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03 Vibration of string

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M
Maged Mostafa

This document outlines the key concepts and objectives for understanding vibration of continuous structures like strings and cables. It derives the wave equation for a string under tension as a second order PDE. It then shows how to solve for natural frequencies and mode shapes by separating variables. Examples are given of the first mode shape and calculating the natural frequency of a piano wire. The assignment asks students to solve cable problems with different boundary conditions.

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Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Vibration of Continuous Structures
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Course Contents  SDOF  M-DOF • Cables/String • Bars • Shafts • Vibration Attenuation • Beams • FEM for Vibration • Plates • Aeroelasticity
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Derive the equation of motion for Cable/ String • Estimate the Natural Frequencies • Understand the concept of mode shapes • Apply BC’s and IC’s to obtain structure response
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com String and Cables
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Derive the equation of motion for Cable/String • Estimate the Natural Frequencies • Understand the concept of mode shapes • Apply BC’s and IC’s to obtain structure response
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Strings and Cables • This type of structures does not bare any bending or compression loads • It resists deformations only by inducing tension stress • Examples are the strings of musical instruments, cables of bridges, and elevator suspension cables
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com The string/cable equation • Start by considering a uniform string stretched between two fixed boundaries • Assume constant, axial tension t in string • Let a distributed force f(x,t) act along the string f(x,t) t x y
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Examine a small element of string xtxf t txw xFy   ),(sinsin ),( 2211 2 2 tt    • Where  is the mass per unit length of the cable • Force balance on an infinitesimal element • Now linearize the sine with the small angle approximate sin(x) = tan(x) = slope of the string 1 2 t2 t1 x1 x2 = x1 +x w(x,t) f (x,t)
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com                   )( :about/ofseriesTaylortheRecall 2 1 112 xO x w x x x w x w xxw xxx   t     t   t t x t txw xtxf x txw x txw xx             2 2 ),( ),( ),(),( 12      t   t 2 2 ),( ),( ),( t txw txf x txw x      t         x t txw xtxfx x txw x x       2 2 ),( ),( ),( 1      t  
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com 0,0),(),0( 0at)()0,(),()0,( , ),(),( 00 2 2 22 2 ��   ttwtw txwxwxwxw c x txw tc txw t    t     Since t is constant, and for no external force the equation of motion becomes: Second order in time and second order in space, therefore 4 constants of integration. Two from initial conditions: And two from boundary conditions: , wave speed
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Physical quantities • Deflection is w(x,t) in the y-direction • The slope of the string is wx(x,t) • The restoring force is twxx(x,t) • The velocity is wt(x,t) • The acceleration is wtt(x,t) at any point x along the string at time t Note that the above applies to cables as well as strings Subscript denotes differentiation w.r.t. to that parameter
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Derive the equation of motion for Cable/String • Estimate the Natural Frequencies • Understand the concept of mode shapes • Apply BC’s and IC’s to obtain structure response
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Modes and Natural Frequencies     2 2 2 2 2 2 2 2 2 )( )( )( )( 0 )( )( , )( )( )( )( =and=where)()()()( )()(),(                   tTc tT xX xX xX xX dx d tTc tT xX xX dt d dx d tTxXtTxXc tTxXtxw    Solve by the method of separation of variables: Substitute into the equation of motion to get: Results in two second order equations coupled only by a constant:
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Solving the spatial equation:             n aX aX XX tTXtTX aaxaxaxX xXxX n           equationsticcharacteri 1 2 2121 2 0sin 0sin)( 0)0( ,0)(,0)0( 0)()(,0)()0( nintegratioofconstantsareand,cossin)( 0)()( Since T(t) is not zero an infinite number of values of  A second order equation with solution of the form: Next apply the boundary conditions:
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com The temporal solution         1 22 )sin()cos()sin()sin(),( )sin()cos()sin()sin( sincossinsin),( )conditionsinitialfrom(getnintegratioofconstantsare, cossin)( 3,2,1,0)()( n nn nn nnnnnnn nn nnnnn nnn x n ct n dx n ct n ctxw x n ct n dx n ct n c xctdxctctxw BA ctBctAtT ntTctT         Again a second order ode with solution of the form: Substitution back into the separated form X(x)T(t) yields: The total solution becomes:
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Using orthogonality to evaluate the remaining constants from the initial conditions                       010 0 1 0 2 0 )sin()sin()sin()( )0cos()sin()()0,( :conditionsinitialtheFrom 2,0 , )sin()sin( dxx m x n ddxx m xw x n dxwxw mn mn dxx m x n n n n n nm    
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com             3,2,1,)sin()( 2 )0cos()sin(c)( 3,2,1,)sin()( 2 3,2,1,)sin()( 2 0 0 1 0 0 0 0 0            ndxx n xw cn c x n cxw ndxx n xwd nm mdxx m xwd n n nn n m      
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Derive the equation of motion for Cables/Strings • Estimate the Natural Frequencies • Understand the concept of mode shapes • Apply BC’s and IC’s to obtain structure response
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com A mode shape             t c xtxw d ndxx n xd ncxw nxxw Assume n n          cos)sin(),( 1 3,2,0)sin()sin( 2 ,0,0)( 1)=(ioneigenfunctfirsttheiswhich,sin)( 1 0 0 0 Causes vibration in the first mode shape
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Plots of mode shapes 0 0.5 1 1.5 2 1 0.5 0.5 1 X ,1 x X ,2 x X ,3 x x sin n 2 x     nodes
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com String mode shapes Video 1 String mode shapes Video 2
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Example :Piano wire: L=1.4 m, t=11.1x104 N, m=110 g. Compute the first natural frequency.   110 g per 1.4 m = 0.0786 kg/m 1  c l   1.4 t    1.4 11.1104 N 0.0786 kg/m  2666.69 rad/s or 424 Hz
Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Assignment 1. Solve the cable problem with one side fixed and the other supported by a flexible support with stiffness k N/m 2. Solve the cable problem for a cable that is hanging from one end and the tension is changing due to the weight  N/m

