- The document discusses issues with achieving a true free-free boundary condition when experimentally testing structures. A true free-free condition allows all parts of the structure to move freely without constraints.
- It analyzes a beam model supported on springs at both ends to simulate a free-free condition. Having very soft springs results in rigid body modes of motion that match a true free-free condition. Stiffer springs distort the modes away from free-free.
- Measured frequencies from the beam model match theoretical free-free frequencies when the ratio of the first elastic mode frequency to highest rigid body frequency is above 10. Below this, the first elastic mode is sensitive to the boundary conditions imposed by the springs.
FOR HUMANITY: (V4) A BREAKTHROUGH IN TOKAMAK APPLIED PHYSICS GRAVITATIONAL WA...GLOBAL HEAVYLIFT HOLDINGS
To whom it may concern: (Note: Abbreviated acknowledgement narrative by Dr. Andrew W. Beckwith )
The author, Dr. Andrew Beckwith, authorizes Myron D. Stokes, Publisher, eMOTION! REPORTS.com, a legacy automotive/aerospace research and analysis site and Managing Director, Global HeavyLift Holdings, Inc., a Defense Logistics Agency listed Federal Contractor, as to the dissemination of the following Tokamak applied physics notes for Gravitational wave generation, as his acting manager of public release of the aforementioned document. The information within will be peer reviewed , but the basic technology is intended for humankind as far as fundamental physics advancement world wide.
The notes, within, were created by Dr. Beckwith, in the Keyuan Hotel in room 1205, in Chongqing, PRC, as part of a joint USA-PRC endeavor as to GW physics; the notes are correctly identified by Gary Stephenson , as a civilian employee of the US air force, as pertinent to " Higher drift current during fusion burning" as an enabler of high GW amplitudes, of the order of h ~ 10^-25 to 10^-26 which are candidates for testing of GW direct identification technology incurrent development in both PRC and in America. The notes were also written up as of November 4 to November 12, in an eight day period, in room 1205 of the Keyuan hotel as of Chongqing University under the auspices of Chongqing University department of physics, in work which was enabled by the hospitality of Chongqing University which took unusually comprehensive steps as to the proper circumstances for the creation of this work so cited by the Author.
The following individuals should be thanked as far as their discussions and input as to formation of the "Higher drift current during fusion burning" which is crucial to the development of this material.
a. Dr. Fangyu Li, whose interest in Tokamak physics never flagged, as to its utilization. He informed the author during a stay from November 2 to November 13 of his partnership with a Tokamak fusion laboratory as of Hefei, PRC, which would serve as a test bed of GW amplitudes. His physics questions were timelyand very important during the 12 days of stay in Chongqing University.
b. Dr. Fan, Chief scientific administrator of Chongqing University wrote in administrative authorization of Dr. Beckwith's visit to Chongqing University and also as part of a 40 year friendship with Dr. Li, accompanied Dr. Li to the Tokamak fusion laboratory as of Hefei, PRC, which the author saw in photographs as to the facility, and the chief engineering officer who runs the Hefei Tokamak facility
The above written statement should be part of a slide share release of this basic information as set up by Myron D. Stokes, of Global Heavylift Holdings corporation without further delay.
Andrew Beckwith, PhD, written in Setauket, New York, as of 6 PM, November 16 ( November 17, PRC time, Chongqing), 2013
FOR HUMANITY: (V4) A BREAKTHROUGH IN TOKAMAK APPLIED PHYSICS GRAVITATIONAL WA...GLOBAL HEAVYLIFT HOLDINGS
To whom it may concern: (Note: Abbreviated acknowledgement narrative by Dr. Andrew W. Beckwith )
The author, Dr. Andrew Beckwith, authorizes Myron D. Stokes, Publisher, eMOTION! REPORTS.com, a legacy automotive/aerospace research and analysis site and Managing Director, Global HeavyLift Holdings, Inc., a Defense Logistics Agency listed Federal Contractor, as to the dissemination of the following Tokamak applied physics notes for Gravitational wave generation, as his acting manager of public release of the aforementioned document. The information within will be peer reviewed , but the basic technology is intended for humankind as far as fundamental physics advancement world wide.
