CHAPTER – IIIFREE VIBRATION OF SINGLE

DEGREES OF FREEDOM systems

Outlines

Free Vibration of an Undamped Translational System

Free Vibration of an Undamped Torsional System

Free Vibration with Viscous Damping (Translational and Torsional system)

Free Vibration with Coulomb Damping (Translational and Torsional system)

Free Vibration with Hysteretic Damping

Free vibration occurs because there is an initial disturbance to the

system.

The initial disturbance is referred to as an initial condition, which

can be either a displacement of the mass, an initial velocity of the

mass, or both.

Frequency of the system is determined by the mass and spring

constant of the system, initial conditions do not affect the vibration

frequency of an undamped system, therefore it is called the natural

frequency.

Vibration amplitude and the phase angle are also determined by

initial conditions. Theoretically, an undamped free vibration system

will vibrate forever once it is started.

Free Vibration of an Undamped Translational System

A system is said to undergo free vibration when it oscillates only

under an initial disturbance with no external forces acting afterward.

Some examples are the oscillations of the pendulum of a

grandfather clock, the vertical oscillatory motion felt by a bicyclist

after hitting a road bump, and the motion of a child on a swing after

an initial push.

Single-degree-of-freedom (SDOF) system is a system whose motion is

defined just by a single independent co-ordinate (or function) e.g. x which

is a function of time.

SDOF systems are often used as a very crude approximation for a

generally much more complex system.

Figure shows a spring-mass system that represents the simplest possible

vibratory system. It is called a single-degree-of-freedom system,

since one coordinate (x) is sufficient to specify the position of the mass at

any time. There is no external force applied to the mass; hence the motion

resulting from an initial disturbance will be free vibration.

Fig: A spring-mass system in horizontal position

The aim of developing a SDOF mathematical model is to use it in

order to find the position 𝑥(𝑡) of the moving mass m at any instant

of time, also often velocity ሶ𝑥(𝑡) and acceleration ሷ𝑥(𝑡).

Question: How can we drive the equation of motion for this system?

The equation of motion can be derived using:

Newton's 2nd law of motion

D’Alembert's Principle

The principle of virtual displacement

The principle of conservation of energy

Using Newton's 2nd law of motion

∑𝐹 = 𝑚𝑎,−𝑘 𝑥 = 𝑚 Ẍ

Thus, 𝑚Ẍ + 𝑘𝑥 = 0 is the equation of motion.

Using D’Alembert's principle states that

“the sum of all active and reactive forces minus the inertia force

gives the virtual state of equilibrium known as dynamic equilibrium

state.”

∑𝐹 −𝑚 ሷ𝑥 = 0

−𝑘𝑥 −𝑚 ሷ𝑥 = 0

𝑚 ሶ𝑥 + 𝑘𝑥 = 0 is the equation of motion.

9

Using Principle of Virtual Displacements

“If a system that is in equilibrium under the action of a set of forces

is subjected to a virtual displacement, then the total virtual work

done by the forces will be zero.”

Consider spring-mass system as shown in figure, the virtual work done

by each force can be computed as:

xxmW

xkxW

i

S

)( force inertia by the done work Virtual

)( force spring by the done work Virtual

0 xkxxxm

0 kxxm

10

Since the virtual displacement can have an arbitrary value, 𝛿𝑥 ≠ 0,

equation above gives the equation of motion of the spring-mass

system as

When the total virtual work done by all the forces is set equal to zero,

we obtain

Using Principle of conservation of energy

A system is said to be conservative if no energy is lost due to

friction or energy-dissipating non elastic members.

If no work is done on a conservative system by external forces

(other than gravity or other potential forces)

Then the total energy of the system remains constant. Since the

energy of a vibrating system is partly potential and partly kinetic,

the sum of these two energies remains constant.

The kinetic energy T is stored in the mass by virtue of its velocity,

and the potential energy U is stored in the spring by virtue of its

elastic deformation.

Thus the principle of conservation of energy can be

expressed as:

The kinetic and potential energies are given by:

and

Substitute these equations in to the above equation,

Thus, the equation of motion can be:

0)(

constant

UTdt

d

UT

2

2

1xmT

2

2

1kxU

0 kxxm

Response of un damped free vibration

Force acting on the mass due to the spring is – 𝑘𝑥.

