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Kinematics of Agricultural MachinesStanislav Barton Mendel University of Agriculture and Forestry in Brno Zemedelska 1, 613 00 Brno, Czech Rep. barton@mendelu.cz & Karel Englis Unversity in Brno Sujanovo nam. 1, 602 00 Brno, Czech Rep.

This worksheet demonstrates the use of Maple to study the kinetics of an agricultural machine. Maple is used to model the trajectory, velocity and acceleration of machine components, which are plotted and animated.

restart: interface(warnlevel=0): with(plots):

Dung is falling down through a stall grate where it is cleared away by the machine in Figure 1. In this article we shall demonstrate how to use Maple to study the kinematics of such a complicated machine. The propelled end of the supporting bar is connected to the chain running around two teeth wheels and moves ahead with constant velocity. The lead end follows the guiding bars. Scrapers are fixed to the supporting bar. We have to study the trajectory, velocity and acceleration of the business ends of scrapers. Results will be plotted and animated.Figure 1.MFNWtKUb<ob<R=MDLCdNNZlk:lJ>H:XQRht?KR<O`Lo\\jyyyyKF\\KB<LJ\\kyyyyK:::::RycdpEaJFnC_b;_drOugimCpl]A_Ffbg>xyyikWZWPxL@i;qfM>]jYtSHmXXtO@dfvlaF[=n^T>fjPl`>t?>`ZvZ=Nhj>]JXj;FaR^[BIr:`]Fn`FVfeg\\pviyIr\\O^JX_A_]wqyEVj]g_?_u:oZ]vZ^QcLvx_o[^IZsPfpF`AvyD^vDQ`ANv:`]FntmNh>Fl=Nh_XdC^^I>bh`cLAjNg\\Bi\\fviW_][xqyOkEa]<veHVggIw=f_I^q`@bHFvV>lmXmN_mFigyA_A_mV^\\ZGg?FqxnZEGZtn`KAnVAro_f@WZRPyOosbGuC@av_dHFw=W]C?mdYbban<xego^iyw]?OstJKXj;EJWb;CIRYYRUBl=hL]rc=FASb]eByyGJCIFOHfis<QFjaIciB\\sxTcd=IF>=gV[Rhif\\sTKKXYOSC_YoGrswtd]gdgU;yEbqBBeUaQFQuClUX_Ox;uR@SuPIfl=TtwHD_Uo;sgMGOeGn=U=aW<evn]uQOpHHriPRwEM>moPdlulOJ`Tc`XZlo]@UHQp^trIdMLxJZnn>Ob:_rDG[DAsrnfKqsaQwsXbn^h[y^v_kAfhr_aLwlDIgKX^VIk>pc?gsR>\\O_^dOf=ocT@w;WqOwk\\owOW`vOZ`Y\\MWp>ftlycR_bHNvPF[<AgqW^l__dv`YqtKvr?xtlQg_Vc`>k_q]KGamypaVjHAr_f^WNueYlT>soA^lGnBimYo_M`h_AkBFkL@vXWy:ogFn^hOqxPlr^^h`cpQ[OXofxsLQfsXcCX[vOZh>uy>maxyJXql>eDNhpGkuPeYXpwiqBv]<QpDwpe`x[arB>qEXhFF[BIcO^b>pv;nnZXiwpiVFniyiFVfFAeGFmoWighkrv[Tnx=fp^_o;`\\OFjHyvQ@^]HkJGqbibLX`koumpefgeO@kLxqfYfgNjUQZmacrPfYpaV@kMF`;FmJIt=xlm_o=`e`^yeG\\^qcCXbjrnib=ec<EfMIi:Wt;wiJKTQKt[ETBie_kkKPlT<nEPONTSpEYKXV[lw\\YWIXRa\\XCyow@lr@M^aVdXTwqtB]KoHRQtw\\DvVUx[tX\\lSa\\tBQy_yvZXqiLSZUtjtRaMkAEkUmxLtMFUwZXsC]PUiT`DWJAWVXXoQp>iylEsharhaU=qMk<Pw<ljmL\\XnTQJKXSn]ouElM\\LwqKHhs=TSGARmMybUYfmSPaK?