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03 Vibration of string

  • 1. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Vibration of Continuous Structures
  • 2. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Course Contents  SDOF  M-DOF • Cables/String • Bars • Shafts • Vibration Attenuation • Beams • FEM for Vibration • Plates • Aeroelasticity
  • 3. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Derive the equation of motion for Cable/ String • Estimate the Natural Frequencies • Understand the concept of mode shapes • Apply BC’s and IC’s to obtain structure response
  • 4. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com String and Cables
  • 5. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Derive the equation of motion for Cable/String • Estimate the Natural Frequencies • Understand the concept of mode shapes • Apply BC’s and IC’s to obtain structure response
  • 6. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Strings and Cables • This type of structures does not bare any bending or compression loads • It resists deformations only by inducing tension stress • Examples are the strings of musical instruments, cables of bridges, and elevator suspension cables
  • 7. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com The string/cable equation • Start by considering a uniform string stretched between two fixed boundaries • Assume constant, axial tension t in string • Let a distributed force f(x,t) act along the string f(x,t) t x y
  • 8. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Examine a small element of string xtxf t txw xFy   ),(sinsin ),( 2211 2 2 tt    • Where  is the mass per unit length of the cable • Force balance on an infinitesimal element • Now linearize the sine with the small angle approximate sin(x) = tan(x) = slope of the string 1 2 t2 t1 x1 x2 = x1 +x w(x,t) f (x,t)
  • 9. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com                   )( :about/ofseriesTaylortheRecall 2 1 112 xO x w x x x w x w xxw xxx   t     t   t t x t txw xtxf x txw x txw xx             2 2 ),( ),( ),(),( 12      t   t 2 2 ),( ),( ),( t txw txf x txw x      t         x t txw xtxfx x txw x x       2 2 ),( ),( ),( 1      t  
  • 10. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com 0,0),(),0( 0at)()0,(),()0,( , ),(),( 00 2 2 22 2    ttwtw txwxwxwxw c x txw tc txw t    t     Since t is constant, and for no external force the equation of motion becomes: Second order in time and second order in space, therefore 4 constants of integration. Two from initial conditions: And two from boundary conditions: , wave speed
  • 11. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Physical quantities • Deflection is w(x,t) in the y-direction • The slope of the string is wx(x,t) • The restoring force is twxx(x,t) • The velocity is wt(x,t) • The acceleration is wtt(x,t) at any point x along the string at time t Note that the above applies to cables as well as strings Subscript denotes differentiation w.r.t. to that parameter
  • 12. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Derive the equation of motion for Cable/String • Estimate the Natural Frequencies • Understand the concept of mode shapes • Apply BC’s and IC’s to obtain structure response
