|       Chapter 5   Analytical
  techniques and solutions for linear elastic solids       5.10 Rayleigh-Ritz method for estimating
  natural frequency of an elastic solid  We
  conclude this chapter by describing an energy based method for estimating the
  natural frequency of vibration of an elastic solid.   5.10.1 Mode shapes and natural frequencies;
  orthogonality of mode shapes and Rayleighs Principle   It
  is helpful to review the definition of natural frequencies and mode shapes
  for a vibrating solid.  To this end,
  consider a representative elastic solid   say a slender beam that is free at both
  ends, as illustrated in the figure. 
  The physical significance of the mode shapes and natural frequencies
  of the vibrating beam can be visualized as follows: 
   Suppose that the
       beam is made to vibrate by bending it into some (fixed) deformed shape   ;
       and then suddenly releasing it.  
       In general, the resulting motion of the beam will be very
       complicated, and may not even appear to be periodic.However, there
       exists a set of special initial deflections   ,
       which cause every point on the beam to experience simple harmonic motion
       at some (angular) frequency   ,
       so that the deflected shape has the form   .The special
       frequencies    are called the natural frequencies
       of the system, and the special initial deflections    are called the mode shapes. ÂA continuous system
       always has an infinite number of mode shapes and natural frequencies.
       The vibration frequencies and their modes are conventionally ordered as
       a sequence    with   .  The lowest frequency of vibration is
       denoted   .
       The mode shapes for the lowest natural frequencies tend to have a long
       wavelength; the wavelength decreases for higher frequency modes.  If you are curious, the exact mode
       shapes and natural frequencies for a vibrating beam are derived in
       Section 10.4.1.In practice the lowest natural frequency of the system is of
       particular interest, since design specifications often prescribe a
       minimum allowable limit for the lowest natural frequency.   We
  will derive two important results below, which give a quick way to estimate
  the lowest natural frequency: 1.       The mode shapes are orthogonal, which means
  that the displacements associated with two different vibration modes    and    have the property that   
 2.       We will prove Rayleigh’s principle, which can be
  stated as follows.  Let    denote any kinematically admissible
  displacement field (you can think of this as a guess for the mode shape),
  which must be differentiable, and must satisfy    on   .  Define measures of potential energy    and kinetic energy    associated with    as   
 Then   ,
  and    if and only if   
 The result is useful because the fundamental
  frequency can be estimated by approximating the mode shape in some convenient
  way, and minimizing   .     Orthogonality of mode shapes   We
  consider a generic linear elastic solid, with elastic constants    and mass density   .
  Note that 
   External forces do
       not influence the natural frequencies of a linear elastic solid, so we
       can assume that the body force acting on the interior of the solid is
       zero.Part of the boundary
          may be subjected to prescribed displacements.  When estimating vibration frequencies,
       we can assume that the displacements are zero everywhere on   The remainder of the
       boundary    can be assumed to be traction free.   By
  definition the mode shapes and natural frequencies have the following
  properties: 
   The displacement
       field associated with this vibration mode is   The displacement
       field must satisfy the equation of motion for a linear elastic solid
       given in Section 5.1.2, which can be expressed in terms of the mode
       shape and natural frequency as   
 
   The mode shapes must satisfy    on    to meet the displacement boundary
       condition, and    on    to satisfy the traction free boundary
       condition.   Orthogonality of the mode
  shapes can be seen as follows. 1.       Let     and    be two mode shapes, with corresponding
  vibration frequencies     and   .
  Since both mode shapes satisfy the governing equations, it follows that   
 2.       Next we show that   
 To
  see this, integrate both sides of this expression by parts.  For example, for the left hand side,    
 where we have used the divergence theorem, and noted
  that the integral over the surface of the solid is zero because of the
  boundary conditions for    and   .  An exactly similar argument shows that   
 Recalling
  that    shows the result. 3.       Finally, orthogonality of the mode shapes follows by
  subtracting the second equation in (1) from the first, and using (2) to see
  that   
 If m and n are two distinct modes with
  different natural frequencies, the mode shapes must be orthogonal.     Proof of Rayleigh’s principle 1.       Note first that any kinematically admissible
  displacement field can be expressed as a linear combination of mode shapes as   
 To see the formula for the coefficients   ,
  multiply both sides of the first equation by   ,
  integrate over the volume of the solid, and use the orthogonality of the mode
  shapes. 2.       Secondly, note that the mode shapes satisfy   
 To
  see this, note first that because    satisfies the equation of motion, it follows
  that   
 Next,
  integrate the first term in this integral by parts (see step (2) in the poof
  of orthogonality of the mode shapes), and use the orthogonality of the mode
  shapes to see the result stated. 3.       We may now expand the potential and kinetic energy
  measures    and    in terms of sums of the mode shapes as
  follows   
 where
  we have used the result given in step (2) and orthogonality of the mode
  shapes. 4.       Finally, we know that    for   ,
  which shows that   
 We
  see immediately that   ,
  with equality if and only if    for          5.10.2 Estimate of natural frequency of vibration
  for a beam using Rayleigh-Ritz method   The
  figure illustrates the problem to be solved: an initially straight beam has
  Young’s modulus    and mass density   ,
  and its cross-section has area A and
  moment of area   .  The left hand end of the beam is clamped,
  the right hand end is free.  We wish to
  estimate the lowest natural frequency of vibration.   The
  deformation of a beam can be characterized by the deflection    of its neutral section.  The potential energy of the beam can be
  calculated from the formula derived in Section 5.7.4, while the kinetic
  energy measure T can be
  approximated by assuming the entire cross-section displaces with the
  mid-plane without rotation, which gives        
 The
  natural frequency can be estimated by selecting a suitable approximation for
  the mode shape   ,
  and minimizing the ratio   ,
  as follows: 1.       Note that the mode shape must satisfy the boundary
  conditions   .  We could try a polynomial   ,
  where C is a parameter that can be
  adjusted to get the best estimate for the natural frequency. 2.       Substituting this estimate into the definitions of V and T and evaluating the integrals gives   
 3.       To get the best estimate for the natural frequency,
  we must minimize this expression with respect to C.  It is straightforward
  to show that the minimum value occurs for   .
  Substituting this value back into (2) gives    4.       For comparison, the a formula for exact natural
  frequency of the lowest mode is derived in Section 10.4.1, and gives   .     |