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diff --git a/Engineering_Mechancis,_Schaum_Series_by_McLean/chapter19.ipynb b/Engineering_Mechancis,_Schaum_Series_by_McLean/chapter19.ipynb new file mode 100755 index 00000000..98ce73cc --- /dev/null +++ b/Engineering_Mechancis,_Schaum_Series_by_McLean/chapter19.ipynb @@ -0,0 +1,503 @@ +{
+ "metadata": {
+ "name": "chapter 19.ipynb"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+ {
+ "cells": [
+ {
+ "cell_type": "heading",
+ "level": 1,
+ "metadata": {},
+ "source": [
+ "CHAPTER 19: MECHANICAL VIBRATIONS"
+ ]
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-2, Page No:465"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "#Initilization of variables\n",
+ "k=18 #lb/in\n",
+ "g=386 #in/s**2\n",
+ "W=35 #lb\n",
+ "\n",
+ "#Calculations\n",
+ "f=(1/(2*pi))*sqrt((k*g/W)) #cps\n",
+ "period=1/f #s\n",
+ "\n",
+ "#Result\n",
+ "print'The period of vibration is',round(period,2),\"s\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The period of vibration is 0.45 s\n"
+ ]
+ }
+ ],
+ "prompt_number": 4
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-11, Page No:471"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "ds=0.2 #m\n",
+ "ts=0.05 #m\n",
+ "rhos=7850 #kg/m**3 density of steel\n",
+ "dw=0.002 #m\n",
+ "lw=0.9 #m\n",
+ "G=80*10**9 #Pa\n",
+ "\n",
+ "#Calculations\n",
+ "#Torsional Constant\n",
+ "K=(pi*dw**4*G)/(32*lw) #m/rad\n",
+ "#Mass Calculations\n",
+ "m=(4**-1)*pi*(ds**2)*ts*rhos #kg\n",
+ "#Moment of Inertia\n",
+ "Io=(0.5)*m*(ds*2**-1)**2 #kg.m**2\n",
+ "#Frequency\n",
+ "f=(1*(2*pi)**-1)*(sqrt(K*Io**-1)) #Hz\n",
+ "\n",
+ "#Result\n",
+ "print'The natural frequency of the system is',round(f,2),\"Hz\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The natural frequency of the system is 0.24 Hz\n"
+ ]
+ }
+ ],
+ "prompt_number": 6
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-13, Page No 472"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "m=120 #kg\n",
+ "k=0.3 #m\n",
+ "ls=0.6 #m\n",
+ "ds=0.05 #m\n",
+ "G=80*10**9 #Pa\n",
+ "\n",
+ "#Calculations\n",
+ "#Polar Moment of Inertia\n",
+ "J1=m*k**2 #kg.m**2\n",
+ "J2=J1 #kg.m**2\n",
+ "J=(32**-1)*pi*(ds**4) #m**4\n",
+ "#Frequency\n",
+ "f=(1/(2*pi))*(sqrt((J*G*(J1+J2))/(ls*J1*J2))) #Hz\n",
+ "\n",
+ "#Result\n",
+ "print'The natural frequency of the torsional oscillation is',round(f,1),\"Hz\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The natural frequency of the torsional oscillation is 19.6 Hz\n"
+ ]
+ }
+ ],
+ "prompt_number": 2
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-14, Page No: 473"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "ds=2 #in\n",
+ "L=15 #in\n",
+ "Wf1=300 #lb\n",
+ "k1=6 #in\n",
+ "Wf2=100 #lb\n",
+ "k2=4 #in\n",
+ "G=12*10**6 #Pa\n",
+ "g=386 #in/s**2\n",
+ "\n",
+ "#Calculations\n",
+ "#Moment of inertia of flywheel\n",
+ "Jf=(Wf1*g**-1)*k1**2 #lb-s**2-in\n",
+ "#Moment of inertia of the rotor\n",
+ "Jr=(Wf2*g**-1)*k2**2 #lb-s**2-in\n",
+ "#Moment of inertia of the shaft cross section\n",
+ "J=(32**-1)*pi*ds**4 #in**4\n",
+ "#Frequency\n",
+ "f=((pi*2)**-1)*(sqrt((J*G*(Jf+Jr))*(L*Jf*Jr)**-1)) #cps\n",
+ "\n",
+ "#Result\n",
+ "print'The natural frequency of the system is',round(f,1),\"cps\"\n",
+ "\n",
+ "#The answer may wary due to decimal point descrepancy"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The natural frequency of the system is 93.9 cps\n"
+ ]
+ }
+ ],
+ "prompt_number": 14
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-15, Page No: 473"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "W=10 #lb\n",
+ "A=2 #in**2\n",
+ "#Calculations\n",
+ "\n",
+ "wn=sqrt(((A*144**-1)*5*62.4*5)/2.59) #rad/s\n",
+ "\n",
+ "#Result\n",
+ "print'The frequency of oscillation is',round(wn,2),\"rad/s\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The frequency of oscillation is 2.89 rad/s\n"
