Explanation Given: The spring constant is k . Magnitude of periodic force is P m sin ω f t . Amplitude of vibration should be three times of static deflection. Concept used: Write the expression for the natural frequency of the system. ω n = k m ...... (1) Here, ω n is the natural frequency of system, k is the spring constant and m is the mass of collar. Write the expression for static deflection in the system caused by the constant force of magnitude P m as. δ s t = P m k ...... (2) Here, δ s t is the static deflection and P m is the constant force. Write the expression for the amplitude of forced vibration. x m = δ s t ( 1 − ω f 2 ω n 2 ) ...... (3) Calculation: Case 1: Consider the amplitude of forced vibration exceeds positively three times the static deflection. x m > 3 δ s t Substitute δ s t ( 1 − ω f 2 ω n 2 ) for x m in the above expression. δ s t ( 1 − ω f 2 ω n 2 ) > 3 δ s t 1 ( 1 − ω f 2 ω n 2 ) > 3 ( 1 − ω f 2 ω n 2 ) < 1 3 ω f ω n > 2 3 Simplify the above expression for ω f . ω f > ω n 2 3 ...... (4) Case 2: Consider the amplitude of forced vibration exceeds negatively three times the static deflection. x m < − 3 δ s t Substitute δ s t ( 1 − ω f 2 ω n 2 ) for x m in the above expression

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

See similar textbooks

Concept explainers

Forced undamped vibrations

In mechanical engineering, vibration is expressed as the term that occurs due to an object's backward and forward movement about a point of equilibrium. Generally, there are three types of vibrations which are,

Videos

Question

Chapter 19.4, Problem 19.101P

To determine

The range of values of ωf for which the amplitude of the vibrations exceeds three times the static deflection.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 19.4, Problem 19.100P

Chapter 19.4, Problem 19.102P

Students have asked these similar questions

A 180-kg motor is bolted to a light horizontal beam. The unbalance of its rotor is equivalent to a 28-g mass located 150 mm from the axis of rotation, and the static deflection of the beam due to the weight of the motor is 12 mm. The amplitude of the vibration due to the unbalance can be decreased by adding a plate to the base of the motor. If the amplitude of vibration is to be less than 60 μm for motor speeds above 300 rpm, determine the required mass of the plate.

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 2.3-kg object is suspended using two linear springs (k1 = 45 N/m and k2 = 21 N/m), as shown below. Due to aerodynamic effects and other losses, you observe that the amplitude of the vibration is halved after 5 complete cycles. If the initial mass was displaced 270 millimetres below the equilibrium position (i.e. negative initial displacement) and then released from rest, determine the position of the mass after 11 seconds. Give your answer in millimetres rounded to one decimal place. Enter positive values if the mass is above the equilibrium position and negative values if it is below the equilibrium position. k k

Chapter 19 Solutions

Vector Mechanics For Engineers

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - A 20-lb block is initially held so that the...Ch. 19.1 - Prob. 19.7P Ch. 19.1 - A simple pendulum consisting of a bob attached to...Ch. 19.1 - Prob. 19.9P Ch. 19.1 - A 5-kg fragile glass vase is surrounded by packing...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The range of values of ω f for which the amplitude of the vibrations exceeds three times the static deflection.

Solution Summary: The author analyzes the range of values of omega, where the amplitude of the vibration exceeds positively three times the static deflection.

Concept explainers

Videos

The range of values of ωf for which the amplitude of the vibrations exceeds three times the static deflection.

Want to see the full answer?

Chapter 19 Solutions