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Dynamical systems have been a research topic since newton's time due to their significant role in the sciences. Dynamical systems answer many questions like, How does the solution behave as time goes to infinity? Describe turbulence. System dynamics engineering courses emphasize hands-on learning and a strong mathematics foundation. Due to it, most students need homework help but can’t find a reliable service. TutorBin's tutors provide the best system dynamics solutions to aid those students.

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Subjects Our System Dynamics Tutor Covers!

Our system dynamics tutor covers these topics:

Bifurcation Theory:

It addresses the issue of how parameter changes impact solution behavior. This subject has a solid algebraic/analytic bent. The learner will get knowledge of a variety of bifurcations. The Hopf bifurcation, in which a stable, steady-state solution transforms into an unstable periodic solution, is a crucial example.

Hamiltonian systems:

In these systems, at least one conserved quantity exists (energy). One type of similar transformation keeps a second quantity, such as area, volume, or a symplectic form. Such dynamical systems are standard and have unique characteristics.

Systems with mild initial conditions dependencies and chaotic systems:

This expands on a group of intriguing examples of hyperbolic dynamical systems that you can thoroughly examine using symbolic dynamics. Smale's Horseshoe is a significant classical example.

Ergodic theory:

Here, one learns about ergodic theorems (Birkhoff, Von Neumann), averages, invariant measures, and their significant generalization, the multiplicative ergodic theorem of Oscledec.

Theory of perturbation:

Since the sun is the most significant mass in planet motion, it alone essentially controls how each particular planet moves. The Keppler solutions, which result in elliptical planetary orbits, are comparable. In reality, other worlds with minor masses about the sun and possibly great distances from the planet of interest also affect how a single planet moves. As a result, it is possible to examine the variations in planetary velocity from a Keppler orbit in terms of power series that take into account the masses and distances of the other planets. It was at this point that perturbation theory was born. There are also significantly more advanced and contemporary techniques, such as the Kolmogorov-Arnold-Moser (KAM) theory.

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Recently Asked System Dynamics Questions

