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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  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:

  1. Suppose that the beam is made to vibrate by bending it into some (fixed) deformed shape u i = u i (0) ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWG1bWaa0baaSqaaiaadMgaaeaacaGGOaGaaGimaiaacMca aaGccaGGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4b WaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG4m aaqabaGccaGGPaaaaa@4064@ ; 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.
  2. However, there exists a set of special initial deflections u i (0) = U i (n) ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaqhaaWcbaGaamyAaaqaaiaacI cacaaIWaGaaiykaaaakiabg2da9iaadwfadaqhaaWcbaGaamyAaaqa aiaacIcacaWGUbGaaiykaaaakiaacIcacaWG4bWaaSbaaSqaaiaaig daaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGSaGa amiEamaaBaaaleaacaaIZaaabeaakiaacMcaaaa@4291@ , which cause every point on the beam to experience simple harmonic motion at some (angular) frequency ω n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaWGUbaabeaaaa a@3374@ , so that the deflected shape has the form u i ( x k ,t)= U i (n) ( x k )cos ω n t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGcca GGOaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcacaWG0bGaaiyk aiabg2da9iaadwfadaqhaaWcbaGaamyAaaqaaiaacIcacaWGUbGaai ykaaaakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaiGa cogacaGGVbGaai4CaiabeM8a3naaBaaaleaacaWGUbaabeaakiaads haaaa@475A@ .
  3. The special frequencies ω n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaWGUbaabeaaaa a@3374@  are called the natural frequencies of the system, and the special initial deflections U i (n) ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaqhaaWcbaGaamyAaaqaaiaacI cacaWGUbGaaiykaaaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqa aOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamiEam aaBaaaleaacaaIZaaabeaakiaacMcaaaa@3D59@  are called the mode shapes.  
  4. A 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 ω 1 , ω 2 , ω 3 ... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyYdC3aaSbaaSqaaiaaigdaaeqaaO GaaiilaiabeM8a3naaBaaaleaacaaIYaaabeaakiaacYcacqaHjpWD daWgaaWcbaGaaG4maaqabaGccaGGUaGaaiOlaiaac6caaaa@3DB5@  with ω n+1 > ω n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyYdC3aaSbaaSqaaiaad6gacqGHRa WkcaaIXaaabeaakiabg6da+iabeM8a3naaBaaaleaacaWGUbaabeaa aaa@3A89@ .  The lowest frequency of vibration is denoted ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyYdC3aaSbaaSqaaiaaigdaaeqaaa aa@34B6@ . 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.
  5. 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 U i (k) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyvamaaDaaaleaacaWGPbaabaGaai ikaiaadUgacaGGPaaaaaaa@3640@  and U i (j) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyvamaaDaaaleaacaWGPbaabaGaai ikaiaadQgacaGGPaaaaaaa@363F@  have the property that

V U i (k) ( x n ) U i (j) ( x n )dV=0(kj) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaa8quaeaacaWGvbWaa0baaSqaaiaadM gaaeaacaGGOaGaam4AaiaacMcaaaGccaGGOaGaamiEamaaBaaaleaa caWGUbaabeaakiaacMcaaSqaaiaadAfaaeqaniabgUIiYdGccaWGvb Waa0baaSqaaiaadMgaaeaacaGGOaGaamOAaiaacMcaaaGccaGGOaGa amiEamaaBaaaleaacaWGUbaabeaakiaacMcacaWGKbGaamOvaiabg2 da9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caGGOaGaam4AaiabgcMi5kaadQgacaGGPaaaaa@6466@

2.      We will prove Rayleigh’s principle, which can be stated as follows.  Let U ^ i ( x k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadwfagaqcamaaBaaaleaacaWGPbaabe aakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaaaa@3612@  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 U ^ i ( x k )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadwfagaqcamaaBaaaleaacaWGPbaabe aakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaiabg2da 9iaaicdaaaa@37D2@  on S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaaaaa@3247@ .  Define measures of potential energy V ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmOvayaajaaaaa@32ED@  and kinetic energy T ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmivayaajaaaaa@32EB@  associated with U ^ i ( x k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadwfagaqcamaaBaaaleaacaWGPbaabe aakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaaaa@3612@  as

