THEORY OF VIBRATION

Introduction

There are two general classes of vibrations - free and forced. Free vibration takes place when a system oscillates under the action of forces inherent in the system itself, and when external impressed forces are absent. The system under free vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamic system established by its mass and stiffness distribution.

Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered, and dangerously large oscillations may result. The failure of major structures such as bridges, buildings, or airplane wings is an awesome possibility under resonance. Thus, the calculation of the natural frequencies of major importance in the study of vibrations.

Vibrating systems are all subject to damping to some degree because energy is dissipated by friction and other resistances. If the damping is small, it has very little influence on the natural frequencies of the system, and hence the calculation for the natural frequencies are generally made on the basis of no damping. On the other hand, damping is of great importance in limiting the amplitude of oscillation at resonance.

The number of independent coordinates required to describe the motion of a system is called degrees of freedom of the system. Thus, a free particle undergoing general motion in space will have three degrees of freedom, and a rigid body will have six degrees of freedom, i.e., three components of position and three angles defining its orientation. Furthermore, a continuous elastic body will require an infinite number of coordinates (three for each point on the body) to describe its motion; hence, its degrees of freedom must be infinite. However, in many cases, parts of such bodies may be assumed to be rigid, and the system may be considered to be dynamically equivalent to one having finite degrees of freedom. In fact, a surprisingly large number of vibration problems can be treated with sufficient accuracy by reducing the system to one having a few degrees of freedom.

 

Harmonic Motion

Oscillatory motion may repeat itself regularly, as in the balance wheel of a watch, or display considerable irregularity, as in earthquakes. When the motion is repeated in equal intervals of time T, it is called period motion. The repetition time t is called the period of the oscillation, and its reciprocal, ,is called the frequency. If the motion is designated by the time function x(t), then any periodic motion must satisfy the relationship .

Harmonic motion is often represented as the projection on a straight line of a point that is moving on a circle at constant speed, as shown in Fig. 1. With the angular speed of the line o-p designated by w , the displacement x can be written as

(1)

Figure 1 Harmonic Motion as a Projection of a Point Moving on a Circle

The quantity w is generally measured in radians per second, and is referred to as the angular frequency. Because the motion repeats itself in 2p radians, we have the relationship

(2)

where t and f are the period and frequency of the harmonic motion, usually measured in seconds and cycles per second, respectively.

The velocity and acceleration of harmonic motion can be simply determined by differentiation of Eq. 1. Using the dot notation for the derivative, we obtain

(3)

(4)

 

Vibration Model

The basic vibration model of a simple oscillatory system consists of a mass, a massless spring, and a damper. The spring supporting the mass is assumed to be of negligible mass. Its force-deflection relationship is considered to be linear, following Hooke's law, , where the stiffness k is measured in newtons/meter.

The viscous damping, generally represented by a dashpot, is described by a force proportional to the velocity, or . The damping coefficient c is measured in newtons/meter/second.

Figure 2 Spring-Mass System and Free-Body Diagram

 

Equation of Motion : Natural Frequency

Figure 2 shows a simple undamped spring-mass system, which is assumed to move only along the vertical direction. It has one degree of freedom (DOF), because its motion is described by a single coordinate x.

When placed into motion, oscillation will take place at the natural frequency fn which is a property of the system. We now examine some of the basic concepts associated with the free vibration of systems with one degree of freedom.

Figure 2 Spring-Mass System and Free-Body Diagram

Newton's second law is the first basis for examining the motion of the system. As shown in Fig. 2 the deformation of the spring in the static equilibrium position is D , and the spring force kD is equal to the gravitational force w acting on mass m

(5)

By measuring the displacement x from the static equilibrium position, the forces acting on m are and w. With x chosen to be positive in the downward direction, all quantities - force, velocity, and acceleration are also positive in the downward direction.

We now apply Newton's second law of motion to the mass m :

and because kD = w, we obtain :

(6)

It is evident that the choice of the static equilibrium position as reference for x has eliminated w, the force due to gravity, and the static spring force kD from the equation of motion, and the resultant force on m is simply the spring force due to the displacement x.

By defining the circular frequency w n by the equation

(7)

Eq. 6 can be written as

(8)

and we conclude that the motion is harmonic. Equation (8), a homogeneous second order linear differential equation, has the following general solution :

(9)

where A and B are the two necessary constants. These constants are evaluated from initial conditions , and Eq. (9) can be shown to reduce to

(10)

The natural period of the oscillation is established from , or

(11)

and the natural frequency is

(12)

These quantities can be expressed in terms of the static deflection D by observing Eq. (5), . Thus, Eq. (12) can be expressed in terms of the static deflection D as

(13)

Note that , depend only on the mass and stiffness of the system, which are properties of the system.

 

Viscously Damped Free Vibration

Viscous damping force is expressed by the equation

(14)

where c is a constant of proportionality.

Symbolically. it is designated by a dashpot, as shown in Fig. 3. From the free body diagram, the equation of motion is .seen to be

(15)

The solution of this equation has two parts. If F(t) = 0, we have the homogeneous differential equation whose solution corresponds physically to that of free-damped vibration. With F(t) ¹ 0, we obtain the particular solution that is due to the excitation irrespective of the homogeneous solution. We will first examine the homogeneous equation that will give us some understanding of the role of damping.

