Mechanical vibration is the oscillatory motion of a system about an equilibrium position. The key characteristics of vibration include the period, which is the time for one full cycle; frequency, which is the number of cycles per unit time; and amplitude, which is the maximum displacement from equilibrium. Vibrations can be caused by restoring forces alone (free vibration) or by periodic external forcing (forced vibration). Damping due to friction causes vibrational energy to dissipate over time.
Introduction to oscillations and simple harmonic motionMichael Marty
Physics presentation about Simple Harmonic Motion of Hooke's Law springs and pendulums with derivation of formulas and connections to Uniform Circular Motion.
References include links to illustrative youtube clips and other powerpoints that contributed to this peresentation.
In Class 12 Physics, students explore Alternating Current (AC), a fundamental concept in electrical engineering. Unlike Direct Current (DC), AC periodically reverses the flow of electric charge, generating sinusoidal waveforms. Topics include waveform representation, frequency, voltage, and current analysis. AC finds applications in power systems, electronics, and appliances. Through theory and experiments, students gain vital insights for future studies in electrical engineering.
For more information, visit- www.vavaclasses.com
Introduction to oscillations and simple harmonic motionMichael Marty
Physics presentation about Simple Harmonic Motion of Hooke's Law springs and pendulums with derivation of formulas and connections to Uniform Circular Motion.
References include links to illustrative youtube clips and other powerpoints that contributed to this peresentation.
In Class 12 Physics, students explore Alternating Current (AC), a fundamental concept in electrical engineering. Unlike Direct Current (DC), AC periodically reverses the flow of electric charge, generating sinusoidal waveforms. Topics include waveform representation, frequency, voltage, and current analysis. AC finds applications in power systems, electronics, and appliances. Through theory and experiments, students gain vital insights for future studies in electrical engineering.
For more information, visit- www.vavaclasses.com
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Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
2. Mechanical vibration is the motion of a particle or body which oscillates
about a position of equilibrium.
Time interval required for a system to complete a full cycle of the motion is the period
of the vibration.
Number of cycles per unit time defines the frequency of the vibrations.
Maximum displacement of the system from the equilibrium position is the amplitude
of the vibration.
When the motion is maintained by the restoring forces only, the vibration is
called free vibration. When a periodic force is applied to the system, the
motion is described as forced vibration.
When the frictional dissipation of energy is neglected, the motion is said to
be undamped. Actually, all vibrations are damped to some degree.
3. • If a particle is displaced through a distance xm
from its equilibrium position and released with no
velocity, the particle will undergo simple harmonic
motion,
0
kx
x
m
kx
x
k
W
F
ma st
• General solution is the sum of two particular
solutions,
t
C
t
C
t
m
k
C
t
m
k
C
x
n
n
cos
sin
cos
sin
2
1
2
1
• x is a periodic function and n is the natural
circular frequency of the motion.
• C1 and C2 are determined by the initial conditions:
t
C
t
C
x n
n
cos
sin 2
1
0
2 x
C
n
v
C
0
1
t
C
t
C
x
v n
n
n
n
sin
cos 2
1
4.
t
x
x n
m sin
n
n
2 period
2
1 n
n
n
f natural frequency
2
0
2
0 x
v
x n
m amplitude
n
x
v
0
0
1
tan phase angle
• Displacement is equivalent to the x component of the sum of two vectors
which rotate with constant angular velocity
2
1 C
C
.
n
0
2
0
1
x
C
v
C
n
5.
t
x
x n
m sin
• Velocity-time and acceleration-time curves can
be represented by sine curves of the same
period as the displacement-time curve but
different phase angles.
2
sin
cos
t
x
t
x
x
v
n
n
m
n
n
m
t
x
t
x
x
a
n
n
m
n
n
m
sin
sin
2
2
6. • Results obtained for the spring-mass system can
be applied whenever the resultant force on a
particle is proportional to the displacement and
directed towards the equilibrium position.
for small
angles,
g
l
t
l
g
n
n
n
m
2
2
sin
0
:
t
t ma
F
• Consider tangential components of acceleration
and force for a simple pendulum,
0
sin
sin
l
g
ml
W
7. 0
sin
l
g
An exact solution
for
leads to
2
0
2
2
sin
2
sin
1
4
m
n
d
g
l
which requires numerical solution.
g
l
K
n
2
2
8. • If an equation of motion takes the form
0
or
0 2
2
n
n x
x
the corresponding motion may be
considered as simple harmonic motion.