@JsDlTMW^YyL`u]UL`qUoMoueLftVY<vVeuwIMRiRuDlUHkEPTsxKaPTDurjdl>psAmmvMkATXHUKm]oaLvxtYMdrVyr]ToUHxxiwouO^qUw`M\\tuApufLX[MOqmNhLuNAN_inn<mIyQ`ILqQmdAU[TXvPOqERV]jFiYx`OquwkmumYQgyXYLKddM]`PC]qNAtKYn;<sgIkUaKf=PmuutayZXuPEXkUoUXl^Em[Xx^lswIPsdJq@xgxpWePl\\LcPnIEYwalHuWE]ng`uGqOgtvtdMYLNdMnHUjslqnIvd=lpasspwH\\uyUY?pwuypeXSwYTshSKEuNpkM<l?ioxhmsXupYsb\\vbAxHqK]dkgdunQypHPL]Sq\\vETpKDQTHkUaL_]kZtyQejryYTyMIDvf<TdHQgIP\\iuidPshwtIlALvGxWolObqSYqu_LLm@sgHKedru\\SxYXslwJmKCXjAaOHPXfpp\\tthqmjUPNLU[xQbqvY\\gQamipq^whiHZx>xJpdYPw`fdXvgWI]lW^mGk`GhEHnYqdeAjSVesgxbfuZGcl?nkGbm@a<gvhNcVPiVa^VAsuOaJG`Envs`bHgnEhd]XPyB\\qUfSXHofA]fIsDkGvVOR^[UQuh`IDAEb]?yNArwIWD_cmIwRug]orwwH=?XGmhKwg>yiNKFgQb>UfgsXH=TlWrNIDAohvkFXqwxIGvKXjywsKV>?IsKesYYJkHfUxJWEIesk_ueKTp]iiMg`uieSvWogOIufErSMV^qusgYSKWwmuIud\\shBwTJYByUUEUB[?tAEyo;SVKVR_VgcGOGicUE@QBWCUlyt=yfMIfE]SSesWCXiauMMvXor]SYC_E^qyVYGWqT^;I`WtqEGb=XlYRJAiAiTHwR`qXhit[cXVMSqICFAchiRoQvuwt>IVeaEMmrdQVHGWHysDuRQ=rwaxDGTcqxS=gQGFOeCpQb<Wi>aRWWubSrqkfmEVZWb^ava_XPWstWRLybSMv:]u=qcOGGgey;uvAoYZIUeUSewVaOEdAUKkr?]btIE<iC@mNDQlXmqHuR:uxeqMmPPQ<ol=NhLOKyNpMkMuu?DnWItLTsOmN=qJsXyOuvePmclmhQYQ@s_IqOqmMuvpIsehQUaPKQSayMBIPI]O^]ofPNAqSjITcqkUttPEjD\\ue`XtXR>ajrlTYLvXmXDLrihpsmutdoRhL:euhlRJaNnELJioYpr\\AcPnpsGsuWoIqwj^bZ_hc_wHYhTIunf\\eGuWXcYis>ouuVm`PdPxcMxlegufwbLPsjqhRYi>o_MPs_@wVilmNqNOq]I\\>A]_XxBydKnrwN\\EiabYeQ>w^PaC`uQI_LOskxdlIyZ>k]?[nibHPbsYtPygpQaBYpBIm_FjRAak`_QGmB>\\QgpDifKIas`yuXcQgwA?\\m^mEItZnk=wklN^QHjsHy^^yhnnui]bV`AhbHnhbqvHgZkgxd`epHs^ppuhtAXdpQmOGw<v\\A?hwFc[_vNo_??y<wbA^uHX[f_oJIgB_]A_mYxd_AySXeUohTO`sPi<OsDf]BXshPhCiefPxGNlwgmHQi>FleaZ\\f^pn^Uq_[YrDXx=nsv>^:@]WG[\\Vb:ymehfU?wPisZ^aDwnma``v\\DNpcxxG?kugkMhhQqimXs^QxBNvvo_AXvB`b?YvtaxrF[GYjVW[kNwDfmmA[\\VZngppid]G\\`@qqOkDodYQjgqitNq]``SNaWxhpyn>v_IvkM?dmvbqWhdxc>yxkfuWIyxapRpo`i`ppj>poshnZYabqvW^cuhnsXvs`]xX\\B@pBIqR^jVYreGwqxt[`iOpwpwp_ahSwff^h:y[dvh@pwbydI^lDa^_Xj^Ip>^o]WaG>hp`csp[cGs_^sH`jH^r_wsRWumN^]ogbfy_phywZvibB^hwixO>l]OliOnsGs>Qqb?xqQe@h`oXt;wlyGgjqr_yd=f]eYjsYjqygXa_S@eXOqshxwPauoxD@wiXwnIv:fm_GcxoqEAuvAn=QlCYZ\\vdYNhQf_BVgRayj?u`NhmqqoO]\\?nMNagQh[?eAHw=qcManO_vx_xGyd>hawWyeYw>ootfl]G\\sHyCwpZXjmguXnkZ@[_Xt;IZZ^_WgvTGqE`h\\ggyGjHa^lWwsYrCyrxW]M^hb@`iYsNYyQFq@OtAF\\sHk]wpm?iqoyjnvOAftN_eAj_N^QHiS`jw`xi^hE_^AwfYVwWxk@puI_y<`w[nnCXfih[uqiagjHamLydtfZGflvXZKN_Qw`UomEo^DpkrH]M>kK?vx>_B`]MQ^CYZC^jBouHXirXs^g\\rXdUOpVgf<VoRgvrah;QhgxwsA[\\Fe\\VdxXfi@\\sn`^Qkyvf_OywWmG_myX:gfkstRCudKbRogcedEQGlUrcQUCKCOqGwexceyHEdEgEV?xi_DHovrsrhmUVkuX;RnwH`cyc;uU;cE_W]]cDiVlegsgRhAvXEsCGTawb;QENEilOIQsIrmrVAGWaxmUXQqdmOtC?R_WbBiTXSskUcVyWukvkUwVafn]hXaxtegSIxDuBKWvPEiVKHAah?