Sistema Continua Y Un Sistema Discreto

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 aCircular 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 ContinuousSystems

• 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 Simplerand Accurate compared to MDOF System • Strings, Bars, Rods & Beams

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Example of a Pen Stock

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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 WaveEquation-Forced Vibrations Free Vibrations of a Beam Vibration of Membranes
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Systems Governed by Wave Equation
Stretched String Potential Function Lagrangian Function Hamilton’s Principle U - StrainEnergy T – Kinetic Energy W – External Work Done

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Systems Governed by Wave Equation
Stretched String

Governing Equation

w,tt = a w, xx
2

a is the Wave Propagation
Velocity

p a =m
2

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Systems Governed by Wave Equation
Axial Vibrations of a Bar Energy Components

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Systems Governed by Wave Equation

Governing Equation

a =
2

E

ρ

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SystemsGoverned by Wave Equation
Torsional 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 Equation
Torsional Vibration of a CircularRod Kinetic Energy External Work Done by Moment

Governing Equation

a2 =

G

ρ

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Continuous Systems
Topics: • Introduction to Continuous Systems • Example of a Penstock • Systems...