Mechanics of Machines - vibration

 

6 Vibration

 

 

Introduction  Vibration 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 evaluation. 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:

Figure 1 

Figure 2  K.A. 

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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 converted 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 models for systems shown in Figure 1 and Figure 2 (among others), and finding solutions to these mathematical models.

 

 

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Basic Concepts  Various quantities, e.g. the position

of a particle, undergo

more or less regular changes over time. The processes involved 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 time intervals

will have the same value at regular

(Figure 3), during an oscillatory process: [1]

If the oscillatory process involves movement, the motion is called a periodic vibration. The time PERIOD

is referred to as the

of the vibration.

Figure 3 

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The quantity 1

[2]

is called the FREQUENCY of the vibration. It represents the number 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 quantity

is given by a cosine or sine function: .

or

.

[3]

Figure 4  K.A. 

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:  Angular/circular frequency 

 or   

:  Amplitude of vibration 

1⁄ , it follows that:

2 (cf. Figure 4) and

Since

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 0

.

the initial conditions are:

0

0 Harmonic vibrations with arbitrary initial conditions can always be represented by .

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:  Amplitude 

 

:  Phase angle (cf. Figure 5) 

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[5]

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Figure 5  Harmonic vibrations [5] can also be obtained through a superposition of two vibrations of the form of [3]. Using the trigonometric formula . .

[6]

.

and setting .

.

and

[7]

we obtain .

.

[8]

The representations of harmonic motions by equations [5] and [8] are therefore equivalent and interchangeable.

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A harmonic oscillation can be generated by a point sition

(initial po-

) 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 vibration.

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 necessary to completely define the position and orientation of the body in space.  

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Free Vibrations  Free 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 ).

Figure 6  The system has 1 DOF, given by the horizontal position of the block.

Figure 7: Free body diagram (FBD) showing horizontal forces 

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

0 be the equilibrium position

(cf. Figure 7). Let

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 (

 

–

3. Apply Newton’s 2nd law (∑

from the spring ): 0

[9]

Setting [10] equation [9] is re-written as: 0

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[11]

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Equation [11] is a 2nd order, linear homogeneous differential equation having constant coefficients. It’s general solution is given by . where

and

.

[12]

are integration constants.

4. Use initial conditions to determine integration constants: 0 0 Equation [12] becomes

.

.

[13]

Equation [12] is equivalent to .

[14]

With

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According to equation [14], the block performs a harmonic vibration with circular frequency

. The natural fre-

quency ( ) of free vibrations can be determined from the relationship

2

. The natural frequency of vibrations of a

system is also commonly known as its eigenfrequency.  

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Problem 6.1 Find the equation of motion of a block of mass by a linear spring (spring constant

, suspended

), which is displaced

downwards from its equilibrium position and released with no initial velocity.

 

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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 released?

Figure 8 

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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 below). 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.

 

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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 acceleration 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]

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Figure 9   

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)

 

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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.

Figure 10 

[17]

 

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Perpendicular-axis theorem

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Radius of gyration The moment of inertia of a body about a specified axis is sometimes calculated using the radius of gyration, . This is a geometrical property which has units of length. For a given solid, if its mass and radius of gyration are known, the body’s 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.

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

                

                     

   

              

 

             

 

 

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Problem 6.5 Determine the moment of inertia rod's density

for the slender rod. The

and cross-sectional area

are constant. Ex-

press the result in terms of the rod's total mass

 

.

 

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

. Determine the radius of gyration

density of the material is

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5

⁄

. The

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Problem 6.7 The right circular cone shown is formed by revolving the shaded area around the x axis. Determine the moment of inertia

and express the result in terms of the total mass

of the

cone. The cone has a constant density .

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Problem 6.8 The body of arbitrary shape has a mass and a radius of gyration about amount

of

, mass center at

,

. If it is displaced a slight

from its equilibrium position and released, determine

the natural period of vibration.

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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.

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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 elongation ∆

downwards. The bar

provides a restoring force oppose

the

downward

to dis-

placement of the mass and es-

Figure 11 

tablish an equilibrium condition. A force equal in magnitude and opposite in direction acts

on

the

bar

(action-

reaction).

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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 downwards. Under equilibrium conditions, a restoring force provided by the cantilever beam balances the weight of the mass

Figure 12 

. 3

(cf. “Deflection curves of beams”)

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Thus, we obtain the spring constant for a cantilever beam as 3

III.

[22] 

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:

Figure 13  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 described by 0

[24] 

 

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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).

Figure 14  The two springs (spring constants

and

) undergo the

same elongation when the mass is displaced. They can be replaced by an equivalent single spring with the spring constant .

[25]  Therefore,

for

(spring constants

a

system

of

parallel

springs

), the spring constant of the equivalent

spring is given by the sum of the individual spring constants:

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[26] 

 

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Springs in series Consider two springs in series (Figure 15).

Figure 15  Find the stiffness of the equivalent spring

.

In the case of arbitrarily many springs in series, the spring constant of the equivalent spring is found from The flexibility (compliance) of a spring is defined as the inverse of the stiffness:

1

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Problem 6.10 An elastic beam (flexural rigidity ports a box (mass

) with negligible mass sup-

) as shown below. Find the natural circu-

lar frequency of the system.

 

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

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