Sensitivity of Simple and Non-Simple Eigenvalues to Perturbation
This project will: involve application of composite materials to a vibratory model for a detailed parametric and sensitivity analysis of simple and non-simple eigenvalues of vibrating systems.
Primary Faculty co-Advisors
Su-Seng Pang, Mechanical Engineering Department (Composite Materials)
Yitshak M. Ram, Mechanical Engineering Department (Mechanical Vibration)
Ioan I. Negulescu, Human Ecology Department (Nonwoven Materials)
Off-campus Participant: Guoqiang Li (Southern University/LSU joint faculty)
Technical Proposal:
Bases for Current Research
Introduction
The sensitivity of the natural frequencies of a vibrating system to change in the system parameters is of significant theoretical and practical importance. It plays a major role in the determination of structural modification that is required to achieve prescribed dynamic characteristics.
We have developed a formula for determining the sensitivity of the eigenvalues of a damped vibratory system due to perturbation. We first present the classical result that holds for simple eigenvalues, and then extend the result to the case of multiple eigenvalues. The example demonstrates the use of the newly developed equation in the case of a system with repeated eigenvalues.
Sensitivity of Eigenvalues
Free oscillations of linear vibratory systems are governed by a set of second order differential equations of the form
,
(1)
, 
where the mass matrix
is symmetric positive definite, the damping matrix
and the stiffness matrix
are symmetric positive semi-definite matrices, and where dots
denote derivatives with respect to time
. The system of differential equations (1) leads to the
eigenvalue problem
,
(2)
where the eigenvector
is a constant vector and the eigenvalue
.
In some cases, the system
matrices depend on a parameter 
, i.e., 
, 
and 
. Then multiplying (2) from the left by
gives
.
(3)
Differentiating (3) with respect
to yields
(4)
where primes denote derivatives
with respect to . The above equation has significant applications in
structural modification and in the theory of inverse problems in vibrations (see
e.g., [1]).
Repeated Eigenvalues
If the eigenvalue
is a repeated eigenvalue with multiplicity
, the corresponding eigenvector is not uniquely determined, up
to a scale factor. In fact, there is a subspace
of dimension 
for which each vector in
is an eigenvector of the eigenvalue problem (2), see e.g.,
[2]. In general, for different eigenvectors associated with the repeated
eigenvalue , the equation (4) will produce different values for the
sensitivity. It is therefore concluded, as in [3] for example, that the use of
equation (4) is restricted to the case where the eigenvalue under consideration
is not repeated.
If
is an eigenvalue of (2) with multiplicity
then the sensitivity of the two eigenvalues are determined by
the roots 
of the quadratic equation
,
(5)
where
. This result can be extended to the case where the repeated
eigenvalue is of higher dimension, 
. In that case the sensitivity of the repeated eigenvalues is
determined by the solution of the following polynomial of degree
,
(6)
where
is the binominal coefficient.
Obviously, equation (5) is a
special case, , of the general equation (6). We note that (4) is also a
special case of (6) since for 
(6) gives
(7)
which is the same as (4) in its equivalent form
.
(8)
Example
Let
,
(9)
and
,
(10)
which can be interpreted as the
eigenvalue problem associated with adding a spring of constant
between the first two masses of the three-degree-of-freedom
system shown in Figure 1.
Figure 1
Then
(11)
which has a repeated eigenvalue
of multiplicity when 
. For this case equation (5) gives
(12)
Substituting
and 
in (12) yields
,
(13)
i.e.,
or 
. With 
the modified system has eigenvalues
and 
. Using 
, where 
is the modified eigenvalue, confirms the result.
Conclusion
The sensitivity
of repeated eigenvalues to change in the system parameter can be obtained via
equation (6). The result holds for a general damped mechanical system with
repeated eigenvalues. It has been shown that the classical
result expressed in equation (4), which is not applicable when
, is a special case of our general expression (6). The
results can be applied for composite materials, as long as the formulation (1)
holds.
