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+{
+"cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 8: SPRINGS"
+ ]
+ },
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.10: S8_10.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-10\n",
+"clc;\n",
+"clear;\n",
+"A=1783;\n",
+"m=0.190;\n",
+"d=1.5;\n",
+"D=15;\n",
+"M=300;\n",
+"E=20800;\n",
+"k=30;\n",
+"//sigult= ultimate strength of the material\n",
+"// sigy= yield strength of the material\n",
+"sigult=A/(d^m);\n",
+"sigy=0.7*sigult;\n",
+"//siga= allowable yield strength of the material\n",
+"siga=sigy/2;\n",
+"C=D/d;\n",
+"Ki=(4*(C^2)-C-1)/(4*C*(C-1));\n",
+"Z=%pi*(d^3)/32;\n",
+"//sigb=bending strength of the material;\n",
+"sigb=Ki*M/Z;\n",
+"while (sigb>=siga) \n",
+" d=d+0.15;\n",
+" D=15;\n",
+" C=D/d;\n",
+" sigult=A/(d^m);\n",
+"sigy=0.7*sigult;\n",
+"siga=sigy/2;\n",
+"Ki=(4*(C^2)-C-1)/(4*C*(C-1));\n",
+"Z=%pi*(d^3)/32;\n",
+"sigb=Ki*M/Z;\n",
+"end\n",
+"d=2;// rounding off the value of the diameter.\n",
+"D;\n",
+"Na=(d^4)*E/(64*D*k);\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('d is %0.1f mm ',d);\n",
+" printf('\n D is %0.1f mm ',D);\n",
+" printf('\n Na is %0.2f mm ',Na);"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.11: S8_11.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-11\n",
+"clc;\n",
+"clear;\n",
+"L=1180;\n",
+"W=40*(10^3);\n",
+"Nf=2;\n",
+"Ng=8;\n",
+"E=207*(10^3);\n",
+"//sigut is ultimate strength\n",
+"sigut=1400;\n",
+"FOS=2;\n",
+"//siga= allowable yield strength of the material\n",
+"siga=1400/2;\n",
+"//sigbf=bending strength in full length\n",
+"sigbf=700;\n",
+"b=75;\n",
+"t=((4.5*W*L)/(((3*Nf)+(2*Ng))*sigbf))^(0.5);\n",
+"t=14;\n",
+"I=(Nf*b*(t^3))/12;\n",
+"Wf=(3*Nf*W)/((3*Nf)+(2*Ng));\n",
+"del=(Wf*(L^3))/(48*E*I);\n",
+"\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('t is %0.0f mm ',t);\n",
+" printf('\n Wf is %0.0f N ',Wf);\n",
+" printf('\n I is %0.0f mm^4 ',I);\n",
+" printf('\n del is %0.1f mm ',del);"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.12: S8_12A.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-12A\n",
+"clc;\n",
+"clear;\n",
+"W=80000;\n",
+"sigbfr=500;\n",
+"L=1100;\n",
+"Nf=3;\n",
+"Ng=10;\n",
+"N=Nf+Ng;\n",
+"t=((1.5*W*L)/(N*6*sigbfr))^(1/3);\n",
+"t=15;\n",
+"b=6*t;\n",
+"E=207*10^3;\n",
+"deli=(W*(L^3))/(8*E*N*b*(t^3));\n",
+"Wi=(W*Nf*Ng)/(N*((3*Nf)+(2*Ng)));\n",
+"\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('t is %0.1f mm ',t);\n",
+" printf('\n deli is %0.1f mm ',deli);\n",
+" printf('\n Wi is %0.0f N ',Wi);"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.13: S8_13.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-13\n",
+"clc;\n",
+"clear;\n",
+"//ultimate strength=sigut\n",
+"sigut=1500;\n",
+"C=7;\n",
+"d=3;\n",
+"D=C*d;\n",
+"Ks=1+(0.5/C);\n",
+"Kw=(((4*C)-1)/((4*C)-4))+(0.615/C);\n",
+"Pmax=120;\n",
+"Pmin=40;\n",
+"Pm=80;\n",
+"Tm=(Ks*8*Pm*D)/(%pi*(d^3));\n",
+"Ta=(Kw*8*Pmin*D)/(%pi*(d^3));\n",
+"Tse=0.22*sigut;\n",
+"Tys=0.45*sigut;\n",
+"x=(Tys-(0.5*Tse))/(0.5*Tse);\n",
+"y=((x)*Ta)+Tm;\n",
+"FOS=(Tys/y);\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('Tm is %0.2f MPa ',Tm);\n",
+" printf('\n Ta is %0.1f MPa ',Ta);\n",