V ^ = V 1 2 C ijkl U ^ k x l U ^ i x j dV ω 2 T ^ = V 1 2 ρ ω 2 U ^ i U ^ i dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadAfagaqcaiabg2da9maapefabaWaaS aaaeaacaaIXaaabaGaaGOmaaaacaWGdbWaaSbaaSqaaiaadMgacaWG QbGaam4AaiaadYgaaeqaaOWaaSaaaeaacqGHciITceWGvbGbaKaada WgaaWcbaGaam4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaa dYgaaeqaaaaaaeaacaWGwbaabeqdcqGHRiI8aOWaaSaaaeaacqGHci ITceWGvbGbaKaadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG 4bWaaSbaaSqaaiaadQgaaeqaaaaakiaadsgacaWGwbGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyY dC3aaWbaaSqabeaacaaIYaaaaOGabmivayaajaGaeyypa0Zaa8quae aadaWcaaqaaiaaigdaaeaacaaIYaaaaiabeg8aYjabeM8a3naaCaaa leqabaGaaGOmaaaakiqadwfagaqcamaaBaaaleaacaWGPbaabeaaki qadwfagaqcamaaBaaaleaacaWGPbaabeaaaeaacaWGwbaabeqdcqGH RiI8aOGaamizaiaadAfaaaa@6D49@

Then

V ^ T ^ ω 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaaceWGwbGbaKaaaeaaceWGub GbaKaaaaGaeyyzImRaeqyYdC3aa0baaSqaaiaaigdaaeaacaaIYaaa aaaa@391D@ , and V ^ T ^ = ω 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaaceWGwbGbaKaaaeaaceWGub GbaKaaaaGaeyypa0JaeqyYdC3aa0baaSqaaiaaigdaaeaacaaIYaaa aaaa@385D@  if and only if U ^ i = U i (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmyvayaajaWaaSbaaSqaaiaadMgaae qaaOGaeyypa0JaamyvamaaDaaaleaacaWGPbaabaGaaiikaiaaigda caGGPaaaaaaa@391F@

The result is useful because the fundamental frequency can be estimated by approximating the mode shape in some convenient way, and minimizing V ^ / T ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmOvayaajaGaai4laiqadsfagaqcaa aa@3489@ .

 

 

Orthogonality of mode shapes

 

We consider a generic linear elastic solid, with elastic constants C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaaaaa@353A@  and mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@ . Note that

  1. 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.
  2. Part of the boundary S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaaaaa@3247@  may be subjected to prescribed displacements.  When estimating vibration frequencies, we can assume that the displacements are zero everywhere on S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaaaaa@3247@
  3. The remainder of the boundary S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGOmaaqabaaaaa@3248@  can be assumed to be traction free.

 

By definition the mode shapes and natural frequencies have the following properties:

  1. The displacement field associated with this vibration mode is u i ( x k ,t)= U i (n) ( x k )cos ω n t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGcca GGOaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcacaWG0bGaaiyk aiabg2da9iaadwfadaqhaaWcbaGaamyAaaqaaiaacIcacaWGUbGaai ykaaaakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiykaiGa cogacaGGVbGaai4CaiabeM8a3naaBaaaleaacaWGUbaabeaakiaads haaaa@475A@
  2. 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

C ijkl 2 u k x i x l =ρ 2 u j t 2 C ijkl 2 U k (n) x i x l +ρ ω n 2 U j (n) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKaaalaadoeakm aaBaaajeaWbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGcdaWcaaqa aiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaam 4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGa eyOaIyRaamiEamaaBaaaleaacaWGSbaabeaaaaqcaaSaeyypa0Jaeq yWdiNcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadwha daWgaaWcbaGaamOAaaqabaaakeaacqGHciITcaWG0bWaaWbaaSqabe aacaaIYaaaaaaakiaaykW7caaMc8UaaGPaVlaaykW7cqGHshI3caaM c8UaaGPaVlaaykW7caaMc8EcaaSaam4qaOWaaSbaaKqaahaacaWGPb GaamOAaiaadUgacaWGSbaabeaakmaalaaabaGaeyOaIy7aaWbaaSqa beaacaaIYaaaaOGaamyvamaaDaaaleaacaWGRbaabaGaaiikaiaad6 gacaGGPaaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaa kiabgkGi2kaadIhadaWgaaWcbaGaamiBaaqabaaaaOGaey4kaSscaa SaeqyWdiNaeqyYdCNcdaqhaaWcbaGaamOBaaqaaiaaikdaaaGccaWG vbWaa0baaSqaaiaadQgaaeaacaGGOaGaamOBaiaacMcaaaGccqGH9a qpcaaIWaaaaa@8256@

  1. The mode shapes must satisfy U i (n) ( x k )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaqhaaWcbaGaamyAaaqaaiaacI cacaWGUbGaaiykaaaakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqa aOGaaiykaiabg2da9iaaicdaaaa@3A0F@  on S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaaaaa@3247@  to meet the displacement boundary condition, and C ijkl ( U k (n) / x l ) n i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGcdaqadaqaaiabgkGi2kaadwfadaqhaaWcbaGa am4AaaqaaiaacIcacaWGUbGaaiykaaaakiaac+cacqGHciITcaWG4b WaaSbaaSqaaiaadYgaaeqaaaGccaGLOaGaayzkaaGaamOBamaaBaaa leaacaWGPbaabeaakiabg2da9iaaicdaaaa@4494@  on S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGOmaaqabaaaaa@3248@  to satisfy the traction free boundary condition.

