Dynamic analysis considers mass and acceleration effects that are neglected in static analysis. When loads are suddenly applied or variable, a solid body will vibrate elastically about its equilibrium position. This periodic motion is called free vibration. The document then provides formulations for modeling dynamic systems using kinetic and potential energy and Lagrange's equations. It derives the mass matrices for various finite elements, including bars, trusses, plane stress/strain elements, axisymmetric elements, beams, and frames. Consistent mass matrices are developed using the elements' shape functions and densities.
CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE EL...IAEME Publication
There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. the consistent / coupled mass matrix and the lumped mass matrix. Correct determination of natural frequencies is extremely important and forms the basis of any further NVH (Noise vibration and harshness) calculations and Impact or crash analysis. It has been thought by the finite element community that the consistent mass matrix should not be used as it leads to a higher computational cost and this opinion has been prevalent since 1970. We are of the opinion that in today’s age where computers have become so fast we can use the consistent mass matrix on relatively coarse meshes with an advantage for better accuracy rather than going for finer meshes and lumped mass matrix
Adaptive Control Machining systems, Introduction, Where to use adaptive control ?.Adaptive Control, Elements of an adaptive control machining system, Types of Adaptive controls, Benefits of AC
CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE EL...IAEME Publication
There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. the consistent / coupled mass matrix and the lumped mass matrix. Correct determination of natural frequencies is extremely important and forms the basis of any further NVH (Noise vibration and harshness) calculations and Impact or crash analysis. It has been thought by the finite element community that the consistent mass matrix should not be used as it leads to a higher computational cost and this opinion has been prevalent since 1970. We are of the opinion that in today’s age where computers have become so fast we can use the consistent mass matrix on relatively coarse meshes with an advantage for better accuracy rather than going for finer meshes and lumped mass matrix
Adaptive Control Machining systems, Introduction, Where to use adaptive control ?.Adaptive Control, Elements of an adaptive control machining system, Types of Adaptive controls, Benefits of AC
Vibration Analysis and Modelling of a Cantilever Beam Muhammad Usman
This report in cooperates the techniques, adopted for the evaluation of vibration analysis of a cantilever beam using both techniques i.e. theoretical as well as the practical ones. Computer based analysis of a beam were also performed using Solid Works and Mat Lab software. These techniques helped a lot in finding the natural frequencies and in making the vibrational characteristic behavior of a cantilever beam thus steel used as a material.
As per VTU CAD/CAM/CIM Module -4 18ME72-Part-A
Computer Numerical Control: Introduction, components of CNC, CNC programming, manual part programming, G Codes, M Codes, programming of simple components in turning, drilling and milling systems, programming with canned cycles. Cutter radius compensations.
Robot Technology: Robot anatomy, joints and links, common robot configurations, robot control systems, accuracy and repeatability, end effectors, sensors in robotics.
Net shape forging process means producing a component with exact shape and dimension in one blow with no flash and gutter. i.e no need for post forging operations and increases the production rate.
Vibration Analysis and Modelling of a Cantilever Beam Muhammad Usman
This report in cooperates the techniques, adopted for the evaluation of vibration analysis of a cantilever beam using both techniques i.e. theoretical as well as the practical ones. Computer based analysis of a beam were also performed using Solid Works and Mat Lab software. These techniques helped a lot in finding the natural frequencies and in making the vibrational characteristic behavior of a cantilever beam thus steel used as a material.
As per VTU CAD/CAM/CIM Module -4 18ME72-Part-A
Computer Numerical Control: Introduction, components of CNC, CNC programming, manual part programming, G Codes, M Codes, programming of simple components in turning, drilling and milling systems, programming with canned cycles. Cutter radius compensations.
Robot Technology: Robot anatomy, joints and links, common robot configurations, robot control systems, accuracy and repeatability, end effectors, sensors in robotics.
Net shape forging process means producing a component with exact shape and dimension in one blow with no flash and gutter. i.e no need for post forging operations and increases the production rate.
