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diff --git a/MECHANICS_OF_SOLIDS_by_S.S._Bhavikatti/Chapter5_m75AW4e.ipynb b/MECHANICS_OF_SOLIDS_by_S.S._Bhavikatti/Chapter5_m75AW4e.ipynb new file mode 100644 index 00000000..87ac24ec --- /dev/null +++ b/MECHANICS_OF_SOLIDS_by_S.S._Bhavikatti/Chapter5_m75AW4e.ipynb @@ -0,0 +1,774 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter5-FRICTION" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.1" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "P= 1250.0 N\n", + "P= 1210.36288071 N\n" + ] + } + ], + "source": [ + "import math\n", + "Wa=1000.0 #weight of block a\n", + "Wb=2000.0 #weight of block b\n", + "uab=1.0/4.0 #coefficient of friction between A and B\n", + "ubg=1.0/3.0 #coefficient of friction between ground and B\n", + "#When P is horizontal\n", + "#considering equilibrium of block A\n", + "N1=Wa #Normal Reaction on block A from block B\n", + "F1=uab*N1 #limiting Friction between A and B\n", + "T=F1 #tension\n", + "#considering equilibrium of block B\n", + "N2=N1+ Wb #Normal Reaction on block B from Ground\n", + "F2=ubg*N2 #limiting Friction between A and ground\n", + "P=F1+F2\n", + "print \"P=\",P,\"N\"\n", + "#When P is inclined at angle o\n", + "o=30.0*3.14/180.0\n", + "#considering equilibrium of block A\n", + "N1=Wa #Normal Reaction on block A from block B\n", + "F1=uab*N1 #limiting Friction between A and B\n", + "T=F1 #tension\n", + "#considering equilibrium of block B\n", + "#from\n", + "#N2+Psin30=N1+Wb\n", + "#Pcos30=F1+F2\n", + "#F1=ubg*N2\n", + "N2=(N1+Wb-F1*math.tan(o))/(1+ubg*math.tan(o))\n", + "P=(N1+Wb-N2)/math.sin(o)\n", + "print \"P=\",P,\"N\"\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.2" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "29.0693410161 °\n" + ] + } + ], + "source": [ + "import math\n", + "Wa=300.0 #weight of upper block \n", + "Wb=900.0 #weight of lower block \n", + "u1=1.0/3.0 #coefficient of friction between upper block and lower block\n", + "u2=1.0/3.0 #coefficient of friction between ground and lower block\n", + "#using \n", + "#N1=Wacoso Normal Reaction\n", + "#F1=u1*N1 Friction\n", + "#N2=Wbcoso+N1\n", + "#F2=u2*N2\n", + "o=math.atan((u1*Wa+u2*Wb+u2*Wa)/Wb)*180/3.14\n", + "print o,\"°\"\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.3" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Angle of inclination is 30.0152164356\n", + "coefficient of friction is 0.1\n" + ] + } + ], + "source": [ + "import math\n", + "W=500.0 #weight of block\n", + "F1=200.0 #force up the inclined plane when block is moving down\n", + "F2=300.0 #force up the inclined plane when block is at rest\n", + "#When block starts moving down the plane\n", + "#sum of all forces perpendicular to the plane = 0\n", + "#N =Wcoso\n", + "#sum of all forces parallel to the plane = 0\n", + "#Fr+F1=Wsino\n", + "#sino-ucoso=F1/w 1\n", + "#When block starts moving up the plane\n", + "#sum of all forces perpendicular to the plane = 0\n", + "#N =Wcoso\n", + "#sum of all forces parallel to the plane = 0\n", + "#Wsino+Wucoso=F2\n", + "#using these equations\n", + "o=math.asin((F1*0.5/W)+(F2*0.5/W)) #angle of inclination\n", + "print \"Angle of inclination is \",(o*180/3.14)\n", + "#using 1\n", + "u=math.sin(o)-F1/W\n", + "print \"coefficient of friction is\",round(u,3)\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.4" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Angle of Inclination 21.8124674778\n" + ] + } + ], + "source": [ + "import math\n", + "uag=0.5 #coefficient of friction between block A and the plane\n", + "ubg=0.2 #coefficient of friction between block B and the plane\n", + "Wb=500.0 #weight of block B\n", + "Wa=1000.0 #weight of block A\n", + "#Considering equilibrium of block A,\n", + "#sum of all forces along the plane is 0\n", + "#N1=Wacoso ,Fr=uagN1\n", + "#sum of all forces perpendicaular to the plane is 0\n", + "#T=uagWacoso-Wasino\n", + "#Considering equilibrium of block A,\n", + "#sum of all forces along the plane is 0\n", + "#N2=Wbcoso ,Fr=uagN2\n", + "#sum of all forces perpendicaular to the plane is 0\n", + "#T=Wbsino-ubgwbsino\n", + "o=math.atan((uag*Wa+ubg*Wb)/(Wa+Wb))*180.0/3.14\n", + "print \"Angle of Inclination\",o;\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.5" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "853.305553493 N\n" + ] + } + ], + "source": [ + "import math\n", + "Wl=750.0 #weight of lower block\n", + "Wu=500.0 #weight of upper block\n", + "o1=60.0*3.14/180.0 #angle of inclined plane\n", + "o2=30.0 *3.14/180.0 # anlge at which pull is applied\n", + "u=0.2 #coefficient of friction\n", + "#for 750 N block\n", + "#Σ Forces normal to the plane = 0 \n", + "N1=Wl*math.cos(o1)\n", + "F1=u*N1\n", + "#Σ Forces parallel to the plane = 0\n", + "T=F1+Wl*math.sin(o1)\n", + "#Σ Forces horizontal to the plane = 0\n", + "P=(T+u*Wu)/(math.cos(o2)+u*math.sin(o2))\n", + "print P,\"N\"\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.6" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Least Weight is 266.34090474 N\n", + "Greatest Weight is 969.473014916 N\n" + ] + } + ], + "source": [ + "import math\n", + "o1=60.0*3.14/180.0 #angle of inclination of plane AC\n", + "o2=30.0*3.14/180.0 #angle of inclination of plane BC\n", + "Wbc=1000.0 #weight of block on plane BC\n", + "ubc=0.28 #coefficient of friction between the load and the plane BC \n", + "uac=0.20 #coefficient of friction between the load and the plane AC\n", + "#for least weight \n", + "N1=Wbc*math.cos(o2) #Normal Reaction\n", + "F1=ubc*N1 #frictional Force\n", + "T=Wbc*math.sin(o2)-F1 #Tension\n", + "#for block on plane AC\n", + "#N2=Wcoso1\n", + "#F2=uac*N2\n", + "#T=F2+W sino2\n", + "W=T/(uac*math.cos(o1)+math.sin(o1))\n", + "print \"Least Weight is\",W,\"N\"\n", + "#for greatest weight \n", + "N1=Wbc*math.cos(o2) #Normal Reaction\n", + "F1=ubc*N1 #frictional Force\n", + "T=Wbc*math.sin(o2)+F1 #Tension\n", + "#for block on plane AC\n", + "#N2=Wcoso1\n", + "#F2=uac*N2\n", + "#T=F2+W sino2\n", + "W=T/(-1*uac*math.cos(o1)+math.sin(o1))\n", + "print \"Greatest Weight is\",W,\"N\"\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.7" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Weight 10498.172578 N\n" + ] + } + ], + "source": [ + "import math\n", + "u=0.4 #The coefficient of friction on the horizontal plane\n", + "oi=30 #angle of inclined plane\n", + "o=20.0 #The limiting angle of friction for block B on the inclined plane\n", + "wb=5000.0 #weight of block b\n", + "ub=math.tan(o*3.14/180.0) #coefficcient of friction on plane\n", + "#for block B\n", + "#N1 N2 N3 are normal reaction\n", + "#F1 F2 are frictional forces\n", + "#F1=ub*N1 \n", + "#N1 sinoi + F1 cos oi=wb\n", + "N1=wb/(math.sin(oi*3.14/180.0)+ub*math.cos(oi*3.14/180.0))\n", + "F1=ub*N1\n", + "C=N1*math.cos(oi*3.14/180.0)-F1*math.sin(oi*3.14/180.0)\n", + "\n", + "#force balance on A in horizontal balance\n", + "F2=C\n", + "N2=F2/u\n", + "#force balance on A in vertical balance\n", + "W=N2\n", + "print \"Weight \",W,\"N\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.8" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Force = 23812.7516422 N\n" + ] + } + ], + "source": [ + "import math\n", + "w=20000.0 #weight of upper block\n", + "o=15.0 #The angle of friction for all surfaces of contact\n", + "u=math.tan(o) #coefficient of friction\n", + "#R1 R2 are forces\n", + "Or1=15.0 #angle force R1 makes with x axis\n", + "Or2=35.0 #angle force R2 makes with Y axis\n", + "R2=w*math.sin((90-Or1)*3.14/180.0)/math.sin((90+Or1+Or2)*3.14/180.0)\n", + "#applyig lamis theorem on block B\n", + "Or1=15.0 #angle force R3 makes with Y axis\n", + "Or2=35.0 #angle force R2 makes with Y axis\n", + "P=R2*math.sin((180-Or1-Or2)*3.14/180.0)/math.sin((90+Or1)*3.14/180.0)\n", + "print \"Force =\",P,\"N\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.9" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "P= 66.26 KN\n" + ] + } + ], + "source": [ + "import math \n", + "w=160.0 #weight of block,KN\n", + "u=0.25 #coefficient of friction\n", + "phi=math.atan(u)\n", + "\n", + "#The free body diagrams of wedges A, B and block C .The problem being symmetric, the reactions R1 and R2 on wedges A and B are equal. The system of forces on block C andon wedge A are shown in the form convenient for applying Lami’s theorem\n", + "R1=w*math.sin(math.pi-(16*math.pi/180)-phi)/math.sin(2*(phi+math.pi*16/180))\n", + "#consider the equillibrium of the wedge A ,Ny lamis's theorem,we get\n", + "P=R1*math.sin(math.pi-phi-phi-(16*math.pi/180))/math.sin((math.pi/2)+phi)\n", + "print\"P=\",round(P,2),\"KN\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.10" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Force required is 62.0836173323 N\n" + ] + } + ], + "source": [ + "import math\n", + "l=4.0 #length of ladder\n", + "u1=0.2 #coefficient of friction between the wall and the ladder\n", + "w=200.0 #weight of ladder\n", + "u2=0.3 #coefficient of friction between floor and the ladder\n", + "wm=600.0 #weight of man\n", + "lm=3.0 #distance of man\n", + "o=3.14*60.0/180.0 #angle made by ladder with floor\n", + "#sum of all moment about A =0\n", + "Nb=(w*l/2*math.cos(o)+wm*lm*math.cos(o))/(l*(math.sin(o)+u1*math.cos(o))) # normal reaction from wall\n", + "Fb=u1*Nb #friction from wall\n", + "#force balance in vertical direction\n", + "Na=(w+wm-Fb) # normal reaction from ground\n", + "Fa=u2*Na #friction from ground\n", + "P=Nb-Fa\n", + "print \"Force required is \",P,\"N\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.11" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Angle of inclination is 71.6013500101 degrees\n" + ] + } + ], + "source": [ + "import math\n", + "l=6.0 #length of ladder\n", + "u1=0.4 #coefficient of friction between the wall and the ladder\n", + "w=200.0 #weight of ladder\n", + "u2=0.25 #coefficient of friction between floor and the ladder\n", + "wl=900.0 #weight of load\n", + "ll=5.0 #distance of load\n", + "#force balancing\n", + "#Na Nb normal reaction at A and B\n", + "#Fa Fb friction at A and B\n", + "#Fa=u2*Na \n", + "#Fb=u1*Nb\n", + "#Na+Fb=w+wl\n", + "#Fa=Nb\n", + "Nb=(wl+w)*u2/(1+u2*u1)\n", + "Na=Nb/u2\n", + "Fa=u2*Na\n", + "Fb=u1*Nb\n", + "#sum of all moments about a is =0\n", + "temp=((w*l*0.5)+(wl*ll)-(Fb*l))/(Nb*l)\n", + "o=math.atan(temp)*180/3.14\n", + "print \"Angle of inclination is \",o,\"degrees\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.12" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "length will 0.5 times\n" + ] + } + ], + "source": [ + "import math\n", + "o=45.0*3.14/180.0 #angle of inclination \n", + "u=0.5 #coefficient of friction\n", + "r=1.5 #ratio of mans weight to ladders weight\n", + "o1=45.0*math.pi/180.0 #angle of inclination\n", + "#from law of friction\n", + "#Fa = μNa\n", + "#Fb = μNb\n", + "#Fa – Nb = 0 \n", + "#Na + Fb = W + r W\n", + "#ΣMA = 0\n", + "o=(((u*u+u)*(1+r)/((1+u)))-1.0/2.0)/r\n", + "print \"length will\",o,\"times\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.13" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Maximum weight is 6277.60420331\n", + "Minimum weight is 57.3467183245\n" + ] + } + ], + "source": [ + "import math\n", + "n=1.25 #number of turns\n", + "o=2*3.14*n #angle of contact\n", + "u=0.3 #coefficient of friction\n", + "t=600.0 #force at the other end of the rope\n", + "#if the impending motion of the weight be downward.