+" printf('\n FOS is %0.3f ',FOS);"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.14: S8_14.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-14\n",
+"clc;\n",
+"clear;\n",
+"Tse=360;\n",
+"Tys=660;\n",
+"d=25;\n",
+"P=0.03;\n",
+"m=40;\n",
+"Pmin=((%pi*(d^2)*P)/4)+(m*9.81/1000);\n",
+"k=6;\n",
+"//Additional load= Padd=k*further compression in spring\n",
+"Padd=k*10;\n",
+"Pmax=Padd+Pmin;\n",
+"Pm=(Pmax+Pmin)/2;\n",
+"Pa=(Pmax-Pmin)/2;\n",
+"d=2;\n",
+"D=12;\n",
+"C=6;\n",
+"Ks=1+(0.5/C);\n",
+"Ks=1.083;\n",
+"Kw=(((4*C)-1)/((4*C)-4))+(0.615/C);\n",
+"Ta=(Kw*8*Pa*D)/(%pi*(d^3));\n",
+"Tm=(Ks*8*Pm*D)/(%pi*(d^3));\n",
+"x=(Tys-(0.5*Tse))/(0.5*Tse);\n",
+"y=((x)*Ta)+Tm;\n",
+"FOS=(Tys/y);\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('Tm is %0.2f MPa ',Tm);\n",
+" printf('\n Ta is %0.3f MPa ',Ta);\n",
+" printf('\n FOS is %0.2f ',FOS);"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.1: S8_1.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-1\n",
+"clc;\n",
+"clear;\n",
+"d=5;\n",
+"D=30;\n",
+"G=84*(10^3);\n",
+"Na=15;\n",
+"//Axial Load W\n",
+"W=300;\n",
+"//Spring index C\n",
+"C=30/5;\n",
+"//Shear stress Augmentation factor Ks\n",
+"Ks=((2*C)+1)/(2*C);\n",
+"//Wahl's factor Kw\n",
+"Kw=(((4*C)-1)/((4*C)-4))+(0.615/C);\n",
+"//Curvature correction factor Kc\n",
+"Kc=Kw/Ks;\n",
+"//Spring stiffness k\n",
+"k=(G*(d^4))/(8*(D^3)*Na);\n",
+"//Axial deflection delta\n",
+"delta=W/k;\n",
+"\n",
+" // printing data in scilab o/p window\n",
+"printf('Ks is %0.4f ',Ks);\n",
+"printf('\n Kw is %0.4f ',Kw);\n",
+"printf('\n Kc is %0.3f ',Kc);\n",
+"printf('\n The Spring Stiffness is %0.1f N/mm',k);\n",
+"printf('\n The Axial deflection is %0.3f mm',delta);"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.2: S8_2.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-2\n",
+"clc;\n",
+"clear;\n",
+"W=196.2;\n",
+"lenthofscale=50;\n",
+"k=196.2/50;\n",
+"C=8;\n",
+"Ks=(1+(0.5/C));\n",
+"\n",
+"// Let us choose oil tempered wire 0.6-0.7 %C. Refer to Table 8-4 for constants A and m, relating strength wire \n",
+"//diameter.\n",
+"G=77.2*(10^3);\n",
+"A=1855;\n",
+"m=0.187;\n",
+"// equating Tmax=0.5*sig(ut).\n",
+"// Ks*(8*W*D/(pi*(d^3)))=0.5*A/(d^2)\n",
+"d1=(Ks*(8*W*C/(%pi*A*0.5)));\n",
+"d=d1^(1/1.813);\n",
+"D=C*d;\n",
+"Na=G*(d^4)/(8*(D^3)*k);\n",
+"//Solid length = SL\n",
+"SL=(Na-1)*d\n",
+"\n",
+"\n",
+" // printing data in scilab o/p window\n",
+"printf(' wire diameter is %0.3f mm ',d);\n",
+"printf('\n mean diameter is %0.3f mm ',D);\n",
+"printf('\n Number of acting coils are %0.3f ',Na);\n",
+"\n",
+"//The difference in the values of d,D and Na is due to rounding-off the digits."
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.3: S8_3.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-3\n",
+"clc;\n",
+"clear;\n",
+"d=1.626;\n",
+"A=2211;\n",
+"m=0.145;\n",
+"rm=3;\n",
+"ri=(rm-(d/2));\n",
+"sigma=A/(d^m);\n",
+"W=(sigma*%pi*(d^3)*ri)/(32*(rm^2));\n",
+"\n",
+" // printing data in scilab o/p window\n",
+"printf('Ultimate tensile Strength is %0.1f MPa ',sigma);\n",
+"printf('\n Force at which the spring hook fails is %0.1f N ',W);\n",
+"\n",
+"//The difference in the values of sigma and W is due to rounding-off the digits."
