Vibration Engineering

Vibration Engineering

Damped and Undamped Free Vibrations Of Torsional Members Torsional Vibration

Torsional VibrationTorsional VibrationMass Moment of InertiaThis mass analog is called the moment of inertia, I, of the object

r = moment armSI unit are kg-m2

Shell ElementDisk Element

Example Example

When the object is made up of point masses you calculate moment of inertia using:

L

L

R

R2R2

Rab

ab

R

RMoments of inertia for some common geometric solidsParallel Axis TheoremThe moment of inertia about any axis parallel to and at distance d away from the axis that passes through the centre of mass is:

WhereIG= moment of inertia for mass centre Gm = mass of the bodyd = perpendicular distance between the parallel axes.

Radius of GyrationFrequently tabulated data related to moments of inertia will be presented in terms of radius of gyration.

Mass Center

Example

10 Ib10 IbEx: Moment of Inertia of a Uniform Thin Hoop for Axis perpendicular to the plane at the Center of the HoopAssume r is constant

Example 1An aluminium disk is shrunk onto a steel shaft to form the rearrangement of simple torsional system of cantilever type. The length L of the shaft is 80 in., the diameter d of the shaft is 2 in., the shear modulus G is 12x106 psi, and the shaft is assumed to be weightless. If the mass moment of inertia I of the disk is 2.8 in.lbs2, determine the torsional frequency of the one degree disk-shaft system and its period of vibration.

SOLUTION Example 2The disk and gear undergo torsional vibration with the periods shown. Assume that the moment exerted by the wire is proportional to the twist angle.Determine a) the wire torsional spring constant, b) the centroidal moment of inertia of the gear, and c) the maximum angular velocity of the gear if rotated through 90o and released.

SOLUTION Using the free-body-diagram equation for the equivalence of the external and effective moments, write the equation of motion for the disk/gear and wire.With the natural frequency and moment of inertia for the disk known, calculate the torsional spring constant.With natural frequency and spring constant known, calculate the moment of inertia for the gear.Apply the relations for simple harmonic motion to calculate the maximum gear velocity.SOLUTION

SOLUTION Using the free-body-diagram equation for the equivalence of the external and effective moments, write the equation of motion for the disk/gear and wire.

With the natural frequency and moment of inertia for the disk known, calculate the torsional spring constant.

Solution With natural frequency and spring constant known, calculate the moment of inertia for the gear.

Apply the relations for simple harmonic motion to calculate the maximum gear velocity.

Example 3A cylinder of weight W is suspended as shown.Determine the period and natural frequency of vibrations of the cylinder.

kSOLUTION From the kinematics of the system, relate the linear displacement and acceleration to the rotation of the cylinder.Based on a free-body-diagram equation for the equivalence of the external and effective forces, write the equation of motion.Substitute the kinematic relations to arrive at an equation involving only the angular displacement and acceleration.

SOLUTION From the kinematics of the system, relate the linear displacement and acceleration to the rotation of the cylinder.

Based on a free-body-diagram equation for the equivalence of the external and effective forces, write the equation of motion.

SOLUTION Substitute the kinematic relations to arrive at an equation involving only the angular displacement and acceleration.

Damped Free Vibration of Rotational Bodies

IDamped Free Vibration of Rotational BodiesExample 1A simple disk-shaft of cantilever type is subjected to a torsional oscillatory motion that is under the influence of viscous damping. If Kt = 20,000 Nm/rad and J = 12.6 Nms2, determine the value of the viscous damping constant c so that the damped torsional frequency d of the disk shaft system is 80% of its free, undamped, torsional frequency . Also calculate the values of , d, , and d.

Example 2A viscously damped system has a stiffness of 5,000 N/m, critical damping constant of 0.2 N-s/mm, and a logarithmic decrement of 2.0. If the system is given an initial velocity of 1 m/s, determine the maximum displacement of the system.

ASSIGNMENT 1An aluminium shaft is shrunk onto a steel shaft to form the rearrangement shown in Fig. 2.22. The two portions of the shaft have the same length L = 60 in. and the same diameter d = 2in. The steel shear modulus G = 12 x 106 psi, and the shaft is assumed to be weightless. If the moment of inertia J of the disk is 2.8 in.lbs2 , determine the torsional frequency of vibration of the one-degree disk-shaft system and its period of vibration.

ASSIGNMENT 2 A simple disk-shaft of cantilever type is subjected to a torsional oscillatory motion that is under the influence of viscous damping with damping constant c = 450.0 N ms/rad. If the length of the shaft is 4.0m and J = 14.50 Nms2, determine the diameter d of the shaft when the damped torsional frequency of vibration is 32.0 rps. The shear modulus G of the steel shaft is 80 x 109 N/m2.

Assignment 3 The needle indicator of an electronic instrument is connected to a torsional viscous damper and a torsional spring. If the rotary inertia of the needle indicator about its pivot point is 25 kg-m2 and the spring constant of the torsional spring is 100 N-m/rad, determine the damping constant of the torsional damper if the instrument is to be critically damped.

Assignment 4A viscously damped system has a stiffness of 5,000 N/m, critical damping constant of 0.2 N-s/mm, and a logarithmic decrement of 2.0. If the system is given an initial velocity of 1 m/s, determine the maximum displacement of the system.

Assignment 5A torsional pendulum has a natural frequency of 200 cycles/min when vibrating in a vacuum. The mass moment of inertia of the disc is 0.2 kg-m2. It is then immersed in oil and its natural frequency is found to be 180 cycles/min. Determine the damping constant of the disc, when placed in oil, is given an initial displacement of 2o, find its displacement at the end of the first cycle.

Assignment 6 A simple pendulum is set into oscillation from its rest position by giving it an angular velocity of 1rad/s. It is found to oscillate with an amplitude of 0.5rad. Find the natural frequency and length of the pendulum.

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