Advanced Computational Analysis

Transcription

1 Advnced Computtionl Anlysis REPORT REPORT NO: S Revision A Title: Closed-Form Anlysis Of Forces And Moments In Bungee Trmpoline Structure Client: Mr Jmes Okey Author: Dr M Lcey BSc PhD CEng F I Mech E ADIPS Registrtion No.: Dte: 26 th Februry A, Min Rod, Gedling, Nottinghm NG4 3HP Tel (0115) e-mil: info@c-consultnts.co.uk

2 Summry The results of the closed-form clcultions for the lods trnsferred to the system from the bungee ropes substntite those predicted by the finite element results. Whilst the forces nd bending moments predicted by this nlysis for the luminium poles re slightly different from those predicted by the finite element nlysis, it is cler tht the finite element model is significntly more ccurte nd provides close pproximtion to the true system. Pge 2 of ACA S Revision A

3 Index Summry... 2 Method Of Anlysis... 4 Results... 5 Conclusions... 6 Figures... 7 Clcultions Pge 3 of 16

4 Method Of Anlysis The nlysis detiled below provides closed-form estimte of the forces nd moments which occur in bungee trmpoline structure, when single pssenger bounces verticlly with pre-defined verticl ccelertion. This pplied force is ssumed to be best estimte of the mximum force, since t the bottom of the descent the pssenger is rrested by trmpoline. The forces nd moments derived in the nlysis detiled below re bsed on liner system of constrints nd stiffness nd number of ssumptions hve to be mde in order to provide n estimte of these forces nd moments. Assumptions i) The structurl system is liner nd ny frictionl resistnce is excluded. ii) The motion of the pssenger is purely verticl. iii) Any restrint ropes offer no resistnce to the deflexion of the structurl frme. iv) The self-weight of the structure is omitted. v) The nlysis ssumes tht only one pssenger is ctive on the overll structure. Any dditionl lod from other pssengers bouncing simultneously with the first pssenger cn be derived from kinemtic summtion of the forces in clcultion sheets 1 to 4. The forces nd moments in the luminium poles re verified in clcultion sheets 1 to 7 below. The results of the closed form clcultions re compred with finite element results in tble 1.0 below Pge 4 of 16

5 Results Item Closed-Form Result Finite Element Result Verticl Rection At Pulley R V (N) Horizontl Rection At Pulley R HA (N) Horizontl Rection At Pulley R BA (N) Verticl Force At Motor R VM (N) Horizontl Force At Motor R VM (N) Axil Force In Pole R X (N) Bending Moment In Pole About z-z xis M ZZMAX (Nm) Bending Moment In Pole About y-y xis M YYMAX (Nm) Tble 1 Summry Of Results For Stresses, Utilistion Fctors Deflexions And Bse Rection Forces Pge 5 of ACA S Revision A

6 Conclusions The results of the closed-form clcultions show tht the resolved forces t the top of the luminium poles re within 10 % of those predicted by the finite element nlysis. In ddition to this the resolved forces t the motor re within 20 % of ech other. The smll discrepncy in the results is due to the method used to model the pulleys t the top of the luminium poles in the finite element nlysis. The finite element nlysis uses torsionl spring elements with low stiffness vlue, which enbles the solution to converge. In using the torsionl springs some of the tension in the bungee rope is removed from the bungee rope nd trnsferred to the luminium poles vi bending moment. However, since the results re within 20% of ech other it is cler tht the finite element model is close pproximtion of the true sitution. Whilst the closed-form results for the resolved forces t the pulley nd motor substntite those predicted by the finite element nlysis, the xil force nd bending moments predicted in the luminium poles re significntly different. The discrepncy is primrily due to the ssumptions mde bove in tht the system cts linerly nd constrints re kinemticlly sufficient i.e. the effect of the cbles hs been neglected. In relity these cbles would effectively ct s prop for the luminium pole thereby significntly reducing the bending moments predicted by the closed-form clcultions. In ddition to this, since the cbles re ngled downwrds the tension in the cbles would increse the xil force in the luminium poles, s predicted by the finite element nlysis In conclusion, the results of the closed-form clcultions for the lods trnsferred to the system from the bungee ropes substntite those predicted by the finite element results. Whilst the forces nd bending moments predicted by this nlysis for the luminium poles re different from those predicted by the finite element nlysis, it is cler tht the finite element model is significntly more ccurte nd provides close pproximtion to the true system. Dr M. Lcey Pge 6 of ACA S Revision A

7 Figures Figure 1.1 Typicl View Of Bungee Trmpoline Pge 7 of ACA S Revision A

8 b R HA R V RHB T c T R HMB R VM d R HMA P Figure 2.1 Key To Loction Of Symbols For Bungee Trmpoline Arm Pge 8 of 16

9 e R HA R V RHB h f i x y z g Figure 2.2 Key To Loction Of Symbols For Bungee Trmpoline Arm Pge 9 of 16