_VuoB<]Xi_X@IuRgiLIWq[XGKDaMSE]R^SIfaskkcLyfbShVsVWoudYWwOh=EG]SRYyRaqWmaCW[HhSc]eGnOIQsxU]sYWSWweUyuZexiey[iYUmYhitaoG?qXeWuq]rn?DAcTLSskQcVqw;KwOii>]xu?yqqesCh<uvQEslCiOyX][c[_TneS[MW]WyAghaug@SHX_reAIHqdvOiTSDloSa[X;_woUiwGIo?fbODL[H\\kU_aXxgXnkdl?WxSsouYsQeluUkmIQ;wWgiHyHbkhLssGIDiEUWWSj]gf]sC_SImrc=xrUB[aCWwRUmuqMBKmwuIce]VLMVskF`AUAqVY=EsaujCI;ufw]Yuwr`ksWqs[aYJIDU]GqUbUOIkyrlUcWMTtArlMuC]v]Etq=I`;wvMu\\QTtgyIUX<uRsQHPUIVMwEKvcisweRmkT@oS_iEOGWGmWmMUo]T[;tcubBIiZWhgqiEiBWCHKWYvWBlqHpgTMyBLsw`kSGigLUDQYe[CcVUduifdoF^]UxCueCCYoitosXugpYtOaWnwU^CdLEGaawF]iyeehyG^UTx;tdaBZiHeoygsUSmGp[HN;IYcvZ]hQeEfexxMXy=XDowf=cmIY;EunuCcMT=oxHgt>=GqMbmAXg?r`_fdGHEiSdyWPWY`oB=oxhoW@sxTWFc=xuUFXMY@CHX]F<eFlsTpIbriDrSRQkIsyCeaXQiHvKRquuPkXbAsjwvGkyWgdRqdk=V[\\jmesv]NIPNq@SbDP;uVU<jeYR?pwpYXYeV<]W<uRcqqsdO]ek^mOnLnMDQhmuw<TDEXw@ycUsd`XH\\wyxsXeYTQrxmOAhTbQRXUPUPO@TNMMxWlXKUmuYnImUSEQ;uV[itxMMFYYlAX=updAQOdJHxU_Ql@LsmIO^xmb\\QSprcIw`@PIlj?mqsEwHENr`Th\\o]PlamPaqv_dw]yOmhMuaLFIpLxru\\qPpp>\\PCHPI@VEMUEuqomkquqVpMhaQG@VLxo<dQsdUC@oIdLXQtlhyKpinovchqSHwox]sxfHpqBo[>ofTAi?ocwafG_ghiluPldQvJI`hXsOviopv[Iw=IflarQw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To describe its trajectory we shall use a parametric description, [x(t), y(t)]. The trajectory consists from the four separate segments - two line segments and two semicircles. In this case it is best to use piecewise functions. The whole problem can be solved analytically, but Maple's outputs will be quite long, so we shall assign numerical values to the variables. The SI units will be used. The origin of the coordinate system is in the center of the first teeth wheel. o = turns of the first wheel in one second and pi = numerical value of NiMlI1BpRw==.Su:=[Cx=0.64,Cy=0, r=0.1925,o=2.98,Xo=2.01,Yo=-.2,X1=1.0,Y1=-.18,phi=35*pi/180,S=1.3, Ph[1]=0.1, Ph[2]=0.3, Ph[3]=1.085, Lh=0.4, R=0.1, pi=evalf(Pi)]:assign(Su); omega = angular velocity of the first teeth wheel and v = peripheral velocity of the first wheel = sliding velocity of the chainomega:=2*pi*o; v:=omega*r; At first we shall prepare partial functions describing trajectory of the propeller end as a parametric function of time L = length of of the individual trajectory segments and later cumulative length of the trajectory segments L:=[Cx,r*pi,Cx,r*pi]; L:=[seq(sum(L[i],i=1..j),j=1..4)];T = times of the transition from one segment to the otherTau:=map(u->u/v,L); Tf = duration of one periodTf:=evalf(subs(Su,Tau[4])); motion to rightFx[1]:=v*t; Fy[1]:=r;right semicircleFx[2]:=Cx+r*cos(pi/2-omega*(t-Tau[1])); Fy[2]:=r*sin(pi/2-omega*(t-Tau[1])); motion to leftFx[3]:=Cx-v*(t-Tau[2]); Fy[3]:=-r; left semicircleFx[4]:=r*cos(3/2*pi-omega*(t-Tau[3])); Fy[4]:=r*sin(3/2*pi-omega*(t-Tau[3])); Now we can build piecewise functions Xi(t) and H(t) describing trajectory. Later we shall handle with them very simply - as with symbols.Xi:=piecewise(seq([t<=Tau[j],Fx[j]][],j=1..4));Eta:=piecewise(seq([t<=Tau[j],Fy[j]][],j=1..4));This plot command we can use as a graphical test of the correctness.plot([Xi,Eta,t=0..Tf],scaling=constrained);

