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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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• Bifurcation Theory:
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• 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

Expert help when you need it
• Q1:Consider the closed-loop system shown below:See Answer
• Q2: Problem 2. (a) A tachometer has an analog display dial graduated in 5 rpm increments.The user manual states an accuracy of 1% of reading. Estimate the uncertainty in the reading at 10 rpm, 500 rpm and 5000 rpm. (b) A certain obstruction type flow meter (orifice, venturi, nozzle), shown in the following figure is used to measure the mass flow rate of air at low velocities. The relationship describing the flow rate is: \dot{m}=C A \sqrt{\left[\frac{2 g_{c} p_{1}}{R T_{1}}\left(p_{1}-p_{2}\right)\right]} where, * C = empirical-discharge coefficient. * A = flow area * T1 = upstream temperature * R = gas constant for air * Pi and p2 = upstream and downstream pressures, respectively. Calculate the percent uncertainty in the mass flow rate for the following conditions:See Answer
• Q3: A non linear dynamic system is described by the following three state variable equations: \dot{q}_{1}=q_{2}-q_{3} \dot{q}_{2}=u+q_{2}+\frac{1}{2}\left|q_{1}\right| q_{1} \dot{q}_{3}=3 q_{1}+6 q_{3} Assume that \bar{q}_{3}=2 What is the value of \bar{q}_{1} ?See Answer
• Q4: 1. Given the electrical circuit shown below (a) Obtain an equivalent rotational mechanical system free body diagram. (b) Obtain an equivalent translational mechanical system free body diagram. (c) Obtain the state variable form of the system in terms of electrical elements. See Answer
• Q5: What is the value of \bar{q}_{2}See Answer
• Q6: If the equations are linearized around the normal operating point, they can be put into matrix form as \dot{\hat{q}}=A \hat{q}+B \hat{u} where q=\left[q_{1} q_{2} q_{3}\right]^{T} What is the value of the A matrix? See Answer
• Q7: 1. Consider the closed-loop system shown below: where K, > 0, K, > 0 and T > 0. Determine stability of the closed-loop system using Routh's stability criterion.See Answer
• Q8: - Consider the closed-loop system shown below: Find the sensitivity function (i.e. S(s) = E(s)/R(s)) (5 points) and calculate the steady-state error with the unit ramp reference (i.e. r(t) = t) (5 points) using final value theorem.See Answer
• Q9: Problem 5. In the Problem 4, calculate the steady-state errors with the unit step (r(t) = 1)(5 points) and the unit ramp (r(t)3Dt) (5 points) references.See Answer
• Q10: Problem 1. (a) A first order instrument is described by the following mathematical model: 2 \frac{d y}{d t}+30 y=120 The time constant of the system is 15 seconds. (b) The following spring-mass-damper system is: * Zero order * 1st Order * 2nd Order * 3rd Order * YES * NO (c) The natural frequency of the above system is: \omega_{\mathrm{n}}=\sqrt{\frac{M}{K}} * YES * NO (d) The damping ratio of the above system is: \zeta=\frac{f_{v}}{2 \sqrt{K M}} * YES *NO (e) A thermometer is an analog device (f) A pressure gauge is a digital device * YES * NO (g) The following signal has a frequency of 10-Hz. * YES * NO (h)A design stage uncertainty analysis is performed when there is a finite set of data available. * TRUE * FALSESee Answer
• Q11: Below are three differential equations. From each one, find a (i) transfer function and (ii)the state space representation. The values that you'll use for the coefficients are different for each student and can be accessed in the accompanying excel document. For both the TF and SS, use f(t) as the input and x as the output. a_{1} \ddot{x}+b_{1} \ddot{x}+c_{1} \dot{x}+d_{1} x=f(t) b_{2} \ddot{x}+c_{2} \dot{x}+d_{2} x=f(t) c_{3} \dot{x}+d_{3} x=f(t) See Answer
• Q12: a.Develop a model, i.e. differential equation(s), that describes the behavior of this system (10 points) b- What is one assumption you're making about the system? (5 points c. We are treating this as a linear time invariant system.What is a reason that the real system would not be linear? What is a reason that the real system would not be time invariant? (5 points)See Answer
• Q13: What flow rate, win, is needed to maintain a steady state level of 0.2 m (7.5 points)? 1 Solve for how long it will take for the system to reach 63.2% of its steady state value (you do not have to solve the DE to find this!) (5 points) See Answer