Figure 3 Viscously Damped Free Vibration

With the homogeneous equation :

(16)

the traditional approach is to assume a solution of the form :

(17)

where s is a constant. Upon substitution into the differential equation, we obtain :

which is satisfied for all values of t when

(18)

Equation (18), which is known as the characteristic equation, has two roots :

(19)

Hence, the general solution is given by the equation:

(20)

where A and B are constants to be evaluated from the initial conditions and .

Equation (19) substituted into (20) gives :

(21)

The first term, , is simply an exponentially decaying function of time. The behavior of the terms in the parentheses, however, depends on whether the numerical value within the radical is positive, zero, or negative.

When the damping term (c/2m)2 is larger than k/m, the exponents in the previous equation are real numbers and no oscillations are possible. We refer to this case as overdamped.

When the damping term (c/2m)2 is less than k/m, the exponent becomes an imaginary number, . Because

the terms of Eq. (21) within the parentheses are oscillatory. We refer to this case as underdamped.

In the limiting case between the oscillatory and non oscillatory motion , and the radical is zero. The damping corresponding to this case is called critical damping, cc.

(22)

Any damping can then be expressed in terms of the critical damping by a non dimensional number z , called the damping ratio:

(23)

and

(24)

(i) Oscillatory Motion (z < 1.0) Underdamped Case :

(25)

The frequency of damped oscillation is equal to :

(26)

Figure 4 shows the general nature of the oscillatory motion.

Figure 4 Damped Oscillation z < 1

(ii) Non oscillatory Motion (z > 1.0) Overdamped Case :

(27)

The motion is an exponentially decreasing function of time as shown in Fig. 5.

Figure 5 Aperiodic Motion z > 1

(iii) Critically Damped Motion (z = 1.0) :

(28)

Figure 6 shows three types of response with initial displacement x(0).

Figure 6 Critically Damped Motion z = 1

 

Forced Harmonic Vibration

Harmonic excitation is often encountered in engineering systems. It is commonly produced by the unbalance in rotating machinery. Although pure harmonic excitation is less likely to occur than periodic or other types of excitation, understanding the behavior of a system undergoing harmonic excitation is essential in order to comprehend how the system will respond to more general types of excitation. Harmonic excitation may be in the form of a force or displacement of some point in the system.

We will first consider a single DOF system with viscous damping, excited by a harmonic force , as shown in Fig. 7. Its differential equation of motion is found from the free-body diagram.

(29)

Figure 7 Viscously Damped System with Harmonic Excitation

The solution to this equation consists of two parts, the complementary function, which is the solution of the homogeneous equation, and the particular integral. The complementary function. in this case, is a damped free vibration.

The particular solution to the preceding equation is a steady-state oscillation of the same frequency w as that of the excitation. We can assume the particular solution to be of the form :

(30)

where X is the amplitude of oscillation and f is the phase of the displacement with respect to the exciting force.

The amplitude and phase in the previous equation are found by substituting Eqn. (30) into the differential equation (29). Remembering that in harmonic motion the phases of the velocity and acceleration are ahead of the displacement by 90° and 180°, respectively, the terms of the differential equation can also be displayed graphically, as in Fig. 8.

Figure 8 Vector Relationship for Forced Vibration with Damping

It is easily seen from this diagram that

(31)

and

(32)

We now express Eqs (31) and (32) in non-dimensional term that enables a concise graphical presentation of these results. Dividing the numerator and denominator of Eqs. (31) and (32) by k, we obtain :

(33)

and

(34)

These equations can be further expressed in terms of the following quantities:

The non-dimensional expressions for the amplitude and phase then become

(35)

and

(36)

These equations indicate that the non dimensional amplitude , and the phase f are functions only of the frequency ratio , and the damping factor z and can be plotted as shown in Fig 9.

Figure 9 Plot of Eqs. (35) and (36)

Rotating Unbalance

Unbalance in rotating machines is a common source of vibration excitation. We consider here a spring-mass system constrained to move in the vertical direction and excited by a rotating machine that is unbalanced, as shown in Fig. 10. The unbalance is represented by an eccentric mass m with eccentricity e that is rotating with angular velocity w . By letting x be the displacement of the non rotating mass (M - m) from the static equilibrium position, the displacement of m is :

Figure 10 Harmonic Disturbing Force Resulting from Rotating Unbalance

The equation of motion is then :

which can be rearranged to :

(37)

It is evident that this equation is identical to Eq. (29), where is replaced by , and hence the steady-state solution of the previous section can be replaced by :

(38)

and

(39)

These can be further reduced to non dimensional form :

(40)

and

(41)

Example

A counter rotating eccentric weight exciter is used to produce the forced oscillation of a spring-supported mass as shown in Fig. 11. By varying the speed of rotation, a resonant amplitude of 0.60 cm was recorded. When the speed of rotation was increase considerably beyond the resonant frequency, the amplitude appeared to approach a fixed value of 0.08 cm. Determine the damping factor of the system.

Figure 11

Solution :

From Eqn. (40), the resonant amplitude is :

When w is very much greater than , the same equation becomes

By solving the two equations simultaneously, the damping factor of the system is

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