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Mecway
FEA
Modelling of an Drawn Archery Bow
I'm new to Mecway, but have used many other FE codes in the past.
I am trying to apply pre-tension to a model of a simple Bow (as in Bow and arrow). At the moment I'm trying to get sensible results prior to trying to apply pre-tension to the bow string. Any ideas on how to apply pretension would be appreciated.
I attach several files below, most using the internal solver and one uses the CCX solver. None of these have any pre-tension
I am getting some strange results. Some of these are discussed below. Note the material properties and dimensions are just guesses to get some results! This analysis is just experimental before I switch over to the main problem I wish to look at.
Using the Internal solver
Case 1) Using a Tension only Truss property for the bow string, the solver gives the following message:
00:00 Iteration 2
00:00 Assembling matrix
00:00 Applying loads
00:00 Applying constraints
00:00 Solving matrix
00:00 Error: P-4 May be underconstrained or have zero material property values.
00:00 Failed
I can't work out how to correct this - bow case 1 Truss linear.liml
Case 2) I converted the bow string to 2 bending beam elements. The connection to the bow uses flexible joints to remove any rotational support. This model gives results that look sensible apart from where fingers pull the string due to the internal beam bending action
As the string is modelled by just two beam elements, Mecway doesn't show any internal beam deflection.
bow case 1 linear.liml
If I increase the number of elements in the bow string the resulting total deflection is the same (which is as expected) and this displays a very curved bow string!
To avoid the error of internal beam bending at the finger pull position, I tried two cases:
Case 3) I added a flexible joint at the finger pull position, but the results were very silly.
Case 4) I separated the string at the finger pull position into two nodes, so the bow string was not continuous at the finger pull position. The pull force was distributed to these two nodes and a constraint equation was applied to ensure both dX values were the same. The resulting deflections are very silly, and I can't see what I've done wrong.
bow case 2 node string linear.liml and see figure below:
Case 5) Finally I have tried Case 2 with the CCX solver, still linear static 3D.
Using line2 elements gives very silly results.
Switching to line3 elements for everything gives similar results to the Case 2 model. This uses bending elements for the string. The modelling of the bow to string connection is via constraint equations at adjacent nodes at the ends of the bow and string.
bow case 1 linear ccx.liml
Modelling the middle of the bow string as non-continuous (as case 4) with constrained nodes in X gives silly results.
Also I have to constrain all nodes for out of plane actions (Z direction) otherwise there is significant out of plane deformation despite every node have Z=0 exactly. I'm guessing this is numerical accuracy issue, but not sure?
Any advice on what I have done wrong would be appreciated.
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Comments
Case 1) You don't need tension-only if you know it will always be in tension. In either case, truss elements won't work because the pulling point is unconstrained in the direction of the applied force so no linear static solution is possible (Newton's 3rd law). In linear analysis, the deformed shape is not used to find the stiffness so the string is effectively always vertical.
> Case 3) I added a flexible joint at the finger pull position, but the results were very silly.
Cases 3 and 4 would have the same unconstrained problem as truss elements.
Some ways to avoid the lack of X constraint:
- Use a prescribed displacement instead of a force then read the force back out of the reaction force in the solution.
- Use beam elements like case 2 but with section properties that give it low bending stiffness and correct tensile stiffness. Eg. a small cross-section area and large Young's modulus. You can't make the bending stiffness exactly zero because it'll again be underconstrained.
However, a linear analysis will still be wrong because it neglects the stiffening of the string as its orientation changes which is the important effect. You have to use non-linear analysis and CCX. Those same constraint issues and possible solutions occur there too.
The bow is under a combination of bending moment and Compressive force.
I also agree with Victor that’s a nonlinear problem.
Second order elements can capture better the end surface normal. That’s important for example if you consider Only compression or symmetry BC.
Even Nonlinear can fail if you apply a horizontal load on a midspam node of a perfectly vertical member without bending stiffness. That’s because the member must develop infinite tensile force to counteract the perpendicular force. To solve it just give it a small offset to the midspam node.
This would be my proposal for the setup with beams and 2 spring elements for the string. (Althought I would solve the arch with solid elements).
I had forgotten about the application of a normal force to a truss element issue!
disla: Would it be possible to see you input file? I've tried to follow your approach from the figure, but have failed.