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.4: S8_4.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-4\n",
+"clc;\n",
+"clear;\n",
+"Do=25;\n",
+"// mean coil diameter D=25-d\n",
+"W=150;\n",
+"T=800;\n",
+"G=81000;\n",
+"// Substituting values in equation T=8*W*D/(%pi*(d^3))\n",
+"// therefore, the equation becomes d^3 + 0.477*d = 11.936\n",
+"//consider d=2.2mm, (d can be taken between 2.2-2.3mm)\n",
+"d=2.337; //(nearest available wire gauge)\n",
+"C=9.5;\n",
+"D=22.2; \n",
+"Do=D+d;\n",
+"Ks=1+(0.5/C);\n",
+"Tmax=Ks*8*W*D/(%pi*(d^3));\n",
+"// check for safety- Tmax<T;\n",
+"Lo=100;\n",
+"Ls=40;\n",
+"//Lo=Ls+delta+0.15*delta\n",
+"delta=(Lo-Ls)/1.15;\n",
+"delta=50;\n",
+"k=150/50;\n",
+"Na=(G*d^4)/(8*(D^3)*k);\n",
+"\n",
+"N=Na+2;\n",
+"Ls=N*d;\n",
+"Lo=Ls+(1.15*delta);\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('d is %0.3fmm ',d);\n",
+" printf('\n D is %0.2f mm',D);\n",
+" printf('\n Ls is %0.2f mm',Ls);\n",
+" printf('\n Lo is %0.2f mm',Lo);\n",
+" if (Do<=25)\n",
+" disp ('The diameter is within space constraints'); \n",
+"end"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.5: S8_5.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-5A\n",
+"clc;\n",
+"clear;\n",
+"Di=15;\n",
+"Do=20;\n",
+"d=2.3;\n",
+"D=17.5;\n",
+"C=D/d;\n",
+"Ks=1+(0.5/C);\n",
+"Wmax=100;\n",
+"Tmax=Ks*8*Wmax*D/(%pi*(d^3));\n",
+"G=81000;\n",
+"delmax=67.7/2.366;\n",
+"k=100/28;\n",
+"Na=G*(d^4)/(8*k*(D^3));\n",
+"Ls=Na+1; //(for plain ends)\n",
+"delmax=28;\n",
+"//TL= total working length\n",
+"TL=Ls+delmax+(0.15*delmax);\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('d is %0.1fmm ',d);\n",
+" printf('\n C is %0.1f ',C);\n",
+" printf('\n Na is %0.1f ',Na);"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.6: S8_6.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-6\n",
+"clc;\n",
+"clear;\n",
+"// 18 SWG=1.219MM in dia\n",
+"d=1.219;\n",
+"E=198.6*10^3;\n",
+"G=80.7*10^3;\n",
+"m=0.19;\n",
+"A=1783;\n",
+"sig=A/(d^m);\n",
+"Tys=(0.4*sig);\n",
+"Do=12.5;\n",
+"D=Do-d;\n",
+"C=D/d;\n",
+"Ks=((2*C)+1)/(2*C);\n",
+"W=(Tys*%pi*(d^3))/(8*D*Ks);\n",
+"Nt=13.5;\n",
+"Na=Nt-2;\n",
+"del=(8*W*(D^3)*Na)/(G*(d^4));\n",
+"Ls=(Nt-1)*d;\n",
+"Lo=Ls+del+(0.15*del);\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('Tys is %0.1f MPa ',Tys);\n",
+" printf('\n W is %0.1f N ',W);\n",
+" printf('\n del is %0.3f mm ',del);\n",
+" printf('\n Ls is %0.4f mm ',Ls);\n",
+" printf('\n Lo is %0.2f mm ',Lo);\n",
+" \n",
+" //Answers in the book for Torsional yeild strength have been rounded-off to the nearest whole number."
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.7: S8_7.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-7\n",
+"clc;\n",
+"clear;\n",
+"d=1.016;\n",
+"A=2211;\n",
+"m=0.145;\n",
+"G=81000;\n",
+"Nt=16;\n",
+"Na=16-2;\n",
+"sig=A/(d^m);\n",
+"Tys=0.45*sig;\n",
+"Do=12.6;\n",
+"D=Do-d;\n",
+"C=D/d;\n",
+"Ks=1+(0.5/C);\n",
+"W=(Tys*%pi*(d^3))/(8*D*Ks);\n",
+"k=(G*(d^4))/(8*(D^3)*Na);\n",
+"del=W/k;\n",
+"Ls=(Nt-1)*d;\n",
+"Lo=Ls+(1.15*del);\n",
+"\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('Tys is %0.1f MPa ',Tys);\n",
+" printf('\n Do is %0.1f N ',Do);\n",
+" printf('\n W is %0.1f N ',W);\n",
+" printf('\n k is %0.3f N ',k);\n",
+" printf('\n del is %0.2f mm ',del);\n",
+" printf('\n Ls is %0.2f mm ',Ls);\n",
+" printf('\n Lo is %0.3f mm ',Lo);\n",
+" \n",
+" if ((Lo/D)>=5.26)\n",
+" disp ('The spring will fail under buckling'); \n",
+"end\n",
+"\n",
+"//Values after the decimal point has not been considered for answer of Torsional yeild strength in the book, whereas answers for deflection and free-length is different as entire value of variables is taken for calculation in the code."