The notes, within, were created by Dr. Beckwith, in the Keyuan Hotel in room 1205, in Chongqing, PRC, as part of a joint USA-PRC endeavor as to GW physics; the notes are correctly identified by Gary Stephenson , as a civilian employee of the US air force, as pertinent to " Higher drift current during fusion burning" as an enabler of high GW amplitudes, of the order of h ~ 10^-25 to 10^-26 which are candidates for testing of GW direct identification technology incurrent development in both PRC and in America. The notes were also written up as of November 4 to November 12, in an eight day period, in room 1205 of the Keyuan hotel as of Chongqing University under the auspices of Chongqing University department of physics, in work which was enabled by the hospitality of Chongqing University which took unusually comprehensive steps as to the proper circumstances for the creation of this work so cited by the Author.
The following individuals should be thanked as far as their discussions and input as to formation of the "Higher drift current during fusion burning" which is crucial to the development of this material.
a. Dr. Fangyu Li, whose interest in Tokamak physics never flagged, as to its utilization. He informed the author during a stay from November 2 to November 13 of his partnership with a Tokamak fusion laboratory as of Hefei, PRC, which would serve as a test bed of GW amplitudes. His physics questions were timelyand very important during the 12 days of stay in Chongqing University.
b. Dr. Fan, Chief scientific administrator of Chongqing University wrote in administrative authorization of Dr. Beckwith's visit to Chongqing University and also as part of a 40 year friendship with Dr. Li, accompanied Dr. Li to the Tokamak fusion laboratory as of Hefei, PRC, which the author saw in photographs as to the facility, and the chief engineering officer who runs the Hefei Tokamak facility
The above written statement should be part of a slide share release of this basic information as set up by Myron D. Stokes, of Global Heavylift Holdings corporation without further delay.
Andrew Beckwith, PhD, written in Setauket, New York, as of 6 PM, November 16 ( November 17, PRC time, Chongqing), 2013
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
Grab CBSE sample paper for class 11 Physics & practice diligently to secure apt marks. Download the free PDF now & join Studymate to accentuate your graph. Visit http://www.studymateonline.com/sample-papers/cbse-sample-papers-for-class-11-physics/
Validation of Polarization angles Based Resonance Modes IJERA Editor
The symmetry, tilt and elongation degrees are figures of merit which can be used to describe the radar target
shape once incorporated with the target resonance modes. Through optimization of the second moments of the
quadrature-polarized residues matrix, the angles are determined by the optimum co-null polarization states. The
approach is tested and validated against low signal-to-noise ratio and also the late-time onset selection when
extracting the mode set. A wire plane model is used and the results show that with ensemble averaging it
possible to have robust polarization angle set, even with small number of sample set
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
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
»Over the last two decades several patents and research papers have reported purported practical methods to extract useful energy from the vacuum. I describe the inventions and analyze the underlying physics. From an analysis based on first principles it is clear that most of the inventions have fundamental errors and cannot work. The basic concept of harvesting zero-point energy remains viable, and at least one patented concept might work.
The vacuum is filled with a high density of zero-point energy, in the form of modes (vibrational patterns) of electromagnetic field. Over the last eight decades it has become clear that this zero-point field (ZPF) vacuum energy is not simply a mathematical formalism, but produces demonstrable effects on physical systems. Along with that realization has come the desire to extract energy from the ZPF.
One set of methods use nonlinear elements to convert the ZPF into a usable form. A rectifier (used to convert AC to DC) is a strongly nonlinear element. One patent makes use of antennas to capture the ZPF. This energy is then rectified and used. Another set of inventions simply rectify fluctuations (noise) in electronic elements as an extraction method. Using a detailed balance argument, I show that these methods cannot work.
