Q2. A mass weighing 16 pounds stretches a spring 10 inches. (a) Derive the differential equation that would represent the above undamped mass-spring system. (b) Determine the period of motion if the mass is initially released from a point 1 foot below the equilibrium position with an upward velocity of 3 ft/s. (c) Now assume that the mass is attached to a dashpot device that offers a damping force numerically equal to 3 times the instantaneous velocity. Determine the motion (i.e., the solution) of the damped mass-spring system with the same conditions stated in (b).

Q2. A mass weighing 16 pounds stretches a spring 10 inches. (a) Derive the differential equation that would represent the above undamped mass-spring system. (b) Determine the period of motion if the mass is initially released from a point 1 foot below the equilibrium position with an upward velocity of 3 ft/s. (c) Now assume that the mass is attached to a dashpot device that offers a damping force numerically equal to 3 times the instantaneous velocity. Determine the motion (i.e., the solution) of the damped mass-spring system with the same conditions stated in (b).

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Similar questions

A 0.450kg mass oscillates at the end of a spring with the stiffness of 250 N/m. Thedamping force is proportional to the velocity –av and a= 8.50 kg/s. what is the frequencyof oscillation? What is the damping constant for a critically damped system?
+ b 3. A tuning fork is an example of a "resonant system", that is, one that has a low damping ratio. A model of a (half) tuning fork is rod of mass m with air drag modeled as a damper connected approximately at its middle, and a torsional spring at the base of the cantilever. Assume the cantilever rotates back and forth about its pivot by a small angle o. Because other forces are much larger, you can reasonably neglect the effect of gravity in this model. The moment of inertia of a cantilevered beam pinned at one end is equal to ml2, where I is its length. (a) What is the natural frequency of this system if m= 0.1 kg and k = 2548 Nm/rad, and 1 = 0.1 m. Please give the value in rad/s and Hz. (b) Find an upper bound on the damping coefficient b that insures that the damping ratio of the tuning fork is no greater than = 0.01. (c) For this damping ratio, what is the damped natural frequency of the system? (d) For this damping ratio, what is the decay rate (the size of the exponent…
A mass of 1 slug is attached to a spring whose constant is 25 /4 lb /ft. Initially the mass is released 1 ft above the equilibrium po-sition with a downward velocity of 3 ft/ sec, and the subsequent motion takes place in a medium that offers a damping force nu-merically equal to 3 times the instantaneous velocity. An external force ƒ(t) is driving the system, but assume that initially ƒ(t) = 0.Formulate and solve an initial value problem that models the given system. Interpret your results.
Fig. P4.6 4.22. Derive the differential equation of motion of the simple vehicle model shown in Fig. P4.7. The vehicle is assumed to travel over the rough surface with a constant vehicle speed v. Obtain also the steady state response of the vehicle. For this system, let m = 10kg, k = 4 x 103 N/m, c = 150N · s/m, i = 2 m, and the amplitude Yo = 0.1 m. Determine the maximum vertical displacement of the mass and the corresponding vehicle speed. Determine also the maximum dynamic force transmitted to the mass at the resonant speed. m 2πη y = Y, sin y Fig. P4.7
A mass weighing 4 pounds is attached to a spring whose constant is 2 ib/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equlibrium position with a downward velacity of 12 t/s. Determine the time (in s) at which the mass passes through the equilibrium position. (Use g -32 /s for the acceleration due to gravity.) Find the time (in s) after the mass passes through the equilibrium position at which the mass attains its extreme displacement from the equilibrium position. What is the position (in t) of the mass at this instant? ft
A 25-kg mass is suspended from a spring with a constant of 2 N/mm, which is in turn suspended at the end of a steel cantilever beam with a thickness of 3mm, a width of 20 mm, and a length of 250 mm. Determine the natural frequency of the motion of the weight. + Hints: Please use stainless steel for the cantilever beam in this problem. Table 5 attached at the end of this assignment may come in handy. m The natural frequency w₂ can be found as wn = where k is the spring stiffness of the system and m is the mass.
P3. The peak amplitude of a mass-spring-damper system, under a harmonic excitation, is observed to be 1 cm. The undamped natural frequency of the system is 5 Hz, and the static deflection of the mass under the maximum force is 0.5 cm. Now, instead of a harmonic excitation, a sudden impulse of 10 N.s is applied. Determine the response of the system. 4 z 0.5 cm
4.3. In a single degree of freedom undamped spring-mass system, the mass m = 3 kg, and the stiffness coefficient k = 2 x 103 N/m. The system is subjected to a harmonic forcing function which has an amplitude 60 N and frequency 30 rad/s. The initial conditions are such that xn = 0.0m and io = 1 m/s. Determine the displacement, velocity, and acceleration of the mass after t = 1 s.
A rotating machine of 400 kg is similar to the system shown below. It operates at 3600 rpm (note: 1 rpm = 2π/60 rad/s). The machine is unbalanced such that its effect is equivalent to a 4 kg mass located 20 cm from the axis of rotation. An isolator with a spring stiffness of 8x106 N/m and a damping constant of 2x104 Ns/m is placed between the machine and the foundation. Determine the steady state response of the system. Find the force transmitted to the foundation and transmissibility of the isolator. Find the damping ratio of the system ζ . Find the transmissibility of the system, Tf. Find the frequency ratio of the system, β. Find the amplitude of the harmonic excitation force of the system, Fo in Newton (N). Find the displacement amplitude of the steady state response of the system, X in millimeters (mm). Find the damped frequency of the system, ωd in rad/s. Find the force transmitted to the foundation, FT in Newton (N). Find the frequency of the harmonic…
The maximum permissible recoil distance of a gun is specified as 0.5 m. If the initial recoil velocity is to be between 8 m/s and 10 m/s, find the mass of the gun and the spring stiffness of the recoil mechanism. Assume that a critically damped dashpot is used in the recoil mechanism and the mass of the gun has to be at least 500 kg.
A 4.0kg body is suspended from a spring of constant k = 3.4kN/m. At time t = 0, it has a downward velocity of 0.5m/s as it passes through the point of static equilibrium. Determine the natural frequency of the system in rad/s. m Answer: Next page rious page P Type here to search
Q.4 A damped single degree of freedom mass-spring system is excited at resonance by a harmonic forcing function which has an amplitude of 40 N. The system has mass m of 3 kg, a stiffness coefficient k of 2700 N/m, and a damping coefficient c of 20 N·s/m. If the initial conditions are such that xo = 5 cm, and xo = 0, determine the displacement, velocity, and acceleration of the mass after t = 0.2 s.