Program Bike * Program to compute the path of a bicycle along a specified path as given * by drag, friction, and rider power (with a maximum force allowed at low * velocity). * Alternative Solution: 1-D integration along specific path implicit none * Parameters need for program (Physical constants) and the "constants" * in the program include 'FBike.h' integer*4 max_iter ! Maximum number of iterations allowed parameter ( max_iter = 1000 ) * MAIN PROGRAM VARIABLES real*8 Trav_time, Trav_time1, Trav_time2 ! Travel time (sec) real*8 step ! Integration step size (secs). We initially ! set this value but check that it is OK and ! reset as needed. real*8 Pos, Vel, XPos ! Distance and velocity of bike along path and ! X-position of bike. **** Let the user know what this program does: write(*,120) 120 format(/,'12.010 Program to solve Bike Motion problem',/, . 'Given track characteristics and rider/bike properties', . ' the time, energy and path are computed') **** Find out from the user the necessary inputs. We use a variety * of input schemes here. The characteristics of the object can * be put into a file thus allowing easy runs with different * distances and launch angles. call read_input ***** Report to the user the setup parameters call report_setup **** Run test runs to see what integration step size we need. step = 1.0d0 ! Initially set step size to 1-sec (tested in int_soln) **** Write out the solution outT = .false. Trav_time1 = 0 Trav_time2 = 1.e3 * Check that step size matchs tolerance terr. (Tolerance is in meters * so convert to time with 10 m/s velocity assumption) * Assume a speed of 10 m/s convert distance into time difference do while ( abs(Trav_time1-Trav_time2).gt.terr/10.0 .and. . step .gt.1.d-3 ) Pos = 0.0 Vel = 0.0 Xpos = 0.0 Trav_time1 = 0.d0 call Int1D(step, Trav_time1, Pos, Vel, Xpos ) step = step/2 Pos = 0.0 Vel = 0.0 Xpos = 0.0 Trav_time2 = 0.d0 call Int1D(step, Trav_time2, Pos, Vel, Xpos ) write(*,160) step, Trav_time1,Trav_time2, . abs(Trav_time1-Trav_time2)*1000.d0*10 160 format('Step size ',E12.3,' s, Times ',2F10.4, ' s ', . ' Error ',F10.2,' mm') end do outT = .true. if( out_int.le.0 ) outT = .false. Pos = 0.0 Vel = 0.0 Xpos = 0.0 Trav_time = 0.d0 call Int1d(step, Trav_time, Pos, Vel, Xpos ) write(*,220) Track_len/1.d3, Trav_time, Trav_time/3600.d0, . tot_energy, . tot_energy/cal_to_joule, . Mass*abs(Vel)**2/2.d0 220 format(/,'Time to travel ',F6.3,' km, ',F8.2,' seconds, ',F6.2, . ' hrs',/, . 'Rider Energy ',F12.2,' Joules, ',F12.3, ' Calories',/, ' 'Kinetic ',F12.2,' Joules',/) write(*,240) Vel 240 format('Final Velocity ',F6.3,' m/sec') **** Thats all end CTITLE REPORT_SETUP subroutine report_setup implicit none * Routine to report the parameters being used in the run * Parameters need for program (Physical constants) include 'FBike.h' * write(*,120) 120 format(/,'PROGRAM PARAMETERS',/, . '++++++++++++++++++') write(*,140) Track_len/1.0d3, Terr*1.d3 140 format('Length of track ',F7.3,' (km) and error ',F6.1, '(mm)') write(*,160) Slope, As, Bs, Lambda/1.0d3 160 format('Track Slope ',F5.3,' Sin and Cos amplitudes ',2(F5.2,1x), . '(m) and wavelenghth ',F6.2,' (km)') write(*,180) Mass, Area, Cd, Cr 180 format('Rider/Bike Mass ',F6.2,'(kg), Area ',F6.3,' (m**2),', . 'Drag and rolling Coefficient ',F6.2,1x,F6.4) write(*,200) P_rider, F_max 200 format('Rider Power ',F6.2,' (Watts) and max force ',F6.2,' (N)') write(*,220) 220 format(60('+')) ***** Thats all return end CTITLE READ_INPUT subroutine read_input implicit none * Routine to read the input parameters of the program include 'FBike.h' integer*4 ierr ! IOSTAT error reading user inputs * LOCAL VARIABLES real*8 terr_mm ! Integration accuracy mm real*8 lambda_km ! wavelength km real*8 Track_len_km ! Track length in km **** Set default values. User can then get defaults by typing / Mass = 80.d0 ! Mass of object (kg) Cd = 0.90d0 ! Drag coefficient of Bike