+ ]
+ }
+ ],
+ "prompt_number": 15
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-16, Page No:474"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "f=50 #cps\n",
+ "g=386 #in/s**2\n",
+ "E=30*10**6 #lb/in**2\n",
+ "l=4 #in\n",
+ "I=2.08*10**-6 #in**4\n",
+ "\n",
+ "#Calculations\n",
+ "W=(3*E*I*g)/(((f*2*pi)**2)*l**3) #lb\n",
+ "\n",
+ "#Result\n",
+ "print'The value of W is',round(W,3),\"lb\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The value of W is 0.011 lb\n"
+ ]
+ }
+ ],
+ "prompt_number": 16
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-19, Page No:478"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "F=10 #lb\n",
+ "v=20 #in/s\n",
+ "g=386 #in/s\n",
+ "W=12 #lb\n",
+ "k=20 #lb/in\n",
+ "\n",
+ "#Calculations\n",
+ "#Coefficient of damping\n",
+ "c=F*(v**-1) #lb-s/in\n",
+ "#Natural Frequency\n",
+ "wn=sqrt((k*g)/W) #rad/s\n",
+ "#Critical Damping coefficient\n",
+ "cr=(2*W*(g**-1))*wn #lb-s/in\n",
+ "#Damping Coefficient\n",
+ "d=c*(cr**-1)\n",
+ "#Frequency of damped vibrations\n",
+ "wd=sqrt(1-d**2)*wn #rad/s\n",
+ "\n",
+ "#Result\n",
+ "print'The frequency of damped vibrations is',round(wd,1),\"rad/s\"\n",
+ "\n",
+ "# The answer is off by 0.1 units"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The frequency of damped vibrations is 24.0 rad/s\n"
+ ]
+ }
+ ],
+ "prompt_number": 18
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-20, Page No 478"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "wn=25.4 #rad/s\n",
+ "t=0.261 #s\n",
+ "d=0.316\n",
+ "\n",
+ "#Calculations\n",
+ "delta=d*t*wn #logarithmic decay\n",
+ "\n",
+ "#Result\n",
+ "print'The rate of decay is',round(delta,3)\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The rate of decay is 2.095\n"
+ ]
+ }
+ ],
+ "prompt_number": 19
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-24, Page No 483"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "F=9 #N\n",
+ "m=5 #kg\n",
+ "k=6000 #N/m\n",
+ "f1=1 #Hz\n",
+ "f2=5.4 #Hz\n",
+ "f3=50 #Hz\n",
+ "\n",
+ "#Calculations\n",
+ "#Natural Frequency\n",
+ "fn=((pi*2)**-1)*(sqrt(k/m)) #Hz\n",
+ "deltaf=F*(k/1000)**-1 #mm\n",
+ "#Part(a)\n",
+ "r1=f1*fn**-1\n",
+ "amp1=deltaf*(1-r1**2)**-1 #mm amplitude\n",
+ "#Part (b)\n",
+ "r2=f2*fn**-1\n",
+ "amp2=deltaf/(1-r2**2) #mm amplitude\n",
+ "#Part (c)\n",
+ "r3=f3*fn**-1\n",
+ "amp3=deltaf/(1-r3**2) #mm amplitude\n",
+ "\n",
+ "#Result\n",
+ "print'The amplitudes in part (a),(b) and (c) respectively are',round(amp1,3),\"mm ,\",round(amp2,1),\"mm and\",round(amp3,3),\"mm\"\n",
+ "\n",
+ "# The answer for amp2 is incorrect in textbook"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The amplitudes in part (a),(b) and (c) respectively are 1.551 mm , 36.9 mm and -0.018 mm\n"
+ ]
+ }
+ ],
+ "prompt_number": 27
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-25, Page No 483"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of vraiables\n",
+ "g=386 #in/s**2\n",
+ "W=20 #lb\n",
+ "w=600 #rpm\n",
+ "ratio=12**-1\n",
+ "\n",
+ "#Calculations\n",
+ "r=sqrt((1*ratio**-1)+1) \n",
+ "fn=((w/60)/r) #cps\n",
+ "k=((fn*2*pi)**2*W)/(g) #lb/in\n",
+ "\n",
+ "#Result\n",
+ "print'The value of k is',round(k,1),\"lb/in\"\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The value of k is 15.7 lb/in\n"
+ ]
+ }
+ ],
+ "prompt_number": 28
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 19.19-28, Page No 487"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "import math\n",
+ "\n",
+ "#Initilization of variables\n",
+ "X=12 #mm\n",
+ "me_M=1.3 #mm\n",
+ "\n",
+ "#Calculations\n",
+ "d=(me_M)/(2*X)\n",
+ "\n",
+ "#Result\n",
+ "print'The damping ratio is',round(d,3)\n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The damping ratio is 0.054\n"
+ ]
+ }
+ ],
+ "prompt_number": 29
+ }
+ ],
+ "metadata": {}
+ }
+ ]
+}
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