GEO-SLOPE International Ltd, Calgary, Alberta, Canada Vibrating Beam

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1 GEO-SLOPE Internatonal Ltd, Calgary, Alberta, Canada Introducton Vbratng Beam Ths example looks at the dynamc response of a cantlever beam n response to a cyclc force at the free end. Closed form solutons are avalable for ths case, whch can be used to verfy the QUAKE/W formulaton and code. Problem descrpton The beam s fxed at the left end, and s free to move on the rght end. A cyclc force s appled at the rght end, as llustrated n Fgure. Fgure Forced vbraton of a cantlever beam Based on closed form solutons for a case lke ths by Weaver (990), the steady state response of the beam can be expressed as: y Pl X ( X ) xl snt kl where: P s the magntude of vbraton force, L s the length of the beam, E s the stffness of the beam, I s the moment of nerta, X are the characterstc functons representng the normal modes of vbraton of the beam, β are the magnfcaton factors and k l are the roots of the system frequency equaton that relate to the crcular frequences of the beam. The characterstc functons for the above system can be expressed as: X = cosh kx - cos kx- (snhx- snx) and: cosh kl cos kl snh k l sn k l where snh and cosh are hyperbolc functons, and x s the dstance from the left to the rght end of the beam. The frequency equaton for the system s: QUAKE/W Example Fle: Vbratng beam.docx (pdf) (gsz) Page of 5

2 cos kl cosh kl GEO-SLOPE Internatonal Ltd, Calgary, Alberta, Canada The roots or solutons for the equaton are: Solutons of frequency equaton for the beam The magnfcaton factor s: / p 5 6 k I Where P s the crcular frequences of the system that can be expressed as: p k A where A s the cross sectonal area of the beam and s the unt mass of the beam. The vertcal dsplacement on the rght end of the beam s: y xl Pl snt X xl k l where (X)x= represents a value of dsplacement at the rght end of the beam for a normal mode. In ths case: X x l Therefore, the vertcal dsplacement at the rght end of the beam can be re-wrtten as: y xl Pl snt k l Pl... sn t Pl snt QUAKE/W Example Fle: Vbratng beam.docx (pdf) (gsz) Page of 5

3 GEO-SLOPE Internatonal Ltd, Calgary, Alberta, Canada The closed-form solutons usng these equatons are as shown n the next table. Beam parameters and numerc results Parameter and Soluton Case Case Weaver s Soluton QUAKE/W Weaver s Soluton QUAKE/W Force magntude, P Force frequency Young s modulus, E 0,000 0,000 0,000 0,000 Posson s rato Length of the beam, l Area of cross-secton, A Inerta moment, I / / / / Unt mass Dampng rato p p p Ampltude of (y) at x= peak average after 5 seconds 0.08 peak average 0.00 wth phase shft at every 7 seconds QUAKE problem defnton Fgure shows the fnte element mesh. It s a plane stran and undamped dynamc analyss. The force s defned by a cyclc "Y-force vs. tme" boundary functon, as shown n Fgure. Fle Name: Vbratng beam.gsz Elevaton Dstance - m Fgure Vbratng beam problem QUAKE/W Example Fle: Vbratng beam.docx (pdf) (gsz) Page of 5

4 GEO-SLOPE Internatonal Ltd, Calgary, Alberta, Canada Y Force functon Y-Boundary Force (kn) Tme (sec) Fgure Forcng functon Case In ths Case, the fundamental frequency of the beam s about.06 rad/sec, whch s a lttle more than double the forcng frequency of 6.8 rad/sec. Ths makes the beam vbrate at ponts other than at resonant ponts. Fgure shows the beam vbraton at the rght end for Case. The ampltude from the hand calculatons s The QUAKE/W ampltude after 5 seconds s between and , or about Y-dsplacement w th tme Y-Dsplacement (m) Tme (sec) Fgure Case beam vbraton at the rght end 5 Case The second case s where the fundamental frequency s about 7.0 rad/sec, whch s slghtly more than the forcng frequency. Ths makes the beam vbrate close to the frst resonant pont and results n a response pattern that more clearly shows the phase shft behavor wthn a forced vbraton. QUAKE/W Example Fle: Vbratng beam.docx (pdf) (gsz) Page of 5

5 GEO-SLOPE Internatonal Ltd, Calgary, Alberta, Canada In the second case, the stffness E s reduced from 0,000 kpa (G = 6,5) to 0,000 kpa (G =,5) (ths s the only change requred to re-run the problem for the second case). The closed form ampltude s The QUAKE/W average ampltude for three successve peaks s about 0.00, as s evdent n Fgure 5, wth a phase shft approxmately every 7 seconds. 0.0 Y-dsplacement w th tme Y-Dsplacement (m) Tme (sec) Fgure 5 Case beam vbraton at the rght end 6 Concluson As s evdent from ths comparson, the QUAKE/W results are n good agreement wth the closed form solutons, ndcatng that QUAKE/W s correctly formulated and coded for ths smple vbraton problem. QUAKE/W Example Fle: Vbratng beam.docx (pdf) (gsz) Page 5 of 5

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