Technological Institute of the Philippines938 A. Boulevard, Cubao, Quezon City

Written Report in Vibration Engineering“FORCED VIBRATION”

Submitted by: Submitted to:NOEL, Kevin Louis V. Engr. Lito SazonTORRES, Mark John A.MANNAG, Gian Wilson E.CACA, Ismael P.

What is Forced Vibration?

When a periodically varying force is applied on an object, the object vibrates in accordance to the frequency applied force. The frequency of the object is the same as the frequency of the applied force. This is called forced vibration. when the force is removed, the object will not vibrate with this frequency, but it will vibrate with a frequency peculiar to the object called natural, every object has its own natural frequency. Thus to make a body to vibrate with its natural frequency, we apply some varying force and remove that force, on removing the force, the object will vibrate with its natural frequency. Of course due to damping eventually the vibration will cease after certain time. Suppose that the frequency of the applied force coincides with natural frequency in other words if the frequency of the forced vibration coincides with natural frequency of the object, the amplitude of vibration is the maximum, and this is called resonance.

Forced Harmonic Excitation

Harmonic excitation is often encountered in engineering systems. It is commonly produced by the unbalance in rotating machinery. Although pure harmonic excitation is less likely to occur than periodic or other types of excitation, understanding the behavior of a system undergoing harmonic excitation is essential in order to comprehend how the system will respond to more general types of excitation. Harmonic excitation may be in the form of a force or displacement of some point in the system.

We will first consider a single DOF system with viscous damping, excited by a harmonic force , as shown in Figure. Its differential equation of motion is found from the free-body diagram.

Where: =amplitude at frequency

If:

c/m = 2n Where: =amplitude at resonance

k/m= p2

p/m q ωp = p

Where: ωp= max forced amplitudethen…

Phase Response

Tan α =

Steady State Response Formula

Where:

= forcing frequency

= damping coefficient

Frequency Response

Where:

= max deflection = max value of steady state response

= static deflection

Example Problem (Harmonic Type excitation)

The equation of motion of a single degree of freedom system is given by:

4 ωt , in consistent units. Find all the quantities involved

Solutionm = 4, c = 4, k=16, P=8

a.) p2 = : p = 2 units

b.) = 2n = 1 : n = .5 unit

c.) c = 2mp = 2*2*4 = 16 units

d.) = = = .25 units

e.) Xs = = = .5 units

f.) Xr = = = 1 unit

g.) Xp = = = 1.032 units

h.) corresponding to Xp = = = .935 units

i.) α = tan-1 = = tan-1 .5 = 26.3°

Centrifugal Forced Type Excitation

Unbalance in rotating machines is a common source of vibration excitation. We consider here a spring-mass system constrained to move in the vertical direction and excited by a rotating machine that is

unbalanced, as shown in Fig. 10. The

unbalance is represented by

an eccentric mass m w ith

eccentricity e t

hat is rotating with angular velocity w . By letting x be the displacement of the non rotating mass (M - m) from the static equilibrium position, the displacement of m is :

General Formula

=

=

Let:M= total mass including the eccentric mass me= eccentricityx = vertical displacement of the mass

= angular velocity of the machine

Example Problem (Centrifugal forced type excitation)

A small motor driving a compressor weighs 30kg and causes each of the rubber isolators to deflect b 4mm. the motor runs at a constant speed of 1800rpm. The compressor piston has a 50mm stroke. The piston and reciprocating parts weigh .5 kg and can be assumed to perform a simple

harmonic motion. Assume for rubber to be .25, determine the amplitude

of vertical motion at the operating speed.

Solution:

p= 49.5 rad/sec

= = 60π rad/sec

= = = 3.8

= 0.25

= = 1.06

And so,

X = = .4416cm

System Damping

Quality Factor for Viscous Damping

In the study of forced vibration, the stiffness of the resonance is measured by a quantity Q called quality factor. Quality factor is defined as the reciprocal of the bandwidth at half power point. It is characterized a resonator’s bandwidth relative to its center frequency.

Q=

Where = damping coefficient

Quality Factor for Hysteretic Damping

Q=

Where r = height/spring constant

Quality factor is a dimensionless parameter that describes how underdamped an oscillator is. It is defined in terms of the ratio of the energy stored in the resonator to the energy lost per cycle to keep the amplitude constant at a resonant frequency. Higher quality factor indicates a lower energy lost.

For example, a pendulum suspended from a high quality bearing oscillation in air has a higher Q while a pendulum immersed in oil has a lower Q because oscillators with high quality factor have a lower damping so that they ring longer. Damping is an effect that reduces the amplitude of oscillation in an oscillatory system.

Behavior of the System in terms of Quality Factor

Low Quality Factor (Q < ½) – overdamped

- The system returns to equilibrium without oscillating. It does not oscillate at all but when displaced, it returns to its position approaching steady state value asymptotically.

High Quality Factor (Q > ½) – underdamped

- The system oscillates with the amplitude gradually decreasing to zero. As the Q increases, relative amount of damping decreases.

Intermediate Quality Factor (Q = ½) – critically damped

- The system returns to equilibrium as quickly as possible without oscillation. Critical damping results in the fastest response approaching to the final value.

