This project aims to analyze the dynamic stability of beams subjected to periodic loads. The student will obtain governing equations in the form of a generalized energy equation and introduce modal damping to study its effects. So far, the critical loads and frequencies have been found for different beam support conditions. The total potential energy equation will be used to determine the non-dimensional frequency and instability regions will be obtained by plotting this against the dynamic load factor. The expected outcomes are to characterize the buckling and vibration behavior of beams and show damping can stabilize dynamics and reduce instability regions.

1. PROJECT REVIEW PHASE -I
on
Department ofMechanical
Engineering.
Course: Engineering Design.
Presented By:
M. Harsha.
17881D9503..
DYNAMIC STABILITY OF BEAMS WITH DAMPING.
Guide : Dr. B. SUBBARATNAM.
Professor & HOD.
Dept of Mechanical Engineering.
Vardhaman College of Engineering.
Vardhaman College of Engineering
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2. CONTENTS
 Abstract.
 Literature Review.
 Methodology.
 Introduction to Stability.
 Dynamic Stability.
 Details of Project Till Now.
 Expected Outcome.
 References.
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3. ABSTRACT
This Project is to attempt the clear understanding of dynamic stability of beams
subjected to periodic loads . Prediction of dynamic stability behaviour of structural
members is necessary for assessing the intensity of a structure.
This Study is necessary for the case of Rockets, missile and civil structures as a
whole or its constituent structural members which are very light in weight. As an
ideal case only static analysis cannot determine the failure of the structure, so by
analyzing dynamic stability we can have better prediction on a structure.
The dynamic stability of a structures must be analyzed at least for a constant part
of the load and periodic load with highest period as this is the most critical part for
all practical purposes.
The governing equations are to be obtained in the form of generalized energy
equation, which is same for all structural members.
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4.  For the case of Combination resonance of beams subjected to periodic and
concentrated loading case modal damping is introduced to study the effects of
damping, which is for secondary effects.
 Then the combination resonance of a beam without damping and with
damping is compared. By introducing Damping the dynamic instability region is
reduced.
 The dynamic stability of beams for isoperimetric element is well established.
Then the obtained values are compared with computational results.
Then Analytical and computational values of combination resonance are to
be justified and graph is drawn between frequency and dynamic load factors,
which shows the reduction in dynamic stability region. Thus the intensity of a
beam can be identified.
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5. LITERATURE REVIEW
An article By N.M Bailev published in 1924 is the first work to dynamic stability.
In this the dynamic stability of beam was examined and the boundaries of
principal regions of instability was determined.
Later Krylov and Bogoliubov has examined the problem and examined the case
of support conditions.
The above works have the common characteristics in that problem of dynamic
stability is reduced to approximately second order D.E with periodic coefficients
They have found out for primary functions.
Now by using both Primary functions and secondary functions we can exactly
find out the dynamic stability boundaries.
 International Journal on Master Dynamic stability on structural members where
primary functions are found. American Institute of Aeronautics and Astronautics
Journal(AIAA), Vol.46, No.2, 2008, pp. 537-540.
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6. Journal on Aerospace Structures on Master Formula for Geometrically Nonlinear
Dynamic Instability of Shear Flexible Beams. In this Rayleigh Ritz method for
continuum approximation method.
“Dynamic Stability of beams subjected to end periodic and static tensile axial
load”, Journal of Structural Engineering, Vol.39, No.2, June-July 2012, pp.263-268.
In this for primary functions for Periodic loads and static load are founded.
Journal on Institution of Engineers on “Development of Three Simple Master
Dynamic Stability Formulas for Structural Members Subjected to Periodic Load”,
Journal of The Institution of Engineers (India), Vol .92, May 2011, pp. 9-14. In this
various relations are found out.
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7. Stability Instability
Measure of a Performance of the system. Unpredictable Behaviour of the system or
the system undergoing Erratic changes.
Small Input leads to an output that must
not diverge.
Small Input Produces Large Output and
Diverges.
Introduction:
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8. Defining a system w.r.t. Linear Time Invariant System.