 

Orthogonality of the mode shapes can be seen as follows.

1.      Let  U i (m) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyvamaaDaaaleaacaWGPbaabaGaai ikaiaad2gacaGGPaaaaaaa@3642@  and U i (n) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyvamaaDaaaleaacaWGPbaabaGaai ikaiaad6gacaGGPaaaaaaa@3643@  be two mode shapes, with corresponding vibration frequencies ω m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyYdC3aaSbaaSqaaiaad2gaaeqaaa aa@34ED@   and ω n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyYdC3aaSbaaSqaaiaad6gaaeqaaa aa@34EE@ . Since both mode shapes satisfy the governing equations, it follows that

V ( C ijkl 2 U k (n) x i x l U j (m) +ρ ω n 2 U j (n) U j (m) ) dV=0 V ( C ijkl 2 U k (m) x i x l U j (n) +ρ ω m 2 U j (m) U j (n) ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaykW7daWdrb qaamaabmaabaqcaaSaam4qaOWaaSbaaKqaahaacaWGPbGaamOAaiaa dUgacaWGSbaabeaakmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYa aaaOGaamyvamaaDaaaleaacaWGRbaabaGaaiikaiaad6gacaGGPaaa aaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaakiabgkGi2k aadIhadaWgaaWcbaGaamiBaaqabaaaaOGaamyvamaaDaaaleaacaWG QbaabaGaaiikaiaad2gacaGGPaaaaOGaey4kaSscaaSaeqyWdiNaeq yYdCNcdaqhaaWcbaGaamOBaaqaaiaaikdaaaGccaWGvbWaa0baaSqa aiaadQgaaeaacaGGOaGaamOBaiaacMcaaaGccaWGvbWaa0baaSqaai aadQgaaeaacaGGOaGaamyBaiaacMcaaaaakiaawIcacaGLPaaaaSqa aiaadAfaaeqaniabgUIiYdGccaWGKbGaamOvaiabg2da9iaaicdaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aa8 quaeaacaaMc8+aaeWaaeaajaaWcaWGdbGcdaWgaaqcbaCaaiaadMga caWGQbGaam4AaiaadYgaaeqaaOWaaSaaaeaacqGHciITdaahaaWcbe qaaiaaikdaaaGccaWGvbWaa0baaSqaaiaadUgaaeaacaGGOaGaamyB aiaacMcaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaO GaeyOaIyRaamiEamaaBaaaleaacaWGSbaabeaaaaGccaWGvbWaa0ba aSqaaiaadQgaaeaacaGGOaGaamOBaiaacMcaaaGccqGHRaWkjaaWcq aHbpGCcqaHjpWDkmaaDaaaleaacaWGTbaabaGaaGOmaaaakiaadwfa daqhaaWcbaGaamOAaaqaaiaacIcacaWGTbGaaiykaaaakiaadwfada qhaaWcbaGaamOAaaqaaiaacIcacaWGUbGaaiykaaaaaOGaayjkaiaa wMcaaaWcbaGaamOvaaqab0Gaey4kIipakiabg2da9iaaicdaaaa@AFBE@

2.      Next we show that

V C ijkl 2 U k (n) x i x l U j (m) dV = V C ijkl 2 U k (m) x i x l U j (n) dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaykW7daWdrb qaaKaaalaadoeakmaaBaaajeaWbaGaamyAaiaadQgacaWGRbGaamiB aaqabaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadw fadaqhaaWcbaGaam4AaaqaaiaacIcacaWGUbGaaiykaaaaaOqaaiab gkGi2kaadIhadaWgaaWcbaGaamyAaaqabaGccqGHciITcaWG4bWaaS baaSqaaiaadYgaaeqaaaaakiaadwfadaqhaaWcbaGaamOAaaqaaiaa cIcacaWGTbGaaiykaaaakiaadsgacaWGwbaaleaacaWGwbaabeqdcq GHRiI8aOGaeyypa0Zaa8quaeaajaaWcaWGdbGcdaWgaaqcbaCaaiaa dMgacaWGQbGaam4AaiaadYgaaeqaaOWaaSaaaeaacqGHciITdaahaa WcbeqaaiaaikdaaaGccaWGvbWaa0baaSqaaiaadUgaaeaacaGGOaGa amyBaiaacMcaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaae qaaOGaeyOaIyRaamiEamaaBaaaleaacaWGSbaabeaaaaGccaWGvbWa a0baaSqaaiaadQgaaeaacaGGOaGaamOBaiaacMcaaaaabaGaamOvaa qab0Gaey4kIipakiaadsgacaWGwbaaaa@7229@