  • 13. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Modes and Natural Frequencies     2 2 2 2 2 2 2 2 2 )( )( )( )( 0 )( )( , )( )( )( )( =and=where)()()()( )()(),(                   tTc tT xX xX xX xX dx d tTc tT xX xX dt d dx d tTxXtTxXc tTxXtxw    Solve by the method of separation of variables: Substitute into the equation of motion to get: Results in two second order equations coupled only by a constant:
  • 14. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Solving the spatial equation:             n aX aX XX tTXtTX aaxaxaxX xXxX n           equationsticcharacteri 1 2 2121 2 0sin 0sin)( 0)0( ,0)(,0)0( 0)()(,0)()0( nintegratioofconstantsareand,cossin)( 0)()( Since T(t) is not zero an infinite number of values of  A second order equation with solution of the form: Next apply the boundary conditions:
  • 15. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com The temporal solution         1 22 )sin()cos()sin()sin(),( )sin()cos()sin()sin( sincossinsin),( )conditionsinitialfrom(getnintegratioofconstantsare, cossin)( 3,2,1,0)()( n nn nn nnnnnnn nn nnnnn nnn x n ct n dx n ct n ctxw x n ct n dx n ct n c xctdxctctxw BA ctBctAtT ntTctT         Again a second order ode with solution of the form: Substitution back into the separated form X(x)T(t) yields: The total solution becomes:
  • 16. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Using orthogonality to evaluate the remaining constants from the initial conditions                       010 0 1 0 2 0 )sin()sin()sin()( )0cos()sin()()0,( :conditionsinitialtheFrom 2,0 , )sin()sin( dxx m x n ddxx m xw x n dxwxw mn mn dxx m x n n n n n nm    
  • 17. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com             3,2,1,)sin()( 2 )0cos()sin(c)( 3,2,1,)sin()( 2 3,2,1,)sin()( 2 0 0 1 0 0 0 0 0            ndxx n xw cn c x n cxw ndxx n xwd nm mdxx m xwd n n nn n m      
  • 18. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Derive the equation of motion for Cables/Strings • Estimate the Natural Frequencies • Understand the concept of mode shapes • Apply BC’s and IC’s to obtain structure response
  • 19. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com A mode shape             t c xtxw d ndxx n xd ncxw nxxw Assume n n          cos)sin(),( 1 3,2,0)sin()sin( 2 ,0,0)( 1)=(ioneigenfunctfirsttheiswhich,sin)( 1 0 0 0 Causes vibration in the first mode shape
  • 20. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Plots of mode shapes 0 0.5 1 1.5 2 1 0.5 0.5 1 X ,1 x X ,2 x X ,3 x x sin n 2 x     nodes
  • 21. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com String mode shapes Video 1 String mode shapes Video 2
  • 22. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Example :Piano wire: L=1.4 m, t=11.1x104 N, m=110 g. Compute the first natural frequency.   110 g per 1.4 m = 0.0786 kg/m 1  c l   1.4 t    1.4 11.1104 N 0.0786 kg/m  2666.69 rad/s or 424 Hz
  • 23. Dynamics of Continuous Structures Maged Mostafa #WikiCourses http://WikiCourses.WikiSpaces.com Assignment 1. Solve the cable problem with one side fixed and the other supported by a flexible support with stiffness k N/m 2. Solve the cable problem for a cable that is hanging from one end and the tension is changing due to the weight  N/m