Numerical and Continuum Analysis2_Continuous

7/29/2019 Numerical and Continuum Analysis2_Continuous

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Continuous Systems

Topics: Introduction to Continuous Systems

Example of a Penstock

Systems Governed by Wave Equation Stretched String Axial Vibrations of a Bar

Torsional Vibration of a Circular Rod

Solution of Wave Equation-Free Vibrations Solution of Wave Equation-Forced Vibrations

Free Vibrations of a Beam

Vibration of Membranes


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Introduction Continuous Systems

Any Mechanical System is Continuous in Mass and Stiffness Properties

Some Systems eg Turbine Blade Better as Distributed then Lumped

Partial Differential Equations

Solutions are Simpler and Accurate compared to MDOF System

Strings, Bars, Rods & Beams


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Example of a Pen Stock


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Continuous Systems









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Systems Governed by Wave EquationStretched String

Potential Function

Lagrangian Function

Hamiltons Principle

U - Strain EnergyT Kinetic Energy

W External Work Done


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Systems Governed by Wave EquationStretched String

2

, ,tt xxw a w=

2 pa

m=

Governing Equation

a is the Wave PropagationVelocity


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Systems Governed by Wave EquationAxial Vibrations of a Bar

Energy Components


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Systems Governed by Wave Equation

Governing Equation

2 Ea

=


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Systems Governed by Wave EquationTorsional Vibration of a Circular Rod

Strain Energy

Ip is Second Polar

Moment of Area of C/S


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Systems Governed by Wave EquationTorsional Vibration of a Circular Rod

Kinetic Energy External WorkDone by Moment

2 Ga

=

Governing Equation


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Continuous Systems









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Solution of Wave Equation-Free VibrationsGoverning Equation

Assumed Solution


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Solution of Wave Equation-Free Vibrations

For String Fixed at x = 0 & x = L

Natural Frequencies


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Wave Shape

General Solution Variable Separable Form


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ith Mode

jth Mode


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Continuous Systems

Topics:

Introduction to Continuous Systems








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Solution of Wave Equation-Forced VibrationsNon homogeneous Differential Equation


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Continuous Systems

Topics:



Systems Governed by Wave Equation Stretched String

Axial Vibrations of a Bar






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Free Vibrations of Beam

Governing Differential Equation


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Free Vibrations of Beam

General Solution


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Continuous Systems

Topics:



Systems Governed by Wave Equation Stretched String

Axial Vibrations of a Bar






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A Membrane is a Plate Subjected to Tension and has Negligible Bending Resistance


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Assignment

Show that the governing equation of motion of a shear beam is alsoWave equation

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Consider a bar with one end fixed and with a concentrated mass M atThe Free end. Determine the frequency for axial vibrations of thesystem and establish the Orthogonality conditionsFind the Influence of the end mass on the natural frequencies and

plot the result using nondimensional parameters

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Assignment

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Documents

Numerical and Continuum Analysis2_Continuous