The lead end moves along two lines. The first one is described by two points. The first point is the point of intersection of both lines - here the lead end changes direction of its movement. The second point describes end's position of the first leading line. At first we have to know the equation describing this line,line - general equationp:=y=k*x+q; p1 = parameters k and q describing the first linep1:=solve({subs(x=Xo,y=Yo,p),subs(x=X1,y=Y1,p)},{k,q}); The second line goes through the point of intersection and forms an angle NiMlJHBoaUc= with the first line. We have to determine its parameters.k=subs(tan(alpha)=k,p1,expand(tan(phi-alpha)));q=solve(subs(%,x=Xo,y=Yo,p));p2 = parameters k and q describing the first line p2:={%,%%}; We shall use again a piecewise function to describe the position of the lead end as a parametric function of time. At first we have to determine when the lead end crosses the vertex, because at this time functions will be changed. The simplest method is a numerical solution, based on graphical output. The length S of the supporting bar is always constant. As the propelled end moves along its trajectory [Xi(t), H(t)] sometimes the distance of the vertex and propelled end has to be equal to the bars' length. So we can compare the distance of the propelled end from the vertex with the bar's length and to plot this as a function of time. e = function comparing distance with bar's lengthe:=(Xi-Xo)^2+(Eta-Yo)^2-S^2:zero points of this graph are times of crossing thru vertexplot(e,t=0..Tf); numerical solution. At time interval Float(2129733337, -10) <= tau <= Float(3254225287, -10) the lead end moves along second line.tau:=[fsolve(e,t=0.2..0.3),fsolve(e,t=0.3..0.4)]; Now we have to find analytical functions describing the lead end's position. Generally this end moves along the line described by the parameters k and q, which will be substituted by the parameters from the variable p1 or p2. The propelled end has position [X,Y], which will be later substituted by Xi(t) and H(t) and the distance of both ends = length of the bar has to be s, later substituted by S. If the lead end has unknown coordinates [x, y], these coordinates has to satisfy the following equationcondition of the constant length, the second condition is the