To see this, integrate both sides of this expression by parts.  For example, for the left hand side,

V C ijkl 2 U k (n) x i x l U j (m) dV = V x i ( C ijkl U k (n) x l U j (m) ) dV V C ijkl U k (n) x l U j (n) x i dV S ( C ijkl U k (n) x l U j (m) ) n i dA V C ijkl U k (n) x l U j (m) x i dV = V C ijkl U k (n) x l U j (m) x i dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaGPaVp aapefabaqcaaSaam4qaOWaaSbaaKqaahaacaWGPbGaamOAaiaadUga caWGSbaabeaakmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaO GaamyvamaaDaaaleaacaWGRbaabaGaaiikaiaad6gacaGGPaaaaaGc baGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaakiabgkGi2kaadI hadaWgaaWcbaGaamiBaaqabaaaaOGaamyvamaaDaaaleaacaWGQbaa baGaaiikaiaad2gacaGGPaaaaOGaamizaiaadAfaaSqaaiaadAfaae qaniabgUIiYdGccqGH9aqpdaWdrbqaamaalaaabaGaeyOaIylabaGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGcdaqadaqaaKaaal aadoeakmaaBaaajeaWbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGc daWcaaqaaiabgkGi2kaadwfadaqhaaWcbaGaam4AaaqaaiaacIcaca WGUbGaaiykaaaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamiBaaqa baaaaOGaamyvamaaDaaaleaacaWGQbaabaGaaiikaiaad2gacaGGPa aaaaGccaGLOaGaayzkaaaaleaacaWGwbaabeqdcqGHRiI8aOGaamiz aiaadAfacqGHsisldaWdrbqaaKaaalaadoeakmaaBaaajeaWbaGaam yAaiaadQgacaWGRbGaamiBaaqabaGcdaWcaaqaaiabgkGi2kaadwfa daqhaaWcbaGaam4AaaqaaiaacIcacaWGUbGaaiykaaaaaOqaaiabgk Gi2kaadIhadaWgaaWcbaGaamiBaaqabaaaaOWaaSaaaeaacqGHciIT caWGvbWaa0baaSqaaiaadQgaaeaacaGGOaGaamOBaiaacMcaaaaake aacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaakiaadsgacaWG wbaaleaacaWGwbaabeqdcqGHRiI8aaGcbaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8+aa8quaeaadaqadaqaaKaaalaadoeakmaaBaaajeaWbaGaamyAai aadQgacaWGRbGaamiBaaqabaGcdaWcaaqaaiabgkGi2kaadwfadaqh aaWcbaGaam4AaaqaaiaacIcacaWGUbGaaiykaaaaaOqaaiabgkGi2k aadIhadaWgaaWcbaGaamiBaaqabaaaaOGaamyvamaaDaaaleaacaWG QbaabaGaaiikaiaad2gacaGGPaaaaaGccaGLOaGaayzkaaGaamOBam aaBaaaleaacaWGPbaabeaakiaadsgacaWGbbaaleaacaWGtbaabeqd cqGHRiI8aOGaeyOeI0Yaa8quaeaajaaWcaWGdbGcdaWgaaqcbaCaai aadMgacaWGQbGaam4AaiaadYgaaeqaaOWaaSaaaeaacqGHciITcaWG vbWaa0baaSqaaiaadUgaaeaacaGGOaGaamOBaiaacMcaaaaakeaacq GHciITcaWG4bWaaSbaaSqaaiaadYgaaeqaaaaakmaalaaabaGaeyOa IyRaamyvamaaDaaaleaacaWGQbaabaGaaiikaiaad2gacaGGPaaaaa GcbaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGccaWGKbGa amOvaaWcbaGaamOvaaqab0Gaey4kIipaaOqaaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eyypa0JaeyOeI0Yaa8quaeaajaaWcaWGdbGcdaWgaaqcbaCaaiaadM gacaWGQbGaam4AaiaadYgaaeqaaOWaaSaaaeaacqGHciITcaWGvbWa a0baaSqaaiaadUgaaeaacaGGOaGaamOBaiaacMcaaaaakeaacqGHci ITcaWG4bWaaSbaaSqaaiaadYgaaeqaaaaakmaalaaabaGaeyOaIyRa amyvamaaDaaaleaacaWGQbaabaGaaiikaiaad2gacaGGPaaaaaGcba GaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGccaWGKbGaamOv aaWcbaGaamOvaaqab0Gaey4kIipaaaaa@A596@

