Concept explainers
(a)
Then the distance ‘d’ to maximise the frequency of oscillation when a small initial displacement is given.
Answer to Problem 19.50P
Distance
Explanation of Solution
Given information:
Mass of collar
Mass of rod
Length of rod
The forces corresponding to the rod and collar are shown in the free body diagram below:
Now, from the equation of motion taking moment about A,
And, moment of inertia of rod is
Compare the above equation with un-damped equation of vibration:
Natural frequency:
To maximise the frequency, we need to take the derivative with respect to d and set it equal to zero.
By solving the above equation we get,
Conclusion:
The distance
(b)
The period of oscillation.
Answer to Problem 19.50P
Period of vibration,
Explanation of Solution
Given information:
Mass of collar
Mass of rod
Length of rod
The forces corresponding to the rod and collar are shown in the free body diagram below:
Now, from the equation of motion taking moment about A,
And, moment of inertia of rod is
Compare the above equation with un-damped equation of vibration;
Natural frequency:
To maximise the frequency, we need to take the derivative with respect to d and set it equal to zero.
By solving the above equation we get,
Thus,
Now, natural frequency:
Then, Time period
Want to see more full solutions like this?
Chapter 19 Solutions
Vector Mechanics For Engineers
- (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…arrow_forwardDetermine the period of small oscillations of a small particle that moves without friction inside a cylindrical surface of radius R.arrow_forwardA section of uniform pipe is suspended from two vertical cables attached at A and B. Determine the frequency of oscillation when the pipe is given a small rotation about the centroidal axis OO’ and released.arrow_forward
- The 8-kg uniform bar AB is hinged at C and is attached at and A to a spring of constant==k = 500 N/m. If end A is given a small displacement and released, determine (a) the frequency of small oscillations, (b ) the smallest value of the spring constant k for which oscillations will occur.arrow_forward2) A massless bar of length L carrying a tip mass m is rigidly connected to a homogenous disk of radius R and mass m as shown below. The disk rolls without slipping on the ground. Note that gravity acts down and that a spring of stiffness k and a dashpot of constant c connect the center of the disk to an oscillating wall and a fixed one respectively. Determine the amplitude of response of the angle 0(t).arrow_forwardTwo springs of constants K1 and K2 are connected in series to a block A that vibrates in simple harmonic motion with a period of 5 s. When the same two springs are connected in parallel to the same block, the block vibrates with a period of 2 s. Determine the ratio K1/K2 of the two spring constants.arrow_forward
- 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?arrow_forwardLarge amplitude of vibration is experienced when an engine having mass (1500 + 2) kg operates at a speed of 1500 rpm. An absorber having stiffness k;equal to 1500 N/m and mass m; is attached to the primary system to absorb the vibration. Find: The magnitude of the mass of the absorber. ii. i. The natural frequencies of the system after the addition of the absorber. The values of the stiffness k; and mass m; of the absorber in order to have the natural frequencies of the two degrees of freedom system 15% away from the forcing frequency.arrow_forwardA 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.arrow_forward
- A thin cylindrical rod of uniform mass m and length L is suspended from the ends by two massless springs with constants k1 and k2 (Distance L1 and L2 on either side of the center of mass of the rod). The motion of the center of mass is constrained to move up and down parallel to the vertical y-axis. It also experiences rotational oscillations around an axis perpendicular to the rod and passing through the center of mass (I is the moment of inertia with respect to said axis). be y1 and y2 the displacements of the two ends from their equilibrium positions, as shown in Fig. a) Find the motion's equation (consider k1=k2=k)arrow_forwardTwo masses m1 and m2 each connected by two springs of stiffness k, are connected by a rigid massless horizontal rod of length l as shown in Fig. 5.22. (a) Derive the equations of motion of the system in terms of the vertical displacement of the C.G. of the system, x(t), and the rotation about the C.G. of the system, θ(t) (b) Find the natural frequencies of vibration of the system for m1=50 kg, m2=200 kg and k = 1000 N/m.arrow_forward5. A mass hangs from a helical spring. The periodic time of free vibration in a vertical direction is 1.25sec. When the mass is at rest the upper end of the spring is made to move with an upward displacement of such that y=5 sin 27t cm, († being the time in second measured from the beginning of motion). Through what height is the mass raised in the first 0.4 second? Find also the amplitude of motion of the mass for steady state vibration.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY