Mechanics of Machines - vibration

6

Vibration

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IntroductionVibration plays an important role in nature and engineering.

Many engineering products (automobiles, jet engines, rockets,

bridges...) require a good insight in the concept of vibration for

efficient product design, development and performance evalua-

tion. Vibrations are fluctuations of a mechanical or structural

system about an equilibrium position.

Vibrations are initiated when an inertia element is displaced

from its equilibrium position due to energy being imparted to the

system through an external source (cf. Figure 1 and Figure 2).

A restoring force or moment then proceeds to pull the element

back towards equilibrium.

Examples:

Figure1

Figure2


When work is done on the block of Figure 1 to displace it from

its equilibrium position, potential energy is developed in the

spring. When the block is released, the spring force pulls the

block towards equilibrium with the potential energy being con-

verted to kinetic energy. In the absence of dissipative effects,

e.g. friction, this transfer of energy (potential-kinetic-potential) is

continual, causing the block to oscillate about its equilibrium

position indefinitely.

The study of vibration involves developing mathematical mod-

els for systems shown in Figure 1 and Figure 2 (among others),

and finding solutions to these mathematical models.


BasicConceptsVarious quantities, e.g. the position of a particle, undergo more or less regular changes over time. The processes in-

volved in these changes are called vibrations or oscillations.

Examples:

waves of the oceans

movement of a piston in an engine

vibrations in an electrical circuit

Frequently, a quantity will have the same value at regular time intervals (Figure 3), during an oscillatory process:

[1]

If the oscillatory process involves movement, the motion is

called a periodic vibration. The time is referred to as the PERIOD of the vibration.

Figure3


The quantity

1 [2]

is called the FREQUENCY of the vibration. It represents the num-

ber of cycles per unit time (a CYCLE is the motion completed

during 1 period). The dimension of frequency is 1/time. Its unit

is Hertz (abbreviated Hz):

1 1

Harmonic Vibrations

Harmonic vibrations are characterized by the fact that the quan-

tity is given by a cosine or sine function:

. or . [3]

Figure4


: Angular/circularfrequencyor : Amplitudeofvibration

Since 2 (cf. Figure 4) and 1 , it follows that:

2 2 [4]

Harmonic vibrations that are represented by pure cosine or

pure sine curves are subjected to special initial conditions.

For . , we have the initial conditions: 0 0 0

Similarly, for . the initial conditions are: 0 0 0

Harmonic vibrations with arbitrary initial conditions can always

be represented by

. [5]

: Amplitude : Phaseangle(cf.Figure 5)


Figure5

Harmonic vibrations [5] can also be obtained through a super-

position of two vibrations of the form of [3]. Using the trigono-

metric formula

. . . [6]

and setting

. and . [7]

we obtain

. . [8]

The representations of harmonic motions by equations [5] and

[8] are therefore equivalent and interchangeable.


A harmonic oscillation can be generated by a point (initial po-sition ) which moves on a circular path (radius C) with con-stant angular velocity (Figure 5). The projection on a verti-cal straight line (or on any diameter) performs a harmonic vibra-

tion.

Undamped vibration Vibration of constant amplitude

Damped vibration

Unstable vibration

Degrees of Freedom (DOF)

The degrees of freedom of a body are the set of variables nec-

essary to completely define the position and orientation of the

body in space.


FreeVibrationsFree vibrations occur naturally with no energy being added to

the vibrating system.

Undamped free vibrations

Consider a block (mass ) that moves on a smooth surface (Figure 6). It is connected to a wall with a linear spring (spring

constant ).

Figure6

The system has 1 DOF, given by the horizontal position of the

block.

Figure7:Freebodydiagram(FBD)showinghorizontalforces


To derive the equation of motion of the block, we introduce the

coordinate (cf. Figure 7). Let 0 be the equilibrium position of the block (unstressed spring).

1. Block is displaced ( units to the right from the equilibrium position

2. Draw a FBD showing the different forces acting on the block:

Neglecting friction from the surface with which the block is in

contact, the only horizontal force acting on the block is the

restoring force ( from the spring

3. Apply Newtons 2nd law ( ):

0 [9]

Setting

[10]

equation [9] is re-written as:

0 [11]


Equation [11] is a 2nd order, linear homogeneous differential

equation having constant coefficients. Its general solution is

given by

. . [12]

where and are integration constants.

4. Use initial conditions to determine integration constants:

0 0

Equation [12] becomes

. . [13]

Equation [12] is equivalent to

. [14]

With


According to equation [14], the block performs a harmonic vi-

bration with circular frequency . The natural fre-

quency () of free vibrations can be determined from the rela-tionship 2. The natural frequency of vibrations of a system is also commonly known as its eigenfrequency.


Problem 6.1

Find the equation of motion of a block of mass , suspended by a linear spring (spring constant ), which is displaced downwards from its equilibrium position and released with no

initial velocity.


Problem 6.2

A small mass "" is fastened to a vertical wire that is under tension . What will be the natural frequency of vibration of the mass if it is displaced laterally a slight distance and then re-

leased?

Figure8


Problem 6.3

A rod (length , with negligible mass) carries a mass at its upper end. It is supported by a linear spring (cf. drawing be-

low).