(Quality Factor = 0) – undamped

- The system oscillates at its natural frequency just like a free vibration.

Amplitude at Resonance, Xr =

Amplitude at Cut Off, Xc = where =

Getting the ratio of Xr over Xc, (α)

α = =

we could arrive at damping coefficients,

Where = frequency ratio

To determine the 1 , 2, and α , we need the resonant frequency and the amplitude resonance. Assuming that constant force amplitude type of excitation is acting:

Sample Problem:

The resonance curve for an aircraft engine mounting system was obtained experimentally by attaching a rotating unbalance weight and measuring the vertical amplitude at the propeller shaft due to the forced pitching motion by engine. The speed at the peak amplitude is 1650rpm. The peak amplitude and the amplitude at cut off point are 0.1 units and 0.051 units respectively. Sideband frequencies correspond to the amplitude at cut off point are 1325rpm and 1915rpm.

Determine the damping coefficient in the rubber mounting of the system where the excitation is found to be of the Psinwt type.

Solution:

Assume

Resonant Frequency,

Frequency Ratios are:

= = 0.795 = = 1.150

Resonant Amplitude, Xr

α = = = 1.951

Solving for the damping coefficients,

The damping coefficient is 0.103

Steady State Vibration Isolation

Vibration isolation is the process of isolating an object, such as a piece of equipment, from the source of vibrations. It refers to vibration isolation or mitigation of vibrations by passive techniques such as rubber pads or mechanical springs, as opposed to active vibration isolation or electronic force cancellation employing electric power, sensors, actuators, and control systems.

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Force Isolation

It is used to limit the forces transmitted to the surroundings of the equipment in which shock originates. Shock can be defined as a transient condition where a single impulse of energy is transferred to a system in a short period of time with a large acceleration.

The force acting on the machine is transferred to the foundation through the spring and dashpot. The ratio of the force transmitted to the force impressed is called transmissibility ratio, TR.

TR =

Displacement Isolation

It is used to prevent harmful vibrations that may be present in the base or panel of the instrument from being transmitted.

TR = =

is the ratio of the displacement motion transmitted to the displacement impressed.

For vibration isolation

Sample Problem:

An industrial machine weighing 445 kg is supported on a spring with a static deflection of 0.5cm. if the machine has a rotating imbalance of 25 kg cm, determine the force transmitted at 1200 rpm and the dynamic amplitude.

Solution:

Dynamic amplitude, X

Force transmitted, Ft

Ft = kx = 910 kg/cm (0.442cm)= 402 kgf

Ft = 402 kg force

Forced Vibration

Musical instruments and other objects are set into vibration at their natural frequencywhen a person hits, strikes, strums, plucks or somehow disturbs the object. For instance, a guitar string is strummed or plucked; a piano string is hit with a hammer when a pedal is played; and the tines of a tuning fork are hit with a rubber mallet. Whatever the case, a person or thing puts energy into the instrument by direct contact with it. This input of energy disturbs the particles and forces the object into vibrational motion - at its natural frequency.

If you were to take a guitar string and stretch it to a given length and a given tightness and have a friend pluck it, you would hear a noise; but the noise would not even be close in comparison to the loudness produced by an acoustic guitar. On the other hand, if the string is attached to the sound box of the guitar, the vibrating string is capable of forcing the sound box into vibrating at that same natural frequency. The sound box in turn forces air particles inside the box into vibrational motion at the same natural frequency as the string. The entire system (string, guitar, and enclosed air) begins vibrating and forces surrounding air particles into vibrational motion. The tendency of one object to force another adjoining or interconnected object into vibrational motion is referred to as a forced vibration.

Periodic Motion

A vibrating object is wiggling about a fixed position. Like the mass on a spring in the animation at the right, a vibrating object is moving over the same path over the course of time. Its motion repeats itself over and over again. If it were not for damping, the vibrations would endure forever (or at least until someone catches the mass and brings it to rest). The mass on the spring not only repeats the same motion, it does so in a

regular fashion. The time it takes to complete one back and forth cycle is always the same amount of time.

Shock

The duration of the forcing function or excitation is small compared to the natural time period or frequency of the system. Example of a shock is when you apply a sudden blow with a hammer to a concrete wall.

General Formula Used:

Sample Problem

In the vibration testing of structure, an impact hammer with a load cell to measure the impact force is used to cause excitation as shown in Figure below, assuming m = 5 kg, k = 2000 N/m, C = 10 N-s/m and F = 20 N-s. Find the response of the system.

Solution

Assuming that the impact is given at t = 0, we find the response of the system as

Whenever an external energy is supplied to a mechanical or structural system during vibration the phenomena is considered as Forced Vibration

The external energy maybe classified as External force or displacement excitation, the manner could be harmonic, non-harmonic but periodic, non-periodic or random in motion

Resonance a phenomena where the frequency of excitation coincides the natural frequency of the system causing a very large response or very large amplitude

Beating happens when the forcing frequency is close but not exactly equal to the natural frequency of the system, in this kind of vibration, the amplitude builds up and then diminishes in regular pattern.