Lecture-10 2DOF Free

TWO-DEGREE-OF-FREEDOM

SYSTEMS EQUATIONS OF MOTION AND FREE

VIBRATION RESPONSE

MEMB443 Mechanical Vibrations

LEARNING OBJECTIVES

Upon completion of this lecture, you should be able to:

Understand the difference between generalized and principal coordinates.

Determine the equations of motion of a two-degree-of-freedom system.

Express the equations of motion in matrix form.

Determine the characteristics equation, undamped natural frequencies and associated mode shapes.


INTRODUCTION

Most engineering systems are continuous and have an infinite number of degrees of freedom.

For simplicity of analysis, continuous systems are often approximated as multi-degree of freedom systems, which

requires the solution of a set of ordinary differential

equations.

There is one equation of motion for each degree of freedom.

There are n natural frequencies, each associated with its own mode shape, for a system having n degrees of

freedom.


MATHEMATICAL MODELING

A continuous system can be approximated as a multi-degree of freedom system by substituting the distributed

masses or inertia with a finite number of lumped masses or

rigid bodies.

The lumped masses are assumed to be connected by massless elastic and damping members.

Linear or angular coordinates are used to describe the motion of the lumped masses or rigid bodies.

The minimum number of coordinates necessary to describe the motion of the lumped masses or rigid bodies defines

the number of degrees of freedom of the system.


DEGREES OF FREEDOM

Number of degrees of freedom = Number of masses in the system x number of possible types of motion of each mass.


EQUATIONS OF MOTION

There are n equations of motion for an n degrees of freedom system, one for each degree of freedom.

These equations are generally in the form of coupled differential equations; each equation involves all the

coordinates.

If a harmonic solution is assumed for each coordinate, the equations of motion lead to a frequency equation that gives

n natural frequencies.

During free vibration at one of the natural frequencies, the amplitudes of the n degrees of freedom are related in a

specific manner and the configuration is called a normal

mode, principal mode, or natural mode of vibration.


EQUATIONS OF MOTION (cont.)

If an arbitrary initial condition is given to a system, the resulting free vibration will be a superposition of the

normal modes of vibration.

If the system vibrates under the action of an external harmonic force, the resulting forced harmonic vibration

takes place at the frequency of the applied force.

Under harmonic excitation, resonance occurs when the forcing frequency is equal to the natural frequencies of the

system (i.e. the amplitudes of the motion will be at a

maximum).


EQUATIONS OF MOTION (cont.)

The configuration of a system can be specified by a set of independent coordinates such as length, angle, or some physical parameters.

Any such set of coordinates is called generalized coordinates.

There exist a particular set of coordinates which uncouples the equations of motion (each equation of motion contains only one coordinate).

Such a set of coordinates is called principal coordinates.


FORCED VIBRATION EQUATIONS OF

MOTION

Forced vibration of a two-degree-of-freedom system.



MOTION (cont.)

The equations of motion for forced vibration of a two-degree-of-freedom system is given by:

When , the equations become uncoupled, which implies that the two masses are not

physically connected.

2232122321222

1221212212111

Fxkkxkxccxcxm

Fxkxkkxcxccxm

022 kc



MOTION (cont.)

In matrix form, the equations of motion can be expressed as:

tFtxktxctxm

322

221

322

221

2

1

0

0

kkk

kkkk

ccc

cccc

m

mm Mass Matrix

Damping Matrix

Stiffness Matrix



MOTION (cont.)

The displacement vectors and force vectors for the matrix formulation of the equations of motion are given by:

tFtxktxctxm

tF

tFtF

tx

txtx

2

1

2

1

Displacement Vectors

Force Vectors


UNDAMPED FREE VIBRATION

ANALYSIS

For free vibration analysis, the forces and are set to zero.

Assuming that it is possible to have harmonic motion of and at the same frequency and the same phase

angle , the solutions are:

0

0

2321222

2212111

txkktxktxm

txktxkktxm

tXtx

tXtx

cos

cos

22

11

2m1m

tF1 tF2



ANALYSIS (cont.)

Substituting the solutions into the equations of motion, we obtain:

Since the above equation must be satisfied for all values of the time , the terms between brackets must be zero. t

0cos

0cos

2322

212

221212

1

tXkkmXk

tXkXkkm

0

0

2322

212

221212

1

XkkmXk

XkXkkm



ANALYSIS (cont.)

The previous equations can be expressed in matrix form as given below:

0

0

2

1

32

2

22

221

2

1

X

X

kkmk

kkkm



ANALYSIS (cont.)

It can be seen that the previous equations are satisfied by the trivial solution , which implies that there

is no vibration.

For a non-trivial solution of and , the determinant of the coefficients of and must be zero.

021 XX

1X 2X

1X 2X

0det32

2

22

221

2

1

kkmk

kkkm



ANALYSIS (cont.)

The previous equations can be expanded and expressed as given below:

This equation is called the Characteristic Equation or the frequency equation.

The solution of the characteristic equation yields the frequencies or the characteristics values of the system.

0223221

2

132221

4

21

kkkkk

mkkmkkmm



ANALYSIS (cont.)

The roots of the characteristic equation are given by:

5.0

21

2

23221

2

21

132221

21

1322212

2

2

1

42

1

2

1,

mm

kkkkk

mm

mkkmkk

mm

mkkmkk



ANALYSIS (cont.)

This shows that it is possible for the system to have a non-trivial harmonic solution of the following form when is

equal to and .

and are the natural frequencies of the system.

tXtx

tXtx

cos

cos

22

11

21

1 2



ANALYSIS (cont.)

The values of and depends on the natural frequencies and .

Denoting the values of and corresponding to as and and those corresponding to as

and , we obtain: 2

1 21X 2X

1X 2X 1

11X 21X

22X

12X

32222

2

2

21221

21

22

2

32212

2

2

21211

11

12

1

kkm

k

k

kkm

X

Xr

kkm

k

k

kkm

X

Xr



ANALYSIS (cont.)

Normal modes of vibration corresponding to and is expressed as:

The vectors and , which define the normal modes of vibration, are known as the modal vectors of the

system.

2

12

2

1X

2X

212

21

22

212

111

11

12

111

Xr

X

X

XX

Xr

X

X

XX



ANALYSIS (cont.)

The free vibration solution of the motion in time can be expressed as:

222

12

222

12

2

212

111

11

111

11

2

111

cos

cos

cos

cos

tXr

tX

tx

txtx

tXr

tX

tx

txtx


FREE VIBRATION ANALYSIS OF A

TORSIONAL SYSTEM

Forced vibration of a two-degree-of-freedom torsional system.



TORSIONAL SYSTEM (cont.)

The previous figure shows a torsional system consisting of two discs mounted on a shaft.

The three segments of the shaft have rotational spring constants , , and .

The discs have mass moments of inertia and , the applied torques are and and the rotational

degrees of freedom and .

2tk 3tk1tk

1J 2J

2tM1tM

21



TORSIONAL SYSTEM (cont.)

The differential equations of rotational motion for the discs and can be expressed as:

For the free vibration analysis of this system, the above equation reduces to:

1J 2J

22312222

11221111

ttt

ttt

MkkJ

MkkJ

0

0

2321222

2212111

ttt

ttt

kkkJ

kkkJ


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