Engineering Principles and Analysis of a Pendulum Clock System

This report provides a comprehensive analysis of the engineering principles behind the operation of a pendulum clock. It begins with an introduction to the pendulum clock, explaining its basic structure and the role of gravity in its function as a harmonic oscillator. The report then delves into the power mechanism, wheel train (including the center, third, and escape wheels), and escapement wheel, illustrating how these components work together to regulate time. The core of the report includes accurate calculations of pendulum length, discussing the formula and method used to determine the acceleration due to gravity. It also differentiates between calculated and measured lengths and describes the calibration process, including factors such as temperature and gravity that affect accuracy. The report further explores the motion of the clock, the impact of pendulum length and amplitude, the influence of mass, and the effects of friction and vibration. Finally, it includes personal reflections on the project and references supporting the analysis.

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Table of Contents
Introduction.................................................................................................................................................2
Accurate calculation of length of Pendulum................................................................................................3
How pendulum clock was calibrated...........................................................................................................6
The motion of the clock..........................................................................................................................8
Length of the pendulum clock.................................................................................................................9
The amplitude of the clock......................................................................................................................9
Mass of the pendulum.............................................................................................................................9
High Vibration......................................................................................................................................10
Reflections.................................................................................................................................................10
REFERENCES..............................................................................................................................................11

Introduction
A pendulum clock is a structured pole like rod clock that uses pendulum and hangs vertically
from the top all the end, it has a weight swing that goes sideways because of the force gravity.
Through the force of gravity it exhibits, the pendulum is able to constantly and precisely swing
sideways or back and forth hence making it to be harmonic oscillator (Agarana & Bishop 2015).
Power mechanism
Power mechanism of the pendulum clock is what helps in keeping the clock working or rather
running. The pendulum clock works through the conversion of energy simultaneously. The
maximum the energy that is stored is at when the nod is at the highest point and this is called the
possible potential energy. When the rod of the clock runs downwards near the ground, the energy
is released and it’s converted to kinetic energy (von der Wense , Seiferle & Thirolf 2018). The
nob goes up again to the conversion to potential energy and the cycle continues back and forth.
Though the cycle is influenced by strength of the gravity as well as the length or the distance to
be covered, it takes the same time to finish the swing as the kinetic and the potential energy
keeps swinging from the one form of energy to the other. An oscillation is anything that moves
in harmonic motion and works in a similar way as pendulum is called harmonic oscillator.
(Huygens, 2015)
Wheel train

Wheel train of the pendulum clock is also known as gear train and it enhances improvement of
speed on power mechanism to enable the pendulum to use it. The ratios of the gear train in the
wheel train splits the revolution frequency to provide power to the wheels that alternates once
every hour in twelve hours. A Pendulum clocks is usually made up of three wheels, they include:
the center wheel, the third wheel and the escape wheel.
The rotation and cycling of the Centre wheel is once in an hour, third wheel once in one minute
while the escape wheel ensures that the mainspring unwinds speed is acknowledged and then
regulated. (Huygens, 2015)
Escapement wheel
Escapement wheel-An escapement is an automated connection line in clocks or watches that
helps in ensuring the release and discharges of impulses of the timekeeping elements on the
clock.
Oscillators
One of the most commonly used oscillators is the pendulum of a clock. If you push on a
pendulum to start it swinging, it will oscillate at some frequency -- it will swing back and forth a
certain number of times per second (Whitestone 2012). The length of the pendulum is the main
thing that controls the frequency.
Pendulum clock is widely used by the past and the present generation globally as it is eases the
tracks keep of time through observing the swing rod completion.

Accurate calculation of length of Pendulum
Calculating the acceleration due to the mass and gravity. My pendulum clock length is 70.0 cm
and the period taken by the pendulum is 1.7357 s.
Approach
Am going to find the potential gravity as the period and the length of the pendulum.
Formular is:
T=2π√LgT=2πLg
Assuming my pendulum angle of deflection is less than 10º.
Solution
Square T=2π√LgT=2πLg and solve for g:
g=4π2LT2g=4π2LT2.

Substitute known values into the new equation:
g=4π20.700000 m (1.7357 s)
2g=4π20.700000 m (1.7357 s) 2.
Calculate to find g:
g = 9.002 m/s2.
Discussion
Determination of g using this method is a precise and accurate method that’s why have given the
length of the pendulum is given in five digits in this example. For the accuracy of the
approximation sinθ ≈ θ while the displacement angle is kept below 1.0º.
Difference between the calculated and measured lengths of pendulum
In the measured pendulum length, one deals with the actual length he or she has measured in the
pendulum while in the theoretical or calculated length, one deals with an estimated length hence
the accuracy and precision of the length is not upheld as the length is uncertain.
In calculated lengths, the period taken by the pendulum to swing back and forth is assumed while
in the measured lengths, the period is actually measured and determined.
Lastly, in the measured length, the accuracy is not limited as much as the accuracy in the
calculated lengths.