Describe the motion of the rod if it is displaced from its vertical

position (small displacement to the left) and then released with

no initial velocity.


Mass Moment of Inertia

The study of rotational dynamics requires the computation of

mass moments of inertia. Two situations normally arise:

Point mass rotating about an axis Rigid body rotating about an axis

Mass moment of inertia is an important property of a body. It is

involved in the analysis of any body which has rotational accel-

eration about a given axis.

Just as the mass of a body is a measure of the resistance to translational acceleration, the moment of inertia is a measure of resistance to rotational acceleration of the body.

The moment of inertia of an object is defined as the integral of

the "second moment" about an axis of all the elements of mass

which compose the body. For example, the body's moment of inertia about the in Figure 9 is

[15]


Figure9

The "" is the perpendicular distance from the to the arbitrary element . Since the formulation in-volves , the value of the moment of inertia is different for each axis about which it is computed.

For a point mass, the moment of inertia is calculated from

equation [16] (where is the perpendicular distance between the point mass and the axis of rotation).

. [16]

(for a point mass)


Parallel-axis theorem

If the moment of inertia of the body about an axis passing

through the body's mass center is known, then the moment of

inertia about any other parallel axis can be determined by using

the parallel-axis theorem.

Figure10

[17]


Perpendicular-axis theorem


Radius of gyration

The moment of inertia of a body about a specified axis is some-

times calculated using the radius of gyration, . This is a geo-metrical property which has units of length. For a given solid, if

its mass and radius of gyration are known, the bodys moment

of inertia is obtained using equation [18]

[18]

Composite bodies

If a body consists of a number of simple shapes such as disks,

spheres, and rods, the moment of inertia of the body about any

axis can be determined by adding algebraically the moments of

inertia of all the composite shapes computed about the axis.


Problem 6.4 Determine the moment of inertia for each of the following

situations:


Problem 6.5

Determine the moment of inertia for the slender rod. The rod's density and cross-sectional area are constant. Ex-press the result in terms of the rod's total mass .


Problem 6.6 The paraboloid shown is formed by revolving the shaded area

around the . Determine the radius of gyration . The density of the material is 5 .


Problem 6.7 The right circular cone shown is formed by revolving the

shaded area around the x axis. Determine the moment of iner-

tia and express the result in terms of the total mass of the cone. The cone has a constant density .


Problem 6.8

The body of arbitrary shape has a mass , mass center at , and a radius of gyration about of . If it is displaced a slight amount from its equilibrium position and released, determine the natural period of vibration.


Problem 6.9

The bent rod shown has a negligible mass and supports a

collar at its end. If the rod is in the equilibrium position shown,

determine the natural period of vibration for the system.


Spring constants of elastic systems

The force and the elongation of a linear spring are related by the relationship . .

The spring constant is therefore characterized by

[19]

Similar force-deformation relationships can be established for

many systems containing elastic components.

I. Massless bar (length l, axial rigidity EA)

A massless bar carries a mass

at its free end (Figure 11). Under the action of the mass,

the bar undergoes an elonga-

tion downwards. The bar provides a restoring force to oppose the downward dis-

placement of the mass and es-

tablish an equilibrium condi-

tion. A force equal in magni-

tude and opposite in direction

acts on the bar (action-

reaction).

Figure11


The force and the elongation are related by the equation

[20]

By analogy with the elastic spring, we can determine a

spring constant or stiffness for an elastic bar:

[21]

II. Massless cantilever beam (length l, flexural rigidity EI)

A massless cantilever beam

carries a mass at its free end (Figure 12). The mass causes the end of the initially

straight beam to move down-

wards. Under equilibrium con-

ditions, a restoring force pro-

vided by the cantilever beam

balances the weight of the

mass .

3 (cf. Deflection curves of beams)

Figure12


Thus, we obtain the spring constant for a cantilever beam as

3 [22]

III. Massless shaft (length l, torsional rigidity GJ)

The relationship between the

angle of twist and the torque

in a shaft is governed by the

following equation:

Figure13

The spring constant equivalent for this configuration is obtained

as

[23]

If a disk (moment of inertia ) is fixed to the end of the shaft and undergoes torsional vibrations, then the motion is de-

scribed by

0 [24]


Spring assembly Equivalent spring

Springs in parallel

The motion of a mass may cause elongations of several

springs in a system. Consider the case of two springs in parallel

(Figure 14).

Figure14

The two springs (spring constants and ) undergo the same elongation when the mass is displaced. They can be re-

placed by an equivalent single spring with the spring constant

.

[25]

Therefore, for a system of parallel springs

(spring constants ), the spring constant of the equivalent spring is given by the sum of the individual spring constants:


[26]


Springs in series

Consider two springs in series (Figure 15).

Figure15

Find the stiffness of the equivalent spring . In the case of arbitrarily many springs in series, the spring con-

stant of the equivalent spring is found from

The flexibility (compliance) of a spring is defined as the inverse

of the stiffness:

1 [27]


Problem 6.10

An elastic beam (flexural rigidity ) with negligible mass sup-ports a box (mass ) as shown below. Find the natural circu-lar frequency of the system.


Problem 6.11 Determine the natural circular frequency of the system(s)

shown below.

Mechanics of Machines - vibration

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