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 U i (m) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyvamaaDaaaleaacaWGPbaabaGaai ikaiaad2gacaGGPaaaaaaa@3642@  and U i (n) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyvamaaDaaaleaacaWGPbaabaGaai ikaiaad6gacaGGPaaaaaaa@3643@ .  An exactly similar argument shows that

V C ijkl 2 U k (m) x i x l U j (n) dV = V C ijkl U k (m) x l U j (n) x i dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaykW7daWdrb qaaKaaalaadoeakmaaBaaajeaWbaGaamyAaiaadQgacaWGRbGaamiB aaqabaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadw fadaqhaaWcbaGaam4AaaqaaiaacIcacaWGTbGaaiykaaaaaOqaaiab gkGi2kaadIhadaWgaaWcbaGaamyAaaqabaGccqGHciITcaWG4bWaaS baaSqaaiaadYgaaeqaaaaakiaadwfadaqhaaWcbaGaamOAaaqaaiaa cIcacaWGUbGaaiykaaaakiaadsgacaWGwbaaleaacaWGwbaabeqdcq GHRiI8aOGaeyypa0JaeyOeI0Yaa8quaeaajaaWcaWGdbGcdaWgaaqc baCaaiaadMgacaWGQbGaam4AaiaadYgaaeqaaOWaaSaaaeaacqGHci ITcaWGvbWaa0baaSqaaiaadUgaaeaacaGGOaGaamyBaiaacMcaaaaa keaacqGHciITcaWG4bWaaSbaaSqaaiaadYgaaeqaaaaakmaalaaaba GaeyOaIyRaamyvamaaDaaaleaacaWGQbaabaGaaiikaiaad6gacaGG PaaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGcca WGKbGaamOvaaWcbaGaamOvaaqab0Gaey4kIipaaaa@73A4@

Recalling that C ijkl = C klij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaam4qamaaBaaaleaacaWGPbGaamOAai aadUgacaWGSbaabeaakiabg2da9iaadoeadaWgaaWcbaGaam4Aaiaa dYgacaWGPbGaamOAaaqabaaaaa@3C76@  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

( ω n 2 ω m 2 ) V ( U j (n) U j (m) ) dV=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaykW7daqada qaaKaaalabeM8a3PWaa0baaSqaaiaad6gaaeaacaaIYaaaaOGaeyOe I0scaaSaeqyYdCNcdaqhaaWcbaGaamyBaaqaaiaaikdaaaaakiaawI cacaGLPaaadaWdrbqaamaabmaabaGaamyvamaaDaaaleaacaWGQbaa baGaaiikaiaad6gacaGGPaaaaOGaamyvamaaDaaaleaacaWGQbaaba Gaaiikaiaad2gacaGGPaaaaaGccaGLOaGaayzkaaaaleaacaWGwbaa beqdcqGHRiI8aOGaamizaiaadAfacqGH9aqpcaaIWaaaaa@52E6@

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

U ^ i = n=1 α n U i (n) α m = V U ^ i U i (m) dV V U i (m) U i (m) dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmyvayaajaWaaSbaaSqaaiaadMgaae qaaOGaeyypa0ZaaabCaeaacqaHXoqydaWgaaWcbaGaamOBaaqabaGc caWGvbWaa0baaSqaaiaadMgaaeaacaGGOaGaamOBaiaacMcaaaaaba GaamOBaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoakiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHXoqy daWgaaWcbaGaamyBaaqabaGccqGH9aqpdaWcaaqaamaapefabaGabm yvayaajaWaaSbaaSqaaiaadMgaaeqaaOGaamyvamaaDaaaleaacaWG PbaabaGaaiikaiaad2gacaGGPaaaaOGaamizaiaadAfaaSqaaiaadA faaeqaniabgUIiYdaakeaadaWdrbqaaiaadwfadaqhaaWcbaGaamyA aaqaaiaacIcacaWGTbGaaiykaaaakiaadwfadaqhaaWcbaGaamyAaa qaaiaacIcacaWGTbGaaiykaaaakiaadsgacaWGwbaaleaacaWGwbaa beqdcqGHRiI8aaaaaaa@77DA@

To see the formula for the coefficients α m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqySde2aaSbaaSqaaiaad2gaaeqaaa aa@34BF@ , multiply both sides of the first equation by U i (m) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyvamaaDaaaleaacaWGPbaabaGaai ikaiaad2gacaGGPaaaaaaa@3642@ , integrate over the volume of the solid, and use the orthogonality of the mode shapes.

