EML 5223 Structural Dynamics HW 10. Gun Lee(UFID )

Transcription

1 E 5 Structural Dynamcs HW Gun ee(ufid895-47)

2 Problem 9. ubular shaft of radus r ( ) r[ + ( )/ ], thcknesst, mass per unt volume ρ and shear modulus G. t r( ). Shaft s symmetrc wth respect to /. ass moment of nerta and area polar moment of nerta I( ) πρtr ( ), J tr ( ) π ( ) I I tr ( ) πρ ( ) /,,,..., I I.7ρtr I I.96ρtr 9 I I.5ρtr 8 I I.6ρtr 4 7 I I.98ρtr 5 6 Equvalent torsonal sprng constants GJ ( /4) k / GJ [( ) ] k,,,..., GJ ( /4) k / k k k k 8.69 k k k4 k8.5 k5 k k6.785 oreover, the correspondng lumped torques are ( t) m(, t),,,..., he equatons of moton can be derved by means of agrange s equatons d V +,,,..., n dt θ θ θ where I θ () t s the knetc energy and

3 V k[ θ ( t) θ ( t)], θ( t) θ( t) s the potental energy. Hence, the equatons of moton can be wrtten n the matr form θ() t + Kθ() t () t where θ() t [ θ() t θ() t θ()... t θ()] t t () [ () t () t ()... t ()] t mass matr: dag[ I I... I] dag[ ] ρtr k+ k k k k k k + k k+ k4 k4 k4 k4 + k5 k5 k5 k5 + k6 k6 K k6 k6 + k7 k7 k7 k7 + k8 k8 k8 k8 + k9 k9 k9 k9 + k k k k + k he correspondng egenvalue problem s KΘ ω Θ Normalzed accordng to ω ρ / G ω he results are as follows: Frst three modal vectors are ω.7, ω 6.45, ω , ω.697, ω ω 6.6, ω 7.87, ω 9.7, ω , ω

4 Θ [ ] [ ] [ ] Θ Θ Solve egenvalue problem by usng B I.7;I.96;I.5;I4.6;I5.98; I.7;I9.96;I8.5;I7.6;I6.98; [I ; I ; I ; I4 ; I5 ; I6 ; I7 ; I8 ; I9 ; I] k5.787;k8.69;k98.74;k4.5;k ;k6.785; k5.787;k8.69;k998.74;k8.5;k ; K[k+k -k ;-k k+k -k ; -k k+k4 -k4 ; -k4 k4+k5 -k5 ; -k5 k5+k6 -k6 ; -k6 k6+k7 -k7 ; -k7 k7+k8 -k8 ; -k8 k8+k9 -k9 ; -k9 k9+k -k; -k k+k] N; Rchol(); R' nv()knv(') [,W]eg() vnv(') for :N, w()sqrt(w(,)); end [w,i]sort(w) for :N U(:,)v(:,I()) end 9

5 Problem 9. he lumped-parameter method usng nfluence coeffcents. + B B ( ξ ) s the appled unt torque. and B are the reacton moments. orsonal dfferental equatons: dθ ( ) GJ ( ), < < ξ dθ ( ) GJ ( ), ξ < < Both dfferental equatons must yeld the same rotaton θ ( ξ) at ξ, so that ntegratng ξ dς dς θξ ( ) B GJ ( ς) ξ GJ ( ς) ξ dς dς ( ) B GJ ( ς) ξ GJ ( ς) whch yelds dς ξ GJ ( ς ) ξ dς GJ ( ς ) he fleblty nfluence functon aξ (, ) s defned as the angular dsplacement at due to a unt torque at ξ. Hence, dς a (, ξ ), < < ξ GJ ( ς ) dς a (, ξ) ( ), ξ < < GJ ( ς ).6 4 { ξ 7.85ξ.645ξ ( + ξ ξ ) ξ + ξ + ξ ξ ( ) tanh [.447( )]} dς.98 6 a (, ξ ) GJ ( ς) ( + ) ( + ξ ξ ) 4 4 { ( + ) tanh [.447( )]} 4 4 ξ { ξ 7.85ξ.645ξ ξ ( + ξ ξ ) tanh [.447( )]}, < < ξ 6 dς.98 a (, ξ ) ( ) GJ ( ς) ( + ) ( + ξ ξ ) 4 4 ξ { ξ ξ.649ξ ξ ( + ξ ξ ) tanh [.447( )]} 4 4 { ( + ) tanh [.447( )]}, ξ < < ettng and ξ (,,,...,)

6 he Fleblty matr symmetrc he mass matr (Problem 9.) mass matr: dag[ I I... I ] dag[ ] ρtr he correspondng egenvalue problem s Θ λθ, λ ω Normalzed accordng to ω ρ / G ω he results are as follows: ω.754, ω , ω , ω.6775, ω Frst three modal vectors are Θ 4 5 ω 6.89, ω , ω , ω 9.79, ω [ ] [ ] [ ] Θ Θ

