1. An undamped spring-mass system is shown below. (a) Starting from the equations of motion, find the natural frequencies of the system if m₁ = m, m₂ = 2m, k₁=k, and k₂ = 2k. (b) Find the natural frequencies and mode shapes (i.e. modal vectors) of the system if m₁ = m₂ = 1 kg, k₁ = 2000 N/m, and k₂ = 6000 N/m. (c) Given the properties in (b), determine the response of the system x₁(t) and x₂(t) if the initial conditions are x₁(0) = 0.2 m, ₁(0) = x2(0) = 0, and ₂(0) = 5 m/s. Base k₁ 5 00000 m₁ NO k₂ m₂ x₁(1) x₂ (1) Undamped 2 DOF spring-mass system.

1. An undamped spring-mass system is shown below. (a) Starting from the equations of motion, find the natural frequencies of the system if m₁ = m, m₂ = 2m, k₁=k, and k₂ = 2k. (b) Find the natural frequencies and mode shapes (i.e. modal vectors) of the system if m₁ = m₂ = 1 kg, k₁ = 2000 N/m, and k₂ = 6000 N/m. (c) Given the properties in (b), determine the response of the system x₁(t) and x₂(t) if the initial conditions are x₁(0) = 0.2 m, ₁(0) = x2(0) = 0, and ₂(0) = 5 m/s. Base k₁ 5 00000 m₁ NO k₂ m₂ x₁(1) x₂ (1) Undamped 2 DOF spring-mass system.

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

Determine the natural frequency of the system shown beside using Motion equation
Find the two natural frequencies of the system and describe its two natural modes of oscillation. See Attachment for assistance m1 = m2 = 1; k1 = 0, k2 = 2, k3 = 0
Consider a spring-mass system with mass equal to 1 kg, spring constant equal to 25 Newton/meter.Which damping constant b causes critical damping?If the damping constant b in the above system is set to 3 N ∙ sec/m, then what can be said about the number of timesdoes the object pass through its equilibrium position?If the damping constant b in the above system is set to 8 N ∙ sec/m, then what is the interval of time between thesecond time the object returns to its equilibrium position and the third time it returns to its equilibrium position?
For the system illustrated below, ﬁnd the natural frequency.
Two masses of weight 300 N each are keyed to the opposite ends of a shaft. The shaft length is 70 cm. The shaft is 7 cm diameter for the first 30 cm of its length, 13 cm diameter for the next 20 cm, and 10 cm diameter for the remaining of its length. Find the frequency of free torsional vibrations of the system and the position of the node. The radius of gyration of the masses is 30 cm.
A mass m is attached to two identical springs having constant k as shown in the figure. What is the natural frequency ∞ of this single degree of freedom system?
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?
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
Figure Q3(b) shows a uniform bar AB of mass = 8 kg hinged at point C.Point A is connected to a spring to maintain the bar in vertical direction, and the stiffness k = 500 N/m. If point A is displaced counter-clockwise by a small angle θ = 3.5 degree and released, With the free body diagram and kinetic diagram, determine the initialhorizontal displacement of A. Then determine the period of vibration, the maximum velocity and acceleration at point A.
A spring-mass system with m = 50kg and k = 4000N/m shown in Figure Q3(a).If the mass is released from its equivalent position and results in simple harmonic motion (SHM),(i) Starting from a free body diagram, derive the equation of motion.(ii) Determine the natural frequency. (iii) Explain on how you can reduce the natural frequency of the spring-mass system.
An object of mass 2 kg is attached to a spring of stiffness 200 N / m on a horizontal frictionless surface . Then it is extend by 10 cm and then let free to oscillate . 1- How long will it take to make one complete oscillation ? 2- Determine the amplitude and give an expression for the displacement as a function of time . 3- Calculate its displacement , velocity , acceleration and force acting on it after sec . 4- Calculate its velocity by the time it is extended by 0.5 cm
The static deflection of a spring under gravity, when a mass of 1 kg is suspended from it, is 1 mm. Assume the acceleration due to gravity g =10 m/s2 . Calculate the natural frequency of this spring-mass system (in rad/s).