Consider a mass-and-spring system containing two masses m₁ = 1 and m₂ = 1 whose displacement functions x(t) and y(t) satisfy the differential equations below. (a) Describe the two fundamental modes of free oscillation of the system. (b) Assume that the two masses start in motion with the initial conditions x(0)=21, x'(0)=4, and y(0) = 33, y'(0) = 2 and are acted on by the same force F₁ (t)= F₂(t)= 1080 cos (7t). Describe the resulting motion as a superposition of oscillations at three different frequencies. X'= -7x+6y y' =9x-22y (a) The lower frequency mode has ₁ : amplitude of the oscillation of m₂ is The masses oscillate in times the amplitude of the oscillation of m₁. and the

Consider a mass-and-spring system containing two masses m₁ = 1 and m₂ = 1 whose displacement functions x(t) and y(t) satisfy the differential equations below. (a) Describe the two fundamental modes of free oscillation of the system. (b) Assume that the two masses start in motion with the initial conditions x(0)=21, x'(0)=4, and y(0) = 33, y'(0) = 2 and are acted on by the same force F₁ (t)= F₂(t)= 1080 cos (7t). Describe the resulting motion as a superposition of oscillations at three different frequencies. X'= -7x+6y y' =9x-22y (a) The lower frequency mode has ₁ : amplitude of the oscillation of m₂ is The masses oscillate in times the amplitude of the oscillation of m₁. and the

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) A mass suspended from a helical spring of stiffness s, is displaced by a distance x from its equilibrium position and allowed to vibrate. Show that the motion is simple harmonic. (b) A vertical helical spring having a stiffness of 1540 N/m is clamped at its upper end and carries a mass of 20 kg attached to the lower end. The mass is displaced vertically through a distance of 120 mm and released. Find : 1. Frequency of oscillation ; 2. Maximum velocity reached ; 3. Maximum acceleration; and 4. Maximum value of the inertia force on the mass. (c) A machine of mass 75 kg is mounted on springs and is fitted with a dashpot to damp out vibrations. There are three springs each of stiffness 10 N/mm and it is found that the amplitude of vibration diminishes from 38.4 mm to 6.4 mm in two complete oscillations. Assuming that the damping force varies as the velocity, determine : 1. the resistance of the dashpot at unit velocity ; 2. the ratio of the frequency of the damped vibration to the…
A rigid and weightless rod is restricted to oscillations in the vertical plane as shown in the figure below.Determine the natural frequency of the system of mass m.
A 30 kg object is undergoing lightly damped harmonic oscillations. If the maximum displacement of the object from its equilibrium point drops to 1/2 its orginal value 2.2 s, value of the damping constant b?
A spring, with a 54 lb/in spring constant, is mounted on a vertical wall to support a 64.4 lb mass. At t=0 and an initial velocity of 9 in/sec. to the left, the mass is released from a position 2 inches to the right of the equilibrium position. Determine: a. Angular frequency
Consider two springs that are hung in series from a horizontal support. The top-most spring has a spring constant k1 and equilibrium length L1 and the bottom spring has a spring constant k2 and equilibrium length L2. There is a mass M1 connected to both springs where they meet and a mass M2 connected to the bottom of the lower spring. (a) Find the equilibrium positions of M1 and M2.(b) Find the normal mode frequencies of vertical oscillations of M1 and M2 around this equilibrium point.
A uniform rigid mass of 2 lb is hinged at its centre and laterally supported at one end by a spring of constant k=300 N/m. If the equation obtained for frequency (ωn) in this system is ωn2 =k/m, what is the natural frequency?
spring-mass system with m = 0.01 Mg and k = 5 kN/m is subjected to a harmonic force of amplitude 0.25 kN and frequency ωo. If the maximum amplitude of the forced motion of the mass (particular solution or steady state) is observed to be 100 mm, find the value of ωo
A spring, with a 54 lb/in spring constant, is mounted on a vertical wall to support a 64.4 lb mass. At t=0 and an initial velocity of 9 in/sec. to the left, the mass is released from a position 2 inches to the right of the equilibrium position. Determine: a.Amplitude, C b. Period, Ƭ c. Natural frequency
1. Explain_why_the_natural_frequency_measurements_should_better_reflect_the_system’s_true_undamped natural_frequency_(and_therefore_correlate_better_with_calculated_values)_than_the_frequency_of_oscillation measurements. 2. For_a Linear Spring-Mass System, Torsional Spring-Mass System, and Simple Pendulum System: identify_the_necessary_measurements_required_to_identify_the_natural_frequency_and the damping_ratio_for_the_linear_SMD_and_the_simple_pendulum.
The following mass-and-spring system has stiffness matrix K. For the mks values, find the two natural frequencies of the system and describe its two natural modes of oscillation. m1 m2 k1 k2 k3 x1 x2 K= −k1+k2 k2 k2 −k2+k3 m1=1, m2=5; k1=3, k2=10, k3=15 Let ω1<ω2be the two natural frequencies. ω1= ω2= (Type exact answers, using radicals as needed.)
How is constant amplitude sustained in forced oscillations?
A block of mass m hangs from a spring with a spring constant k. The block is initially held so that the spring is unstretched. If the natural circular frequency of the system is 10 radians/sec, how many cycles of vibration will the block experience in five seconds after it is released?