Consider the second order ordinary differential equation: d²x(t) dt2 dx(t) + 12- dt + 8x(t) = 8u(t) determine the following: a) The transfer function g(s) = x(s)/u(s) b) The natural frequency of oscillation (wn), c) The damping ratio (3), d) The damped frequency of oscillation (wa),
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- How can the Bode plot be used to analyze and design control systems in mechanical engineering applications?The motion of a damped spring-mass system is described by the following ordinary differential equation m * d2x/dt2 + c * dx/dt + k*x = 0 where x = displacement from equilibrium position (in meters), t = time (seconds), m = 20-kg mass, and c = the damping coefficient (N . s/m). Take the damping coefficient c = 5 and the spring constant k = 20 N/m. The initial velocity (dx/dt) is zero, and the initial (t=0) displacement is x = 1 m. Solve analytically.A certain mass is driven by base excitation through a spring (Figure P4.13). Its parameter values are m = 100 kg, c = 1000 N * s/m, and k = 10,000 N/m. Determine its peak frequency w_p, it’s peak M_p, and its bandwidth.
- A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): 1. What is the order of this system?A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): A. Use Laplace transform of the differential equation to determine the transfer function of the system.In this exercise we show that in the general case, exact recovery of a linear compression scheme is impossible. a. Let A ∈ Rn,d be anarbitrary compression matrix where n ≤ d−1. Show that there exists u,v ∈ Rd,u= v, such that Au = Av. Hint: Show that there exists u= 0,v = 0 such that Au = Av = 0. Hint: Consider using the rank-nullity theorem. b. Conclude that exact recovery of a linear compression scheme is impossible.
- A suspension system of Volvo articulated dump truck is to be modelled mathematically given the following data: keq = 10.5 MN/m, m = 30 tons, and c = 300 N-s/m. If the harmonic forcing function is 2000 cos 5t N with the initial displacement and velocity to be 0 and 30 m/s respectively, determine the steady state and total response of the system.A block spring system oscillates in a simple harmonic motion on africtionless horizontal table, its displacement varie with times according to x(t)=0,2cos(2t_3.14/4) the earliest time the particle reaches position x=0,1m isSuppose a spring with spring constant 7 N/m is horizontal and has one end attached to a wall and the other end attached to a 2 kg mass. Suppose that the friction of the mass with the floor (i.e., the damping constant) is 1 N⋅s/m a) Set up a differential equation that describes this system. Let x to denote the displacement, in meters, of the mass from its equilibrium position, and give your answer in terms of x,x′,x′′. Assume that positive displacement means the mass is farther from the wall than when the system is at equilibrium. b) Find the general solution to your differential equation from the previous part. Use c1 and c2 to denote arbitrary constants. Use t for independent variable to represent the time elapsed in seconds. Your answer should be an equation of the form x=… c) Enter a value for the damping constant that would make the system critically damped. ?Ns/m
- A 1.5-kg mass attached to an ideal massless spring with a spring constant of 20.0 N/m oscillates on a horizontal, frictionless track. At time t = 0.00 s, the mass is released from x = 0.0 cm with a velocity of 0.370 m/s to the left. Find the followingA. Time periodB. Total mechanical energy of the massC. AmplitudeD. Phase constant of motion. Discuss the two possible values of phase constant you get and explain how you arrived at the correct answer. Write the equation of motion.E. Maximum acceleration of the mass. (Acceleration is maximum when it ispositive). How long after the release does the maximum acceleration occur?F. Draw the position-time graph for one cycle of motion.Derive the governing differential equation for each system with the chosen generalized coordinate. SEE THE IMAGE BELOW Answers: 1. GDE: (5/2) mẍ + (5/4) kx = 0 2. GDE: (7/48) mL² ϴ [note: theta symbol has two dots above) + (3/8) cL² ϴ [ note: theta symbol has one dot above] + 5 kL² ϴ = 0A cantilever beam of length 700 mm, width 55 mm and thickness 15 mm is made of steel with Young’s modulus 200 GPa and mass density 7800 kg/m3. The displacement model is assumed to be w(x,t)=q1(t)x2L2w ( x , t ) = q 1 x 2 L 2 where L is the beam length. The damping in the beam is negligible. The beam is excited at location 300 mm from the clamped end with a translational force (in the more flexible direction) of magnitude 10 N at a frequency of 21 Hz (the end of the lecture on adding discrete elements to beam problems gives more detail about including external forces). The maximum steady state response of the end of the beam (in mm)