My project is on dynamic stability of beams with damping. We are going to find out the dynamic instability region for a beam with and without damping cases. So that we can able to find whether a beam can undergo any response within its boundary limits and we are going to find upto what extent it can be minimised by introducing damping.

1. PROJECT REVIEW PHASE -II
on
Department ofMechanical
Engineering.
Course: Engineering Design.
Presented By:
M. Harsha.
17881D9503..
DYNAMIC STABILITY OF BEAMS WITH DAMPING
SUBJECTED TO PERIODIC LOADS.
Guide : Dr. B. SUBBARATNAM.
Professor & HOD.
Dept of Mechanical Engineering.
Vardhaman College of Engineering.
Vardhaman College of Engineering
1

2. CONTENTS
 Abstract
Dynamic Stability
Literature Review
 Problem Formulation
 Work to be Done
 Expected Outcome
 References
2

3. ABSTRACT
 This Present work is aimed to find out the dynamic instability regions of
different beam structures. And to have 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.
 The dynamic stability of a structures must be analyzed for periodic loads with
highest period as this is the most critical part for all practical purposes.
 The equations of motion is developed and solved by Energy Method.
3

4.  Dynamic stability of Euler beam having fixed-free, pinned-pinned, fixed-fixed
and fixed-pinned boundary conditions are applied to the governing equation.
Modal damping is introduced in the system to study the effects of damping.
 The results obtained by including damping effects are compared with those
of existing values. The region of dynamic instability with and without damping
are compared.
It has been observed that by incorporating damping, the instability regions
are decreasing.
The present results also converge well with those of experimental study.
4

5. Dynamic Stability of Beams.
5
 If a structure is subjected to transverse loads , and if the amplitude of the load is
less than the static buckling value then the structure experiences longitudinal
vibrations.
 If static loading causes a loss of static stability, then vibrational loading will also
cause loss of dynamic stability.
 Loading on structural members to analyse that types of structure we go for
dynamic stability.
The equation of motion of a beam is derived according to Euler Beam.
 Governing equation of motion is derived using Energy method.
 By applying Boundary conditions we get formulation for dynamic stability.

6. 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 that problem of dynamic stability is
reduced to approximately second order D.E with periodic coefficients.
B.S Ratnam, G.V Rao 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.
6

7. G.V. Rao, B. P Shastry Journal on sound & vibrations Large amplitude
vibrations of beams under elastically restrained ends. In this displacements at
both ends are suppressed and by using Finite element formulation is
developed.
 G.V.Rao, B.S Ratnam Vibration, stability & frequency axial load relation of
short beams. In this effects of shear deformation & rotary inertia is
considerable when beams are short.
G.V.Rao, B.P Shastry Free vibration of short beams. In this Euler beam theory
is no longer valid as the effect of shear deformation.
7

8. 8
PROBLEM FORMULATION
1. Pinned Ends 2. Fixed Ends 3. Fixed Free 4. Fixed Pinned
End Constraints:
Determining the Critical Load (P
cr
) =
∏2
EI
Le
2

9. 9
By substituting Le = effective length of the element.
Le
= L for pinned ends
Le
= L/2 for Fixed ends
Le
= 2L2
for fixed free ends
Le
= √2L for pinned fixed ends
Pcr
=
∏2
EI
L2
Pcr
=
4∏2
EI
L2
Pcr
=
∏2
EI
4L2
Pcr
=
2∏2
EI
L2
For pinned ends
For fixed ends
Fixed free ends
Pinned fixed ends
λcr
=
Pcr
L2
E I
λcr
= ∏2
λcr
= 4∏2
λcr
=
∏2
4
λcr
= 2∏2
Boundary Conditions :
Pinned ends ; w= b sin ∏x
L
Fixed ends ; w = b ( 1- cos ∏x
2L )
Fixed fixed ; w = b ( 1- cos 2∏x
L
)
Pinned Fixed ; w = b ( cos 3∏x- cos 2∏x
2L 2L
)

