MODELING OF DYNAMIC

SYSTEM

NAME : Wanigasooriya S.L.D.A.I

COURSE : B.Sc. Engineering

INDEX NO : 150661X

GROUP : C3-G4

DATE OF PER : 25/04/2016

DATE OF SUB : 09/05/2016

EXPERIMENT NO: ME-1032-P-1

INSTRUCTED BY:

Mr. Aruna Rathnayake

GROUP MEMBERS:

Vidulath B. V. S. C.

Vimantha G. C.

Vindana H. L. P.

Vithana W. V. D. D. S. K.

Vithanage L. S.

Vitharama A. V. R. S.

Walisundara W. M. C. S. B

Wanasinghe A. T.

Wanigasekara D. C.

Wanigasooriy S. L. D. A. I

……………………

INTRODUCTION

VIBRATION

When we displace a body or a system of connected bodies from a position of equilibrium, if there is

a restoring force towards that equilibrium point, then it will begin to oscillate about the equilibrium

point. This periodic motion is called as vibration. Vibrations are common in nature. From atomic

level to large planetary level vibrations are caused in different strengths. Both the nano size

oscillations in solid crystals and the large destructive

earthquakes share this same phenomena. Most

experiences of vibrations in machines, equipment and

structures are considered undesirable. Vibration

caused by earthquakes, aerodynamic forces induced by

wind flow, explosions are disastrous. The collapse of

Tacoma bridge in Washington, USA, in 1940 a newly

constructed suspension bridge of 853m span and

collapsed due to the large deflection created by

resonance; an aspect of vibration.

However there are positive and useful aspects of

vibrations too. Vibrato-separator machines, Vibrator

motors in mobile phones and sensors, packing

machines which use vibrating tables, vibrator

hammers and etc. Though it says the sounds created

by vibrations are undesirable, controlled and

processed vibrations can be rhythmical and very

pleasant to hear. Those sounds we term as music.

Vibration of diaphragms, strings or column of air creates these beautiful sounds. And even our voice

is generated because of the vibrations made in our voice box.

FREE VIBRATION: Vibration of a system in its’ natural frequency without any external force is

free vibration

NATURAL FREQUANCY: The fundamental frequency of an object depends on its mass and

stiffness distribution.

FORCED VIBRATION: Vibration of a system under a periodic excitation external force.

RESONANCE: In a forced vibration if the frequency of parodic force is supplied in such a way that

is same as the natural frequency of object. The system tends to oscillate in large amplitudes.

DAMPING: Because of the friction and air resistance a free vibrating system reduces its’ amplitude

and comes to rest.

THEORY

The system used here consists two springs and a cubic mass connect in series pattern, “spring-

mass-spring” respectively.

Assume that the system possess negligible damping,

Using Newton’s Second Law on Mass,

↑ −𝐾1(𝑥 − 𝑦) − 𝐾2𝑥 = 𝑚

�̈�𝐾1(𝑥 − 𝑦) + 𝐾2𝑥 + 𝑚�̈� = 0

𝑥 = 𝐵𝑠𝑖𝑛(𝑝. 𝑡) �̇�=−𝐵 𝑃 𝑐𝑜𝑠(P.t) �̈�=B P2 𝑠𝑖𝑛 (𝑝. 𝑡) 𝑦 = 𝐴sin (𝑝. 𝑡)

Here 𝛷=0,

-−𝑚(𝐵𝑃2 sin(𝑃. 𝑡) + 𝐾1(𝐵 sin(𝑃. 𝑡) − 𝐴𝑠𝑖𝑛(𝑃. 𝑡)) + 𝐾2( 𝐵 𝑠𝑖𝑛(𝑝. 𝑡)) = 0

−𝑚(𝐵 𝑃2) + (𝐾1 + 𝐾2)𝐵 = 𝐾1𝐴

𝐵

𝐴=

𝐾1𝑚⁄

𝜔𝑛2 − 𝑃2

𝜔𝑛 = √𝐾

𝑚

𝐾 = 𝐾1 + 𝐾2

A Amplitude of the ground

B Amplitude of the mass

P Excitation frequency (P=2πf)

