EXPERIMENT 2 WOODEN PENDULUM

ABSTRACT

This experiment was conducted in order to determine the mass moment of inertia at the centre of

gravity, IG and at the suspension points, IO1 and IO2 by oscillation. From the experiment conducted, the

finding is that there are some differences between the values of IO and IG from the experiment data and

also from theoretical value. The potential factors that cause to the differences in values are further

discussed. The finding is that the wooden pendulum oscillates in non-uniform motion especially when

it is suspended at IO2. Based on the experiment, it is found out that the value of IG and IO from both

suspension points is totally different although they share the same value of mass of the wooden

pendulum. The period is also different for both points setting. After the data was taken, the period of

oscillation, T1 and T2 are obtained from the two different suspension points. Hence, after getting T

value, then the value of IG and IO can be measured. The errors that occur might be due to disturbing

from surrounding and human error. The time for 10 oscillations was taken manually by using

stopwatch.  By the end of this experiment, the values of IG and IO are able to be calculated by using the

theory

INTRODUCTION

`A simple pendulum consists of a point-mass hanging on a length of a string assumed to be

weightless. A small weight hanging by a string from a retort stand illustrates this condition. If the

mass is displaced slightly from its equilibrium position, the mass will perform simple harmonic

oscillation. An extended solid object that is free to swing on an axis is called a physical pendulum,

whose period is now dependant on the mass moment of inertia about the rotational axis and it distance

from the centre of mass. A pendulum is a weight suspended from a pivot so that it can swing freely.

When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force

due to gravity that will accelerate it back toward the equilibrium position. When released, the

restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium

position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is

called the period. A pendulum swings with a specific period which depends mainly on its length.

From its discovery around 1602 by Galileo Galilei, the regular motion of pendulums was used for

timekeeping, and was the world's most accurate timekeeping technology until the 1930’s. Pendulums

are used to regulate pendulum clocks, and are used in scientific instruments such accelerometers and

seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in

geophysical surveys, and even as a standard of length. The word 'pendulum' is new Latin, from the

Latin pendulus, meaning hanging.

THEORY

The simple pendulum is an example of mathematical model due to weight at one end of any supports

which hanged at a pivot (fixed) point. It will exhibits a swing of backward and forward with constant

amplitude due to the gravitational force as it gives an initial release at 10o freedom (angle of

displacement). The amplitude of swings would tend to declines compared with the real pendulum

which is mainly subject to friction and surrounding ambient

On this experiment we had used a wooden pendulum as rigid body suspended from some point which

is it was acted as a physical pendulum which is located on its centre of mass.it is a simply rigid object

which is swings freely above some pivot point. The physical pendulum may be compare with a simple

pendulum, which is consists of a small mass suspended by a string. For this situation the pendulum is

stick with a pivot at a fixed point at the centre of mass. Via this experiment, an equation of motion

was developed in order to find the mass moment of inertia whether at centre of gravity, IG or at

suspension point, IO by oscillation with the deflection angle of 10o

Free Body Diagram of Wood Free Body Diagram of Wood Pendulum

For small displacement, the period of T of a physical pendulum is independent of its amplitude which

is

Ax

G

IA

800mm

T=2 πw

W is represent as a harmonic oscillator

Where I is the rotational inertia of the pendulum about its rotation axis, m is the total mass of the

pendulum, g is the acceleration of gravity and r is the distance from the rotation.

The formula used for the calculation of mass moment of inertia (experimental)

To determine L

To find RG

m = mass of the body (Kg) L = length of the composite body (m) d = distance from x axis to the centroid location (m) I = mass moment of inertia (kg.m2)

I3= 1/12 m3l32 +m3d3

2I2 = 1/4 m2r22 + m2d2

2I1 = 1/12 m1L12 + m1d1

2

T=2√ Lg

RG=x (L2−x )

L1+L2−2 x

W = √ Imgr

To calculate Io

To find IG1

To find the value of IG2

EXPERIMENT 3 FORCE VIBRATION

L1=I G1−m ¿¿

T = Time taken to complete one oscillation (period), t (sec) L = Length, l (m) g = Gravitational acceleration (ms-2) RG = Distance from suspension point to centre of gravity (m) X = Fix length, 0.7m Io = Moment of inertia about point O at suspension point 1 & 2 IG = Moment of inertia about the centre of gravity (kg.m2)