Cr = 0.007d0 ! Rolling (friction) coefficient of Bike Area = 0.67d0 ! area of rider (m**2) P_rider = 100.0d0 ! Nominal constant power for rider (watts) F_max = 20.0d0 ! Maximum force the rider can supply (Newton) Terr = 0.010d0 ! Error on hitting distance (m) (default 1 cm) Slope = 0.001d0 ! Slope of track (radians) As = 5.0d0 ! Sin amplitudes (m) Bs = 0.0d0 ! Cos amplitudes (m) lambda = 2.d3 ! Wavelenth of oscillations (m) Track_len = 10.d3 ! Length (m) out_int = 100.d0 ! Output interval (sec). **** Get input from user: write(*,120) 120 format(/,'Program Parameter Input. [Defaults are printed ', . 'use / to accept default') Terr_mm = terr*1000.d0 lambda_km = lambda/1.0d3 Track_len_km = Track_len/1.0d3 write(*,140) Track_len_km, Terr_mm 140 format('Length of track (km) and error (mm) [',F6.3,' km ', . F5.1,' mm] ',\$) read(*,*,iostat=ierr) Track_len_km, Terr_mm call report_error(ierr,'Reading track length and error') terr = terr_mm/1000.d0 Track_len = Track_len_km*1000.d0 write(*,160) Slope, As, Bs, Lambda_km 160 format('Track Slope, Sin and Cos amplitudes (m) and ', . 'wavelenghth (km) [Defaults ',F5.3,2(F5.2,' m '),1x, . F6.2,' km] ',\$) read(*,*,iostat=ierr) Slope, As, Bs, Lambda_km call report_error(ierr,'Reading track slope and oscillations') write(*,180) Mass, Area, Cd, Cr 180 format('Rider/Bike Mass (kg), Area (m**2), ', . 'Drag and rolling Coefficient ,', . '[',F6.2,' kg, ',F6.3,' m^2, ', F6.2,' and ', . F6.4,'] ',\$) read(*,*,iostat=ierr) Mass, Area, Cd, Cr call report_error(ierr,'Reading Mass, Xarea, Cd and Cr') write(*,200) P_rider, F_max 200 format('Rider Power (W) and max force (N) [',F6.2,' Watts,', . F6.2,' N] ',\$) read(*,*,iostat=ierr) P_rider, F_max call report_error(ierr,'Reading rider power and max force') write(*,220) out_int 220 format('Output interval, zero for none (default ',F8.2,' s) ',\$) read(*,*,iostat=ierr) out_int call report_error(ierr,'Reading output interval') return end CTITLE INT1d subroutine Int1d(step,int_time, Pos, Vel, Xpos ) implicit none * Runge-Kutta intergration routine. * Parameters need for program (Physical constants) include 'FBike.h' * PASSED VARIABLES real*8 step ! Integration step size (secs). real*8 int_time ! Integration time. If this is zero initially ! integrate on position, otherwise integrate on ! on time. real*8 Pos, Vel, Xpos ! position and velocity along track and Xposition * LOCAL VARIABLES real*8 P, V, X ! Intermediate position and velocities needed real*8 k1, k2, k3, k4 ! Coefficients needed for Runge Kutta integration real*8 accel ! Function that return x acceleration in real part ! and z-acceleration imaginary part real*8 time,T ! Time into integration real*8 tpast ! Time past the zero point for the last step. real*8 dist ! Final distance traveled in X-direction real*8 ustep ! Used step for integration. As the object approaches the ! ground, we take smaller steps in the last interval. real*8 AP1, AP2, AP3 ! Actual power supplied by rider at each ! point in the integration. This is ! integrated to get energy used. real*8 Th1, Th2, Th3 ! Slopes at points in integration (angle theta) logical done ! Set true when the integration is complete based either ! on position or flight time integer*4 n ***** Initialize position time = 0.d0 tot_energy = 0.d0 done = .false. ustep = step if( step.eq.0 ) then print *,' Zero step ' step = 0.05d0 ustep = step end if n = 0 if( outT ) then write(*,110) 110 format('O* Time X_pos S_pos ', . ' S_vel Energy',/, . 