffness value. Plot all results, then compare the velocity, towing force, and elongation for each set of conditions.

7:02 1 Done 5 -dub-prod.instructure.com AA ○ 20 points The figure below illustrates a two-degree-of- freedom system. Mass m₁ = 10 kg is located at the end of a clamped beam with a mass of my = 10 kg, a Flexural rigidity of EI = 4000 Nm², and a length of L = 2 m. Mass m₁ is connected to mass m2 = 20 kg using two springs, k₁ = 1000 N/m and k2 = 3000 N/m, as shown in the figure. Determine the second mode shape vector? beam mp (-3.75) (9) (-4.24) (-5.56) Return (-2.251 Submit 7:02 1 Done -dub-prod.instructure.com AA ○ freedom system. Mass m₁ = 10 kg is located at the end of a clamped beam with a mass of m, 10 kg, a Flexural rigidity of EI = 4000 Nm², and a length of L2 m. Mass m₁ is connected to mass m₂ = 20 kg using two springs, k₁ = 1000 N/m and k23000 N/m, as shown in the figure. Determine the second mode shape vector? (-3:75) (71²¹) (-4,24) (-5.56) E (-2,25) Return Clear my selection Submit - beam mp/n8:16 1 Done -dub-prod.instructure.com Tastrate a Live augice Un Mass m1 = Return 10 kg is located at the eam with a mass of my = 10 kg, of EI = 4000 Nm², and a length m₁ is connected to mass ; two springs, k1 = 1000 N/m /m, as shown in the figure. ond mode shape vector? beam mp (-3.75) (¹) (-4:24) (-5.56) (-2:25) Clear my selection .. k₁ k₂ 4G O AA Submit m₁ ↓ www X1 m₂ ↓ X2

2.) Derive the relationship between the output, the potential difference across the resistor R (vR),and the input v for the series LCR circuit shown in Fig 17.23 pp 437. Solve the same problem if the output is considered to be the voltage drop across the inductor. [20 pts]

1.) Derive an equation relating the input force F, with the output, displacement x, for the systems described in the textbook by Fig 17.20 (both systems a) and b) )on page 437. [20 pts]

4.) Figure 17.26 on page 438 shows a thermal system involving two compartments, with one containing a heater. If the temperature of the compartment containing the heater is T1, the temperature of the other compartment is T, and the temperature surrounding the compartments is T3, develop equations describing how the temperature T,and T2 vary with time. All the walls of the containers have the same thermal resistance and negligible capacity. The two containers have the same thermal capacitance C. [20 pts]

Consider the differential equation below. m \ddot{y}=k(z(t)-y)+b \frac{d}{d t}(z(t)-y) where y(t) = Yejwt and z(t) = Zejwt Ya) Solve for the frequency response, i.e., the ratio Y/Z ) Assuming that the damping ratio is 3 = 0.05 and that m = 1, plot the magnitude of the transfer function vs. frequency of the response of part (a), i.e., plot Y/Z vs.(W/WN). Use a logarithmic scale for the x-axis. ) Use the plot in part (b) to explain for which range of frequencies one can ignore z(t) in the original differential equation (i.e. set z(t) to zero). Assuming that the damping ratio is = 0.05 and that m = 1, plot the phase angle of the transfer function vs. frequency of the response of part (a) (i.e. the phase angle of Y/Z vs. (W/WN). Use a logarithmic scale for the x-axis. (e) Assuming that 3 = 0.1, and m = 1, use the plots in parts (b) and (d) to explain for which range of frequencies one can ignore all the derivative terms in the original differential equation. ) With the same assumptions as in part (e), use the solution for part (a) to find y(t)when z(t)10sin(3t).= Consider a car moving at a constant speed Vo on a wavy road where the "waves"have a peak-to-peak distance (wavelength) of L, and an amplitude of Z, inches.Show that this system will result in the same differential equation provided at the beginning of the problem.

1. (8 points) the figure below shows the responses yi(t), y5 (t), ..., yž (t) of eight different processes to a unit impulse input. the transfer functions of the eight processes are listed below in a shuffled order. match the transfer function with the response and justify each using analytical mathematics (poles, time constant, gain, damping coefficient, and the first- and second-order impulse response we derived in class). i.e., while I encourage you to use Controlz.jl to check your answers, "the simulation matched" is not an acceptable justification. G1(s)= 6/3s+1 corresponds to panel (a,....B,....,h) because G2(s) = 5/2s+1 corresponds to panel ( a,b....h)______ because =. G3(s) = 12/6s+1 corresponds to panel (a,b,....h) because . G4(s) = 6 / 9/4s2+3/5s+1 corresponds to panel ( a,b,....h) because G5 (s) 6/9/4s2+3/2s+1 corresponds to panel (a,b,...h) because 6.G6(s) =6/9/4s2+ 9/2s+1 corresponds to panel (a,b, ....,h) because 6.G7(s) = 6/9s2+9s+1 corresponds to panel (a,b,....h) because 6.- G8s(s) = 6/9s2-3/10s+1 corresponds to panel ( a,b,....h) beacuse

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 Mars rover autonomous vehicle includes a robotic arm for collecting rock samples. The dynamics of the robotic arm system have been analysed, and a root locus obtained for changes in a controller gain k varying from 0 to oo. The root locus is shown in the attached Root Locus.pdf document, and available for download from the MEC321 Blackboard course pages. a) Write down the start points and end points of the root locus diagram. b) Describe in detail how the system's transient response changes as k is increased. c) The designers wish to achieve a damping ratio of 0.423 from the system, but with the fastest possible settling time. (i) Use the magnitude condition to determine the required value for k, noting the need for the fastest possible settling time. d) Write down the fastest possible settling time for the system, and briefly explain why this is the maximum possible value.