2.      Secondly, note that the mode shapes satisfy

V C ijkl U k (m) x l U j (n) x i dV ={ ω m 2 V ( U j (m) U j (m) ) dVm=n 0mn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapefabaqcaa Saam4qaOWaaSbaaKqaahaacaWGPbGaamOAaiaadUgacaWGSbaabeaa kmaalaaabaGaeyOaIyRaamyvamaaDaaaleaacaWGRbaabaGaaiikai aad2gacaGGPaaaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGSbaa beaaaaGcdaWcaaqaaiabgkGi2kaadwfadaqhaaWcbaGaamOAaaqaai aacIcacaWGUbGaaiykaaaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGa amyAaaqabaaaaOGaamizaiaadAfaaSqaaiaadAfaaeqaniabgUIiYd GccqGH9aqpdaGabaqaauaabeqaceaaaeaacqaHjpWDdaqhaaWcbaGa amyBaaqaaiaaikdaaaGcdaWdrbqaamaabmaabaGaamyvamaaDaaale aacaWGQbaabaGaaiikaiaad2gacaGGPaaaaOGaamyvamaaDaaaleaa caWGQbaabaGaaiikaiaad2gacaGGPaaaaaGccaGLOaGaayzkaaaale aacaWGwbaabeqdcqGHRiI8aOGaamizaiaadAfacaaMc8UaaGPaVlaa ykW7caaMc8UaamyBaiabg2da9iaad6gaaeaacaaIWaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaad2gacq GHGjsUcaWGUbaaaaGaay5Eaaaaaa@C0E6@

To see this, note first that because U i (m) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyvamaaDaaaleaacaWGPbaabaGaai ikaiaad2gacaGGPaaaaaaa@3642@  satisfies the equation of motion, it follows that

V ( C ijkl 2 U k (m) x i x l U j (m) +ρ ω m 2 U j (m) U j (m) ) dV=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaykW7daWdrb qaamaabmaabaqcaaSaam4qaOWaaSbaaKqaahaacaWGPbGaamOAaiaa dUgacaWGSbaabeaakmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYa aaaOGaamyvamaaDaaaleaacaWGRbaabaGaaiikaiaad2gacaGGPaaa aaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaakiabgkGi2k aadIhadaWgaaWcbaGaamiBaaqabaaaaOGaamyvamaaDaaaleaacaWG QbaabaGaaiikaiaad2gacaGGPaaaaOGaey4kaSscaaSaeqyWdiNaeq yYdCNcdaqhaaWcbaGaamyBaaqaaiaaikdaaaGccaWGvbWaa0baaSqa aiaadQgaaeaacaGGOaGaamyBaiaacMcaaaGccaWGvbWaa0baaSqaai aadQgaaeaacaGGOaGaamyBaiaacMcaaaaakiaawIcacaGLPaaaaSqa aiaadAfaaeqaniabgUIiYdGccaWGKbGaamOvaiabg2da9iaaicdaaa a@66EA@

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 V ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmOvayaajaaaaa@32ED@  and T ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmivayaajaaaaa@32EB@  in terms of sums of the mode shapes as follows