To be a Stable System in LTI, System must satisfy Two conditions.
 BIBO Stability concept- Bounded Input Bounded Output.
 If there is No Input Output must be Zero irrespective of Initial
Conditions.
Defining a system w.r.t. Damping
 A System is said to be stable if Damping Ratio > 0.
A system is unstable if Damping Ratio < 0.
Defining a system w.r.t Response.
 Response should decrease to zero as time approaches Infinity.
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9. Beam is subjected to a periodic load in x-direction and compressive load in
y- direction. Static load factors and dynamic load factors are determined.
Periodic load and For a beam, using Critical loading conditions we are going
to find out for all 4 beams.
Simple supported beam.
cantilever beam.
Fixed Fixed.
Pinned Fixed.
Then we are going to find out the values of frequencies for all cases.
By using end conditions, the Total Potential Energy of a Beam is to be
known By U-W-T. By getting the total potential energy, we get angular
frequency and we plot graph for the angular frequency and dynamic stability
factors.
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10. STATIC & DYNAMIC STABILITY
Static Stability:
The Initial tendency of a system when disturbed, to return its original
position.
Dynamic Stability:
The Overall tendency of a system when disturbed, to return its original
position.
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11. Details of Project till now
Using critical load Pcr = ∏2EI
L2
For simple supported Beam.
Pcr = ∏2EI
(2L2)2
For Cantilever Beam
Pcr = ∏2EI
(L/√2)2
For Fixed Fixed.
Pcr = ∏2EI
(L/2)2
Pinned Fixed.
Here, Le
2= L for Simply Supported
2L for Cantilever
L/√2 for Fixed Fixed
L/2. for Pinned Fixed.
.
Frequencies have been found using Pcr.
By using the equation total Potential energy
∏ = U-W-T, ∏ is Determined. By substituting the
known values we get ᶱ/ ᴪ value. By this we get 2
equations. Then graph is drawn between µ and
ᶱ/ᴪ dynamic instability region is obtained.
Pcr = ∏2EI
Le
2
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12. EXPECTED OUTCOME
The Buckling and Vibrational behaviour of a beam are characterized and the beam
is less susceptible to buckling for the loading near the ends of the edges.
The widths of the combination resonance zones are smaller for the loading near
the ends of the edges as compared to those when the loading is extending over the
edge.
The effect of damping on the dynamic instability characteristics is that there is a
finite critical value of the dynamic load factor below which the beam stable. For
nearly uniform loading the critical dynamic load factors for combination resonance
zones become so high.
 As the effect of dynamic stability is reduced when damping systems are
introduced then system does not show any inherent properties.
The effects of damping on the simple resonance characteristics are to be
stabilizing. Damping may show stabilizing effect on the combination resonance
characteristics.
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13. REFERENCES:
 Timoshenko, S.P and Gere, J.M Theory of Elastic Stability 2nd Edition. Mc Graw
Hill, New York, 1961.
 Engel, R.S Dynamic Stability of an axially loaded Beam on elastic foundation with
damping. J. Sound Vibration, 1991, 146, 463-478
 Nayfeh A.H Perturbation Method, Wiley, New York, 1973.
Ostiguy, G.L Samson, L.P and Nguyen, H On the occurrence of simultaneous
resonances in a parametrically excited rectangular plate, Trans ASME 1993, 115,
344-352.
Hutt, J.M and Salam, A. E Dynamic Stability of plates by finite Element Method, J.
Engng Mech Div, ASCE 1971,97,879-899.
 Simple and Combination resonances of rectangular plates subjected to non-
uniform edge loading with damping. P.J Deolasi and P.K Datta.
 G.V Rao, B.S. Ratnam, Jagadish Babu Gunda and G. R. Janardhana, 2011 Master
Formula for Evaluating Vibration Frequencies of structural Members under
Compressive Loads, The IES Journal Part A: Civil & Structural Engineering, Vol.4.
V.V Bolotin 1964 Dynamic Stability of Elastic Systems, Holden Day, San Franciso.
 Periodic In plane Load, Thin walled Structures, Vol.44, No.9 937-942.
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