This document discusses eigenvalue problems and their formulation for parabolic and hyperbolic partial differential equations. It covers: - Eigenvalue problems and their formulation for structural analysis, heat transfer, fluid mechanics etc. - Formulation of the 1D heat equation as an eigenvalue problem using separation of variables. - Natural vibration analysis of beams using Euler-Bernoulli beam theory and Timoshenko beam theory. - Buckling analysis of beams modeled as eigenvalue problems using Euler-Bernoulli and Timoshenko beam theories. - Examples of calculating natural frequencies and buckling loads of beams using the finite element method.

1. BITS Pilani
Hyderabad Campus
Eigenvalue and Time
Dependent Problems
Prof. Srinivasa Prakash Regalla

2. BITS Pilani, Hyderabad Campus
 Eigenvalue Problems
Formulation of eigenvalue problems for parabolic,
hyperbolic equations
Finite element formulation
Natural vibration of a beam
Euler beam
Timoshenko beam
Stability and buckling of beams
 Time dependent problems
Learning Outcomes

3. BITS Pilani, Hyderabad Campus
It is posing of a BV/IV/BIV problem in a form:
What is an eigenvalue
problem?
)
(
)
( u
B
u
A 

and seeking a solution for eigenvalues  through it for non-
trivial eigenvectors, u.
The  and u have different physical meanings for different
mechanical problems. For example, the natural axial
vibration of a bar are transverse vibration of a cable are
represented by:
and here  is the square of the natural frequency of vibration, .

4. BITS Pilani, Hyderabad Campus
Different meanings of
eigenvalues in different problems
Problem Area Physical meaning of 
Structural
deformation/stress problems
Square of natural
frequency of vibration
Buckling problems Buckling load
Heat transfer Amplitude of the Fourier
components
Fluid mechanics Amplitude of the Fourier
components

5. BITS Pilani, Hyderabad Campus
One example application is the 1-D transient heat transfer:
Formulation of eigenvalue
problems: parabolic equation
Here:
u = temperature
k = thermal conductivity
 = density
c = specific heat
A = cross-sectional area
q = heat generation rate per unit length
(1)

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Proposing the homogeneous solution as product of two
functions:
Substituting into the homogeneous form of diff. equ.:
Separation of variables gives:
Equating the two terms to a single constant gives:
The minus sign signifies
that the Temperature
decreases with time.

7. BITS Pilani, Hyderabad Campus
Writing it as two equations:
(2)
(3)
Considering equation (2) and integrating:
t
Ke
T
K
t
T
dt
T
dT
dt
T
dT 


 










 
 )
ln(
)
ln(
For the complete solution to the governing equation (1), we need
solution to (3) also. Solving equation (3) for  (eigenvalues) and
U(x) (eigenvectors) is called as eigenvalue problem.

8. BITS Pilani, Hyderabad Campus
For constant A, c, k and , the general solution of (2) is:
Substitution of boundary conditions:
0
cos
0
sin
cos
1
0




L
L
L






 cannot be zero hence cosL has to be equal to zero.

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For nontrivial solution, the determinant of the coefficient matrix
zero:


 .
,
2
5
,
2
3
,
2
etc
L




The constant K is absorbed into Cn. The constants Cn can be
determined from the initial condition:
If we multiply by sinmx both sides and integrate over the
domain (0,L):

  









L
m
n
n
L
m
n
n
dx
x
u
dx
x
x
C
0
0
1 0
sin
sin
sin 



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For the non-trivial condition of m=n, we obtain:
Notice the orthogonality condition:


L
m
n
dx
x
x
u
L
C
0
0
sin
)
(
2

That way we complete the homogeneous solution:
The complete solution of equation (1) can then be obtained
as sum of homogeneous solution and particular solution:

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The homogeneous solution and separation of variables gives
Formulation of eigenvalue
problems: hyperbolic equation

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Natural Vibration of a Beam – Euler-
Bernoulli Beam Theory (EBT)
For EBT, the governing equation is:
Two example boundary
conditions:
(1) Both ends fixed
(2) One end fixed and the
other end spring
supported

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w(x, t) = frequency of natural vibrations
W(x) = Mode shapes
Substituting:
Natural Vibration of a Beam – Euler-
Bernoulli Beam Theory (EBT)
For EBT, the governing equation is:
Where =2

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j = Hermite cubic polynomial functions
Week Formulation
 = weight function

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The Hermite Shape Functions

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Hermite Shape Functions
Using Local Coordinates
The interpolation functions i can be expressed in terms of local
coordinate 𝑥 = 𝑥 − 𝑥𝑒 as:

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Elemental level FE Equation

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For constant EI and A the stiffness and mass matrices are:
Elemental Matrices

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EBT
Example: Cantilever Beam
TBT
How many elements should we consider?
Once the type of element is decided by the weak form, the number of elements is decided by
the number of eigenvalues we want to determine. The number of eigenvalues, and hence
number of natural frequencies is equal to the net number of unconstrained degrees of
freedom in the meshed system.