10 Clcultions Advnced Computtionl Anlysis 4, Min Rod, Gedling, Nottinghm. NG4 3HP Telephone Client : Airmx Infltble ACA Contrct No : S Dte : 27 th Februry 2013 Description : Closed-Form Clcultions For Triler Mounted 4-Person Bungee Trmpoline ACA Engineering Consultnts Dimensions nd Lods 1) Dimensions m b m c m d m e m f m g m h 4.08 m 2) Lods The combined mss of the pssenger nd hrness is 90 kg. It is ssumed tht the mximum verticl ccelertion of the pssenger is 2 g. Hence the totl verticl lod P into one side of the system is P N Determintion of rection forces t pulley 1) Tension in bungee rope. Referring to figure 2.1 tension in bungee rope is given by Tsin( θ ) P P is hlf the totl lod from the pssenger tn( θ ) T 2 b 2 + sin( θ ) b P cosθ 2 + b 2 b 2 b 2 + T P 2 + b N Prepred By: R. Anderson ACA 2013 Checked By: Dr M. Lcey Section: 1 Sheet: 1 of 7 Pge 10 of ACA S Revision A

11 Advnced Computtionl Anlysis 4, Min Rod, Gedling, Nottinghm. NG4 3HP. Telephone Contrct No. S ) Totl verticl rection t pulley The totl verticl rection t pulley is given by R V T sin( θ ) + T sin( γ ) ACA Engineering Consultnts tn( γ ) c sin( γ ) b 2 + d 2 e R V R V cos( γ ) b 2 d 2 + T + T 2 + b 2 T 2 + b 2 + c c c 2 + b 2 R V P N Prepred By: R. Anderson Checked By Dr M. Lcey ACA 2013 Section: 2 Sheet: 2 of: 7 Pge 11 of 16

12 Advnced Computtionl Anlysis 4, Min Rod, Gedling, Nottinghm. NG4 3HP. Telephone Contrct No. S2149 The horizontl rection R HA t the pulley is given by ACA Engineering Consultnts R HA T cos( γ ) cos( φ ) tn( φ ) b sin( φ ) d b cos( φ ) b 2 + d 2 d b 2 + d 2 T b 2 + d 2 R HA T d R HA d b 2 + d 2 R HA P d 2 + b N Prepred By: R. Anderson Checked By Dr M. Lcey ACA 2013 Section: 2 Sheet: 3 of: 7 Pge 12 of 16

13 Advnced Computtionl Anlysis 4, Min Rod, Gedling, Nottinghm. NG4 3HP. Telephone Contrct No. S2149 The horizontl rection R HB t the pulley is given by ACA Engineering Consultnts R HB T cos( γ ) sin( φ ) + T cos( θ ) R HB T b 2 d 2 + d b 2 d b 2 b 2 + R HB d b T b 2 P 2 b 2 + R HB d + b 2 + b 2 d 2 + b 2 b P N The forces t the motor cn be stted from equilibrium nd the use of symmetry ( ) R VM 2 R V. P R VM 2P c 2 + b N Prepred By: R. Anderson Checked By Dr M. Lcey ACA 2013 Section: 3 Sheet: 4 of: 7 Pge 13 of 16

14 Advnced Computtionl Anlysis 4, Min Rod, Gedling, Nottinghm. NG4 3HP. Telephone Contrct No. S2149 ACA Engineering Consultnts R HMA 2R HA R HMA 2 P d 2 + b R VMB N The pulley forces must now be resolved into the locl co-ordinte system of the bungee pole, to determine the xil force, sher force nd bending moment distribution in the pole. In figure 2.2 the x, y, z co-ordinte system is locl co-ordinte system for the pole 1) Resolving forces into x-direction ( ) sin ω R X R HA sin( α) + R HB cos( α) tn( α) g e sin( α) ( ) + R V cos( ω ) g cos( α) g 2 + e 2 e g 2 + e 2 tn( ω ) g 2 + e 2 f sin( ω ) g 2 + e 2 cos( ω ) g 2 + e 2 + f 2 f g 2 + e 2 + f 2 ( ) R X R HA g + R HB e g 2 + e 2 g 2 + e 2 g 2 + e 2 + f 2 + R V f g 2 + e 2 + f 2 Prepred By: R. Anderson Checked By Dr M. Lcey ACA 2013 Section: 3 Sheet: 5 of: 7 Pge 14 of 16

15 Advnced Computtionl Anlysis 4, Min Rod, Gedling, Nottinghm. NG4 3HP. Telephone Contrct No. S2149 R X R HA g + R HB e + R V f g 2 + e 2 + f 2 ACA Engineering Consultnts N xil force in pole 2) Resolving forces in y direction ( ) cos ω R Y R HA sin( α) + R HB cos( α) ( ) R V sin( ω ) ( ) f R HA g + R HB e R Y g 2 + e 2 g 2 + e 2 + f 2 ( ) f R V g 2 + e 2 g 2 + e 2 + f 2 R HA g + R HB e R V g 2 + e 2 R Y g 2 + e 2 g 2 + e 2 + f 2 ( ) ( ) ( ) 287. N y direction sher in pole Prepred By: R. Anderson Checked By Dr M. Lcey ACA 2013 Section: 3 Sheet: 6 of: 7 Pge 15 of 16

16 Advnced Computtionl Anlysis 4, Min Rod, Gedling, Nottinghm. NG4 3HP. Telephone Contrct No. S2149 4) Bending moment in pole The bending moment in the pole bout the z-z xis is given by M zzmx R Y h ACA Engineering Consultnts h R HA g + R HB f R V g 2 + e 2 M zzmx g 2 + e 2 g 2 + e 2 + f 2 ( ) ( ) 3112 Nm The bending moment in the pole bout the y-y xis is given by. M yymx R Z h ( ) h M yymx R HA e h R HB g g 2 + e 2 ( ) Nm Prepred By: R. Anderson Checked By Dr M. Lcey ACA 2013 Section: 3 Sheet: 7 of: 7 Pge 16 of 16