𝜔n Natural frequency ( 𝜔n=2πf)

Ki Stiffness of springs

t Time

∅ Phase angle

m Mass of the Cubic Mass

Mass (m)

𝑥 = 𝐵𝑠𝑖𝑛(𝑃. 𝑡 + ∅)

𝑦 = 𝐴𝑠𝑖𝑛(𝑃. 𝑡)

PROCEDURE

P 1906 LINEAR VIBRATION APPARATUS

This is a model of P1906 Linear Vibration Apparatus. This equipment mainly use for Bail Chatter

Test, Spring Rate Test, Spring-MM System Test and Double Cantilever Test. The apparatus

comprises a driving unit, vibration generator and the vibrating spring-mass system. Driving unit

consists of an oscillator and a power amplifier which have been designed to provide a sinewave

power output that need for vibration generator to generate vibration. Oscillator is a circuit which gain

provides sinewave input to power amplifier that have the ability to amplifier a signal with a voltage

of +1 and minimal phase shift from DC to 10Hz. The generator consists of a permanent magnet that

produces a magnetic field that produces a magnetic field and oscillatory current coil, which produce

a perpendicular force to the line of flux and direction of current causing vibrations

Test mass

_____ 3

_____ 2

_____ 1

_____ 0

_____ 1

_____ 2

_____ 3

Excess mass

Springs

Amplitude Scale

Ground

Vibrator

Amplitude Scale

Vibrator

PROCEDURE IN STEPS

Set the apparatus as shown in the diagram by loading 23.5g as to coincide the top edge of

the mass with the 0 line of the scale

Adjusted the amplitude of the ground excitation (A) to 1 unit using the controller.

Adjusted the ground excitation frequency (f) to the lowest and switch on the apparatus.

By increasing the frequency using the controller and obtained the natural frequency of the

system roughly.

Started the practical by setting excitation frequency (f) below the observed maximum

amplitude frequency without a system failure.

Increased frequency by 1Hz or 0.5Hz and took 9 readings for amplitude of mass (B) from the

apparatus the scale with the respective frequency.

Repeated the steps with 41.0g, 58.5g, 76.0g, and 93.5g and 111.0g masses and recorded the

observations.

And plotted 5 graphs which shows the behavior of (B/A) against frequency (f /P).

Those graphs showed a similar shape as in this figure.

We obtained the natural frequency of that system respective to mass, using the amplitude we assumed

to be maximum. (Since we cannot reach the maximum frequency without a system failure)

And next plotted the maximum frequency against the reciprocal of relevant mass.

Here the equation of the graph is,

𝜔𝑛 = 𝐾𝑚⁄

This equation is in the form of 𝑦 = 𝑚𝑥,

So from the gradient of the graph we obtained the K. (the resultant string constant)

B/A

f (Hz) fn

𝜔𝑛2(𝑟𝑎𝑑𝑠−2)

1𝑚⁄ (𝑘𝑔−1)

CONCLUTION

The relation given is a hyperbolic equation, giving the same shape the graph proves the relation.

According to the second graph when the frequency reach the natural frequency of the system it gain

the maximum amplitude, proves the equation.

DISCUSSION

VIBRATION MOTORS

Have you ever questioned, how your phone vibrates when

you receive a call or a massage? If so the answer you will

find is a tinny motor that creates vibrations. These tinny

motors were first discovered in 1960, but they were begun to

use widely after 90th century in electronic devices to receive

alerts. There are mainly two types of Vibration Motors. One

is an eccentric rotating mass vibration motor (ERM) which

uses a small unbalanced mass on a DC motor that creates a

force that produces a vibration. They are generally 4mm wide and 10mm long, which may have a

shaft of 1mm diameter. The other type is a linear resonant actuator (LRA) contains a small internal

mass attached to a spring which is driven by an electric oscillator. Not only in mobile phones, these

motors are now popular in PlayStation, in auto-mobile sensors and in medical equipment.