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Example:Cantilever Beam (continued): One 2-nodedlinear1-D
element(with Hermite ShapeFunctions)for EBT:
EBCs
NBCs:
Q3 = Q4 = 0

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Setting the determinant of the above coefficient matrix, we
obtain the characteristic equation and solving it we get
eigenvalues.
Two cases can be examined:
(1) Neglecting the inertia forces
(2) Taking inertia forces into account
The characteristic equation

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If we write 𝜔𝑖 = 𝜔𝑖𝐿2 𝜌𝐴 𝐸𝐼, the exact solution is
𝜔1 = 3.516 and 𝜔2 = 22.035 comparing which with above
values, it is clear that translational vibration frequency is
accurate but rotational vibration frequency is not accurate.
Characteristic equation neglecting
inertia forces – Eigen values

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The eigenvector can be computed from the first of the reduced
equation:
Eigen vectors without inertia
forces
Thus we get one eigenvector for each of two eigenvalues (i):

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Writing 𝜌𝐼 =
𝜌𝐵𝐻3
12
= 𝜌 𝐵 × 𝐻 × 𝐻2 12 = 𝜌𝐴 𝐻2 12 and
take H/L=0.01, the characteristic equation is:
Eigenvalues and eigenvectors
without inertia forces
If we write 𝜔𝑖 = 𝜔𝑖𝐿2 𝜌𝐴 𝐸𝐼, the exact solution is
𝜔1 = 3.5158 and 𝜔2 = 22.0226.

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Determine the smallest natural frequency of a beam with
clamped ends, and of constant cross-sectional area A,
moment of inertia I, and length L. Use the symmetry and
two Euler—Bernoulli beam elements in the half beam.
Exercise-1

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Note that the beam problem is a hyperbolic equation, hence the
eigenvalue is the square of the natural frequency of flexural
vibration, ω. For a mesh of two Euler—Bernoulli elements in a half
beam (i.e., h = L/4), the assembled equations are given by
Solution:

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The determinant of the coefficient matrix yields a cubic
polynomial in ω2. Note that by considering the half beam
we restricted the natural frequencies to those of
symmetric modes. The antisymmetric modes (only) can
be obtained by using U5 = 0 instead of U6 = 0.

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The Governing Equations are:
Timoshenko Beam Theory
Displacement approximation is:

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For linear interpolation functions:

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Determine the smallest natural frequency of a beam with
clamped ends, and of constant cross-sectional area A,
moment of inertia I, and length L. Use the symmetry and
two Timoshenko beam elements in the half beam.
Exercise-2

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Solution:

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Consider a beam (of Young’s modulus E, shear modulus G, area of
cross section A, second moment area about the axis of bending I,
and length L) with its left end (x = 0) clamped and its right end (x =
L) is supported vertically by a linear elastic spring (see Figure P6.5).
Determine the fundamental natural frequency using (a) one Euler-
Bernoulli beam element and (b) one Timoshenko beam (IIE) element
(use the same mass matrix in both elements).
Exercise-3:

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One-element mesh is used. The boundary conditions are:
U1 = U2 = 0 and Q1
3 = −kU3. The eigenvalue problems
are formulated below.
Solution:

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The equation governing the onset of buckling of a column with
an axial compressive load of No is given by:
Stability in Buckling of Beams,
Eigenvalue Problem Modeling using EBT
This represents an eigenvalue
problem with =No where the
smallest value of No is called
as the critical buckling load.

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The finite element model of the above equation is:
The finite element model:
where

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The [G] matrix is called as stability matrix.

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Where W(x) and S(x) represent transverse deflection and
rotation, respectively, in the onset of buckling. The finite
element model with equal interpolation of W and S is:
Stability in Buckling of Beams,
Eigenvalue Problem Modeling using TBT
For the TBT, the equation are:

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Determine the critical buckling load of a cantilever beam (A, I,
L, E) using (a) one Euler—Bernoulli beam element and (b)
one Timoshenko beam element (RIE).
Solution: One element mesh is used. The boundary conditions
are: U1 = U2 = 0.The eigenvalue problems are formulated
below.
Exercise:

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