Consider a simple pendulum (mass m, length l) whose point of support is forced by some external mechanism to jiggle up and down according to xo = R sin wt. Write down the Lagrangian and find the equation of motion for the angle o. [Hint: Be careful writing down the kinetic energy T. A safe way to get the velocity right is to write down the

Consider a simple pendulum (mass m, length l) whose point of support is forced by some external mechanism to jiggle up and down according to xo = R sin wt. Write down the Lagrangian and find the equation of motion for the angle o. [Hint: Be careful writing down the kinetic energy T. A safe way to get the velocity right is to write down the

Power System Analysis and Design (MindTap Course List)

6th Edition

ISBN:9781305632134

Author:J. Duncan Glover, Thomas Overbye, Mulukutla S. Sarma

Publisher:J. Duncan Glover, Thomas Overbye, Mulukutla S. Sarma

Chapter6: Power Flows

Section: Chapter Questions

Problem 6.27P

See similar textbooks

Similar questions

Fig. Q1(b) shows a rotational mechanical system with two masses with moment of inertia, J1 and J2. The T(t) is the applied torque to J1, while the θ1(t) and θ2(t) are the angular displacement of J1 and J2, respectively. Draw the free-body diagrams of mass with moment of inertia J1 and mass with moment of inertia J2, and then develop differential equations (with constant J1, J2, D1, D2, and K in the equations) that represent the mechanical system. Then using the given value of the constants in Figure Q1(b), derive the transfer function G(s) = θ2(s)/T(s) and the block diagram of the system.
In the system shown in the figure below, the disk of inertia J and radius r is constrained tomove only about the stationary axis A. A viscous damping force of translational value fvexists between bodies J & M. If an external force f(t) is applied to the mass, find thetransfer function G(s) = Θ(s)/F(s)
An inclined plane of angle θhas a spring of force constant k fastened securely at the bottom so that the spring is parallel to the surface as shown. A block of mass m is placed on the plane at a distance d from the spring. From this position, the block is projected downward toward the spring with speed v. Calculate By what distance is the spring compressed when the block momentarily comes to rest? develop an equation for x. x when θ=30.0° , k = 1 kN/m, m = 5 kg, d = 0.5 m, and v =1 m/s. If we add another spring in series, by what distance is the spring compressed when the block momentarily comes to rest? If we add another spring in parallel, by what distance is the spring compressed when the block momentarily comes to rest?
Consider the continuous-time model of the overhead crane proposed in Problem 7.10 with mc = 1000 kg, ml = 1500 kg, and l = 8 m. Design a discrete full-order observer state feedback control to provide motion of the load without sway. Problem 7.10 The following differential equations represent a simplified model of an overhead crane:3 where mC is the mass of the trolley, mL is the mass of the hook/load, l is the rope length, g is the gravity acceleration, u is the force applied to the trolley, x1 is the position of the trolley, and x3 is the rope angle. Consider the position of the load y = x1 1 + sinx3 as the output. (a) Determine a linearized statespace model of the system about the equilibrium point x = 0 with state variables x1, x3, the first derivative of x1, and the first derivative of x3. (b) Determine a second statespace model when the sum of the trolley position and of the rope angle is substituted for the rope angle as a third state variable.
A metal rod having a mass m and length l has resistance R. The rod hangs from two vertical metallic wires of length l in a uniform vertical magnetic field B of a horseshoe magnet (see figure). The wires have negligible resistance. An AC voltage source is connected between the wires which provides a voltage U(t) = U0 cos ωt, where U0 and ω are constants. a) Show that for small angular displacements φ the motion of the rod can be described by the equation U(t) = αφ ̈ + βφ ̇ + γφ, and express the constants α, β, γ in terms of quantities m, l, B, R and the gravitational acceleration g. b) At a certain frequency ω = ω0 the amplitude of the oscillations of the rod becomes large and the equation above does not describe themotion anymore. Estimate the frequency ω0.
Find the damping ratio, natural frequency, peak time, % overshoot, and settling time for a system with the following pole plot.
7.Differential equation for a simple wheelchair with a suspension system (bumper) can be written as: ?2???2+??????+????=1???. Fe is the gravitational force exerted by the sitting person, 800 N; “m” is mass of the wheelchair, 25 kg; “Ks” is the stiffness (inverse of compliance) of the spring, 2500 N/m; the damping coefficient of the dashpot “B” is 500N/(m/s). How much is the damping coefficient ?of the system? The original of the problem is attached.
The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N45°W at a speed of 50 km/h. (This means that the direction from which the wind blows is 45° west of the northerly direction.) A pilot is steering a plane in the direction N60°E at an airspeed (speed in still air) of 250 km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane.
Consider the rotational translational mechanical system as shown in Figure Q1(a): Draw the free-body-diagrams of mass with moment of inertia, J, and then develop the differential equation that represent this mechanical system.
Consider an infinitely long wire in the form of y2=4cx, which is in a stationary state. Another infinitely long wire parallel to the y-axis starts moving with an initial velocity of zero at t=0 from the initial position at x=0 with acceleration a ax. Assuming a magnetic flux density (magnetic induction) of B=B0 az throughout the region, determine the induced electromotive force on the moving wire as a function of time using Faraday's law.
Draw the acceleration-time, velocity-time, distance-time graph of a particle that has a constant acceleration of 9m/s^2 for 6 seconds and a starting velocity of 5m/s, what is the final velocity and total distance traveledby the particle. (starting distance traveled=0m)
In the simple, one loop, one turn, 2-pole, DC machine: R=0.5 Ohms, VB = 180 V, r= 0.45 m, B = 0.4 T, and the length of the conductor L= 0.9 m. Find the starting current i and the steady state speed of the loop at no-load.