Another set of patents describe using a Casimir cavity to mechanically extract energy from the ZPF. A Casimir cavity consists of two closely space reflecting plates that exclude ZPF electromagnetic modes having wavelengths larger than twice the gap spacing. The result is that the imbalance in the density of the ZPF inside and outside the cavity causes the plates to be attracted to each other. This attractive potential can be used, but only once. To produce power continuously, a method must be devised to form a reciprocating Casimir engine. The patents purport to switch off the Casimir attraction while the plates are pulled apart, so that they can repeatedly accelerate together and produce power. This approach is shown to be fundamentally flawed, and cannot produce power continuously.
A recently issued patent describes a method by which vacuum energy is extracted from gas flowing through a Casimir cavity. According to stochastic electrodynamics, the electronic orbitals in atoms are supported by ambient ZPF. When the gas atoms are pumped into a Casimir cavity, where long-wavelength ZPF modes are excluded, the electrons spin down into lower orbitals, releasing energy. This energy is harvested in a local absorber. When the electrons exit the Casimir cavity, they are re-energized to their original orbitals by the ambient ZPF. The process is repeated to produce continuous power. This method does not suffer from the fundamental flaws of the other approaches, and might work.«
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
Grab CBSE sample paper for class 11 Physics & practice diligently to secure apt marks. Download the free PDF now & join Studymate to accentuate your graph. Visit http://www.studymateonline.com/sample-papers/cbse-sample-papers-for-class-11-physics/
Validation of Polarization angles Based Resonance Modes IJERA Editor
The symmetry, tilt and elongation degrees are figures of merit which can be used to describe the radar target
shape once incorporated with the target resonance modes. Through optimization of the second moments of the
quadrature-polarized residues matrix, the angles are determined by the optimum co-null polarization states. The
approach is tested and validated against low signal-to-noise ratio and also the late-time onset selection when
extracting the mode set. A wire plane model is used and the results show that with ensemble averaging it
possible to have robust polarization angle set, even with small number of sample set
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
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
»Over the last two decades several patents and research papers have reported purported practical methods to extract useful energy from the vacuum. I describe the inventions and analyze the underlying physics. From an analysis based on first principles it is clear that most of the inventions have fundamental errors and cannot work. The basic concept of harvesting zero-point energy remains viable, and at least one patented concept might work.
The vacuum is filled with a high density of zero-point energy, in the form of modes (vibrational patterns) of electromagnetic field. Over the last eight decades it has become clear that this zero-point field (ZPF) vacuum energy is not simply a mathematical formalism, but produces demonstrable effects on physical systems. Along with that realization has come the desire to extract energy from the ZPF.
One set of methods use nonlinear elements to convert the ZPF into a usable form. A rectifier (used to convert AC to DC) is a strongly nonlinear element. One patent makes use of antennas to capture the ZPF. This energy is then rectified and used. Another set of inventions simply rectify fluctuations (noise) in electronic elements as an extraction method. Using a detailed balance argument, I show that these methods cannot work.
Another set of patents describe using a Casimir cavity to mechanically extract energy from the ZPF. A Casimir cavity consists of two closely space reflecting plates that exclude ZPF electromagnetic modes having wavelengths larger than twice the gap spacing. The result is that the imbalance in the density of the ZPF inside and outside the cavity causes the plates to be attracted to each other. This attractive potential can be used, but only once. To produce power continuously, a method must be devised to form a reciprocating Casimir engine. The patents purport to switch off the Casimir attraction while the plates are pulled apart, so that they can repeatedly accelerate together and produce power. This approach is shown to be fundamentally flawed, and cannot produce power continuously.
A recently issued patent describes a method by which vacuum energy is extracted from gas flowing through a Casimir cavity. According to stochastic electrodynamics, the electronic orbitals in atoms are supported by ambient ZPF. When the gas atoms are pumped into a Casimir cavity, where long-wavelength ZPF modes are excluded, the electrons spin down into lower orbitals, releasing energy. This energy is harvested in a local absorber. When the electrons exit the Casimir cavity, they are re-energized to their original orbitals by the ambient ZPF. The process is repeated to produce continuous power. This method does not suffer from the fundamental flaws of the other approaches, and might work.«
Ultrasonic guided wave techniques have great potential for structural health monitoring applications. Appropriate mode and frequency selection is the basis for achieving optimised damage monitoring performance.