. 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.

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)

: 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.

4. Estimate the friction drag on a 6ft wide and 18ft airplane wing assuming it is a flat plate. The airplane is flying at 300ft/s. Assume air is in compressible and density is0.00204slug/ft and viscosity is 0.364×10ʻlb.s/ft² (Hint:Lecture Notes)

3Pulse Width Modulation a) We will now modulate the 555 output pulse width by applying a voltage level to the control pin of the 555 (at pin 5). Set the input trigger frequency to 1 kHz and use a 0.1 µF capacitor. Record the width of the output pulse as a function of the control voltage input for the following values. Question 5 What happened between control voltages of 3.33, 4, and 5.00 V? Explain this behavior in terms of the equation you just derived. Question 4 Why is the 555 output pulse width the same for a control voltage of 3.33 V as it is when there is no input control voltage? Explain why and how the output pulse width depends on the control voltage. Derive an equation describing the effect of the control voltage on the pulse width and compare your results to theoretical values from the equation. b) Show the output pulse width from simulations when control voltages are at 3.33, 4 and 5.00V.

5 Given the transfer function G(s)=\frac{2 s+3}{3(s+8)} compute the frequency response y (r for the input u(t) = 0,3 cos 5t.

1. (2.5 points) Solve the following ODE using the Laplace Transform approach. \ddot{y}(t)+7 \dot{y}(t)+10 y(t)=4 y(0)=\dot{y}(t)=0 \text { Note: } \mathcal{L}(1)=\frac{1}{s} \text { and } \mathcal{L}\left(e^{-a t}\right)=\frac{1}{s+a} \text {. }

4. Find the inverse Laplace transform of: F(s)=\frac{5(s+2)}{(s+1)(s+3) s^{2}} Show your calculations.

5.1Derive the state-variable equations for the system that is modeled by the following ODES where a, w, and z are the dynamic variables and v is the input. 0.4 \dot{\alpha}-3 w+\alpha=0 0.25 z+4 z-0.5 z w=0 \ddot{w}+6 w+0.3 w^{3}-2 \alpha=8 v

Consider the first-order system given below. T_{1}=\frac{8}{s+8} i) Find the 2% settling time. ii) Find the rise time. iii) Given the quantities above, sketch and label the time response of the system.

IFigure P10.1 shows a general feedback control system with forward-path transfer functions Ge(s) (controller) and Gp(s) (plant) and feedback transfer functions H(s). Given the following transfer functions,determine the closed-loop transfer function T(s) = Y(s)/R(s).

Given the 1/0 equation 2 y+10 y=3 u compute the frequency response y,,() for the input u(r) 18 sin 4r.

Solve the following differential equation by the trial solution: \ddot{y}+25 y=1 It has initial conditions y(0)=1 / 2 and \dot{y}(0)=1 / 10 The answer will have the form A * \cos 5 t+B * \sin 5 t+C What is

A particle which moves in two-dimensional curvilinear motion has coordinates in millimeters which vary with time / in seconds according to x 5i2+4 and y 2r3 +6. For time t= 3 s,determine the radius of curvature p of the particle path and the magnitudes of the normal and tangential accelerations.

s Use MATLAB 10 plot the Bode diagram for the 1-DOF mechanical system in Problem 9.11 (Fig. P9,11). Estimate the frequency response for the position input v) = 0.04 sin 50r m by reading the Bode diagram(indicate the frequency response parameters on the plot of the Bode diagram). Obtain a more accurate answer by using MATLAB's bode command with Ieft-handside arguments for computing magnitude and phase angle.

7. The sports car and the driver have a total mass of 2500kg. The projected front area of the car is 0.78m?. The car is traveling at50km/h when the driver puts the transmission into neutral and allows the car to freely coast until after 150s its speed reaches 40km/h. Determine the drag coefficient for the car, assuming its values is constant. Neglect rolling and other mechanical resistance. (Hint: Watch"Chapter 9-Finding CD Value.mp4")

Problem 5b (10 points total): In class, you have derived the response of a first-ordersystem to a unit-step input. Given a first-order system of the form G(s) = K/(1+ Ts),where T is the time-constant, and K is the constant, find:%3D i) The time-response to a unit-ramp input r(t) = t. (7 points) ii) The steady-state error for error measured as e(t) = r(t) - c(t). (Hint: the steady-state error's measured as t tends to infinity). (3 points)