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

http://www.boatdesign.net/forums/attachments/hydrodynamics-aerodynamics/57602d1306714847-hull-asymmetry-minimum-wave-drag-hull-trim.jpg

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