Modal Transient Analysis of a Beam with Enforced Motion via a Ramp Invariant Digital Recursive Filtering Relationship

Size: px

Start display at page:

Download "Modal Transient Analysis of a Beam with Enforced Motion via a Ramp Invariant Digital Recursive Filtering Relationship"

Franklin Morris
5 years ago
Views:

1 oal ransient Analysis of a Beam ith Enforce otion via a Ramp nvariant Digital Recrsive iltering Relationship By om rvine tomirvine@aol.com December, Variables f f ass matrix Stiness matrix Applie forces orces at riven noes orces at free noes entity matrix ransformation matrix Displacement vector Displacements at riven noes Displacements at free noes Consier a beam hich is moele via the finite element metho as a mlti-egree-of-freeom (DO) system. he homogeneos eqation of motion for an DO system is [ ][] [][] [] () [ ] () f

2 Partition the matrices an vectors as follos f f f f f f (3) he eqations of motions for enforce isplacement an acceleration are given in Appenices A an B, respectively. Create a transformation matrix sch that f (4) (5) f f f f (6) Premltiply by, f f f f (7) Eqation (7) can then be ncople sing the normal moes per the metho in Reference 4. he moal transient analysis can be then be performe on the moal coorinates sing the ramp invariant igital recrsive filtering relationship in Reference 5 for the case here the acceleration of one or moal egrees-of-freeom have been prescribe. he reslting moal responses are then transforme into physical responses per the metho in Reference 4. Next, the transformation in eqation (4) is performe.

3 inally, the physical responses are reassemble in the correct orer if necessary. References.. rvine, he Generalize Coorinate etho for Discrete Systems, Revision, Vibrationata,... rvine, oal ransient Analysis of a System Sbjecte to an Applie orce via Ramp nvariant Digital Recrsive iltering Relationship, Revision B, Vibrationata,. 3.. rvine, ransverse Vibration of a Beam via the inite Element etho, Revision, Vibrationata,. 4.. rvine, he Generalize Coorinate etho for Discrete Systems, Revision, Vibrationata,. 5.. rvine, oal ransient Analysis of a System Sbjecte to an Applie orce via Ramp nvariant Digital Recrsive iltering Relationship, Revision B, Vibrationata,. 3

4 APPENDX A Enforce Displacement Again, the partitione eqation of motion is f f f f f (A-) ransform the eqation of motion to ncople the mass matrix so that the reslting mass matrix is ˆ ˆ (A-) Apply the transformation to the mass matrix f f (A-3) f f f (A-4) f f f f (A-5) 4

5 f f f f f f f f (A-6) (A-7) Let (A-8) f f f f ˆ ˆ (A-9) f (A-) f (A-) f (A-) he transformation matrix is (A-3) ˆ f f (A-4) 5

6 6 ˆ (A-5) f f (A-6) By similarity, the transforme stiness matrix is f f f f kˆ kˆ kˆ kˆ (A-7) f ˆ (A-8) f f ˆ (A-9) f f ˆ (A-) ˆ kˆ kˆ kˆ kˆ ˆ ˆ (A-) ˆ kˆ kˆ ˆ (A-) he eqation of motion is ths kˆ ˆ kˆ ˆ (A-3)

7 7 he final isplacements are fon via f (A-4) f (A-5)

8 APPENDX B Enforce Acceleration Again, the partitione eqation of motion is f f f f f (B-) ransform the eqation of motion to ncople the stiness matrix so that the reslting stiness matrix is ˆ ˆ (B-) f f (B-3) f f f (B-4) f f f f (B-5) 8

9 f f f f f f f f (B-6) (B-7) Let (B-8) f f f f ˆ ˆ (B-9) f (B-) f (B-) f (B-) (B-3) 9

10 f f ˆ (B-4) ˆ (B-5) f f (B-6) By similarity, the transforme mass matrix is f f f f mˆ mˆ mˆ mˆ (B-7) f ˆ (B-8) f f ˆ (B-9) f f ˆ (B-) ˆ ˆ ˆ mˆ mˆ mˆ mˆ (B-)

11 mˆ mˆ ˆ ˆ (B-) he eqation of motion is ths mˆ ˆ ˆ mˆ (B-3) he final isplacements are fon via f (B-4) f (B-5)

12 APPENDX C Enforce Acceleration Example Sample Beam Consier the cantilever beam in igre C-. E, L igre C-. he governing ierential eqation for the isplacement y(x,t) is 4 y E x 4 y t (C-) here E L is the mols of elasticity is the area moment of inertia is the length is the mass ensity (mass/length) Note that this eqation neglects shear eformation an rotary inertia.