In this paper, several important guided wave mode attributes are
introduced in addition to the commonly used phase velocity and group velocity dispersion curves while using the general corrosion problem as an example. We first derive a simple and generic wave excitability function based on the theory of normal mode expansion and the reciprocity theorem. A sensitivity dispersion curve is formulated based on the group velocity dispersion curve. Both excitability and sensitivity dispersion curves are verified with finite element simulations. Finally, a
goodness dispersion curve concept is introduced to evaluate the tradeoffs between multiple mode selection objectives based on the wave velocity, excitability and sensitivity.
Similar to Sem.org imac-xxiv-conf-s19 cp06-the-elusive-free-free-boundary-condition (20)
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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.
For more technical information, visit our website https://intellaparts.com
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
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.
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.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
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.
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
1. The Elusive Free-Free Boundary Condition
by
Kenneth G. McConnell
Professor Emeritus, Vibration Engineering
Department of Aerospace Engineering
Iowa State University of Science and Technology
Ames, IA 50011
(mclken@msn.com)
ABSTRACT:
The application of substructuring concepts requires
the use of free-free FRF’s to describe the test item.
The free-free test environment is easily achieved in
theoretical calculations but is usually compromised
when simulated experimentally due to the
requirement to support the structure against gravity
forces. This paper looks at some interesting testing
issues through an analysis of a free-free beam
mounted on end springs as well as experimental
results from an actual beam. In other words, “Why
can’t I correct for these incorrect boundary conditions
except in a very special case that generally gives
unsatisfactory results?”
INTRODUCTION:
The importance of boundary conditions in
dealing with vibration problems was clearly brought
to my attention when dealing with an interesting
catapult excited vibration of the island structure of an
aircraft carrier as reported by Hagen, et el [1] and
McConnell [2]. In this case, the torsional stiffness of
the supporting sponson structure played an important
role in reducing the island’s fundamental natural
frequency from about 5 Hz for most modern aircraft
carriers to 3.2 Hz for the one in question. It turned
out that the island’s fundamental natural frequency
was highly dependent on the torsional stiffness of the
sponson structure.
Recently, vibration testing of space station
structures requires testing for many natural
frequencies and mode shapes that are below 1.0 Hz.
Foss [3] developed a method for achieving the free-
free boundary condition by suspending the structure
from a single point using a spring to support both the
exciter armature and the test structure and then
measuring the input force that consists of both the
spring support force and the excitation force. The
end result is the achievement of a real free-free
boundary condition.
However, this solution looked too good to
be true and it was for there can be some significant
problems. First, the single support point must be
above the mass center in order to cancel out the
gravity force, which limits the location of the
excitation force. For example, the support point may
not be the point where we need the driving point and
transfer FRF’s. Second, how do I test for the
potential 36 input-output driving point relationships
that may be needed to describe a structure for
connecting to another structure? Third, one may miss
important mode shapes and natural frequencies when
exciting at only one point in a single direction, etc.
THE ALMOST FREE-FREE BEAM
We shall try to explore some of the issues
that surround the so called free-free boundary
condition. A simple model is used as shown in Fig. 1
2. which is a cold rolled steel beam that has been tested
many times by my students over the years. The
characteristics of this beam are: length l = 92.0 inch
(233.7 cm); width b= 1.25 inch (3.18 cm); thickness
h = 1.00 inch (2.54 cm). It is supported by springs at
the left (k1) and right (kN) ends. The different
receptances are calculated using Mathcad software
and an influence coefficient beam formulation. This
calculation method is the same as a finite element
program in that the end springs influence both the
system modes shapes and natural frequencies.
This beam has two potential rigid body
modes of vibration dependent on the spring
constants. When the springs are very soft and equal
(like 0.140 lbs/in), these two rigid body modes are
straight up and down (y) for one mode and rotation
about the mass center g (θ) for the other mode.