Acknowledgment
This work was supported in part by a National Science Foundation (NSF) grant CMS-9978786, NSF/Interactive Graduate Education Research Training and the Louisiana Board of Regents, BoRSF, under agreement NASA/LEQSF (2001-2005)-LaSPACE and NASA/ LaSPACE under grant NGT5-40115 for support during this project.
References
- S. Elhay and Y.M. Ram, An affine inverse eigenvalue problem, Inverse Problems, Vol. 18, pp. 455–466, 2002
- L. Meirovitch, Elements of Vibration Analysis, McGraw-Hill Book Company, New York, 1975
- U. Prells and M.I. Friswell, Calculating derivatives of repeated and nonrepeated eigenvalues without explicit use of eigenvectors, AIAA Journal, Vol. 38, pp. 1426-1436, 2000.
Number of IGERT apprentices to be recruited and probable home departments: Two -- one from Mechanical Engineering (Kanika N. Vessel) and one from Human Ecology (Ronda Nichols)
Consistency with the Macromolecular Education, Research & Training theme: The project will require us to understand nonwoven (biodegradable) polymers, modeling of these polymer systems, and their use in the design of various structural systems. Courses that will be valuable are the MS-I and MS-II courses, where we will learn the basis before advancing.
How does the project form a vector cross-product of existing research themes by the participants?
Existing research directions.
Dr. Su-Seng Pang’s research expertise is in the area of Composite Materials, including: advanced composite piping systems, composite materials, stress analysis and joining technology. An example of his funded projects is entitled “Development of High Pressure/Temperature Piping Systems Using Hybrid Fiber Reinforcement,” sponsored by the Louisiana Board of Regents and EDO Fiber Science/
Dr. Yitshak M. Ram’s research expertise is in the area of Vibrations, including: dynamics, control, and mathematical modeling. An example of his funded projects is entitled “Model Construction and Physical Parameter Identification from Spectral Data,” sponsored by the National Science Foundation.
Dr. Ioan I. Negulescu’s expertise is in the area of non-woven materials, with focus on chemistry and chemical engineering analysis.
New research direction.
With joint supervision of Kanika N. Vessel’s dissertation research, Drs. Su-Seng Pang and Yitshak M. Ram are able to extend their existing research into a new direction. This can be demonstrated by the following two grants that were funded after the co-supervision of Ms. Vessel.
(1) “Development of Innovative Frame Structures and Intermodal Tanks Using Advanced Composite Materials,” Louisiana Board of Regents, Global Container Group, and EDO Fiber Science (PI: S.S. Pang; Co-PIs: G. Li, J.E. Helms, Y.M. Ram, and M.M. Khonsari), 06/01/01 - 06/30/04, $242,000.
(2) “Structural Optimization and Stability,” National Science Foundation, (PI: Y.M. Ram; Co-PIs: M.M. Khonsari and S.S. Pang), 10/01/03 - 09/30/06, $249,972.
How do students benefit from the team-oriented research, beyond what would be available to them from either advisor separately? Structural Dynamics is a developing area. Many industries, such as automotive, space, and agriculture test the vibrations in these systems under different conditions to fully understand how sensitive a parameter is to perturbation. This is necessary before marketing these systems and structures for customer use. The student from mechanical engineering will gain a far deeper understanding from dynamic mechanical measurements from Dr. Negulescu. Similarly, the student from the human ecology department will be able to gain knowledge in composite materials, application in vibratory systems, and determination of sensitivity of eigenvalues.
Briefly describe the support level available to each individual faculty or off-campus participant (i.e., without IGERT)
Listed of Currently-Funded Research/Educational Projects from Drs. S.S. Pang and Y.M. Ram
(1) “Development of Innovative Frame Structures and Intermodal Tanks Using Advanced Composite Materials,” Louisiana Board of Regents, Global Container Group, and EDO Fiber Science (PI: S.S. Pang; Co-PIs: G. Li, J.E. Helms, Y.M. Ram, and M.M. Khonsari), 06/01/01 - 06/30/04, $242,000.