equation p - lead end moves along line. We have two equations in two variables. eK:=(X-x)^2+(Y-y)^2=s^2; we obtain two solutions, we have to select the correct onesol:=[allvalues(solve({eK,p},{x,y}))]; At time Float(2129733337, -10) <= tau <= Float(3254225287, -10) the lead end moves along the second line. We shall substitute t by the midpoint of this intervalsolf:=eval(subs(X=Xi,Y=Eta,s=S,p2,t=(Tau[1]+Tau[2])*0.5,sol)); and the correct solution will be that one satisfying x > Xosolf:=map(u->evalb(subs(u,x-Xo)>0),solf); This is the correct solutionsol:=zip((u,v)->`if`(u,v,NULL),solf,sol)[]; Now we can build piecewise functions describing the position of the lead end as a function of time.xi:=subs(X=X(t),Y=Y(t),s=S,piecewise(t<tau[1],subs(sol,p1,x),t<tau[2],subs(sol,p2,x),subs(sol,p1,x))); eta:=subs(X=X(t),Y=Y(t),s=S,piecewise(t<tau[1],subs(sol,p1,y),t<tau[2],subs(sol,p2,y),subs(sol,p1,y))); Graphical correctness of result.plot(subs(X(t)=Xi,Y(t)=Eta,[xi,eta,t=0..Tf]));

If we know the position of the propelled and lead end of the supporting bar we can use only a few simple relations from linear algebra to derive functions describing their positions as a functions of time. If these functions are known computation of the elements of vectors of velocity and acceleration or their absolute values, it is a simple problem. Because the functions X(t) and H(t) and x(t) and y(t) are quite long and their derivatives will be even more complicated, we shall use the following substitutions to simplify the next computation. But Maple outputs will be still too long, so we shall not print them out.Time derivatives Xi1:=diff(Xi,t): Xi2:=diff(Xi,t,t): Eta1:=diff(Eta,t): Eta2:=diff(Xi,t,t): DSu:=[X(t)='Xi',diff(X(t),t)='Xi1',diff(X(t),t,t)='Xi2',Y(t)='Eta',diff(Y(t),t)='Eta1',diff(Y(t),t,t)='Eta2']; These substitutions will shorten outputs of the velocity and accelerationxi1:=subs(DSu,diff(xi,t)): xi2:=subs(DSu,diff(xi,t,t)): eta1:=subs(DSu,diff(eta,t)): eta2:=subs(DSu,diff(eta,t,t)): dsu:=[x(t)='xi',diff(x(t),t)='xi1',diff(x(t),t,t)='xi2',y(t)='eta',diff(y(t),t)='eta1',diff(y(t),t,t)='eta2'];We have three scrapers, so we use indexed variables. Significance of the variables corresponds to their names. The first index indicates the first scraper seen from the left.Ex:=(x(t)-X(t))/S; Ey:=(y(t)-Y(t))/S; for j from 1 to 3 do; Fx[j]:=X(t)+Ex*Ph[j]+Ey*Lh; Fy[j]:=Y(t)+Ey*Ph[j]-Ex*Lh; Vx[j]:=diff(Fx[j],t); Vy[j]:=diff(Fy[j],t); Ax[j]:=diff(Vx[j],t); Ay[j]:=diff(Vy[j],t); V[j]:=sqrt(Vx[j]^2+Vy[j]^2); A[j]:=sqrt(Ax[j]^2+Ay[j]^2); At[j]:=diff(V[j],t); An[j]:=sqrt(A[j]^2-At[j]^2); od:

