Bisection Method of Solving a Nonlinear Equation-More Examples ...

Chapter 03.03Bisection Method of Solving a Nonlinear Equation-More ExamplesMechanical EngineeringExample 1A trunnion has to be cooled before it is shrink fitted into a steel hub.Figure 1 Trunnion to be slid through the hub after contracting.T fThe equation that gives the temperature to which the trunnion has to be cooled to obtainthe desired contraction is given by1037242f ( Tf) 0.5059810Tf 0.38292 10Tf 0.7436310Tf 0.8831810 0Use the bisection method of finding roots of equations to find the temperature T fto whichthe trunnion has to be cooled. Conduct three iterations to estimate the root of the aboveequation. Find the absolute relative approximate error at the end of each iteration and thenumber of significant digits at least correct at the end of each iteration.SolutionFrom the designer’s records for the previous bridge, the temperature to which the trunnionwas cooled was 108F. Hence assuming the temperature to be between 100Fand 150F , we haveTf , 150F, T f , u 100FCheck if the function changes sign between T and .f ,Tf , u03.03.1

03.03.2 Chapter 03.03HencefffT f 150Tf , f , u 0.5059810 0.7436310 1.290310 f ( 100)103 0.5059810 0.7436310 1.8290 1034104( 150)3 0.38292 10( 150) 0.8831810( 100)3( 100) 0.88318102 0.38292 10277( 150)( 100)33TfT f 150f1001.2903101.829010 0f , f , uSo there is at least one root between T and T ,that is between 150 and 100 .Iteration 1The estimate of the root isTf , Tf , uTf , m2150 ( 100)2 125f T f 125f f , m 0.5059810 0.74363104 2.335610f T f 150104f ,( 125)3f u 0.3829210( 125) 0.883181072( 125)34T f1251.2903102.335610 0f , f , mHence the root is bracketed between T and T ,, that is, between 150 and 125 .f ,f mSo, the lower and upper limits of the new bracket areTf , 150 , T f , u 125At this point, the absolute relative approximate errorhave a previous approximation.Iteration 2The estimate of the root isTf , mTf , Tf , u2150 1252 137.5222acannot be calculated, as we do not

Bisection Method-More Examples: Mechanical Engineering 03.03.3ffT f 137.5f , m 0.5059810 0.74363101044= 5.376210f T f 137.( 137.5)3 0.3829210( 137.5) 0.883181027( 137.5)44T f 125 5.3762 102.335610 0f , m f , u5 Tf , mT f , u125Hence, the root is bracketed between and , that is, betweenSo the lower and upper limits of the new bracket areTf , 137 .5, T f , u 125The absolute relative approximate erroraTnewf , mT Tnewf , moldf , m100aat the end of Iteration 2 is137.5 ( 125)100137.5 9.0909%None of the significant digits are at least correct in the estimated root ofTf , m 137.5as the absolute relative approximate error is greater that 5% .Iteration 3The estimate of the root isTf , Tf , uTf , m2137.5 ( 125)2 131.25f T f 131.25f f , m 0.5059810 0.7436310= 1.5430310f T f 1251044( 131.25)3 0.3829210( 131.25) 0.8831810272( 131.25) and 137.5 .44T f131.25 2.3356101.5430 10 0f , f , mHence, the root is bracketed between T and T ,, that is, between 125 and 131.25 .f ,f mSo the lower and upper limits of the new bracket areT, 131 .25,, 125fT f uThe absolute relative approximate erroraTnewf , mT Tnewf , moldf , m100aat the ends of Iteration 3 is2

03.03.4 Chapter 03.03131.25 ( 137.5)100131.25 4.7619%The number of significant digits at least correct is 1.Seven more iterations were conducted and these iterations are shown in the Table 1 below.Table 1 Root offTx0as function of number of iterations for bisection method.Iteration T , a% 12345678910f ,−150−150−137.5−131.25−131.25−129.69−128.91−128.91−128.91−128.81Tf , u−100−125−125−125−128.13−123.13−123.13−128.52−128.71−128.71f m−125−137.5−131.25−128.13−129.69−128.91−128.52−128.71−128.81−128.76---------9.09094.76192.43901.20480.606060.303950.151750.0758150.037922f T f , m2.3356104 5.37621041.54301053.9065105 5.7760106 9.3826101.4838 10 562.7228106 3.3305107 3.0396104thAt the end of the 10 iteration, a 0.037922%Hence, the number of significant digits at least correct is given by the largest value ofwhich2m 0 .510a2m0 .037922 0.5102m0 .075844 10log0.075844 2 mm 2 log0.0758443. 1201Som 3The number of significant digits at least correct in the estimated root of 128. 76 is 3.NONLINEAR EQUATIONSTopic Bisection Method-More ExamplesSummary Examples of Bisection MethodMajor Mechanical EngineeringAuthors Autar KawDate August 7, 2009Web Site http://numericalmethods.eng.usf.edumfor