\n", + "W=T2=t*2.71**(u*o)\n", + "print \"Maximum weight is \",W\n", + "#if the impending motion of weight be upwards\n", + "W=T1=t*2.71**(-1*u*o)\n", + "print \"Minimum weight is \",W" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.14" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Weight is 136.9599857 N\n" + ] + } + ], + "source": [ + "import math\n", + "ur=0.20 #The coefficient of friction between the rope and the fixed drum\n", + "uo=0.30 #The coefficient of friction between other surfaces\n", + "cosa=4.0/5.0 #cos of angle of inclination\n", + "sina=3.0/5.0 #sin of angle of inclination\n", + "Ww=1000.0 #weight\n", + "o=3.14 #angle of contact of rope with pulley\n", + "#for unknown weight\n", + "#force balance perpendicular to the plane\n", + "#N1 = W cos α\n", + "#fr=uoN1\n", + "#force balance along the plane\n", + "#T1 = F1 + W sin α\n", + "#for 1000 N body\n", + "#force balance perpendicular to the plane\n", + "#N2=N1+Wwcosa\n", + "#fr2=uoN2\n", + "#force balance along the plane\n", + "#T2= Wwsina -F1 -F2\n", + "#T2=T1*e^(ur*o)\n", + "W=(Ww*sina-uo*Ww*cosa)/(((uo*cosa+sina)*(2.71**(uo*o)))+(uo*cosa+uo*cosa))\n", + "print \"Weight is \",W,\"N\"\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.15" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "force P applied at the end of the brake lever 274.480678202\n" + ] + } + ], + "source": [ + "import math\n", + "u=0.3 #coefficient of friction\n", + "r=250 #radius of brake drum\n", + "l=300 #length of lever arm\n", + "M=300000.0 #torque\n", + "o=r*3.14/180.0\n", + "l2=50.0\n", + "#using \n", + "#T2 = T1e^(μθ) T1 and T2 are tension\n", + "#(T2-T1)r=M\n", + "T1=M/(r*(2.71**(u*o)-1))\n", + "T2=(2.71**(u*o))*T1\n", + "#Consider the lever arm. Taking moment about the hinge\n", + "p=T2*l2/l #force P applied at the end of the brake lever\n", + "print \"force P applied at the end of the brake lever\",p\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Example 5.16" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Length of belt is 6972.02507534 mm\n", + "Power Transmitted 3252832.96438 Watt\n" + ] + } + ], + "source": [ + "import math\n", + "d1=500.0 #diameter of a shaft\n", + "d2=100.0 #diameter of a shaft\n", + "D=3000.0 #distance between shafts in mm\n", + "T=1000.0 #Maximum permissible tension in the belt\n", + "U=0.25 #coefficient of friction between the belt and the pulley\n", + "R=220.0 #revlution per minute of larger shaft\n", + "O1=O2=3.14+2*math.asin((d1+d2)/(2*D))\n", + "#Length of belt = Arc length DC + Arc length FE + 2BG\n", + "L=(d1/2+d2/2)*O1+2*D*math.cos(math.asin((d1+d2)/(2*D)))\n", + "print \"Length of belt is \",L,\"mm\"\n", + "T1=T/(2.71**(U*O1))\n", + "Velocity_of_the_belt =d1/2*(R*2*3.14/60.0)\n", + "Power_transmitted=(T-T1)*Velocity_of_the_belt\n", + "print \"Power Transmitted\",Power_transmitted,\"Watt\"\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "# Example 5.17" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Length of belt is 6955.3382782 mm\n", + "Power Transmitted 3035637.41075 Watt\n" + ] + } + ], + "source": [ + "import math\n", + "d1=500.0 #diameter of a shaft\n", + "d2=100.0 #diameter of a shaft\n", + "D=3000.0 #distance between shafts in mm\n", + "T=1000.0 #Maximum permissible tension in the belt\n", + "U=0.25 #coefficient of friction between the belt and the pulley\n", + "R=220.0 #revlution per minute of larger shaft\n", + "O1=3.14+2*math.asin((d1-d2)/(2*D))\n", + "O2=3.14-2*math.asin((d1-d2)/(2*D))\n", + "#Length of belt = Arc length DC + Arc length FE + 2BG\n", + "L=(d1/2*O1+d2/2*O2)+2*D*math.cos(math.asin((d1-d2)/(2*D)))\n", + "print \"Length of belt is \",L,\"mm\"\n", + "T1=T/(2.71**(U*O2))\n", + "Velocity_of_the_belt =d1/2*(R*2*3.14/60.0)\n", + "Power_transmitted=(T-T1)*Velocity_of_the_belt\n", + "print \"Power Transmitted\",Power_transmitted,\"Watt\"\n", + "\n", + "\n", + "\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python [Root]", + "language": "python", + "name": "Python [Root]" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.12" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} |