For the above differential equation we have three possible solutions.

x = 𝐴𝑒 𝒔𝒕 , 𝑥 = 𝐴 sin𝜔𝑡, 𝑥 = 𝐵 cos𝜔𝑡

Also any combination of these solution can be a solution, the most

general form of solution for this equation will be:

The most general form of solution for this equation will be

Where, ωn = Τ𝐾 𝑚, is the natural frequency

A1 and A2 are constants, which can be determined from the

initial conditions of the system.

Thus the solution of the equation subject to the initial conditions is

given by

tAtAtx nn sincos)( 21

02

01

)0(

)0(

xAtx

xAtx

n

tx

txtx n

n

n

sincos)( 00

The above equation is harmonic function of time.

The amplitude and phase angle can be:

16

If initial displacement (𝑥0) is zero,

tx

tx

tx n

n

n

n

sin

2cos)( 00

If initial velocity ( ሶ𝑥0) is zero,

txtx ncos)( 0

Free Vibration of an Undamped Torsional System

l

GIM t

0

17

From the theory of torsion of circular shafts, we have the relation:

Shear modulus

Polar moment of

inertia of cross

section of shaft

Length shaftTorque

32

4

0

dI

18

l

Gd

l

GIMk t

t32

4

0

Polar Moment of Inertia:

Torsional Spring Constant:

Equation of Motion:

00 tkJ

2/1

0

J

ktn

19

Applying Newton’s Second Law of Motion,

Thus, the natural circular frequency:

The period and frequency of vibration in cycles per second are:

2/1

0

2/1

0

2

1

2

J

kf

k

J

tn

t

n

Note the following aspects of this system:

g

WDDhJ

832

44

0

20

1) If the cross section of the shaft supporting the disc is not circular,

an appropriate torsional spring constant is to be used.

2) The polar mass moment of inertia of a disc is given by:

3) An important application: in a mechanical clock

where ρ is the mass density

h is the thickness

D is the diameter

W is the weight of the disc

General solution can be obtained:

tAtAt nn sincos)( 21

00 )0()0( and )0(

tdt

dtt

21

where A1 and A2 can be determined from the initial conditions. If

The constants A1 and A2 can be found:

nA

A

/02

01

Equation above can also represent a simple harmonic motion.

RESPONSE OF DAMPED FREE VIBRATION

A single degree of freedom system consists of a mass, a spring, and a damper if

the system is modeled as a damped system.

The spring is modeled as a linear spring, which provides a restoring force.

The damper is modeled as a viscous damper, which provides a damping force

proportional to a relative displacement and acting in the direction against a

velocity vector.

If there is a driving force acting on the mass, the system vibrates under the

driving force, which is called forced vibration.

Otherwise, the system may vibrate under initial displacement and/or initial

velocity, which is called free vibration.

Physically, there is no vibrating system that vibrates forever, that means there isalways some kind of damping in the system that dissipates energy.

For mathematical simplicity, the damping is modeled as viscous damping.

Depending on the magnitude of damping, a damped system can be underdamped,critically damped or overdamped.

The critical damping coefficient is determined by the system's mass and springconstant.

Under critical damping, the damping ratio is unity. Critical damping separates nooscillatory motion from oscillatory motion.

When the damping ratio is greater than 1, which is called overdamping, thesystem does not oscillate. For a damping ratio less than 1, which is calledunderdamping, the system oscillates with decaying magnitude, as shown in thefigure below.

For most physical system, damping ratios are less than 1. Actually, most physicalsystems have damping ratio less than 0.1. With damping in the free vibrationsystem, the mass always restores its equilibrium position even it is disturbed. Thegreater the damping, the less time it takes to restore its equilibrium position. So inmost cases, adequate damping is desireable.