Table 1 compares the calculated natural frequencies
with those measured when the highest rigid body
natural frequency is just over 1.0 Hz, giving a
frequency ratio greater than 20 to one. It is seen that
good agreement is achieved between the calculated
Table 1. Predicted vs Measured free-free Natural
Frequencies
Mode
Num
fp (Hz)-y
Theoretical
fp (Hz)-y
Measured
fp (Hz) - z
Theoretical
fp (Hz)-z
Measured
1 24.07 24.2 30.1
2 66.3 67.4 82.9 86.5
3 130.1 132.2 162.6 165.3
4 215.0 217.4 268.8 272.1
5 321.2 324.3 401.4 404.4
and measured natural frequencies. The first four
mode shapes are shown in Fig. 2 where the first mode
is Rigid-Body translation while the second mode is
Rigid-Body rotation about the mass center. The third
and fourth mode shapes match those of a free-free
beam. Hence, we obtained a reasonable simulation
of the free-free condition.
Fig. 1. Theoretical free-free beam on springs.
f
2
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92
0.3
0.2
0.1
0
0.1
Mode 1 RB
Mode 2 RB
Mode 3 1st Free Free
Mode 4 2nd Free Free
First Four Mode Shapes
Position x
Amplitude
Fig. 2. First Four Mode Shapes for very soft springs of k1 = kN
= 0.140 lbs/in.
0 9.2 18.4 27.6 36.8 46 55.2 64.4 73.6 82.8 92
0.3
0.2
0.1
0
0.1
0.2
0.3
Mode 1 RB
Mode 2 RB
Mode 3 1st Free-Free
Mode 4 2nd Free-Free
First Four Mode Shapes
Position x
Amplitude
Fig. 3. First Four Mode Shapes for moderate springs
of k1 = kN = 140 lbs/in.
However, if we increase each spring rate by
1000 times to 140 lbs/in., we obtain the mode shapes
shown in Fig. 3. Here it is clear that the first and
second Rigid-Body mode shapes have changed and
are beginning to look like the first and second modes
b
h
/ / / / / /
θ
g
l
/ / / / / /
k1 kN
y
3. for a pinned end beam, which they will become if the
spring constants are increased by an additional factor
of 100.
The first five natural frequencies are shown
in Table 2 as a function of the frequency ratio β (=
f3/f2) where f3 is the beam’s first elastic or free-free
mode and f2 is the higher of the Rigid Body natural
frequencies. Here it is seen that the first elastic beam
mode f3 is quite sensitive to frequency ratio β. When
β decreases from 5.0 to 2.02, f3 increases from 24.75
to 30.57 Hz, while f3 varies from 24.06 (β = 152) up
to 24.75 Hz (β= 5). Hence, one concludes that β
should be on the order of ten or more in order for the
first free-free elastic mode to be reasonably correct.
We also note that the second and third elastic free-
free modes have little change in frequency and/or
mode shape as long as β is on the order of 10 or
more.
Table 2: Natural frequencies as a function of frequency
ratio β*
Freq
Ratio
f1 f2 f3 f4 f5
152 0.091 0.158 24.06 66.31 129.95
15.2 0.981 1.582 24.13 66.33 129.96
10.0 1.393 2.43 24.22 66.36 129.98
5.0 2.08 4.98 24.75 66.56 130.0
2.02 7.01 15.16 30.58 68.88 131.2
* Frequency ratio is first elastic mode natural
frequency divided by highest rigid body natural
frequency, i.e., β = f3/f2 in this table
WHERE DOES THIS CONTAMINATION OF
INFORMATION COME FROM?
Varoto [4] examined the dynamic
interaction between two structures, one called the test
item and the other called the vehicle. The resulting
equations show that the combined structural response
depends on the sum of the connector FRF’s as well as
the test item’s transfer FRF’s. The vehicle’s FRF’s
are as measured on the vehicle when the test item is
removed while all of the test item FRF’s must be
obtained under free-free boundary conditions.
