Explanation Given information: The amplitude of the collar ( x C ) is δ m sin ω f t . Assume the static deflection δ m is small compared with the length l of the pendulum. Calculation: Show the system at before and after moving the collar horizontally as in Figure (1). Refer Figure 1, θ is the angular displacement of the pendulum, m is the mass of the pendulum bob, and mg is the gravitational force due to the mass of the bob. Before giving horizontal movement, the force mg is equal to the tension in the pendulum. The expression for the force balance equation for initial condition as follows: T = m g Calculate the value of sin θ by referring the Figure (1). sin θ = x − x c l The expression for the force balance equation in x -direction in the displaced condition as follows: ∑ F x = m a x Here, ∑ F x is the sum of forces in x-direction and a x is the acceleration in the x -direction. The only force in the x -direction is the tension component T sin θ and it is opposite in direction to that of the acceleration. x ¨ is the acceleration in the x -direction due to the linear displacement. Substitute − T sin θ for ∑ F x and x ¨ for a x . − T sin θ = m x ¨ Substitute mg for T and x − x c l for sin θ . − m g ( x − x c l ) = m x ¨ − g ( x − x c l ) = x ¨ − g x l + g x c l = x ¨ x ¨ + g l x = g l x c Substitute δ m sin ω f t for x c . x ¨ + g l x = g l δ m sin ω f t (1) Here, t is the time. Compare the differential equation (1) with the general differential equation of motion for forced vibration ( x ¨ + ω n 2 x = ω n 2 δ m sin ω f t ) . The expression for the natural circular frequency of forced vibration ( ω n ) as follows: ω n 2 = g l ω n = g l The expression for the amplitude of the forced vibration ( x m ) as follows: x m = δ m 1 − ( ω f ω n ) 2

11th Edition

ISBN: 9781259633133

Author: BEER

Publisher: MCG

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 19.109P

To determine

Find the range of values of (ωf) for which the amplitude of the motion of the bob is less than δm.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 19.4, Problem 19.108P

Chapter 19.4, Problem 19.110P

Students have asked these similar questions

-6- 8 من 8 Mechanical vibration HW-prob-1 lecture 8 By: Lecturer Mohammed O. attea The 8-lb body is released from rest a distance xo to the right of the equilibrium position. Determine the displacement x as a function of time t, where t = 0 is the time of release. c=2.5 lb-sec/ft wwwww k-3 lb/in. 8 lb Prob. -2 Find the value of (c) if the system is critically damping. Prob-3 Find Meq and Ceq at point B, Drive eq. of motion for the system below. Ш H -7~ + 目 T T & T тт +

Q For the following plan of building foundation, Determine immediate settlement at points (A) and (B) knowing that: E,-25MPa, u=0.3, Depth of foundation (D) =1m, Depth of layer below base level of foundation (H)=10m. 3m 2m 100kPa A 2m 150kPa 5m 200kPa B

W PE 2 43 R² 80 + 10 + kr³ Ø8=0 +0 R²+J+ kr200 R² + J-) + k r² = 0 kr20 kr20 8+ W₁ = = 0 R²+1) R²+J+) 4 lec 8.pdf Mechanical vibration lecture 6 By: Lecturer Mohammed C. Attea HW1 (Energy method) Find equation of motion and natural frequency for the system shown in fig. by energy method. m. Jo 000 HW2// For the system Fig below find 1-F.B.D 2Eq.of motion 8 wn 4-0 (1) -5- m

Chapter 19 Solutions

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

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.

Find the range of values of ( ω f ) for which the amplitude of the motion of the bob is less than δ m .

Solution Summary: The author calculates the value of mathrmsintheta by referring to Figure 1.

Concept explainers

Videos

Find the range of values of (ωf) for which the amplitude of the motion of the bob is less than δm.

Want to see the full answer?

Chapter 19 Solutions