'O* (sec) (m) (m) ', . ' (m/s) (Joules)') write(*,160) 'O',time, Xpos, Pos, Vel, tot_energy end if * Itegrate until we get to the track_length do while ( .not.done ) n = n + 1 if( n.gt. 1d6 ) done = .true. ! Put in to make sure program will stop ! at some point **** Compute the accelerations we need k1 = ustep*accel(time, Pos, Vel, AP1, XPos, th1 ) P = Pos + ustep*Vel/2.d0 +ustep*k1/8.d0 V = Vel + k1/2.d0 X = Xpos + (ustep*Vel/2.d0 +ustep*k1/8.d0)*cos(th1) k2 = ustep*accel(time+ustep/2.d0, P, V, AP2, X, th2) P = Pos + ustep*V/2.d0 + ustep*k1/8.d0 V = Vel + k2/2.d0 X = Xpos + (ustep*V/2.d0 + ustep*k1/8.d0)*cos(th2) k3 = ustep*accel(time+ustep/2.d0, P, V, AP2, X, th2 ) P = Pos + ustep*Vel + ustep*k3/2.d0 V = Vel + k3 X = Xpos + (ustep*Vel + ustep*k3/2.d0)*cos(th2) k4 = ustep*accel(time+ustep, P, V, AP3, X,th3) **** Update the time, position and velocity T = time + ustep P = Pos + ustep*(Vel + (k1+k2+k3)/6.d0) V = Vel + (k1+2*k2+2*k3+k4)/6.d0 X = Xpos + (ustep*(Vel + (k1+k2+k3)/6.d0))*cos(th2) ***** Compute total energy used. Use a 3-point Newton-Cotes Methods * (Also called Sumpson's rule) tot_energy = tot_energy + (ustep/2)*(AP1+4*AP2+AP3)/3.d0 **** See if we have passed the end of the track. If we have then * take smaller steps. if( X-Track_len.gt.0 ) then tpast = (X-Track_len)/dble(Vel) ustep = abs(tpast/2.d0) if( abs(X-Track_len).lt. terr/10 ) then time = T Pos = P Vel = V Xpos = X done = .true. if( outT ) then write(*,160) 'O',time, Xpos, Pos, Vel, tot_energy end if endif else * Continue with integration time = T Pos = P Vel = V Xpos = X if( outT ) then if( abs(T-nint(T/out_int)*out_int).le.1.d-3 ) then write(*,160) 'O',time, Xpos, Pos, Vel, tot_energy ! else ! write(*,160) 'T',time, Xpos, Pos, Vel, tot_energy endif 160 format(a,1x,F10.3, 2F15.4, 1F15.4,1x, F12.2) end if end if **** See if we have completed. Based either on position * of time if( int_time.ne.0 ) then * Test on time should be = but to avoid rounding * error we test when is it within step/10. if( abs(time-int_time).lt.step/10 ) done =.true. end if end do **** Return the Distance traveled. Correct for the overshoot i.e., * the last Z-position is below the surface, so compute how many * seconds to get back to surface at last velocity and correct the * horizontal distance for this effect. tpast = Xpos/(Vel*cos(Th2)) c Dist = realpart(Pos) - realpart(Vel)*tpast Dist = Xpos - Vel*cos(Th2)*tpast int_time = time ***** Thats all return end CTITLE ACCEL real*8 function accel( time, P, V, AP, X,Theta) implicit none **** Routine to compute the acceleration of the body at position * P and with Velocity V. Time is passed into the routine but * ir is not needed for the forces considered here. (In a more * general problem with mass wastage we may need time to compute * mass). * Parameters need for program (Physical constants) include 'FBike.h' * PASSED VARIABLES real*8 time ! Integration time (not used in this version) real*8 P, V ! Along track position and velocity real*8 X ! X-coordinate real*8 AP ! Actual power supplied by rider real*8 theta ! Surface slope * LOCAL VARIABLES real*8 gacc, dacc, racc ! Gravity along path, Drag and rolling accelerations real*8 facc ! Rider force acceleration real*8 Vmag ! Magnitude of velocity real*8 Rho_h ! Density of air at our height real*8 F_mag ! Magnitude of rider force (N) ***** Compute the accelations: theta = atan(Slope + . As*cos(2*pi*X/Lambda)*2*pi/Lambda - . Bs*sin(2*pi*X/Lambda)*2*pi/Lambda) * DRAG: Acts opposite to the motion vector (since wind is fixed * relative to motion direction) Vmag = abs(V) Rho_h = rho_air gacc = -g_0*sin(theta) dacc = -(Cd*Rho_h*V**2/2)/Mass*Area ! Note: This is Vmag^2*V_unit_vector racc = -g_0*Cr*cos(theta) * Get the rider power F_mag = F_max if( Vmag.gt.0 ) then F_mag = min(P_rider/Vmag, F_max) endif facc = F_mag/Mass AP = F_mag*Vmag **** Add the acceleration together accel = gacc + dacc + racc + facc ***** Thats all return end CTITLE REPORT_ERROR subroutine report_error(ierr,mess) * Routine to report IOSTAT errors. Initially developed * for 12.010 HW2 Problem 3. * PASSED VARIABLES integer*4 ierr character*(*) mess * LOCAL VARIABLES * None ***** See if the IOSTAT error was non-zero if( ierr.ne.0 ) then write(*,120) ierr, mess 120 format('IOSTAT Error ',i4,' occurred ',a) stop 'HW2_3: IO Error in program' end if return end