13 y(x, t) (t) igre C-. Apply base excitation to the beam as shon in igre C-. here are to caniate methos for applying base excitation at the fixe en of the beam. he first metho is to a a seismic mass hich is attache via a rigi link to the beam s fixe en. he seismic mass ill be given a mass vale hich is consierably greater than the beam s on mass. hen a force ill be applie to the seismic mass to yiel the prescribe acceleration. he secon metho is to apply the acceleration irectly to the fixe en sing the enforce metho from the main text. his is the metho se in this example. inite Element oel oel the cantilever beam ith three elements an for noes sing the mass an stiness matrices from Reference 3. he moel is shon in igre C-3. E E E3 N N N3 N4 igre C-3. (he moel is limite to three elements for brevity. normally be se. ) A larger nmber of elements ol 3

14 here is one translation an one rotational egree-of-freeom at each noe. he isplacement vector for the moel is: y y y3 3 y4 4 Set the beam properties as follos. Cross-Section Bonary Conitions aterial Circlar ixe-ree Alminm Diameter D =.5 inch Cross-Section Area A =.963 in^ Length L = 4 inch Area oment of nertia =.368 in^4 Elastic ols E =.e+7 lbf/in^ Stiness E = 368 lbf in^ ass per Volme v =. lbm / in^3 (.59 lbf sec^/in^4 ) ass per Length =.963 lbm/in (5.8e-5 lbf sec^/in^) ass L =.47 lbm (.E-3 lbf sec^/in) Viscos Damping Ratio =.5 he analysis is carrie ot sing atlab script: beam_base_accel_fea.m. 4

15 he nconstraine mass matrix is.e-3 * he nconstraine stiness matrix is.e+4 * he rotation at the left en is constraine by removing its ros an colmns from the mass an stiness matrices. he y translation at the left en is retaine becase it ill be prescribe in a later step. he constraine mass matrix is.e-3 *

16 he constraine stiness matrix is.e+4 * Create a transformation matrix sch that f (C-) (C-3) he transformation matrix is explicitly given later. he riven isplacement for the sample problem is y he remaining isplacements are y y3 3 y4 4 By sbstittion, f f f f (C-4) 6

17 or the sample problem, =.55e-4 f =.e-3 * f =.e-3 * =.e-3 *

18 = f =.e+3 * f =.e+3 * =.e+4 * he transform matrix is erive as follos. f (C-5) f (C-6) (C-7) 8

19 inv = = = = Premltiply by, f f f f (C-8) 9

20 f f mˆ mˆ mˆ mˆ (C-9) mˆ =. mˆ.e-3 * mˆ =.e-3 * f f ˆ ˆ (C-)

21 ˆ = ˆ =.e+4 * mˆ ˆ mˆ (C-) he base acceleration term is eectively a force. he natral freqencies an mass-normalize moe shapes for the homogeneos form of eqation (C-) are n fn(hz) oeshapes = Again, eqation (C-) can then be ncople sing the normal moes per the metho in Reference 4.

22 he moal transient analysis can be then be performe on the moal coorinates sing the ramp invariant igital recrsive filtering relationship in Reference 5 for the case here the acceleration of one or moal egrees-of-freeom have been prescribe. he igital recrsive filtering relationship for the acceleration is x i exp cos x exp n i n x i exp nsin f f f m i i i (C-) here x f i i n m is the acceleration at step i is the force at step i is the time step is the natral freqency in (raians/sec) is the amping ratio is the mass Note that the mass is eqal to one for the case of a system hich has been ecople sing massnormalize moes. he ampe natral freqency is n - (C-3) he reslting moal responses are then transforme into physical responses per the metho in Reference 4.

23 he final isplacements are fon via f (C-4) (C-5) igre C-4. Again, each moe has 5% amping. he freqency response fnction is shon in igre C-5 for reference. 3

24 igre C-5. No assme that the sample system is sbjecte to a G, 4 Hz sine base excitation. his problem col be solve exactly sing Laplace transforms. Nevertheless, the ramp invariant igital recrsive relationship ill be se. he reslting acceleration at the free en is given in igre C-5. 4

FREQUENCY RESPONSE FUNCTION ANALYSIS OF A MULTI-DEGREE-OF-FREEDOM SYSTEM WITH ENFORCED MOTION

FREQUENCY RESPONSE FUNCTION ANALYSIS OF A MULTI-DEGREE-OF-FREEDOM SYSTEM WITH ENFORCED MOTION REQUENCY RESPONSE UNCON ANALYSS O A UL-DEGREE-O-REEDO SYSE WH ENORCED OON By om rvine Email: tomirvine@aol.com August 6, Variables f Q u u u f i i ass matri Stiness matri Applie forces orces at riven noes