7 Problem 9.7 From Eq. (9.94), Raylegh s quotent for the system of Problem 9. R λ ω V / ma ref dθ( ) Vma ( ) GJ ref I( ) ( ) Θ Θ( ): tral functon n n ( n) ( n) ( ) a φ( ) a sn ( ) GJ ( ) π tr + Θ ( ) I( ) πρtr + symmetrc mass and stffness dstrbutons wth respect to / π from mddle of the shaft, GJ ( ) GJ ( ), I( ) I( ) n/ n/ n/ n/ Vma ks a a + ka a a n/ n/ n/ n/ ref ms a a + ma a a ( n) ( n) ( n) ( n) ( n) ( n) ( n) ( n) ( n) ( n) ( n) ( n) φ / ( ) sn π ( ) sn φ π + We have the stffness coeffcents / ( ) ( ) ( ) n dφ dφ ks GJ( ) / π ( + /)( / ) 4 π ( )( ) cos( ) π cos( ) π,,,,..., n / + / ( ) ( ) d ( ) n dφ φ ka GJ( ) π π ( 4π / + + /)( / ) cos π cos π,,,,..., n/ and mass coeffcents / ( n) s φ φ m I( ) ( ) ( ) / ( + /)( / ) 4πρtr cos( ) π cos( ) π,,,,..., n / + / ( n) a φ φ m I( ) ( ) ( ) / π π ( + /)( / ) 4πρtr cos + π cos π,,,,..., n/

8 he symmetrc and antsymmetrc egenvalue problems have the matr form ( n) ( n) ( n) ( n) K a ω a s s s s K a ω a ( n) ( n) ( n) ( n) a a a a For n4, the stffness and mass matrces for the symmetrc egenvalue problem and the Rtz natural modes are ω ω ω ω K s ρtr s G.746, a ρ s s [ ] G 9.4, a ρ K a a [ ] ρtr G , a ρ s 4 s4 [ ] G.5848, a ρ [ ] Θ ( ).9989sn π( + / ) / sn π( + / ) / Θ ( ).9985sn π( + / ) / sn 4 π( + / ) / Θ ( ).47878sn π( + / ) / sn π( + / ) / Θ ( ).5898sn π ( + / ) / sn 4 π ( + / ) / 4

9 Smlarly, we can obtan the Rtz natural frequences for n, 4, 6, 8, n symmetrc ω G / ρ antsymmetrc ω G / ρ n 4 symmetrc ω.746 G/ ρ, ω 9.4 G/ ρ ω / ρ, ω / ρ antsymmetrc G G n 6 symmetrc ω.764 G/ ρ, ω 9.66 G/ ρ, ω G/ ρ 5 antsymmetrc ω 6.9 G/ ρ, ω G/ ρ, ω G/ ρ n 8 symmetrc ω.789 G/ ρ, ω 9.79 G/ ρ, ω 5.69 G/ ρ, ω.974 G/ ρ 5 7 ω / ρ, ω G/ ρ, ω G/ ρ, ω8 5.8 G/ ρ antsymmetrc G n symmetrc ω.746 G/ ρ, ω 9.99 G/ ρ, ω G/ ρ, ω G/ ρ, ω G/ ρ ω 6.98 / ρ, ω4.75 / ρ, ω G/ ρ, ω G/ ρ, ω.4458 G/ ρ antsymmetrc G G he estmated lowest frequences n Problem 9., 9., and 9.8 are all smaller than the lowest Rtz natural frequency. Note: frequences are same because problem 9. and 9.8 use same lumpng. Problem 9. and 9.8 : ω.7 G/ ρ Problem 9. : ω.754 G/ ρ n8, estmates by the Raylegh-Rta method converges to three decmal places, and ths value s ω.7489 G/ ρ. We can know that estmates by the Raylegh-Rta method s bgger than other method (problem 9., 9., and 9.8)..7 G/ ρ,.754 G/ ρ <.74 G/ ρ he three lowest egenfunctons for n are () Θ ( ).488sn π( + / ) / +.55sn π( + / ) / +.65sn 5 π( + / ) / sn 7 π( + / ) / +.99sn 9 π( + / ) / () Θ ( ).846sn π( + / ) / sn 4 π( + / ) / sn 6 π ( + / ) / +.6sn8 π( + / ) / +.6sn π( + / ) / () Θ ( ).44986sn π( + / ) / sn π( + / ) / +.598sn 5 π( + / ) / +.685sn 7 π( + / ) / +.6sn 9 π( + / ) / () Θ () Θ () Θ hs s normalzed, so, ma value s.