• Q14: 4. Give the differential equation to model the height h1 of the fluid system below. (Include cases where h1 > D and h1 < D.) See Answer
• Q15: 5. For the following fluid system: Give a dynamic-systems model. Assume that h > h1 + h2 and h < h1 + h2 + h3. Assume that the length of the tank is 1 (for example, the area of the base is w1 · 1 = w1). b. Suppose that qmi = 3 kg / s. If h1 = h2 = h3 = 1 and w1 = 2, w2 = 4, w3 = 6, and R = 5. Assume p= 1 and g = 10 m/s. Find the steady-state value of h. Turn in all problem statements with your homework. Box all answersSee Answer
• Q16: 3. Apply MATLAB to the previous problem in order to answer the following: а. Plot h1, h2 and h3 as a function of time when the input is the unit step function. Assume the initial conditions are h1 = 0, h2 = 10, and h2 = 5. Give all commands used to generate your plot. b. Plot the flows across each pipe as a function of time.See Answer
• Q17: 2. For the fluid system below а.Give the dynamic systems model for h1, h2, and h3 in state variable matrix form. b. Determine the height of the water in each tank in steady state, assuming that A1 = 2, A2 = 3,A3 = 1, R1 = R2 = R3 = 1, qmi = 5 kg / s, g = 10 m² / s, p = 0.1 kg / m³. See Answer
• Q18: The thin disk below turns on a massless arm B by way of a smooth bearing that keeps the axes of D and B aligned. The arm is hinged to the shaft driven at a constant angular speed N by a motor. The system is set up so that the arm is horizontal when the disk contacts the ground as shown. Assuming the disk of mass m rolls without slipping, find the force reactions and the moment reactions exerted by the arm on the bearings at point C in terms of the given dimensions, the mass,the normal force acting on the disk, the friction forces acting on the disk and the angular speed N. See Answer
• Q19: Recall, the Eulerian angles that we defined in class as shown below. The axes (i, j, k) are fixed in body frame B and the axes (Î, Ĵ, K)of B with respect to mathF is represented through the angles (ø, 0, v) using a sequence about intermediate z, intermediate y and intermediate z-axis again to obtain body-fixed frame B in the final configuration, from the inertial reference frame F. This is often referred to as 3-2-3 sequence are fixed in the inertial reference frame F. The orientation or z-y-z sequence. You are hired as an control and navigation engineer at a satellite manufacturing firm. The satellite is equipped with thrusters which can rotate it about all the possible axes in the intermediate body-fixed frame at any given instant instead of just the y and z-axes. Now, there is a requirement to use a 2-1-3 sequence or the y-x-z sequence of successive rotations about the intermediate axes, \boldsymbol{F} \stackrel{\hat{J} \text { or } \hat{n}_{12}, \phi}{\longrightarrow} \mathbf{F}_{1} \stackrel{\hat{n}_{11} \text { or } \hat{n}_{21}, \theta}{\longrightarrow} \mathbf{F}_{2} \stackrel{\hat{n}_{23} \text { or } \hat{j}, \psi}{\longrightarrow} \mathbf{B} 1. Write down the individual rotation matrices for each of the Eulerian angles, i.e. Tó, Tạ, Ty. 2. Write the simplest angular velocity vector wB/F in terms of Euler angles according to the2-1-3 rotation sequence. Remember, the simplest expression is always written using basis vectors of intermediate reference frames. 3. Express the angular velocity vector wB/F in the second intermediate reference frame, F2, i.e.in terms of (în21, Ñ22, îÑ23)See Answer
• Q20: 2. Consider the system shown in Fig. 2. A pulley 1 of radius R1 and moment of inertia I1 is connected another pulley A of radius R3 by mean of an unstrecthable belt. Pulleys A and B form a rigid body with a common hub and moment of inertia I2. a. Use the conservation of mechanical energy to find the equivalent system that can be used to replace the pulleys system (pulleys only). b. Write the kinematic equations relating the rotations between the pulleys. c. Use the equations found in a) and b) to determine the equivalent moment of inertia of the pulleys system. d. Determine the moment generated by the mass at pulley B and find the equivalent moment felt at the pulley 1. e. Write the equation of motion in terms of the angular velocity of the pulley 1 in case a moment M is applied at the shaft of the pulley 1. See Answer
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