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.8: S8_8.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-8\n",
+"clc;\n",
+"clear;\n",
+"d=2;\n",
+"Do=20;\n",
+"D=Do-d;\n",
+"C=D/d;\n",
+"Na=9;\n",
+"//Material hard drawn spring steel\n",
+"A=1783;\n",
+"m=0.19;\n",
+"G=81000;\n",
+"sig=A/(d^m);\n",
+"Tys=0.45*sig\n",
+"Kf=1.5;\n",
+"Ta=Tys/Kf;\n",
+"Ks=1+(0.5/C);\n",
+"W=(Ta*%pi*(d^3))/(8*D*Ks);\n",
+"k=(G*(d^4))/(8*(D^3)*Na);\n",
+"del=W/k;\n",
+"Lo=((Na+1)*d)+(1.15*del);\n",
+"p=(Lo-d)/Na;\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('k is %0.3f N/mm ',k);\n",
+" printf('\n W is %0.1f N ',W);\n",
+" printf('\n Lo is %0.3f mm ',Lo);\n",
+" printf('\n p is %0.3f mm ',p);\n",
+" \n",
+" \n",
+" if ((Lo)>=47.34)\n",
+" disp ('The spring will fail under buckling'); \n",
+"end\n",
+"\n",
+"//The answer for value of spring rate 'k' is misprinted in the book. Due to this all subsequent values of del,Lo,p is calucated incorrectly in the book."
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 8.9: S8_9.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"// sum 8-9\n",
+"clc;\n",
+"clear;\n",
+"// for music wire\n",
+"d1=11.5;\n",
+"A=2211;\n",
+"d=1.5;\n",
+"m=0.145;\n",
+"sigut=A/(d^m);\n",
+"sigy=0.78*sigut;\n",
+"Do=16;\n",
+"E=2*(10^5);\n",
+"Nb=4.25;\n",
+"D=Do-d;\n",
+"C=D/d;\n",
+"Ki=((4*(C^2))-C-1)/(4*C*(C-1));\n",
+"Mmax=(sigy*%pi*(d^3))/(32*Ki);\n",
+"kc=((d^4)*E)/(10.8*D*Nb);\n",
+"theta3=Mmax/kc';\n",
+"l1=20;\n",
+"l2=20;\n",
+"Ne=(l1+l2)/(3*%pi*D);\n",
+"Na=Nb+Ne;\n",
+"k=((d^4)*E)/(10.8*Na*D);\n",
+"thetat=Mmax/k';\n",
+"ke=(3*%pi*(d^4)*E)/(10.8*(l1+l2));\n",
+"// angdisp=theta1+theta2=Mmax/ke;\n",
+"angdisp=Mmax/ke;\n",
+"//D1 is final coil diameter\n",
+"D1=(Nb*D)/(Nb+theta3);\n",
+"//IRC=Initial radial clearance\n",
+"IRC=((D-d)-d1)/2;\n",
+"//FRC=Final radial clearance\n",
+"FRC=((D1-d)-d1)/2;\n",
+"\n",
+"\n",
+" // printing data in scilab o/p window\n",
+" printf('Maximum Torque is %0.2f Nmm ',Mmax);\n",
+" printf('\n theta3 is %0.3f turns ',theta3);\n",
+" printf('\n Ne is %0.3f turns ',Ne);\n",
+" printf('\n ke is %0.1f N/mm ',ke);\n",
+" printf('\n theta1+theta2 is %0.4f turns ',angdisp);\n",
+" printf('\n D1 is %0.2f mm ',D1);\n",
+" printf('\n IRC is %0.2f mm ',IRC);\n",
+" printf('\n FRC is %0.2f mm ',FRC);\n",
+" "
+ ]
+ }
+],
+"metadata": {
+ "kernelspec": {
+ "display_name": "Scilab",
+ "language": "scilab",
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generated by cgit (git 2.34.1) at 2025-07-04 04:12:13 +0000