dynamics

Undamped Free Vibration(Week 1)

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April 22, 2023

PowerPoint® Slides

by Dr CHIA Chee Ming

Learning Objective (s)

Students will gain an understanding on the simplest form of vibration, i.e., undamped free vibration.

Students will learn to derive the Equation of Motion for a Single Degree of Freedom (SDOF) system.

Students will learn to define the characteristic of a vibration, i.e., natural frequency, period of vibration and amplitude of vibration.

Students will also be learning the differences between linear vibration and torsional vibration.

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Learning Outcome (s)

Students will be able to define both a Single Degree of Freedom (SDOF) system and a Simple Harmonic Motion (SHM)

Students will be able to derive the Equation of Motion for an undamped free vibration.

Students will be able to define the natural frequency, period and amplitude of an undamped free vibration.

Students will also be able to apply the principle of Equation of Motion on torsional vibration.

April 22, 2023


Vibration

A vibration is a periodic motion of a body or system of connected bodies displaced from a position of equilibrium.

Free vibration occurs when the motion is maintained by gravitational or elastic restoring forces

Forced vibration is caused by an external periodic or intermittent force applied to the system

Both of these types of vibration may be either damped or undamped.

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Undamped Free Vibration

Undamped vibrations can continue indefinitely because frictional effects are neglected in the analysis

In reality, motion of all vibrating bodies is actually damped because of frictional forces present.

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SDOF Undamped Free Vibration

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Free Body Diagram

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The time-dependent path of motion of the block may be determined by applying Equation of Motion (Newton’s 2nd Law of Motion) to the block when it is in the displaced position x

Free Body Diagram

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From the figure, we have the Equation of Motion

The acceleration is proportional to the block’s displacement and such motion is called Simple Harmonic Motion (SHM)

Rearranging the equation of motion gives the ‘standard form’

xmxkmaF xx

;

02 xx n

tBtAx nn cossin


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The constant ωn is called the natural frequency, expressed in rad/s,

is a homogeneous, second-order, linear, differential equation with constant coefficients, and the general solution is

m

kn

02 xx n

)sin(

or

tCx n

Vibration of Motion

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Vibration of Motion

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The maximum displacement of the block from its equilibrium position is defined as the amplitude of vibration

From the figure, the amplitude is C and angle Φ is the phase angle since it represents the amount by which the curve is displaced from the origin when t = 0.

It can be shown that

A

B

BAC

1

22

tan

Vibration of MotionApril 22, 2023


Note that the sine curve completes one cycle in time , hence

The length of time is call a period, which may also be represented as

The frequency, f is defined as the number of cycles completed per unit of time, which is the reciprocal of the period:

The frequency is expressed in cycles/s. This ratio of units is called a hertz (Hz)

tn 2

km 2

m

kf n

2

1

2

1

Example 1

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When a load of mass 10 kg is suspended from a spring, the spring is stretched a distance 100 mm. Determine the natural frequency and the period of vibration for a load of mass 5kg attached to the same spring.

Solution:

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sradn /14s45.0

Example 2

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A spring has stiffness 600 N/m. If a block of mass 4 kg is attached to the spring, pushed a distance 50 mm above its equilibrium position, and released from rest, determine the time-dependent path of motion of the block. Assume that positive displacement is measured downward.

Solution:

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When a body or system of connected bodies is given an initial displacement from its equilibrium position and released, it will vibrate with the natural frequency, ωn.

Provided the body has a SDOF, then the vibrating motion of the body will have the same characteristics as the SHM of the block and spring just presented.

02 xx n

Procedure for Analysis

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Free Body Diagram

Draw the FBD of the body when the body is displaced by a small amount from its equilibrium position

Locate the body with respect to its equilibrium position by using an appropriate coordinate q.

The acceleration of the body’s mass center, a or the body’s angular acceleration, α should have a sense which is in the positive direction of the position coordinate.

Procedure for Analysis

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Equation of Motion

Apply the equation to relate the elastic or gravitational restoring forces and couple moments acting on the body to the body’s accelerated motion

Using kinematics, express the body’s accelerated motion in terms of the second time derivative of the position coordinates.

Substitute the result into the equation of motion and determine ωn by rearranging the terms so that the resulting equation is of the “standard form”,

02 qq n

Example 3

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Determine the period of vibration for the simple pendulum shown. The bob has a mass m and is attached to a cord of length l.

Solution:

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Motion of the system will be related to the position coordinate (q =) θ.

When the bob is displaced by an angle θ, the restoring force acting on the bob is created by the weight component mg sinθ.

at acts in the direction of increasing s (or θ)

Solution:

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Applying the equation of motion in the tangential direction, since it involves the restoring force, yields

. Furthermore, s may be related to θ by the

equation s = lθ, so that

ttt mamgmaF sin ;

lat

0sin Hence, l

g

2

2

dt

sdat

Solution:

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For small displacements, sin θ ≈ θ, in which case

Comparing this equation with , it can be seen that ωn = (g/l)½

The period of time required for the bob to make one complete swing is therefore

0 l

g

02 xx n

g

l

n

2

2

Linear Vibration vs. Torsional Vibration

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I0 = Mass moment of inertia is a measure of an object’s

resistance to changes in its rotation rate (SI unit = kgm2)

kTorque

0IMoment

k = torsional stiffness θ = angular displacement

kxForce k = spring stiffness x = linear displacement

xmForce

Newton’s Second Law:

m = mass of an object

Example 4

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The 10 kg rectangle plate shown is suspended at its center from a rod having torsional stiffness k = 1.5 N.m/rad. Determine the natural period of vibration of the plate when it is given a small angular displacement θ in the plane of the plate.

Given the moment of inertia for the plate = 0.108 kgm2

Solution: FBDApril 22, 2023


Since the plate is displaced in its own plane, the torsional restoring moment created by the rod is M = kθ.

This moment acts in the direction opposite to the angular displacement θ.

0

oo

oo

I

kIk

IM

sk

Io

n

69.15.1

108.022

2 Hence,

Conclusion/Summary

Free vibration – motion maintained by gravitational or elastic restoring force.

Equation of Motion

General solution for Simple Harmonic Motion (SHM)

Frequency – number of cycles completed per unit time, where 1 Hz = 1 cycle per second.

Period – time required to complete a cycle.

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tBtAx nn cossin

02 xx n

n 2

References:

Rao, S. S., 1995, Mechanical Vibrations, Pearson Prentice Hall.

Hibbeler, R. C. 1998, Engineering Mechanics, Vol 2 “Dynamics”, 8th Edition, Prentice-Hall International.

Benham, P.P.; Crawford, R. J. and Armstrong, C. G. 1996, Mechanics of Engineering Materials, 2nd ed, Prentice Hall.

Ferdinand, P. B., Russell, J, Jr, John, T. D., and David, F. M., 2009, Mechanics of Materials, 5th Edition, McGraw-Hill.

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Key terms:


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Term Definition

Vibration A periodic motion of a particle or of an elastic solid about an equilibrium position.

Amplitude The maximum absolute value of a periodically varying quantity

Frequency The number of complete cycles of a periodic process occurring per unit time

Period The time interval between two successive occurrences of a recurrent event or phases of an event; a cycle

Natural Frequency

the frequency at which a system vibrates when set in free vibration

dynamics

Documents

periodic motion

linear vibration

period of vibration

torsional vibration

law of motion

types of vibration

equation of motion newtons

simplest form of vibration