(2) “Structural Optimization and Stability,” National Science Foundation, (PI: Y.M. Ram; Co-PIs: M.M. Khonsari and S.S. Pang), 10/01/03 - 09/30/06, $249,972.
(3) “The transcendental inverse eigenvalue problem and its application to aerospace structures,” NASA/Louisiana Space Consortium, (PI: Y.M. Ram; Other Investigator: K.V. Singh), 03/15/03 - 03/14/04, $24,938.
(4) “Enhancement of SU/LSU Joint Engineering Research and Educational Program in Composite Materials -- Third Hire,” National Science Foundation and Louisiana Board of Regents, (PI: S.S. Pang; Co-PIs: H.P. Mohamadian, G.B. Sinclair, and E. Woldesenbet), 01/17/01 - 01/31/04, $465,000.
(5) “Research Apprenticeship, Community Service and Academic Enhancement Training for LSU Engineering and Mathematics Students,” National Science Foundation (CSEMS Program), (PI: S.S. Pang: Co-PIs: C.L. Peters, G.S. Ferreyra, and I.M. Warner), 07/01/00 - 06/30/04, $396,000.
(6) “Louisiana Alliance for Minority Participation (LAMP) -- Phase II,” National Science Foundation/Louisiana Board of Regents, (Co-Campus Coordinators: S.S. Pang, C.L. Peters, and F.K. Cartledge; Additional Investigator: I.M. Warner), 10/01/00 - 09/30/05, $668,056 (LSU Portion; Subcontract from Southern University, PIs: R.L. Ford, D. Bagayoko, and K. Davidson).
(7) “Graduate Alliance for Education in Louisiana,” National Science Foundation (AGEP Program), (PI: L.A. Lefton; Co-PIs: H.L. Bart, C. Mackie, I.M. Warner, and S.S. Pang; Additional Investigator: S.F. Watkins), 09/01/02 - 08/31/07, $475,245 (LSU Portion; Subcontract from Tulane University).
(8) “Louisiana State University Science, Technology, Engineering, and Mathematics Scholars Program (LA-STEM),” National Science Foundation, (PI: C. Stelly; Co-PIs: I.M. Warner, S.S. Pang, S.Y. McGuire, G. Vincent), 09/01/03 – 08/31/08, $4,000,000.
(9) “Infrastructure Support for the Louisiana State University Science, Technology, Engineering, and Mathematics Scholars Program (LA-STEM),” Research Corporation, (PI: C. Stelly; Co-PIs: I.M. Warner, S.S. Pang, G. Stanley), 09/01/03 – 08/31/09, $474,921.
(10) “Effect of Corrosion Prevention Compounds on Fatigue Crack Propagation in Aluminum Alloy for Aerospace Application,” NASA/Louisiana Space Consortium, (PI: M.A. Wahab; Other Investigator: S.S. Pang), 03/15/03 - 03/14/04, $24,539.
(11) “Development of High Performance Adhesive-Bonded Composite Joints for Cryotanks,” NASA/Louisiana Space Consortium, (PI: S.S. Pang; Co-PI: G. Li), 06/01/03 - 05/31/04, $35,000.
Commitment of faculty & off-campus participants to work side-by-side with apprentices:
Drs. Su-Seng Pang and Yitshak M. Ram, professors of Mechanical Engineering at LSU, also maintain their active research and educational programs. They interact with Ms. Kanika N. Vessel on a weekly basis. Both Pang and Ram are very excited about the possibility of personally working side-by-side with the same student in the “master-apprentice” model presented in this IGERT proposal. This time commitment has been fulfilled throughout the year. In average, all three of them meet two hours a week. Dr. Ioan I. Negulescu, a professor within the Human Ecology Department brings Ms. Vessel to his laboratory and work with his students in Human Ecology. Dr. Guoqiang Li is a joint faculty member in Mechanical Engineering between Southern University and LSU. Li also shares some of the expertise with Pang and Ram. With Li’s participation, Vessel is able to use some of the facilities located at Southern University. While Pang and Ram are co-advisors of Vessel’s Ph.D. dissertation, both Negulescu and Li are members of Vessel’s dissertation committee.