10. 10
Euler Beams Governing equation:
∏= U-W-T Where, U = strain energy
W= Kinetic energy
T= Total energy
∏ =
EI
2 ∫
0
L
( d2
w
dx2 )dx –
P (t)
2
∫
0
L
(dw
dx )
2
dx - m/2 Θ
2
4
∫
0
L
W2
dx.
L
Θ
ω
Similarly for all the cases on solving ;
we get
= 2√(1-α) (1+µ)-
d∏
db
= 0 P(t) = Ps
+Pt
cos Θ t
α+β
2
-( ) Pcr=
µ=
β
2(1-α)
 On solving this equation by substituting
w = b sin ∏x in the below equation-----1
-----------1
Θ
ω
= 2√(1-α)(1+µ)
-
Assuming values of α, µ From 0 to 1.0. values represented in Table 1
For dynamic stability problem:;
Assuming α values ;

11. 11
µ
α= 0.0 α=0.5 α=0.8
Ω1 Ω2 Ω1 Ω2 Ω1 Ω2
0 2.00 2.00 1.40 1.40 0.91 0.91
0.1 1.89 2.09 1.35 1.48 0.85 0.95
0.2 1.79 2.18 1.27 1.54 0.79 0.98
0.3 1.67 2.28 1.19 1.61 0.74 1.04
0.4 1.54 2.37 1.12 1.67 0.69 1.08
0.5 1.41 2.47 1.02 1.76 0.65 1.11
Table 1 : Variation of Ω1 & Ω2 for beams subjected to periodic loads.

12. 12
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1 1.2
β
Θ/W
stability region
Unstable region
Unstable region
--ُ ---α=0.5
stability region
Fig 1 : Ratio of applied frequency to the natural frequency
--ُ --α=0.0

13. 13
Instability region
Θ/w
β
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.1 0.2 0.3 0.4 0.5 0.6
ἁ=0.6 Ω1
ἁ=0.6 Ω2
ἁ=0.7 Ω1
ἁ=0.7 Ω2
Unstable Region
Stability Region
Fig 2: Ratio of applied frequency to the natural frequency

14. 14
 We has founded out Pcr to understand the behaviour of a structure and λcr for all 4 cases.
From the governing equation ∏ = U-W-T by using end constraints for 4 cases we get
formulation infor only uniaxial loads ,for biaxial case we use λb . Generally this is done for plates, shells.
 By simplifying Uniaxial Load equation we obtain dynamic stability equation in non
dimensional form as Θ/w.
This Equation can be treated as Dynamic stability solution for the mentioned loading
condition.
By using Θ/w equation the dynamic stability of a beam with periodic loading .
Then the dynamic instability boundaries Ω1 & Ω2 between which it is dynamically
unstable are given varying β and for ἁ =0.0,0.5,0.8 presented in Table 1. In table 1,
Variation of Ω1 & Ω2 for the value β varying from 0 to 1.0 for a beam with periodic
loads.
The dynamic stability regions of a beam are plotted in the graph.
Details of Project till now.

15. 15
Work to be Done
 Damping is introduced in the Governing Equation
[M]{q}
..
+ [C]{q}. +[K]{q}
 For given values of α, β, Ω, the solution to equation is either bounded or
unbounded. The spectrum of values of these parameters for which the solution is
unbounded gives the region of dynamic instability.
 The boundaries of Simple and combination parametric resonance zones are
obtained.
 It is assumed that the first few normal modes of damped vibration of the beam
are sufficient to define the response for loading.
 Modal Damping matrix is defined. It is reasonable to assume [C] to be a diagonal
matrix which gives desired damping ratio in each mode of vibration.
 By studying the damping effects, the graph is plotted to get the dynamic
instability region.
.

16. EXPECTED OUTCOME
The Buckling and Vibrational behaviour of a beam are determined and the
beam is less susceptible to buckling.
The widths of the instability 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 is that there is a finite critical
value of the dynamic load factor below which the beam stable.
 As the effect of dynamic stability is reduced when damping systems are
introduced then system does not show any irregular properties.
The effects of damping on the resonance characteristics are to be stabilizing.
Damping may show stabilizing effect on the resonance characteristics.
16

17. REFERENCES:
 Timoshenko, S.P and Gere, J.M Theory of Elastic Stability 2nd Edition. Mc Graw
Hill, New York, 1961.
V.V Bolotin 1964 Dynamic Stability of Elastic Systems, Holden Day, San Francisco.
 Mechanical Vibrations by S.S Rao
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.
17

18. 18
THANK YOU