How pendulum clock was calibrated
On a pendulum clock, a weight which is attached to a string steadily falls down. The weight’s
fall causes the possible energy which powers the gears, and in turn pulls at the axle which is
driven by time-keeping gears. Purposely, this setting is to ensure all the needles remain synced.
This means the minute needle moves from one place to the right once the needle for seconds
closes 60 moves, as the minute moves the hand of the hour at 1/60th of its speed (Patel, Moline &
Wagner 2013).
Though, because of the falling weight, the fall should be controlled to some level. If not
controlled, the rapid fall of weight because of gravitational force will make the second needle to
move very fast. This is where the pendulum helps. It’s swinging rocks a lever, making the gear
trains to move forward swinging in very small amount of force. This means the levers locks and
unlocks the gear train to let the weight fall once in a second. The locking and unlocking by the
Lever produces the tick-tock sound characteristic. (Matthys, 2017)
Fig 1: showing Parts of pendulum clock

Initially, the accuracy of mechanical clocks was increased from about 15 minutes to 15 seconds
per day. It was realized later in Paris that an accurate pendulum clock was 2.5 minutes slower,
per day. The conclusion was changes in the gravitational force in different locations.
After some years, Newton guessed that because of the centrifugal force effects on the earth’s
rotation, there was an equatorial bulge which made the poles to be flattened. This was rubbished
by French Royal Observation Cassins, and they travelled to Peru in order to measure the Earth’s
shape. This was the beginning of calibrating pendulum clocks. (Huygens, The Pendulum Clock,
2014)
Some of the main factors that affect the pendulum clock operations are temperature and gravity.
When the material density under the pendulum is high, the gravity will be high, like in the case
of under mountains. This speeds the pendulum a bit. The reverse effect affects oscillation rate of
a pendulum.
Temperatures can equally have an impact on the pendulum clock. Heat usually expands the rod
and cold causes contraction on the rod; hence the period of the swings can be lengthened or
shortened depending on temperature changes. (Matthews, Gauld, & Stinner, 2014)
Calibrating a pendulum clock process includes checking time against an oscillation that is
known. It is recommended to make a correction in the pendulum length in case it is not correct.
The correction will either speed up the period or it slows down in order to match the timing
device that is known. The strategy in calibrating the pendulum is choosing the oscillation
reference.

Factors that affects the limited accuracy of the pendulum clock
The motion of the clock
Pull and pull the clock and then free the clock. The periodic motion and movement tends to
always move and disturbs the pendulum clock as the pull or push of the gravity makes the
pendulum to swing downwards. The principle of periodic motion makes the pendulum to move
The force of gravity pulls the weight, or bob, down as it swings. The pendulum acts like a
falling body, moving toward the center of motion at a steady rate and then returning.
Length of the pendulum clock
The frequency at which the pendulum swings is greatly influenced by the length of the pendulum
itself. Longer lengths tends to reduce the swinging speed of the pendulum, consequently, shorter
lengths increases the potential speed of the pendulum clock.

The amplitude of the clock
The amplitude is angle and the viewpoint in which the pendulum swings, the length in which the
pendulum can reach while swinging. Most of the time, a motion pendulum exhibit a zero degree
angle. The amplitude of the pendulum is determined when the pendulum is started .The degree of
the angle of pendulum is what affects the amplitude of the pendulum
Mass of the pendulum
Mass affects the accuracy of the pendulum, to improve the precision of the pendulum, the bob
weight can be increased as the weight doesn’t influenced the accuracy and movement of the
clock. The weight in turn influences the pulling and pushing back of the pendulum hence
improving the gravity and the mass of the object hence correcting the accuracy of the pendulum
The friction in the clock
The swinging rate is affected and influenced by the resistance of the air. The higher the friction
rate, the lower the swinging of the pendulum. The swinging of the pendulum in an inert motion
initiates eventual stopping of the pendulum.
High Vibration
Proximity of the pendulum vibration affects the swinging rates of the pendulum clock. The
motion initiates the moving back and forth of a pendulum clock. This transmission causes the
identification and comparison of the clocks.

Reflections
Individual working brings about that self- motivation, independence and self-trust that may not
be acquired from teamwork. Also, individual working tend to provide the feeling of self-
association as well as full ownership beneficial of a project that may not come with a communal
or teamwork. Aubel & Fussenegger 2010).
Am a Perfectionism and detailed oriented person, this personality has greatly impacted in the
running and completion of the project in a way that the project was critically thought
through ,accurate and precise data and information recorded, due to this the project took longer
than anticipated to be accomplished and completed.
In my previous engineering projects I have undertaken, my attribute on strengths and weakness
in these projects had a great impact. Being perfectionist and detailed oriented person
significantly helped me to obtain and record an accurate data and information through having a
critical thinking on the ideas on the project. I was able to identify and appreciate my personality
and confidence hence impacting motivation and sense of ownership on the project.
Personal goal for improvement is improving in independent skills. Being able to effectively and
efficiently do individual work confidently and independently .This would certainly help
individuals to effectively solve various problems independently on their own.

REFERENCES
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Stability for Non-linear Differential Equation Governing Pendulum Clock. International Journal
of Applied Engineering Research, 10(24), 44112-44117.
Aubel, D., & Fussenegger, M. (2010). Watch the clock—engineering biological systems to be on
time. Current opinion in genetics & development, 20(6), 634-643.
Drumheller, D. S. (2012). Barometric Compensation of a Pendulum. Journal of applied
mechanics, 79(6).
Llibre, J., & Teixeira, M. A. (2010). On the stable limit cycle of a weight-driven pendulum
clock. European journal of physics, 31(5), 1249.
Lorek, K., Louko, J., & Dragan, A. (2015). Ideal clocks—a convenient fiction. Classical and
Quantum Gravity, 32(17), 175003.
Nijmeijer, H., & Pogromsky, A. Y. (2010). Huijgens' Synchronization: A Challenge.
In Dynamics and control of hybrid mechanical systems (pp. 1-6).
Patel, S., Moline, D., & Wagner, J. (2013, July). Modeling and analysis of an atmospheric driven
Atmos clock with mechanical escapement control. In 2013 European Control Conference
(ECC) (pp. 281-287). IEEE.
Ramirez, J. P., Olvera, L. A., Nijmeijer, H., & Alvarez, J. (2016). The sympathy of two
pendulum clocks: beyond Huygens’ observations. Scientific reports, 6, 23580.
von der Wense, L., Seiferle, B., & Thirolf, P. G. (2018). Towards a 229 Th-based nuclear
clock. Measurement Techniques, 60(12), 1178-1192.