There are potentially 36 required interface
FRF’s for each interface point; i.e.,
⎭
⎬
⎫
⎩
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
=
⎭
⎬
⎫
⎩
⎨
⎧
MDC
BA Fa
α
(1)
Note that a, α, F, and M can have up to three vector
components. Hence, each submatrix A, B, C, and D
has the potential to be 3x3 in size. Dong [5] has
developed an experimental technique to measure the
rotational as well as the linear FRF’s when using a T-
bar attachment.
Let us now look at the output motion at
location p on the beam in Fig. 1 due to an excitation
force of Fq applied at location q. The measured
motion yp (displacement, velocity, or acceleration) is
given by
NqNqqpqqpqp PVPVFVFMy ++== 11 (2)
where Mpq is the measured FRF between points p
and q,
Vpq is the actual free-free beam FRF between
points p and q,
Fq the excitation force at location q,
Vq1 and VqN are the actual free-free transfer
FRF’s between points q and 1 and
q and N,
while P1 = - k1H1q Fq is the spring force at location
1 and
PN = - kN HNqFq is the spring force at
location N due to the actual beam motion at
those points.
Substituting these last two spring forces intoEq. 2 and
canceling the common term Fq gives
4. NqNqNqqpqpq HkVHkVVM −−= 111 (3)
It is clear from Eq. (3) that the measured FRF Mpq is
contaminated by the two spring support terms
qq HkV 111− and NqNqN HkV− a
. It is seen that
this contamination involves both the free-free elastic
transfer FRF’s, Vq1 and VqN that we need to
determine as well as the measured transfer
receptances H1q and HNq that control the boundary
condition forces acting on the structure. Table 3 gives
the required units dependent on the type of FRF that
is measured; i.e., Mpq. In addition we see that all of
the V’s have the same units as Mpq while the H’s
always need to be receptances to go with the
corresponding spring so that the product of ksHsq is
dimensionless. We should suspect that each spring
support would contribute contamination terms like
those in Eq. 3. Hence, we can not always assume
that the contamination is negligible without checking
out the effect of different support conditions; i.e., use
different springs.
Table 3: Required Units that apply to Eq. 3
Mpq Vpq Vq1 VqN H1q HNq
Rec Rec Rec Rec Rec Rec
Mob Mob Mob Mob Rec Rec
Acc Acc Acc Acc Rec Rec
Rec = Receptance, Mob = Mobility,
Acc = Accelerance
EXAMPLE OF TWO DIFFERENT SPRINGS
Now let us consider an example where k1 =
140 lbs/in and kN = 0.140 lbs/in. The resulting mode
shapes are shown in Fig. 4 where it is seen that the
first mode appears like the left end is pinned to a
a
For “s” number of spring support points, these two
terms become ∑s
sqsqs HkV
rigid foundation while the second mode looks more
like a cantilever beam when the base moves up and
down. The third mode looks a lot like the first elastic
beam mode with the left end moving more than the
right end. The fourth mode looks nearly identical to
the second elastic beam mode.
Figure 5 shows the driving point receptance
for the left end at location 1. There is only one Rigid
Body mode showing in this plot since point 1 is
essentially a node point so that the rotational first
rigid body mode is not excited (see Fig. 4 for this
mode shape). When the Rigid Body effects are
subtracted from H11 it is seen that
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92
0.3
0.2
0.1
0
0.1
0.2
0.3
Mode 1
Mode 2
Mode 3 1st Free-Free Mode
Mode 4 2nd Free-Free Mode
First Four Mode Shapes
Position x
Amplitude
Fig. 4. First four mode shapes, k1 = 140 and kN=
0.140 lbs/in
the first free-free natural frequency remains the same
around 30 Hz when the actual free-free natural
frequency is about 24 Hz.