This study deals with the active control of the dynamic response of a string with fixed ends and mass
loaded by a point mass. It has been controlled actively by means of a feed forward control method. A point mass of a
string is considered as a vibrating receiver which be forced to vibrate by a vibrating source being positioned on the
string. By analyzing the motion of a string, the equation of motion for a string was derived by using a method of
variation of parameters. To define the optimal conditions of a controller, the cost function, which denotes the dynamic
response at the point mass of a string was evaluated numerically. The possibility of reduction of a dynamic response
was found to depend on the location of a control force, the magnitude of a point mass and a forcing frequency
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
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
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.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
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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.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
1. DYNAMIC CONSIDERATIONS
Static analysis holds when the loads are slowly applied. When
the loads are suddenly applied, or when the loads are of a
variable nature, the mass and acceleration effects come into
the picture. If a solid body, such as an engineering structure,
is deformed elastically and suddenly released, it tends to
vibrate about its equilibrium position. This periodic motion due
to the restoring strain energy is called free vibration.
1
2. The number of cycles per unit time is called frequency.
The maximum displacement from the equilibrium position is
the amplitude.
FORMULATION
We define the Lagrangean by
L = T - П (11.1)
where T is the kinetic energy and П is the potential energy.
2
3. 2
1
t
t
I Ldt
1 2 , 1 2
, ,..., , ,...,
n n
q q q q q q i i
q = dq /dt,
0 1
i i
d L L
i to n
dt q q
Hamilton’s principle For an arbitrary time interval from t1 to t2,
the state of motion of a body extremizes the functional
If L can be expressed in terms of the generalized variables
where then the equations of
motion are given by
(11.3)
(11.2)
3
4. Example 11.1
Consider the spring-mass system in Fig. E11.1. The kinetic
and potential energies are given by
2 2
1 1 2 2
2
2
1 1 2 2 1
1 1
2 2
1 1
2 2
T m x m x
k x k x x
4
5. Using L = T – П, we obtain the equations of motion
1 1 1 1 2 2 1
1 1
2 2 2 2 1
2 2
0
0
d L L
m x k x k x x
dt x x
d L L
m x k x x
dt x x
which can be written in the form
1 2 2
1 1 1
2 2 2
2 2
0
0
0
k k k
m x x
m x x
k k
0
Mx kx
x
which is of the form
where M is the mass matrix, K is the stiffness matrix, and
and x are vectors representing accelerations and displacements.
(11.4)
5
6. Solid Body with Distributed Mass
Consider a solid body with distributed mass (Fig.11.1). The
potential-energy term has already been considered in Chapter 1.
The kinetic energy is given by
1
2
T
V
T u u dV
(11.5)
6
7.
, ,
T
u u v w
, , .
u v and w
u Nq
where ρ is the density (mass per unit volume) of the material
and
is the velocity vector of the point at x, with components
In the finite element method, we dive the body into elements,
and in each element, we express u in terms of the nodal
displacements q, using shape functions N.
u = Nq (11.7)
In dynamic analysis, the elements of q are dependent on time,
while N represents (spatial) shape functions defined on a
master element. The velocity vector is then given by
(11.8)
(11.6)
7
8. 1
2
T T
e
T q N N dV q
e T
e
m N N dV
1 1
2 2
T e T
e
e e
T T q m q Q MQ
When Eq.11.8 is substituted into Eq.11.5, the kinetic energy Te
in element e is
where the bracketed expression is the element mass matrix
This mass matrix is consistent with the shape functions
chosen and is called the consistent mass matrix. On summing
over all the elements, we get
(11.11)
(11.9)
(11.10)
8
9. 1
2
T T
Q KQ Q F
MQ KQ F
0
MQ KQ
The potential energy is given by
Using the Lagrangean L = T – П, we obtain the equation of
motion:
For free vibrations the force F is zero. Thus,
(11.14)
(11.12)
(11.13)
For the steady-state conditions, starting from the equilibrium
state, we get
sin
Q U t
(11.15)
9
10. 2
KU MU
KU MU
where U is the vector of modal amplitudes of vibration and ω
(rad/s) is the circular frequency (2πf, f = cycles/s or Hz).
