APPLIED MECHANICS

Lecture 05

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The system is excited by a harmonic force of the form

where F0 - amplitude of the forced vibration,

- the forced angular frequencies.

)sin()( 0 tFtF

m

k F(t) = F0sin(t)

x,.x,

..x,

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The equation of motion

)sin(0 tFkxxm )sin(0 tm

Fx

m

kx )sin(2

0 tqxx

The solution of equation

pxtBtAx )cos()sin( 00

The particular solution xp

)sin( tCx pp )sin(2 tCx pp

The constant Cp is determined for 0

220

220 )(

qCqC pp

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The solution

resp.

The constants A and B (C and ) are determined from the initial conditions

)sin()cos()sin(22

000 t

qtBtAx

)sin()sin(22

00 t

qtCx

0

00vx

xxt

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The constants

The solution is

.0 ,0

220

B

qA

)sin()sin( 00

220

ttq

x

The derivative with respect to time

)cos()sin()cos(22

00000 t

qtBtAx

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The displacement is a combined motion of two vibrations: one with the natural frequency 0,

one with the forced frequency The resultant is a nonharmonic

vibration

rad/s 1,0rad/s, 1N/kg, 1 0 q

The amplitude is:

00

20

220

022

0

)1(

1

)(

Ak

F

k

F

m

FqA

where 0

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

Curve of resonance

Resonance - excitating frequency is equal to the natural angular frequency 0 - the resonance phenomenon appears.

Diagram of resonance phenomenon

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - CENTRIFUGAL EXCITING FORCE

Unbalance in rotating machines is a common source of vibration excitation. Frequently, the excited harmonic force came from an unbalanced mass that is in a rotating motion that generates a centrifugal force

)sin()sin()( 200 trmtFtF

m0 is an unbalanced mass connected to the mass m1 with a massless crank of lengths r,

the mass m0 rotates with a constant angular frequency .

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - CENTRIFUGAL EXCITING FORCE

The amplitude of the combined vibration

,11

1

1

1

1

1

00

2

20

2

20

2

20

20

220

Arm

mr

m

m

m

k

rm

m

k

rm

k

FqA

where m = m1 + m0.

02

2

2

01

AA

The magnification factor

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - CENTRIFUGAL EXCITING FORCE

Variation of the magnification factor

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - ARBITRARY EXCITING FORCE

The general case of exciting force is an arbitrary function of time

SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - ARBITRARY EXCITING FORCE

The differential equation of motion

)(tFkxxm

0

00

00 )](sin[)(1

)cos()sin( dtttFm

tBtAx

where is presented in Figure; A, B are constants.

The vibration in this case is described

The integral in equation is called the Duhamel integral.

SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The mechanical model

The equation of motion

m

b

kF(t) = F0sin(t)

x,.x,

..x

)sin(0 tFkxxbxm The following notation is used:

22m

b0

m

kq

m

F0

SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The equation of motion becomes

Case 1:

)sin(2 20 tqxxx

)( crbb or 0

02 20

2 rr

with the roots

diir 220

20

22,1

21 xxx

The characteristic equation

The general solution of differential equation

x1 - solution of the differential homogenous equation, x2 - particular solution of the differential nonhomogeneous equation

SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The solution of the free damped system

The solution of the forced (excited) vibration

Solution of the forced vibration is introduced into equation of motion

)sin())cos()sin(( 11111 teCtBtAex dt

ddt

)cos()sin( 212 tDtDx

D1, D2 are determined by the identification method.

).sin()]cos()sin([

)]sin()cos([)]cos()sin([

tqtDtD

tDtDtDtD

2120

2122

12

2

SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The linear system of algebraic equation

D1, D2 are obtained

.0)(2

,2)(22

021

222

01

DD

qDD

.4)(

2

,4)(

)(

222220

2

222220

220

1

qD

qD

SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The forced vibration x2

or

)]cos(2)sin()[(4)(

22022222

02 tt

qx

)sin( 222 tBx

.2

tan

,4)(

1

2201

22

222220

22

212

D

D

qDDB

The motion of the system

)sin()sin( 221 tBteBx ddt

SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The amplitude of forced vibration

The magnification factor and phase delay

,4)1(

10

222220

2 Am

FqB

p

22221

4)1(

1

p

A22

1

2arctan

p

crp bb 0 - damping ratio

SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

The graphic of the vibration

m 00 x

m/s 2,00 v

rad/s 50

N/kg 1q

rad/s 3,0-1s 1,0

SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE

Resonance

A-F characteristics Phase delay