L2=I G2+m¿¿

I O=I G+mL2

ABSTRACT

The forced vibration experiment was carried out to determine the resonance of Spring-Dashpot

System in different damping conditions. There are 4 data that has been tabulated into table. The

information that has been collected is the value of amplitude according to the frequency that has been

specified. The experiment is carried out by using Universal Vibration System Apparatus TM155. The

frequency that has been used are from 5 Hz to 14 Hz. There are two types of damping system has

been used for this experiment, the damped and no damped system. The damped system that has been

run is set up into two which are closed and open. These two conditions will be give different result in

amplitude readings. The damper also is located into 4 distances which are different. This damper

distance will give impact to the shape of amplitude and its value. The data that has been collected will

be plotted into the graph of amplitude versus frequency. The graph that has been plotted can be

compared with the theoretical graph. The resonance frequency and natural frequency also has been

calculated to achieve the objective of the experiments.

INTRODUCTION

Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium

point. The oscillations may be periodic such as the motion of a pendulum or random such as the

movement of a tire on a gravel road. Vibration is occasionally desirable. For example the motion of a

tuning fork, the reed in a woodwind instrument or harmonica, or mobile phones or the cone of a

loudspeaker is desirable vibration, necessary for the correct functioning of the various devices.

More often, vibration is undesirable, wasting energy and creating unwanted sound noise. For

example, the vibrational motions of engines, electric motors, or any mechanical device in operation

are typically unwanted .The study of sound and vibration are closely related. Sound, or pressure

waves are generated by vibrating structures, these pressure waves can also induce the vibration of

structures. Hence, when trying to reduce noise it is often a problem in trying to reduce vibration.

Free vibration occurs when a mechanical system is set off with an initial input and then

allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and

then letting go or hitting a tuning fork and letting it ring. The mechanical system will then vibrate at

one or more of its natural frequency and damp down to zero. Forced vibration is when a time-varying

disturbance such as load, displacement or velocity is applied to a mechanical system. The disturbance

can be a periodic, steady-state input, a transient input, or a random input. The periodic input can be a

harmonic or a non-harmonic disturbance. Examples of these types of vibration include a shaking

washing machine due to an imbalance, transportation vibration is caused by truck engine, springs,

road, or the vibration of a building during an earthquake. For linear systems, the frequency of the

steady-state vibration response resulting from the application of a periodic, harmonic input is equal to

the frequency of the applied force or motion, with the response magnitude being dependent on the

actual mechanical system.

In physics, resonance is the tendency of a system to oscillate with greater amplitude at some

frequencies than at others. Frequencies at which the response amplitude is a relative maximum are

known as the system's resonant frequencies, or resonance frequencies. At these frequencies, even

small periodic driving forces can produce large amplitude oscillations, because the system stores

vibration energy.

THEORY

Vibration is the one of the type of force that have a particular frequency that has in a periodic input of

force. Force vibration can be divided into two conditions which are in damped situation or undamped

situation. Based on this situation it has a different value that we can get from that experimental value

or in theoretical value. Undamped forced vibration, we can classified it as the most vibration motion

in engineering analysis. Affect from that we can used its principles and operation which is to describe

of various types of machines and structure.

Based on the experiment, we can make a draft which is to describe where its position their length and

also their position that located on that beam

=

Fd

W

Fm

Fs

a

L/2

b

L

Man

Mat

I

Fm = Force motor Fd= Force damping Fs= Force spring M = Moment

Based on that sketch, we can define and derive some equation that we can get the value of the theory

that we can make which is to find the value of the resonance of Spring-Dashpot system in different

damping conditions. It has some of equation that we can get the value of the theoretical for this

experiment.

By defining through that equation we can get the value of the theoretical value. The value of force

time with distance was put into that equation to get that’s value. Finally we can get the value of the

Wb which is the value of the resonance of the spring-dashpot system in different damping conditions.

↶∑ Ma=∑ Ma (refference)

−Fd a+w[ L2 ]+Fm[ L

2 ]−F s b=I¨

θ+¿m at [ L2 ][ L

2 ]¿

13

m L2¨

θ+¿ca θ̈+kb=Fm (L2 )¿

Wb = √ kbmc2

K = Value of the (kb) C = Value of the (ca θ̈)

m = Value of the (13

m L2 θ̈)