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Q1:Problem 6 (40 points) An underwater vehicle's motion in the yaw plane is described by the following simplified set of linearized equations: L m-Y; -Y. 0 0v Y U (Y-m)U 0 0 v Y U2 uudr -Ný I =- N, 0 0 r NU uv NU ur 00 r = N,s.U2 uuSr r (1.1) 0 0 10 y 1 0 0 U y 0 − 0 0 0 1 0 1 0 0 y 0 Yvdot = - 291.6078; Yrdot = - 16.4006; Mass = 265.3515kg; Nvdot = - 16.4006; Izz = 271.4616kg - m2; Nrdot = - 214.2764; Yuv = - 101.8686; Yur =91.6398; Nuv =- 233.5181; Nur =- 159.2130; Yu2dr = 61.8201; Nu2dr = - 44.6410; In compact form, Equation(1.1) can be written as Mx = Cax+BS, =Cax+ BOrudder where x =[v r y y]' . Equation(1.2) can be expressed in the traditional state space form as A11 A12 0 07 bị b (1.2) *= M-'Cax+M 'BS, = Ax+ Bô rudder 11 A21 A22 0 0 b2 0 1 0 0 A34 x+ 8 (1.3) rudder 0 1 0 0 0 A block diagram of a control system is shown in Figure 1. K Srudder Vemd 4 desired 2 e + 5 + Krp 4 H1(s) 4 H2(s) Figure 1: Schematic of yaw loop control for underwater vehicle (20 points) Compute the transfer functions H1 = rudder (s) i(s) and H2 = Smudder(s) ų(s) in terms of the components of the A and B matrices. Keep in variable form. (25 points) For a specified speed U of 20 knots and commanded yaw, l cmd , of 10 deg, verify that the rudder, Srudder, does not exceed a magnitude of 15 degrees for the following gains: K5 =1, Kwp =0.86,Kmp =- 0.3822. Use Simulink for this purpose and comment on the results and include a plot of the yaw command, the yaw response, and the rudder response on the same set of axes. 2See Answer
Q2:4. (40 pts) Consider a spring-mass-damper system shown below, where the input U(t) is displacement input applied to the mass m₁, and X₁(t), X₁(t) are the displacement of each mass, respectively. (Note that the input is displacement, NOT force) k2 W X2 k3 X1 w m2 k1 m1 c1 c2 (a) (8pts) Complete the free-body-diagrams of the two masses in the direction of motion. (b) (8pts) Derive the governing equations of motion of the system in the standard 2nd order differential equation form. (c) (8pts) Using the Cramer's Rule, find the transfer function that relates the motion of m2 (i.e. X2(t)) to the input U(t). (d) (8pts) Given that the system parameter values of m₁ = m2 = 1, c₁= c2= 1, k₁ = k3=1, k2=2, find the poles and the zeros of transfer function found in (c). (e) (8pts) Using the transfer function found in (c), find the time domain response x2 (1) when unit step is applied to the input U(s). Do not evaluate the partial fraction expansion coefficients, the coefficients and the phase in the time domain solution. Can you describe the motion x₂ (t)?See Answer
Q3:1. Given the input/disturbance system shown below: D(s) R(s) G(s) *** C(s) P(s) Figure 1 7 Let G(s) = 5 and P(s) = s+2 Hint: You'll find Section 7.5 of the textbook very useful for completing this problem. a) Calculate the steady-state error by hand for a command input R(s) = ³ with D(s) = 0. b) Verify the result of Part a) using a Simulink model of the system in Figure 1 (include a printout of the scope's output). c) Calculate the steady-state error by hand due to a disturbance input D(s) = with R(s) = 0. d) Verify the result of Part c) using a Simulink model of the system in Figure 1 (include a printout of the scope's output). e) Calculate the total steady-state error by hand due to a command input R(s) == and a disturbance D(s) = -applied simultaneously. f) Verify the result of Part e) using a Simulink model of the system in Figure 1 (include a printout of the scope's output).See Answer
Q4:1. Using the presented above causal loop diagram (Figure 1), develop SD model of new product propagation in the market 1.1 Investigate sensitivity of your model output parameter (time when number of adopters will reach 75% of the market (= potential adopters) to the values of model input parameters a, i, and c. As basis, use values: a = 0.015 (varies between 0.008 and 0.02), i=0.011 (can vary in the range 0.025 0.001), and c = 100 (varies in the range 80-120). In your opinion, which is most important parameter (to which model is most sensitive)?See Answer
Q5:Problem 1 (Points 3): Consider a closed-loop control system shown in Fig. 1 where: Ge(s) = Kp + K¹, Ga(s) = 1, H(s) = 1, G(s) = 1 2s +9 Find the values of the parameters Kp and K, to satisfy the following technical requirements: • The natural frequency is wn = 6rad • The damping ratio of the closed-loop system should be 0.5. Simulate this control system in MATLAB/Simulink. Error Reference Signal Input e 7+ Controller Gc(s) Sensor Signal Control Input U Actuator Ga(s) Sensor H(s) Actuating Signal ua System G(s) Figure 1: Closed-loop control system. Output ySee Answer
