chapter 4-force vibration
DESCRIPTION
vibration,free,damping,dampedTRANSCRIPT
FORCE VIBRATION
CHAPTER 4
What is Force Vibration?
Forced Vibrations: External force continuously driving system
Forcing function is applied and the response of the SDOF is explored in terms of time dependency and frequency dependency and compared to the terms found in the equations of motion
The driving force can be
periodic in time
(force oscillates)
Here: study periodic case: sinusoidal force or displacement in time
General cases: study numerically or analytically building on periodic
solution.
or non-periodic
mfender
Source that producing forcing frequency ()
Real system Modeling in vibration for solution control
Kfender Cfender
In this case we can only target as the fender to vibrate since
the fender is mounted on the base of the tank.
Modeling for solution () Real source of ()
Type of Forcing
External Forcing
Rotor Excitation
Base Excitation
External Forcing models the behavior of a system which has a time varying
force acting on it. An example might be an offshore structure subjected to wave
loading, bridge under wind load.
External Forcing
Bridge Collapse due to wind load Wave loading on offshore structure
Rotor Excitation Rotor Excitation models the effect of a rotating machine mounted on a flexible
floor. The crank with small mass mo rotates at constant angular velocity,
causing the mass m to vibrate.
Base Excitation models the behavior of a vibration isolation system. The
base of the spring is given a prescribed motion, causing the mass to vibrate.
This system can be used to model a vehicle suspension system, or the
earthquake response of a structure.
Base Excitation
When forcing frequency is equal to natural frequency n of the system,
amplitude of the motion becomes very large
At = n , amplitude only limited by damping
This is “Resonance”
Resonance can have catastrophic effects …but is critical to many technologies
Analyze basic system:
mass, spring, dashpot, and harmonic force
2
0 2sin
dx d xkx c F t m
dt dt
Equation of Motion:
)sin( tFo )sin( tFo
2
02sin
d x dxm c kx F t
dt dt
22
22 sin( )o
n n
Fd x dxx t
dt dt m
Finding a Particular Solution
2
1
1
2tan
n
n
Functions only of
Driving frequency
Natural Frequency
n
Note that Displacement that would be caused by static force of Fo oo
n
o Xk
F
m
F2
Displacement:
2222
21
1
nn
n
o
m
FX
22221
1
nn
oX
XM
Define “Magnification Factor”
M= (Dynamic Amplitude of Vibration)/(Static Amplitude)
Depends on
two numbers:
, / n
2222
21
1
nn
n
o
m
FX
n
Magnification Factor:
Large amplification for
1 1~/ n
Little amplification
at low n /
No amplification
at high n /
No amplification
for high
Prob.1
The fan has a mass of 25kg and fixed to the end of a horizontal
beam that has a negligible mass. The fan blade is mounted
eccentrically on the shaft such that it is equivalent to an
unbalanced 0.35kg mass located 100 mm from the axis of
rotation. If the static deflection of the beam is 50 mm as a result
of the weight of the fan, determine the angular speed of the fan at
which resonance will occur.
Prob.2
The fan has a mass of 25kg and fixed to the end of a horizontal
beam that has a negligible mass. The fan blade is mounted
eccentrically on the shaft such that it is equivalent to an
unbalanced 0.35kg mass located 100 mm from the axis of
rotation. If the static deflection of the beam is 50 mm as a result
of the weight of the fan, determine the amplitude of steady-state
vibration of the fan when the angular velocity of the fan is 10
rad/s.
Prob.3
The electric motor has a mass of 50kg and is
supported by four springs, each spring having of
100 N/m. If the motor turns a disk D which is
mounted eccentrically, 20mm from the disk’s
center, determine the angular rotation at which
the resonance occurs.
The tail rotor section of the helicopter consists of four blades, each of mass 2.1
kg, and an engine box of mass 25 kg. The center of gravity of each blade is
170 mm from the rotational axis. The tail section is connected to the main body
of the helicopter by an elastic structure. The natural frequency of the tail section
has been observed as 150 rad/s. During flight the rotor operates at 900 rpm.
Assume the system has a damping ratio of =0.05.
i) Sketch the mathematical model of the system.
ii) During flight a 75-g particle become stuck to one of the blades, 250 mm from
the axis of the rotation. Calculate the amplitude of vibration.
ans. X= 0.36 mm
Prob.4
External Forcing Base Excitation Rotor Excitation
All of these situations are of practical interest. Some subtle but important
distinctions to consider, so we will look at each.
BUT
Strategy is simple: derive Equation of Motion and put into the “Standard Form”
Base Excitation – the Earthquake Problem
Here, base supporting object is subjected to motion.
How does the object respond?
Forces in the spring, dashpot are proportional to
the motion RELATIVE to the base
Draw F.B.D. and get equation of motion….
)sin()( tYty
)(tx
222
2
21
)/2(1
nn
n
Y
X
(Displacement Amplitude of body)/(Displacement Amplitude of Base)
Rough Surface or wavy road
Base Excitation
Displacement
transmission ratio:
222
2
21
)/2(1
nn
n
Y
X
Y
X
n /
1.0
2.0
3.0
0.11.0
Harmonic Base Excitation – Motion
Relative to Base
Sometimes the motion relative to the base is of interest
Introducing the relative displacement z = x – y, the equation of
motion:
or, in terms of relative displacement:
222
2
21nn
n
Y
Z