V ^ = V 1 2 C ijkl ( n=1 α n U j (n) x i ) ( m=1 α m U k (m) x l )= 1 2 ρ m=1 ω m 2 α m 2 V ( U j (m) U j (m) ) dV T ^ = V ρ 2 ( n=1 α n U j (n) ) ( m=1 α m U j (m) )dV= 1 2 ρ m=1 α m 2 V ( U j (m) U j (m) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGceaqabeaaceWGwbGbaKaacqGH9aqpdaWdrb qaamaalaaabaGaaGymaaqaaiaaikdaaaGaam4qamaaBaaaleaacaWG PbGaamOAaiaadUgacaWGSbaabeaakmaabmaabaWaaabCaeaacqaHXo qydaWgaaWcbaGaamOBaaqabaGcdaWcaaqaaiabgkGi2kaadwfadaqh aaWcbaGaamOAaaqaaiaacIcacaWGUbGaaiykaaaaaOqaaiabgkGi2k aadIhadaWgaaWcbaGaamyAaaqabaaaaaqaaiaad6gacqGH9aqpcaaI XaaabaGaeyOhIukaniabggHiLdaakiaawIcacaGLPaaaaSqaaiaadA faaeqaniabgUIiYdGcdaqadaqaamaaqahabaGaeqySde2aaSbaaSqa aiaad2gaaeqaaOWaaSaaaeaacqGHciITcaWGvbWaa0baaSqaaiaadU gaaeaacaGGOaGaamyBaiaacMcaaaaakeaacqGHciITcaWG4bWaaSba aSqaaiaadYgaaeqaaaaaaeaacaWGTbGaeyypa0JaaGymaaqaaiabg6 HiLcqdcqGHris5aaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaI XaaabaGaaGOmaaaacqaHbpGCdaaeWbqaaiabeM8a3naaDaaaleaaca WGTbaabaGaaGOmaaaakiabeg7aHnaaDaaaleaacaWGTbaabaGaaGOm aaaaaeaacaWGTbGaeyypa0JaaGymaaqaaiabg6HiLcqdcqGHris5aO Waa8quaeaadaqadaqaaiaadwfadaqhaaWcbaGaamOAaaqaaiaacIca caWGTbGaaiykaaaakiaadwfadaqhaaWcbaGaamOAaaqaaiaacIcaca WGTbGaaiykaaaaaOGaayjkaiaawMcaaaWcbaGaamOvaaqab0Gaey4k IipakiaadsgacaWGwbaabaGabmivayaajaGaeyypa0Zaa8quaeaada Wcaaqaaiabeg8aYbqaaiaaikdaaaWaaeWaaeaadaaeWbqaaiabeg7a HnaaBaaaleaacaWGUbaabeaakiaadwfadaqhaaWcbaGaamOAaaqaai aacIcacaWGUbGaaiykaaaaaeaacaWGUbGaeyypa0JaaGymaaqaaiab g6HiLcqdcqGHris5aaGccaGLOaGaayzkaaaaleaacaWGwbaabeqdcq GHRiI8aOWaaeWaaeaadaaeWbqaaiabeg7aHnaaBaaaleaacaWGTbaa beaakiaadwfadaqhaaWcbaGaamOAaaqaaiaacIcacaWGTbGaaiykaa aaaeaacaWGTbGaeyypa0JaaGymaaqaaiabg6HiLcqdcqGHris5aaGc caGLOaGaayzkaaGaamizaiaadAfacqGH9aqpdaWcaaqaaiaaigdaae aacaaIYaaaaiabeg8aYnaaqahabaGaeqySde2aa0baaSqaaiaad2ga aeaacaaIYaaaaaqaaiaad2gacqGH9aqpcaaIXaaabaGaeyOhIukani abggHiLdGcdaWdrbqaamaabmaabaGaamyvamaaDaaaleaacaWGQbaa baGaaiikaiaad2gacaGGPaaaaOGaamyvamaaDaaaleaacaWGQbaaba Gaaiikaiaad2gacaGGPaaaaaGccaGLOaGaayzkaaaaleaacaWGwbaa beqdcqGHRiI8aaaaaa@C915@

where we have used the result given in step (2) and orthogonality of the mode shapes.

4.      Finally, we know that ω m ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyYdC3aaSbaaSqaaiaad2gaaeqaaO GaeyyzImRaeqyYdC3aaSbaaSqaaiaaigdaaeqaaaaa@3971@  for m1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyBaiabgwMiZkaaigdaaaa@3575@ , which shows that

V ^ 1 2 ρ ω 1 2 m=1 α m 2 V ( U j (m) U j (m) ) dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmOvayaajaGaeyyzIm7aaSaaaeaaca aIXaaabaGaaGOmaaaacqaHbpGCcqaHjpWDdaqhaaWcbaGaaGymaaqa aiaaikdaaaGcdaaeWbqaaiabeg7aHnaaDaaaleaacaWGTbaabaGaaG OmaaaaaeaacaWGTbGaeyypa0JaaGymaaqaaiabg6HiLcqdcqGHris5 aOWaa8quaeaadaqadaqaaiaadwfadaqhaaWcbaGaamOAaaqaaiaacI cacaWGTbGaaiykaaaakiaadwfadaqhaaWcbaGaamOAaaqaaiaacIca caWGTbGaaiykaaaaaOGaayjkaiaawMcaaaWcbaGaamOvaaqab0Gaey 4kIipakiaadsgacaWGwbaaaa@5465@

We see immediately that V ^ / T ^ ω 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabmOvayaajaGaai4laiqadsfagaqcai abgwMiZkabeM8a3naaDaaaleaacaaIXaaabaGaaGOmaaaaaaa@39C0@ , with equality if and only if α m =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqySde2aaSbaaSqaaiaad2gaaeqaaO Gaeyypa0JaaGimaaaa@3689@  for m>1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyBaiabg6da+iaaigdaaaa@34B7@

 

 

 

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 E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@  and mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@ , and its cross-section has area A and moment of area I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeaaaa@3156@ .  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 w(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaam4DaiaacIcacaWG4bGaaiykaaaa@3554@  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

