APPLIED MECHANICS Lecture 05 Slovak University of Technology Faculty of Material Science and Technology in Trnava
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