Introducing Eq. 11.15 into Eq.11.14, we have
This is the generalized eigen value problem
(11.17)
(11.16)
ELEMENT MASS MATRICES
Treating the material density ρ to be constant over the
element, we have, from Eq. 11.10,
e T
e
m N N dV
(11.18)
10
11.
1 2
1 2
T
q q q
N N N
1 2
1 1
2 2
N N
1
1
2
e T T
e e
e
A
m N NA dx N N d
One-dimensional bar element For the bar element shown in
Fig. 11.2,
where
(11.19)
11
13. 2 1
1 2
6
e e e
A
m
1 2 3 4
1 2
1 2
,
0 0
0 0
T
T
u u v
q q q q q
N N
N
N N
On carrying out the integration of each term in NTN, we find that
Truss element For the truss element shown in Fig. 11.3,
(11.21)
(11.20)
where
1 2
1 1
2 2
N N
13
14. in which ξ is defined from -1 to +1. Then,
2 0 1 0
0 2 0 1
1 0 2 0
6
0 1 0 2
e e e
A
m
(11.22)
FIGURE 11.3 Truss element 14
15. CST element For the plane stress and plane strain conditions
for the CST element shown in Fig. 11.4, we have, from
Chapter 5,
1 2 6
1 2 3
1 2 3
...
0 0 0
0 0 0
T
T
u u v
q q q q
N N N
N
N N N
(11.23)
The element mass matrix is then given by
e T
e
e
m t N N dA
Noting that 2
1 1 2
1 1
, ,
6 12
e e
e e
N dA A N N dA A
etc., we have
15
16. 2 0 1 0 1 0
2 0 1 0 1
2 0 1 0
2 0 1
12
2 0
2
e e e
t A
m
Symmetric
(11.24)
Axisymmetric triangular element For the axisymmetric
triangular element, we have
uT = [ u w]
16
18. where u and w are the radial and axial displacements,
respectively. The vectors q and N are similar to those for the
triangular element given in Eq.11.23. We have
2
e T T
e e
m N N dV N N r dA
(11.25)
1 1 2 2 3 3
2
e T
e
m N r N r N r N N dA
3 2
1 1 2 1 2 3
2 2 2
, , , .
20 60 120
e e e
e e e
A A A
N dA N N dA N N N dA etc
Since r = N1r1 + N2r2 + N3r3, we have
Noting that
18
19. we get
3 2
1
3 2
1
1
2
1
2
3
3
4
2 0 2 0 2 0
3 3 3
4
2 0 2 0 2
3 3 3
4
2 0 2 0
3 3
4
2 0 2
3 3
10
4
2 0
3
4
2
3
e
e
r r
r r r r
r r
r r r r
r
r r r
r
A
r r r
m
Symmetric r r
r r
(11.26)
19
20. 1 2 3
3
r r r
r
1 2 8
1 2 3 4
1 2 3 4
...
0 0 0 0
0 0 0 0
T
T
u u v
q q q q
N N N N
N
N N N N
where
Quadrilateral element For the quadrilateral element for plane
stress and plane strain,
(11.28)
(11.27)
20
21. The mass matrix is then given by
1 1
1 1
det
e T
e
m t N N J d d
This integral needs to be evaluated by numerical integration.