V(w)= 0 L 1 2 EI { d 2 w( x 1 ) d x 1 2 } 2 d x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeGabaa0riaadAfacaGGOaGaam4DaiaacM cacqGH9aqpdaWdXbqaaaWcbaGaaGimaaqaaiaadYeaa0Gaey4kIipa kmaalaaabaGaaGymaaqaaiaaikdaaaGaamyraiaadMeadaGadaqaam aalaaabaGaamizamaaCaaaleqabaGaaGOmaaaakiaadEhacaGGOaGa amiEamaaBaaaleaacaaIXaaabeaakiaacMcaaeaacaWGKbGaamiEam aaBaaaleaacaaIXaaabeaakmaaCaaaleqabaGaaGOmaaaaaaaakiaa wUhacaGL9baadaahaaWcbeqaaiaaikdaaaGccaaMc8UaamizaiaadI hadaWgaaWcbaGaaGymaaqabaaaaa@5071@     T(w)= 0 L 1 2 ρA { w( x 1 ) } 2 d x 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeGabaa0riaadsfacaGGOaGaam4DaiaacM cacqGH9aqpdaWdXbqaamaalaaabaGaaGymaaqaaiaaikdaaaGaeqyW diNaamyqamaacmaabaGaam4DaiaacIcacaWG4bWaaSbaaSqaaiaaig daaeqaaOGaaiykaaGaay5Eaiaaw2haamaaCaaaleqabaGaaGOmaaaa kiaaykW7caWGKbGaamiEamaaBaaaleaacaaIXaaabeaaaeaacaaIWa aabaGaamitaaqdcqGHRiI8aaaa@4B92@

The natural frequency can be estimated by selecting a suitable approximation for the mode shape W ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabm4vayaajaaaaa@32EE@ , and minimizing the ratio V/T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamOvaiaac+cacaWGubaaaa@3469@ , as follows:

1.      Note that the mode shape must satisfy the boundary conditions W ^ =d W ^ /d x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabm4vayaajaGaeyypa0JaamizaiqadE fagaqcaiaac+cacaWGKbGaamiEamaaBaaaleaacaaIXaaabeaakiab g2da9iaaicdaaaa@3B13@ .  We could try a polynomial W ^ = x 1 2 +C x 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGabm4vayaajaGaeyypa0JaamiEamaaDa aaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiaadoeacaWG4bWaa0ba aSqaaiaaigdaaeaacaaIZaaaaaaa@3AEB@ , 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

V T = EI ρA L 4 420(1+3CL+3 C 2 L 2 ) (15 C 2 L 2 +35CL+21) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaacaWGwbaabaGaamivaaaacq GH9aqpdaWcaaqaaiaadweacaWGjbaabaGaeqyWdiNaamyqaiaadYea daahaaWcbeqaaiaaisdaaaaaaOWaaSaaaeaacaaI0aGaaGOmaiaaic dacaGGOaGaaGymaiabgUcaRiaaiodacaWGdbGaamitaiabgUcaRiaa iodacaWGdbWaaWbaaSqabeaacaaIYaaaaOGaamitamaaCaaaleqaba GaaGOmaaaakiaacMcaaeaacaGGOaGaaGymaiaaiwdacaWGdbWaaWba aSqabeaacaaIYaaaaOGaamitamaaCaaaleqabaGaaGOmaaaakiabgU caRiaaiodacaaI1aGaam4qaiaadYeacqGHRaWkcaaIYaGaaGymaiaa cMcaaaaaaa@5410@

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 CL=( 39 12 )/15 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaam4qaiaadYeacqGH9aqpdaqadaqaam aakaaabaGaaG4maiaaiMdaaSqabaGccqGHsislcaaIXaGaaGOmaaGa ayjkaiaawMcaaiaac+cacaaIXaGaaGynaaaa@3C60@ . Substituting this value back into (2) gives V T =12.48 EI ρA L 4 ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaacaWGwbaabaGaamivaaaacq GH9aqpcaaIXaGaaGOmaiaac6cacaaI0aGaaGioamaalaaabaGaamyr aiaadMeaaeaacqaHbpGCcaWGbbGaamitamaaCaaaleqabaGaaGinaa aaaaGccqGHLjYScqaHjpWDdaahaaWcbeqaaiaaikdaaaaaaa@42E5@

4.      For comparison, the a formula for exact natural frequency of the lowest mode is derived in Section 10.4.1, and gives ω 2 =12.36EI/(ρA L 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyYdC3aaWbaaSqabeaacaaIYaaaaO Gaeyypa0JaaGymaiaaikdacaGGUaGaaG4maiaaiAdacaWGfbGaamys aiaac+cacaGGOaGaeqyWdiNaamyqaiaadYeadaahaaWcbeqaaiaais daaaGccaGGPaaaaa@415E@ .