• Analysis objective is to determine n.
mg
W
mb
b
b
m
I
,
2
2
but 2
3
2
2
2
12
1
0
5
3
sin
5
3
b
g
b
g
g
b
b
g
n
n
n
3
5
2
2
,
5
3
then
• For an equivalent simple pendulum,
3
5b
l
• Consider the oscillations of a square plate
I
mb
b
W
sin
9. • Resultant force on a mass in simple harmonic
motion is conservative - total energy is
conserved.
constant
V
T
2
2
2
2
2
1
2
2
1 constant
x
x
kx
x
m
n
• Consider simple harmonic motion of the square
plate,
0
1
T
2
2
1
2
1 2
sin
2
cos
1
m
m
Wb
Wb
Wb
V
2
2
3
5
2
1
2
2
3
2
2
1
2
2
1
2
2
1
2
2
1
2
m
m
m
m
m
mb
mb
b
m
I
v
m
T
0
2
V
0
0 2
2
2
3
5
2
1
2
2
1
2
2
1
1
n
m
m mb
Wb
V
T
V
T
b
g
n 5
3
10. :
ma
F
x
m
x
k
W
t
P st
f
m
sin
t
P
kx
x
m f
m
sin
x
m
t
x
k
W f
m
st
sin
t
k
kx
x
m f
m
sin
Forced vibrations - Occur
when a system is subjected
to a periodic force or a
periodic displacement of a
support.
f
forced frequency
11. t
x
t
C
t
C
x
x
x
f
m
n
n
particular
ary
complement
sin
cos
sin 2
1
2
2
2
1
1 n
f
m
n
f
m
f
m
m
k
P
m
k
P
x
t
k
kx
x
m f
m
sin
t
P
kx
x
m f
m
sin
At f = n, forcing input is in
resonance with the system.
t
P
t
kx
t
x
m f
m
f
m
f
m
f
sin
sin
sin
2
Substituting particular solution into governing
equation,
12. • With viscous damping due to fluid friction,
:
ma
F
0
kx
x
c
x
m
x
m
x
c
x
k
W st
• Substituting x = elt and dividing through by
elt yields the characteristic equation,
m
k
m
c
m
c
k
c
m
2
2
2
2
0 l
l
l
• Define the critical damping coefficient such
that
n
c
c m
m
k
m
c
m
k
m
c
2
2
0
2
2
• All vibrations are damped to some degree by
forces due to dry friction, fluid friction, or
internal friction.
13. • Characteristic equation,
m
k
m
c
m
c
k
c
m
2
2
2
2
0 l
l
l
n
c m
c
2 critical damping coefficient
• Heavy damping: c > cc
t
t
e
C
e
C
x 2
1
2
1
l
l
- negative roots
- nonvibratory motion
• Critical damping: c = cc
t
n
e
t
C
C
x
2
1 - double roots
- non-vibratory motion
• Light damping: c < cc
t
C
t
C
e
x d
d
t
m
c
cos
sin 2
1
2
2
1
c
n
d
c
c
damped
frequency
14.
2
2
2
2
1
2
tan
2
1
1
n
f
n
f
c
n
f
c
n
f
m
m
m
c
c
c
c
x
k
P
x
magnification
factor
phase difference between forcing and
steady state response
t
P
kx
x
c
x
m f
m
sin
particular
ary
complement x
x
x
15. • Consider an electrical circuit consisting of an
inductor, resistor and capacitor with a source of
alternating voltage
0
sin
C
q
Ri
dt
di
L
t
E f
m
• Oscillations of the electrical system are analogous to
damped forced vibrations of a mechanical system.
t
E
q
C
q
R
q
L f
m
sin
1
16. The analogy between electrical and mechanical
systems also applies to transient as well as steady
state oscillations -
• With a charge q = q0 on the capacitor, closing the
switch is analogous to releasing the mass of the
mechanical system with no initial velocity at x =
x0.
• If the circuit includes a battery with constant
voltage E, closing the switch is analogous to
suddenly applying a force of constant
magnitude P to the mass of the mechanical
system.
17. • The electrical system analogy provides a means of
experimentally determining the characteristics of a
given mechanical system.
• For the mechanical system,
0
2
1
2
1
1
2
1
2
1
1
1
1
x
x
k
x
k
x
x
c
x
c
x
m
t
P
x
x
k
x
x
c
x
m f
m
sin
1
2
2
1
2
2
2
2
• For the electrical system,
0
2
2
1
1
1
2
1
1
1
1
C
q
q
C
q
q
q
R
q
L
t
E
C
q
q
q
q
R
q
L f
m
sin
2
1
2
1
2
2
2
2
• The governing equations are equivalent. The
characteristics of the vibrations of the mechanical
system may be inferred from the oscillations of the
electrical system.