VIBRATORY HAMME

Vibrator Hammer is a

tool used to drive piles

in or out of the ground.

They are effectively

used in marine

construction sites,

when constructing

bridges, roads, rail,

walls and many other

type of foundations.

Effectiveness in

driving piles, ability to extract old piles out of ground, ability to use underwater, light in weight,

small and easy in shipping , environment friendliness without any harmful byproduct or much

annoying noises are some advantages of these type of hammers.

There are mainly two types of Vibratory Hammers as Electric Vibratory Hammers and Hydraulic

Vibratory Hammers. The different main difference between these two is that in Electric Vibratory

Hammer, there using a high power diesel generator a large electric motor on the top of the hammer

is rotated to spin the counter-weights. But in Hydraulic Vibrator Hammer a hydraulic motor is used

to spin the counter weights which is powered by a large diesel engine that flow oil in and out to the

motor. Hydraulic Vibratory Hammers has more advantages such as light in weight, easy to handle,

consume less power and faster speed than the other which helps to minimize the vibration

transmission through soil.

But when creating vibrations both of these shares a same phenomenon. Two pairwise eccentric

weight that rotates in a vertical plane generates each a centrifugal force that causes vibration. The

rotation of weights in opposite direction maintaining the same moment produces constant vibration

cycles. The eccentric shafts are mounted in heavy duty roller bearings which minimized the friction.

And an arranged spring yoke at the top of the machine blocks the transmission of vibration to the

carrier.

VIBRATION THERAPY

In 1867. Russian pyscian and inventor Gustav Zander invented a series of machines using systems

of wieghts and pulleys to create sense of vibration for the therapeutic purposes. In 1895, Dr. Johan

Harvey Kellogg use vibration therapy in his health

practices using a vibrating chair made by himself. He

believed that the therapy was good for circulation and

could alleviate constipation. In space programes Russian

physicians noticed that the returning astronauts suffered

from loss of bone mass and bone fractures as their earlier

ages. As a treatment for this they begun to use whole body

vibration to help stenghthen astronaus’ bone mass and

muscles. Today in NASA astronauts wew given a 10 to 20

munite light viraing threatements.

Common form that used to improve muscle soreness

requires staying in a half-squat position with bent knees at

100 degrees for 60 seconds. This causes to produce rapid

muscle contractions, by tricking the pody into thinking it is

falling. Vibration therapy can also be used on separated

body parts using hand held vibrating device.

Vibrator macjine works on the principles of frequance and amplitude, which can be adjust acording

to our needs.Other than the up and down motion there are some machines which create forward and

backward motions and side way motions. But most effective way to create rapid muscle ontractions

is the up and down vibrations.

CONCRETE VIBRATORS

Since improper consolidation of concrete can cause product defects, compromise congrete strenght,

and produce surface blemishes such as bug holes and honeycombing it is important to consolidate

freshly poured comcrete. Concrete vibrator is used to consolidate freshly pured concrete so that

trapped air and excess water are released

and the concrete settles firmly.An

intaernal concrete vibrator consists of a

steel cylinder about sizee of the handle of

a baseball bat,a hose or electric cord

attached to one end, and a vibrator head

which is immersed in wet concrete.The

exterbal concrete vibrator attach using a

bracket or clamp system, to the concrete

forms.External vibrators are avalable in hydraulic, pneumatic or electric power

VIBRATING TABLES

To test product to determine or demonstrate the ability of product to withstand vibrations, the

Vibtaring Tables or Shake Tables are used. These test are commonly done in automotive, aerospace,

and defense industries. These vibrators have the ability to create three different types of vibrations.

They are sine sweep, random vibration, and synthesized shocks. When using all these three types of

applications, one or more accelerometers are instrumented with the part under test to measure

component reponse to the vibration input.