(11.29)
21
22. Beam element For the beam element shown in Fig. 11.5, we
use the Hermite shape functions given in Chapter 8. We have
v = Hq
1
1
2
e T e
e
m H H A d
(11.30)
On integrating, we get
2 2
2
156 22 54 13
4 13 3
156 22
420
4
e e
e e e
e e e
e
e
A
m
Symmetric
(11.31)
22
23. Frame element We refer to Fig.8.9, showing the frame
element. In the body coordinate system x’, y’, the mass matrix
can be seen as a combination of bar element and beam
element. Thus, the mass matrix in the prime system is given
by
2
2 2
'
2
2 0 0 0 0
156 22 0 54 13
4 0 13 3
2 0 0
156 22
4
e e
e e e e
e
e
a a
b b b b
b b b
m
a
Symmetric b b
b
(11.32)
23
24. where
6 420
e e e e
A A
a and b
Using the transformation matrix L is given by Eq. 8.48, we
now obtain the mass matrix me in the global system:
me = LTme’L (11.33)
Tetrahedral element For the tetrahedral element presented in
Chapter 9,
1 2 3 4
1 2 3 4
1 2 3 4
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
T
u u v w
N N N N
N N N N N
N N N N
(11.34)
24
26. Lumped mass matrices Practicing engineers also use
lumped mass techniques, where the total element mass in
each direction is distributed equally to the nodes of the
element, and the masses are associated with translational
degrees of freedom only. For the truss element, the lumped
mass approach gives a mass matrix of
1 0 0 0
1 0 0
1 0
2
1
e e e
A
m
Symmetric
(11.36)
26
27. For the beam element, the lumped element mass matrix is
1 0 0 0
0 0 0
1 0
2
0
e e e
A
m
Symmetric
(11.37)
EVALUATION OF EIGENVALUES AND EIGENVECTORS
KU = λMU (11.38)
We observe here that K and M are symmetric matrices.
Further, K is positive definite for properly constrained
problems.
27
28. Properties of Eigenvectors
For a positive definite symmetric stiffness matrix of size n,
there are n real eigenvalues and corresponding
eigenvectors satisfying Eq.11.38. The eigenvalues may be
arranged in ascending order:
0 ≤ λ1 ≤ λ2 ≤ …≤ λn (11.39)
If U1, U2…Un are the corresponding eigenvectors, we have
KUi = λiMUi (11.40)
28
29. 0
T
i j
U MU if i j
0
T
i j
U KU if i j
1
T
i i
U MU
T
i i i
U KU
The eigenvectors possess the property of being orthogonal
with respect to both the stiffness and mass matrices
(11.41a)
The lengths of eigenvectors are generally normalized so that
The foregoing normalization of the eigenvectors leads to
the relation
(11.42b)
(11.41b)
(11.42a)
29
30. Eigenvalue – Eigenvector Evaluation
The eigenvalue-eigenvector evaluation procedures fall into the
following basic categories:
1.Characteristic polynomial technique
2.Vector iteration methods
3.Transformation methods
Characteristic polynomial From Eq. 11.38, we have
(K - λM)U = 0 (11.43)
30
31. If the eigenvector is to be nontrivial, the required condition is
det(K - λM)= 0 (11.44)
This represents the characteristic polynomial in λ.
Example 11.2
Determine the eigenvalues and eigenvectors for the stepped
bar shown in Fig. E11.2a.
Solution Gathering the stiffness and mass values
corresponding to the degrees of freedom Q2 and Q3, we get
the eigenvalue problem
31
32.
1 2 2
2 1 1 2 2 2 2 2
1 2 2
3 3
2 2 2 2
2 2
2 2
2
6 2
A A A
U A L A L A L U
L L L
E
U U
A L A L
A A
L L
4 2 4
0.283
7.324 10 / .
32.2 12
f
lbs in
g
2 2
6 4
3 3
0.2 0.1 25 2.5
30 10 1.22 10
0.1 0.1 2.5 5
U U
U U
We note here that the density is
Substituting the values, we get
32
35. Note that λ = ω2, where ω is the circular frequency given by
2πf and f = frequency in hertz (cycles/s).
These frequencies are
f1 = 4115 Hz
f2 = 13884 Hz
The eigenvector for λ1 is found from
(K – λ1M)U1 = 0
which gives
2
6
3 1
3.96 3.024
10 0
3.204 2.592
U
U
35
36. The two previous equations are not independent, since the
determinant of the matrix is zero. This gives
3.96 U2 = 3.204U3
Thus,
1 2 2
,1.236
T
U U U
1 1 1
T
U MU
1 14.527 17.956
T
U
2 11.572 37.45
T
U
For normalization, we set
On substituting for U1, we get
The eigenvector corresponding to the second eigenvalue is
similarly found to be
The mode shapes are shown in Fig. E11.2b. 36