Thanks
** Generated by Mecway 24 *NODE 1,-0.1010898679595,-0.5279358078186,0 2,-0.1011483875011,-0.5310025307924,0 3,0,0,0 4,-0.1012028779094,-0.5340692537661,0 5,-0.04410311149671,-0.1385981326659,0 6,-0.100960740244,-0.5218023618711,0 7,-0.1012998792379,-0.5402100743905,0 8,0.001,-0.5740251485476,0 9,-0.07591226768785,-0.2813896655234,0 10,-0.09512112901065,-0.4269994380533,0 11,-0.1015447036299,-0.5740251485476,0 12,-0.1013423712624,-0.5432841720412,0 13,-0.08711485918719,-0.3540175595428,0 14,-0.1003818242047,-0.5003353010549,0 15,-0.1013808157613,-0.5463582696918,0 16,-0.1014152129343,-0.5494323673425,0 17,-0.1014455629603,-0.5525064649931,0 18,-0.1014718659971,-0.5555805626438,0 19,-0.02358830016155,-0.06877489130903,0 20,-0.06154443400548,-0.2094697240707,0 21,-0.1014941221815,-0.5586546602944,0 22,-0.1015123316291,-0.5617287579451,0 23,-0.1015264944347,-0.5648028555957,0 24,-0.1015366106718,-0.5678769532463,0 25,-0.01217833618408,-0.03425640189853,0 26,-0.03422989193243,-0.1035554682315,0 27,-0.05320795885439,-0.1739028846124,0 28,-0.06911253694996,-0.2452986510411,0 29,-0.08191310364701,-0.3176593644717,0 30,-0.09151753430841,-0.3904642507367,0 31,-0.09792564329393,-0.4636231214927,0 32,-0.1012533394667,-0.5371359767399,0 33,-0.101542680393,-0.570951050897,0 34,-0.1010273189818,-0.5248690848449,0 35,-0.1004766220872,-0.5034020240287,0 36,-0.1005673873569,-0.5064687470024,0 37,-0.1006541204842,-0.5095354699761,0 38,-0.1007368219186,-0.5126021929499,0 39,-0.1008154920886,-0.5156689159236,0 40,-0.1008901314015,-0.5187356388974,0 41,-0.1007766608857,-0.5141355544368,0 42,-0.1009259396082,-0.5202690003843,0 43,-0.1009945333539,-0.523335723358,0 44,-0.1010590971699,-0.5264024463318,0 45,-0.1011196313904,-0.5294691693055,0 46,-0.1011761363288,-0.5325358922793,0 47,-0.1012771153352,-0.5386730255652,0 48,-0.1013216312042,-0.5417471232158,0 49,-0.1013620994396,-0.5448212208665,0 50,-0.006089168092038,-0.01712820094926,0 51,-0.01788331817281,-0.05151564660378,0 52,-0.02890909604699,-0.08616517977026,0 53,-0.03916650171457,-0.1210768004487,0 54,-0.04865553517555,-0.1562505086391,0 55,-0.05737619642994,-0.1916863043416,0 56,-0.06532848547772,-0.2273841875559,0 57,-0.07251240231891,-0.2633441582823,0 58,-0.07891268566743,-0.2995245149976,0 59,-0.0845139814171,-0.3358384620073,0 60,-0.0893161967478,-0.3722409051397,0 61,-0.09331933165953,-0.408731844395,0 62,-0.09652338615229,-0.445311279773,0 63,-0.09892836022608,-0.4819792112738,0 64,-0.1012286122776,-0.535602615253,0 65,-0.1015441978209,-0.5724880997223,0 66,-0.1013985202518,-0.5478953185171,0 67,-0.1014308938303,-0.5509694161678,0 68,-0.1014592203433,-0.5540435138184,0 69,-0.1014834999381,-0.5571176114691,0 70,-0.101503732741,-0.5601917091197,0 71,-0.1015199188571,-0.5632658067704,0 72,-0.1015320583705,-0.566339904421,0 73,-0.1015401513444,-0.5694140020717,0 74,-0.1008533155775,-0.5172022774105,0 75,-0.1004297272528,-0.5018686625418,0 76,-0.1005225087687,-0.5049353855155,0 77,-0.1006112579098,-0.5080021084893,0 78,-0.1006959751358,-0.511068831463,0 *ELEMENT,TYPE=B32R 1,40,42,6 2,6,43,34 3,34,44,1 4,1,45,2 5,2,46,4 6,32,47,7 7,7,48,12 8,12,49,15 9,3,50,25 10,25,51,19 11,19,52,26 12,26,53,5 13,5,54,27 14,27,55,20 15,20,56,28 16,28,57,9 17,9,58,29 18,29,59,13 19,13,60,30 20,30,61,10 21,10,62,31 22,31,63,14 23,4,64,32 24,33,65,11 25,15,66,16 26,16,67,17 27,17,68,18 28,18,69,21 29,21,70,22 30,22,71,23 31,23,72,24 32,24,73,33 33,39,74,40 35,14,75,35 36,35,76,36 37,36,77,37 38,37,78,38 39,38,41,39 *ELEMENT,TYPE=SPRINGA 34,8,3 *NSET,NSET=Reaction_Bow 11 *NSET,NSET=Reaction_String 8 *ELSET,ELSET=Single_Spring 34 *ELSET,ELSET=bow_middle 1 2 3 4 5 6 7 8 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 35 36 37 38 39 *ELSET,ELSET=bow_end_taper 9 10 11 12 13 14 15 16 *MATERIAL,NAME=wood_30x10 *ELASTIC,TYPE=ISOTROPIC 10000000000,0.1 *MATERIAL,NAME=wood_20x5 *ELASTIC,TYPE=ISOTROPIC 10000000000,0.1 *BEAM SECTION,ELSET=bow_middle,MATERIAL=wood_30x10,SECTION=RECT 0.03,0.01 0,0,1 *SPRING,ELSET=Single_Spring 3.000000000000E+005 *BEAM SECTION,ELSET=bow_end_taper,MATERIAL=wood_20x5,SECTION=RECT 0.02,0.005 0,0,1 *BOUNDARY 3,3,,0 8,2,,0 8,3,,0 11,3,,0 11,2,,0 11,4,,0 11,6,,0 *AMPLITUDE,NAME=Ax_11_1 0,0 1,-100 *EQUATION 2 8,1,1000,11,1,1000 *STEP,NLGEOM=YES,INC=10000,AMPLITUDE=STEP *STATIC 0.0001,1,0,0.05 *CLOAD,AMPLITUDE=Ax_11_1 11,1,1 *NODE FILE,GLOBAL=YES,OUTPUT=2D U,RF *EL FILE S,NOE *EL PRINT, ELSET=Single_Spring ELSE *END STEPI managed to get that working after recreating the Mecway Analysis parameters etc.