In sine sweep vibration, it starts vibrating at low frequancy and increases in frequancey at a set rate.

And tere is also in the variation of amplitude too. A sine sweep can find the resonant frequancy of

the part. A random vibration test will tests the effects of vibration on the part respective to different

frquencies along a spectrum at different times. A synthesized shock pules is a sum of half-sine waves

coverinf a range of frequencies which

durate in a small time period.

And these vibrating tables also used in

packing processes in material handling

industry by shaking or settling a container

which make container to hold more

products.

VIBRATO-SEPARATOR MACHINE

Vibrato-Separator Machines are very common

application of vibration. These machines are

commonly used for filtration and separation

purposes. To separate unwanted food particles

and impurities in food production, to separate

chemicals (chemical powders or liquids) from

unwanted impurities, separating metal particles in

metal production, in ceramic industry to separate

things like ceramic powder and kaolin clay and

specially in waste disposal to separate waste

particles and waste water treatment, these

machines are widely using. These machines use

physical methods for separation. Most of them have the ability to separate particles according to their

sizes. Most in case different size metal sieves are used to filter and separate particles.

Here the application of vibration is to make the filtration and separation more effective and efficient.

By using vibrator sifting table (screen) is made to vibrate, this vibration makes the particles to move

in different way. These motions make particles to sift from relevant wholes and get separated. Mostly

a motor or a combination of motors are used as the vibration source. They are fixed in different ways

which courses different patterns of motion. Following are commonly used designed for vibrators and

there vibration patterns

Location of motor and Direction of

Rotations

Vibration patterns at

Left end Middle Right end

Single Motor Vibrator

Elliptical Vibration of Varying

Amplitude

Single Motor Vibrator

Elliptical Vibration of Varying

Amplitude

Single Motor Vibrator

Circular Vibration

Single Motor Vibrator

Torsional Movement

Twin Contra Running Motor

Vibrator

Pure Rectilinear Vibrations

2 Motor Vibrators Running in Same

Direction

Torsional Movement

Common problems in these machines are unwanted shocks create by the vibrator, undesirable noise

create when vibrating, dust particles. Proper covered container can solve the dust problem but

noises and shocks cannot be totally illuminated. But by using a damping system.

SYSTEM MODEL OF A VIBRATO-SEPARATOR MACHINE

Side view

Front view

This Vibrator Separator Machine mainly consist of linear vibrating screen which is composed of

screen box, screen frame, vibration motor, shock absorber and machine base. The two motors

provide vibration which fixed at the bottom of screen box. They rotates synchronically in reverse

directions. This situation is quite similar to the Twin Contra Running Motor Vibrator system which

caused rectangular vibration of the screen device supported by shock absorbers. The particles move

Shock absorber (damper)

Outlet Screen Drum

Upper Cover

Transfer Board Support of motor Motor

Inlet

Inlet

Shock absorber (damper)

Feet

Machine Base

forward straightly when tossed upward. By changing the speed of motor the frequency of vibration

can be changed. By using rubber base we can further minimize the shocks and could protect the

ground from cracks that cause by vibrations (shocks).

PRACTICLE ERRORS

Taking values for B (B/A since we adjust A to be 1 unit the value directly reads from the scale gives

the value of B) in high frequency is very difficult as the vibration is so rapid that it is unable to grab

the exact amplitude. Since because of that we always founded with very similar values even for many

frequencies. As a solution for that if we use an indicator or a pointing needle and get the exact place

were the upper edge of mass touches the needle, we could get much accurate value for amplitude.

When drawing the graphs most of the points seems to be more deviated from each other and it is so

difficult to select points to get the necessary hyperbolic curve. And when the frequency get more

close to natural frequency of the system the curves must come parallel to y axis. But with the points

with the point we gained from the experiment were not enough to get that change in the curve. And

that made more difficult in finding the relevant natural frequency for the system. This could be

overcome by selecting wide range of frequencies and get more points. And also if could select a

different scale with a small minimum measurement we could measure much accurate amplitude.