We shall present only a few graphical outputs for the sake of brevity. If we shall create plots in the phase space, we shall use blue color for the x - axis and black color for the y - axis. The most thick line will indicate the first scraper.Phase space plot: [x(t), Vx(t)] and [y(t), Vy(t)].plot(subs(dsu,DSu,[seq([Fx[j],Vx[j],t=0..Tf],j=1..3),seq([Fy[j],Vy[j],t=0..Tf],j=1..3)]),color=[black,black,black,blue,blue,blue],thickness=[3,2,1,3,2,1]); Phase space plot: [x(t), Ax(t)] and [y(t), Ay(t)].plot(subs(dsu,DSu,[seq([Fx[j],Ax[j],t=0..Tf],j=1..3),seq([Fy[j],Ay[j],t=0..Tf],j=1..3)]),color=[black,black,black,blue,blue,blue],thickness=[3,2,1,3,2,1]); Phase space plot: [Vx(t), Ax(t)] and [Vy(t), Ay(t)].plot(subs(dsu,DSu,[seq([Vx[j],Ax[j],t=0..Tf],j=1..3),seq([Vy[j],Ay[j],t=0..Tf],j=1..3)]),color=[black,black,black,blue,blue,blue],thickness=[3,2,1,3,2,1]); x(t) and y(t) - one periodplot(subs(dsu,DSu,[seq(Fx[j],j=1..3),seq(Fy[j],j=1..3)]),t=0..Tf,color=[black,black,black,blue,blue,blue],thickness=[3,2,1,3,2,1]);Vx(t), Vy(t) and |V(t)|.plot(subs(dsu,DSu,[seq(Vx[j],j=1..3),seq(Vy[j],j=1..3),seq(V[j],j=1..3)]),t=0..Tf,color=[black,black,black,blue,blue,blue,red,red,red],thickness=[3,2,1,3,2,1,3,2,1]); Ax(t), Ay(t) and |A(t)|.plot(subs(dsu,DSu,[seq(Ax[j],j=1..3),seq(Ay[j],j=1..3),seq(A[j],j=1..3)]),t=0..Tf,color=[black,black,black,blue,blue,blue,red,red,red],thickness=[3,2,1,3,2,1,3,2,1]); At(t) = tangent acceleration, An(t) = normal acceleration, |A(t)|.plot(subs(dsu,DSu,[seq(At[j],j=1..3),seq(An[j],j=1..3),seq(A[j],j=1..3)]),t=0..Tf,color=[black,black,black,blue,blue,blue,red,red,red],thickness=[3,2,1,3,2,1,3,2,1]); If values of the y - axis will be multiplied by the mass of the scraped dung, we can determine requested power for each scraper or total power. plot(subs(dsu,DSu,[seq(Ax[j]*Vx[j]+Ay[j]*Vy[j],j=1..3),sum(Ax[i]*Vx[i]+Ay[i]*Vy[i],i=1..3)]),t=0..Tf,thickness=[3,2,1,3],color=[black,black,black,blue]); We can see an animation of the working machine. The whole animation is combined from three partial plots.Supporting bar and srapers A1:=display([seq(plot(subs(dsu,DSu,t=tt,{[[Xi,Eta],[xi,eta]], seq([[Fx[j],Fy[j]],[Fx[j]-Ey*Lh,Fy[j]+Ex*Lh]],j=1..3)}), color=brown),tt=[seq(Tf*i/200,i=0..200)])], insequence=true,thickness=3,scaling=constrained):Trajectory of the scrapersA2:=display([seq(plot(subs(dsu,DSu,[seq([Fx[j],Fy[j],t=0..Tf*i/200],j=1..3)]), color=[black,blue,red]),i=1..200)],insequence=true,thickness=3, scaling=constrained): Trajectory of the properelled and lead endA3:=display([seq(plot(subs(dsu,DSu,{[Xi,Eta,t=0..Tf],[xi,eta,t=0..Tf]}), scaling=constrained,color=grey,thickness=3),i=0..200)], insequence=true):AT:=display({A1,A2,A3}):AT;