Q6:1 20 points Consider a single having equation of motion as 21.02 mm mass-spring-damper system * + 2x + 100x = 10t² + fot N where fo = 35 N/s. The response of the system at t = 0.15 sis 6.87 mm 10.97 mm 16.76 mm 1.01 mm #0/nThe figure below shows an equivalent single degree of freedom (SDOF) model of a wind turbine. The rotor and nacelle system mass is M = 2500 kg and the mass per unit length of the tower is m = 500 kg/m. Assume the tower as an uniform beam, with length L = 60 m, moment of inertia I = 0.8 m4, Young's modulus E = 200 GPa. Consider the wind turbine is subjected to a wind gust, modelled as f(t) = Fo (1-) N 0≤t≤ħ₁ 0 N t > ti where Fo= 12 kN and t₁ = 0.18 s. Determine the response of the system at t = 0.36 s. Assume damping ratio of the system is = 0.1./nNacelle Tower, -5.59 mm -0.62 mm -1.24 mm -3.73 mm -2.49 mm Rotor Ke ww x(1) Me F(0)See Answer
Q7:Problem-8: Programming in MATLAB (there are two parts, 8a and 8b, each worth 6 points). NOTE: Throughout the semester, whenever you generate plot(s) in MATLAB, please include appropriate x, y label, title and legend (if needed). You must submit the mfile, answers from the command window, and figure(s). One way to do this is to copy your code, figures and answers from command window and paste it on a word document. Then scan this document along with the rest of your work. 8a. Plot the function, y = 10cos(x)/x for the range 1 ≤ x ≤ 10 using a) fplot and b) "for" loop function. Have both the plots on the same graph. Refer example-1 in manual. 8b. Refer Problem-1/ part-1. Using ode45 function, solve the first order ODE and plot y(t) for 0 ≤t≤ 5 seconds. Use time steps = 0.001 seconds and y(0) = 0. Refer example-2 in manual.See Answer
Q8:Problem 7: The springs are undeformed when xi(t) = x2 = x3 = x4 = 0 at t = 0. Given X₁ > X3. x₁(t) m₁ K voo B1 [ K voo X3 # m4 m3 m₂ B3 B₂ B₁ + K voo K 000 a) Draw the free body diagram of the four masses. b) How many modeling equation(s) are there? Note: No need to find/write modeling equation(s).See Answer
Q9:Problem 6: Textbook Problem 2.16/ Page 43 (Parts a and b only); Take x2 > x1; The mechanical system shown in Figure P2.16 is driven by the applied force fa(t). When x₁ = x₂ = 0, the springs are neither stretched nor compressed. a) Draw the free-body diagrams and write the differential equations of motion for the two masses in terms of x₁ and x₂. b) Find x₁, and x20, the constant displacements of the masses caused by the gravitational forces when fa(t) = 0 and when the system is in static equilibrium. x1 3 vov K M₁ ele M₂ fa(1) K تعصف FIGURE P2.16 K ComSee Answer
Q10:Problem 5: Textbook Problem 2.6/ Page 40; Take x2 > X1; 2.6 In the mechanical system shown in Figure P2.6, the spring forces are zero when x₁ = x₂ = x3 = 0. Let the base be stationary so that x3 (t) = 0 for all values of t. Draw free-body diagrams and write a pair of coupled differential equations that govern the motion when the only input is fa(t). X3(1) K₁ voo B₁ M3 M₁ XXX FIGURE P2.6 x2 - fa(t) 000 K₂ M₂ B₂See Answer
Q11:Problem 3: Find the modeling equation for the system shown below. Let mA and mB be the mass of the two blocks A and B respectively. The blocks are at rest and the spring is undeformed at the instant t=0 when y = 0. Assume the pulleys to be ideal and the cables to be mass-less and inelastic. BSee Answer
Q12:Problem 2: Consider two blocks connected by a massless cable. The masses of blocks A and B are må and m² respectively. The coefficient of kinetic friction between both blocks and the inclined plane is uk = 0.1. The coordinates X₁ and x₂ measure the displacements of the two blocks such that x₁ = x₂ = 0 when the system is at rest. Find a single differential equation of motion for the system in coordinate x₁. Assume ideal pulleys. X1 60° 30% B iz KESee Answer
Q13: 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 ↓ X2See Answer
Q14:1 In 20 points Consider a single mass-spring-damper system having equation of motion as * + 2x + 100x = 10t² + fot N where fo = 35 N/s. The response of the system at t = 0.15 sis 21.02 mm 6.87 mm 10.97 mm 16.76 mm 1.01 mm Clear my selection DOSee Answer
Q15: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.See Answer
Q16:1. 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?See Answer
Q17:Match each one of the responses shown below (Cases 1-4) with one of the following IVPSSee Answer
Q18:For each of the following linear, time-invariant, 2nd order homogeneous ODEs solve for the particular solution that satisfies the initial values given. See Answer
Q19:For each of the following ODEs determine if the eigenvalues are (a) real and distinct, (b) repeated, (c) complex conjugate pairs: See Answer
Q20: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 → +∞See Answer

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