Free Vibration with Viscous Damping

The viscous damping force F is proportional to the velocity ẋ

or v can be expressed by

𝐹 = −𝑐 ሶ𝑥

where c is the damping constant or coefficient of viscous

damping and the negative sign indicates that the damping force is

opposite to the direction of velocity

Contd.,The equation of motion is then

Newton's second law

∑𝐹 = 𝑚𝑎

𝑚 ሷ𝑥 = −𝑐 ሶ𝑥 − 𝑘𝑥

𝑚 ሷ𝑥 + 𝑐 ሶ𝑥 + 𝑘𝑥 = 0

We assume that solution in the form of

𝑥 𝑡 = 𝐶𝑒𝑠𝑡

Insert this equation into the previous equation

𝑚𝑠2 + 𝑐𝑠 + 𝑘 = 0

Then the roots are

𝑠1,2 =−𝑐± 𝑐2−4𝑚𝑘

2𝑚=

−𝑐

2𝑚± (

𝑐

2𝑚)2−

𝑘

𝑚

Contd.,

The roots gives the two solutions

The most general solution in the combination of these solutions

𝑥1 𝑡 = 𝐶1𝑒𝑠1𝑡 and 𝑥2 𝑡 = 𝐶2𝑒

𝑠2𝑡

𝑥 𝑡 = 𝐶1𝑒𝑠1𝑡 + 𝐶2𝑒

𝑠2𝑡

= 𝐶1𝑒 −𝑐

2𝑚+

𝑐

2𝑚

2−

𝑘

𝑚𝑡 + 𝐶2𝑒 −

𝑐

2𝑚+

𝑐

2𝑚

2−

𝑘

𝑚𝑡

Where, 𝐶1 and 𝐶1 are arbitrary constants to be find from initial conditions.

Critical damping constant and damping ratio:

The critical damping is defined as the value of the damping constant 𝐶𝐶for which the radical in Equation becomes zero.

𝑐𝑐2𝑚

2

−𝑘

𝑚= 0

𝑐𝑐 = 2𝑚 ൗ𝑘 𝑚 = 2 𝑘𝑚 = 2𝑚𝜔𝑛

Contd.,For any damped system the damping ratio is defined as the ratio

of damping constant to critical damping constant.

ζ = ൗ𝑐 𝑐𝑐We can write

𝑐

2𝑚=

𝑐

𝑐𝑐.𝑐𝑐2𝑚

= ζ𝜔𝑛

And hence

𝑠1,2 = 𝐶1𝑒−𝜁+ 𝜁2−1 𝜔𝑛𝑡 + 𝐶2𝑒

−𝜁− 𝜁2−1 𝜔𝑛𝑡

Contd.,

Comparison of motions with different types of damping

Logarithmic Decrement

The logarithmic decrement represents the rate at which the

amplitude of a free-damped vibration decreases.

It is defined as the natural logarithm of the ratio of any two

successive amplitudes.

Let t1 and t2 denote the times corresponding to two consecutive

amplitudes

Logarithmic Decrement

There are many methods for measuring the damping of a vibration

system. Logarithmic decrement method and bandwidth method are

introduced here.

Logarithmic decrement method is used to measure damping in time

domain. In this method, the free vibration displacement amplitude

history of a system to an impulse is measured and recorded. A

typical free decay curve is shown as below. Logarithmic decrement

is the natural logarithmic value of the ratio of two adjacent peak

values of displacement in free decay vibration.

Contd.,

Torsion system with viscous damping

• The viscous damping torque for a single dof torsionalsystem

• Where ct is the torsional viscous damping

• The equation of motion is

Free vibration with coulomb damping

• In many mechanical systems, Coulomb or dry-friction dampers areused because of their mechanical simplicity and convenience.

• Coulomb s law of dry friction states that, when two bodies are incontact, the force required to produce sliding is proportional to thenormal force acting in the plane of contact.

• Thus, the friction force F is given by

𝐹 = 𝜇𝑁 = 𝜇𝑊 = 𝜇𝑚𝑔where N is normal force,

µ is the coefficient of dry or kinetic or sliding friction.

• The friction force acts in a direction opposite to the direction ofvelocity.

• Coulomb damping is sometimes called constant damping.

• Since it is independent of the displacement and velocity; it dependsonly on the normal force N between the sliding surfaces.

• Consider a SDOF system with dry friction as shown in Fig. (a).

• Since friction force varies with the direction of velocity, we need to

consider two cases as indicated in Fig.(b) and (c).

Case 1: When x is positive and dx/dt is positive or when x is negative and

dx/dt is positive (i.e., for the half cycle during which the mass moves from

left to right) the equation of motion can be obtained using Newton’s second

law (Fig.b):

• Case 2: When x is positive and dx/dt is negative or when x is

negative and dx/dt is negative (i.e., for the half cycle during which

the mass moves from right to left) the equation of motion can be

derived from Fig. (c):

Case (2):(When Frictional force in positive)

Case (1): SPRING MASS WITH COULOMB DAMPING

(When Frictional force in negative)

Q. What is the solution for this equation of motion?

Damping Material or Solid or Hysteretic Damping