The presented access can be used if the intersection of the circle = e1 - describing all possible positions of the lead end at the leading = e2 can be computed analytically. If not we can use following approach.restart; X(t) and Y(t) are functions decribing position of the properelled end, x(t), and y(t) of the lead end.e1:=(X(t)-x)^2+(Y(t)-y)^2=Lambda^2;Implicit equation describing leadinge2:=F(x,y)=0; #position:=fsolve({subs(t=1.2,e1),e2},{x,y},{x=x1..x2,y=y1..y2}); If t has numeric value, (t =1.2), this commad returns real values of x(t) and y(t), for exampleposition:={x=-.123456789, y=.987654321}these values later can be substituted into velocity or acceleration.x and y are time depending, to compute velocity or acceleration of the lead end we have to convert them into time dependent functionsxyt:=[x=x(t),y=y(t)]; Time derivation of the equation e1e1d:=diff(subs(xyt,e1),t); Time derivation of the equation e2e2d:=diff(subs(xyt,e2),t); Determination of the vector of velocity of lead end. velocity:=solve({e1d,e2d},{diff(x(t),t),diff(y(t),t)}); Su0:=[Delta[X]=X-x,Delta[Y]=Y-y];These substutions will shorten preceeding result.Su1:=[diff(X(t),t)=X[t],diff(Y(t),t)=Y[t], diff(x(t),t)=x[t],diff(y(t),t)=y[t], D[1](F)(x(t),y(t))=F[x],D[2](F)(x(t),y(t))=F[y], X(t)=Delta[X]+x(t),Y(t)=Delta[Y]+y(t)];Velocity:=subs(Su1,velocity);Final velocity of the lead endVelocity:=subs(Su0,collect(Velocity,[F[x],F[y],X[t],Y[t]])); e1dd:=diff(subs(xyt,e1),t,t);e2dd:=diff(subs(xyt,e2),t,t);Acceleration of the lead end, using following substitutions can be shorten.acceleration:=normal(subs(Su1,solve({e1dd,e2dd},{diff(x(t),t,t),diff(y(t),t,t)}))); Su2:=[D[1,1](F)(x(t),y(t))=F[xx],D[2,2](F)(x(t),y(t))=F[yy], D[1,2](F)(x(t),y(t))=F[xy], diff(X[t],t)=X[tt],diff(Y[t],t)=Y[tt], diff(x[t],t)=x[tt],diff(y[t],t)=y[tt]];Acceleration:=subs(Su2,acceleration);Acceleration:=collect(Acceleration,[F[x],F[y]]);Acceleration:=subs(X[t]^2=(X[t]-x[t])^2+2*X[t]*x[t]-x[t]^2,Y[t]^2=(Y[t]-y[t])^2+2*Y[t]*y[t]-y[t]^2,Acceleration);Acceleration:=collect(Acceleration,[F[x],F[y],Delta[X],Delta[Y]]);The final simplifed result.Acceleration:=subs(Su0,Acceleration);

Young H. D., Freedman R. A.: University Physics., Addison Wesley 1996, ISBN 0-201-84769-8Barton S.: Study of kinematics of agricultural machine., PTEE2002 proceedings, Leuven (B), 5-7 June, 2002 Lauriks W., Van Deynse N.: editors, K.U.Leuven. ISBN: 90-5682-359-0 Published on CD-ROM only.

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