If we apply Eq. (3) to this physical situation,
we obtain
11111111111 NNN HkVHkVVH −−= (4)
where H11 is the driving point receptance at the left
end of the beam. Since kN is very small compared to
k1 and V11 is the actual driving point receptance, We
use U11 in order to distinguish the predicted or
corrected receptance from the actual free-free driving
point receptance. Hence, we have that
5. 111
11
11
1 Hk
H
U
−
≅ (5)
Now, we need to evaluate how much the rigid body
modes contaminate the prediction of Eq. 5 by
removing the Rigid Body modes from H11. Let B11
be H11 with the Rigid Body modes removed. Then,
Eq. 5 becomes
111
11
11
1 Bk
B
D
−
≅ (6)
The computed results using Eq. 5 and 6 are shown in
Fig. 6 where it is clearly seen that the first free-free
resonant frequency is essentially 24 Hz in both
corrections. However, it is seen that the correction
based on H11 and contains the Rigid Body modes and
gives strange results below the first elastic free-free
resonance. The correction based on B11, where the
rigid body modes were removed, gives a more
consistent result when compared to the actual
receptance at low frequencies. It is clearly seen in
both Figs. 5 & 6 that the notches are not completely
corrected. This error can have major significance
when the mating vehicle has low values as well in
this frequency range since it is the sum of the two
values that are involved in creating the combined
dynamic behavior.
It should be pointed out that using Eqs. 5
and 6 to correct the driving point receptance on the
right hand side does not cause a significant shift in
the first elastic mode frequency; i.e., f3. This is due
to kN being very soft compared to k1 which has a
major effect on the natural frequencies and mode
shapes. Hence, any simple correction scheme must
be applied with caution.
CONCLUSIONS
I hope that the reader sees that the free-free
boundary condition is difficult to obtain in practice
and we can be significantly fooled by the measured
results. We can achieve a reasonable simulation if
we can separate the first free-free resonance from the
suspension Rigid Body modes by a factor of nearly
10 to one; i.e., β = f3/f2 > 10. Then, removal of rigid
body modes from the measured data should give
reasonable results for driving point FRF’s where the
spring constant is largest. Other physical support
relationships need to be evaluated on an individual
basis. I hope that this little exercise will encourage
some bright person to think about all of these testing
issues, and they will come up with a really neat way
to correct the measured data, like measure all of the
support forces and then use multi-point input output
analysis.
REFERENCES:
1. Hagen, A and K. McConnell, “Catapult
Excited Mast, Island, and Hull Vibration
Studies on Three Aircraft Carriers: USS
America (CVA-66), USS Constellation
(CVA-64), and USSS John F. Kennedy
(CVA-67)”, Naval Ship Research and
Development Center Report 3181,
Washington, DC, 1969.
2. McConnell, K. G., “Tracking the Cause of
Large Shipboard Vibrations During
Catapult Launching,” Proc., SEM Fall
Conference, Savannah, GA, Nov. 1987.
3. Foss, Gary C., Free-Free Modal Testing
Without Suspension Modes, Proc. IMAC-
XIV, P. 437-441, Feb. 1996.
4. Varoto, Paulo S., The Rules for the
Exchange and Analysis of Dynamic
Information, Ph.D. Thesis, Iowa State
University, Ames, IA. 1996.
6. 5. Dong, Jianrong, Extraction Methods for
Multidirectional Driving Point Accelerance
and Transfer Point Accelerance Matrices,
Ph. D. Thesis, Iowa State University, Ames,
IA. 2000.
0 20 40 60 80 100 120 140 160 180 200
1 10
5
1 10
4
1 10
3
0.01
0.1
1
Measured DP Receptance H11
DP Receptance RB removed
Free-Free DP Receptance
Driving Point Receptance 11
Frequency (Hz)
Receptance(in/lb)
Fig. 5. Driving point receptance at point 1 as measured and corrected compared with free-free beam values, k1 = 140 and
kN = 0.140 lbs/in.
0 20 40 60 80 100 120 140 160 180 200
1 10
5
1 10
4
1 10
3
0.01
0.1
1
10
Correced LHS DP Receptance H11
LHS DP receptance B11
Actual Free-Free DP Receptance
Compare Measured and Corr DP Receptance
Frequency (Hz)
Receptance(in/lb)
Fig. 6. Comparison of corrected driving point receptance based on original and modified measured receptance using Eqs.
5 and 6.