ME2353 - FINITE ELEMENT ANALYSIS
UNIT – 4
DYNAMIC ANALYSIS USING FINITE ELEMENT METHOD
PART – A (TWO MARKS)
1. Define Vibration1. What are causes of Vibration2?
1- A motion which repeats itself after an interval of time.
2- Unbalanced force, Elastic nature of the system, Self excitation, External
excitation, Wind and earthquake.
2. Define damping ratio1 and magnification factor2.
1- Ratio of actual damping coefficient to the critical damping coefficient.
2- Ratio of maximum displacement of the forced vibration to the static deflection
under the static force.
3. Define Frequency1 and Resonance2.
1- Number of cycles completed in one second. Unit is Hertz (Hz). (1/ tp).
2- The amplitude of vibration becomes excessively large, when the frequency of
external force is equal to the natural frequency of vibrating body.
4. Define Simple Harmonic Motion1 and write down the differential equation of SHM2.
1- A body is said to have SHM, if it moves or vibrates about a mean position
such that its acceleration is directly proportional to its distance from the mean
position and is directed towards the mean or equilibrium position.
2-
𝑑2 𝑥
𝑑𝑡 2
+ 𝜔2 𝑥 = 0
5. Write down the finite element equation of bar element subjected to axial (or)
longitudinal vibration.
6. Write down the finite element equation of beam element subjected to transverse
vibration.
7. State Lagrange’s equation of motion.
It applies to systems whether or not they conserve energy or momentum, and it
provides conditions under which energy and/or momentum are conserved.
It incorporates the constraints directly by choice of generalized coordinates.
Where:
T-Kinetic energy
Π-Potential energy
8. What is meant by consistent mass matrix1 and lumped mass matrix2?
1- If the same shape function used as same for the stiffness matrix. (Mass is
distributed throughout the element)
2- Mass matrix derived by lumping the mass of the entire elements at its nodes. It
results in a diagonal mass matrix which is easier to solve.
9. Write down the HRZ lumped mass matrix for bar1 and beam element2.
10. What are the solutions for Eigen value problems?
Determinant based methods
Transformation based methods
Vector iteration based methods
11. Define determinant method used to solve Eigen value problems.
In this method, trial value of λ is taken and the determinant I [A] – λ [I] I is
computed. With several trial values a plot is generated. By monitoring the sign
changes in value of determinant Eigen value is iterated.
12. Define mode superposition method1 and Direct integration methods2.
1- It is used to transform the original equations of motion into a set of uncoupled
single degree of freedom oscillator equations in model coordinates. Eigen
values are calculated first.
2- The equations are directly integrated in time without the transformation of
coordinates. No need of calculating the Eigen values first.
13. Write down the algorithm of mode superposition method.
Find all the Eigen pairs
Evaluate {U}T {F} = {f}
Transform initial conditions to initial conditions in modal coordinates.
Evaluate {p} and {p}
Obtain actual displacement and velocity at all points of the structure
Obtain strain and stress as a function of time
14. List out the types of direct integration methods used to solve Eigen value problems.
Central difference method
Average acceleration method
Linear acceleration method
Newmark family method
PART – B (16 MARKS)
1. Derive finite element equation for longitudinal vibration of rod based on weak form.
2. Derive finite element equation for transverse vibration of beam based on weak form.
3. a. Derive finite element equation for vibrating elements using Lagrange’s approach.
b. Write a short note on Determinant method, Transformation method and Vector
iteration method.
4. Derive Consistent mass matrix for bar and beam element using Lagrange’s
approach.
5. Consider a uniform cross-section bar as shown in figure of length L made up of a
material whose Young’s modulus and density are given by E and ρ. Estimate the
natural frequencies of axial vibration of bar using both consistent and lumped mass
matrix(using two element discretization).
6. For the one-dimensional bar shown in figure, determine the natural frequencies of
longitudinal vibration using two elements of equal length. Take E = 2 x 10 5 N/mm2, ρ
= 0.8 x 10-4 N/mm3 and L = 400mm.
7. Determine the natural frequency of vibration for a beam fixed in both the ends as
shown in figure. The beam has mass density ρ, modulus of elasticity E, crosssectional area A, moment of inertia I, and length 2L. For simplicity of the long hand
calculations, the beam is discretised into two elements of length L.
8. Consider the simply supported beam as shown in figure. Let the length L=1m, E = 2
x 1011 N/m2, Area of cross section A = 30 cm2, Moment of inertia I=100 mm4, density
ρ=7800 kg/ m3. Determine the natural frequency using the two types of mass matrix.
9. Determine the natural frequencies for the three degrees of freedom system shown in
figure.
10. Find the Eigen values and Eigen vectors of
4 −20
−2
10
6 −30
−10
4
−13
11. Assembled stiffness and mass matrix are given by
0.360 — 0.180
0
0.052 0.013
0
8
[ K ] = 10 — 0.180
[ m ] = 0.013 0.052
0.013
0.360
0.180
0
0.013
0.026
0
−0.180 0.180
Find the Eigen pairs and natural frequencies if this system using the
simultaneous method.
12. Consider the undamped 2dof system as shown in figure. Find the response of the
system when the first mass alone is given an initial displacement of unity and
released form rest.
The mathematical representation of the system for free, harmonic vibration is
given by
2𝑘
−𝑘
−𝑘
2𝑘
𝑥1
2 𝑚
=
ω
𝑥2
0
0
𝑚
𝑥1
𝑥2
13. Explain about Vector iteration methods.
14. Explain about various types of direct integration methods.