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Module 7

The document outlines a course on Computer Aided Design and Finite Element Analysis, focusing on dynamic analysis using the finite element method. It covers topics such as modal analysis, harmonic analysis, and transient analysis, along with governing equations for axial vibration and natural frequency calculations. Numerical examples are provided to estimate natural frequencies of axial vibrations for different bar configurations.

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renoldelsen
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21 views11 pages

Module 7

The document outlines a course on Computer Aided Design and Finite Element Analysis, focusing on dynamic analysis using the finite element method. It covers topics such as modal analysis, harmonic analysis, and transient analysis, along with governing equations for axial vibration and natural frequency calculations. Numerical examples are provided to estimate natural frequencies of axial vibrations for different bar configurations.

Uploaded by

renoldelsen
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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SCHOOL OF MECHANICAL ENGINEERING

BMEE306L
Computer Aided Design and Finite Element Analysis
Class ID: VL2023240504236
Slot: G2+TG2
Venue: MB212

S RENOLD ELSEN
ASSOCIATE PROFESSOR Sr
DDA/SMEC
Mobile: 9994304360
Mail ID: renoldelsen.s@vit.ac.in
TOPIC

Lecture
Module
Hours

Dynamic analysis using finite element method - Eigen


value and Eigen vectors - 1D Bar and Beam vibration
7 4
problems – Problem solving
Introduction

Modal analysis - Natural frequency and mode shapes

• Harmonic analysis - Forced response of system to a sinusoidal forcing


• Transient analysis - Forced response for non-harmonic loads (impact,
step or ramp forcing etc.)
DYNAMIC CONSIDERATIONS

 Static analysis holds when the loads are slowly applied.

 When the loads are suddenly applied, or when the loads are of a variable
nature, the mass and acceleration effects come into the picture.
Vibration

 If a solid body, such as an engineering structure, is deformed elastically


and suddenly released, it tends to vibrate about its equilibrium position.
 This periodic motion due to the restoring strain energy is called free
vibration.
 The number of cycles per unit time is called frequency.
 The maximum displacement from the equilibrium position is the
amplitude.
Governing Equation of Axial Vibration

[M]{u¨(t)}+[K]{u(t)}={0}

Where:
[M] = Global mass matrix
[K] = Global stiffness matrix
{u(t)} = Nodal displacement vector as a function of time
{ ¨( )} = Nodal acceleration vector as a function of time
Free vibration

([K]−ω2[M]){u}={0}

Where:
ω = Natural angular frequency (ω=2πf)
f = Natural frequency in Hz
Axial Vibration of Rod
Consistent Mass

Lumped Mass
Natural Frequency Calculation

Solve the eigenvalue problem


Determinant([K]−ω2[M])=0

This provides the eigenvalues (ω2) from which natural frequencies (f) are
obtained:

f=ω/2π
Numerical 1
A uniform cross section bar of length 1 m, E = 2 E11 N/m2 , ρ= 7800 kg/m3,
A=30E-6 m2 . Estimate the natural frequency of axial vibration of the bar
using consistent and lumped mass matrix.
Numerical 2
A stepped bar of E = 30E3 N/m2 , ρ= 8500 kg/m3. Estimate the natural
frequency of axial vibration of the bar.

A=2m2 A=1m2

8m 4m

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