1. THIS IS THE DATAPLOT PROGRAM FILE SPRINGS.DP 2. PURPOSE--XXX 3. 4. -----START POINT----------------------------------- 5. 6. PURPOSE--DETERMINE THE NATURAL VIBRATING FREQUENCY 7. OF A VERTICAL SPRING SYSTEM. 8. ANALYSIS TECHNIQUE--DETERMINE THE EIGENVALUES 9. OF A NON-SYMMETRIC MATRIX 10. APPLICATION--SPRING VIBRATION 11. SOURCE (PROBLEM)--FOGIEL, THE LINEAR ALGEBRA PROBLEM SOLVER, 12. RESEARCH AND EDUCATION ASSOCIATION, 1980 13. PAGE 693 14. GIVEN--THE SYSTEM IS VERTICAL, 15. ATTACHED AT TOP AND BOTTOM, 16. IN THE ORDER (FROM TOP TO BOTTOM) 17. SPRING 1, MASS 1, SPRING 2, MASS 2, SPRING 3 18. SPRING 1 CONSTANT K1 = 1 19. SPRING 2 CONSTANT K2 = 4 20. SPRING 3 CONSTANT K3 = 4 21. MASS 1 M1 = 1 22. MASS 2 M1 = 4 23. TO FIND--THE NATURAL FREQUENCIES OF THE SYSTEM 24. NOTE--TO SET UP THE 2 EQUATIONS OF MASS 25. (1 EQUATION FOR EACH OF THE 2 MASSES) 26. WE APPLY NEWTON'S SECOND LAW OF MATION 27. (CHANGE OF MOMENTUM = SUM OF FORCES ACTING ON PARTICLE) 28. AND HOOKE'S LAW 29. (MAGNITUDE OF RESTORING FORCE OF A SPRING 30. IS PROPROTIONAL TO DISPLACEMENT AND 31. TO THE SPRING CONSTANT). 32. NOTE--THE SPECIFICS ARE AS FOLLOWS-- 33. LET Y1 = DISPLACEMENT OF MASS 1 FROM ITS EQUILIBRIUM POSITION 34. LET Y2 = DISPLACEMENT OF MASS 2 FROM ITS EQUILIBRIUM POSITION 35. THE 2 DIFFERENTIAL EQUATIONS ARE THEREFORE 36. M1*Y1'' = -K1*Y1 - K2*(Y1-Y2) 37. M2*Y2'' = K2*(Y1-Y2) - K3*Y2 38. FOR HARMONIC MOTION-- 39. Y1'' = -F**2 * Y1 40. Y2'' = -F**2 * Y2 41. THUS HAVE 42. M1*(-F**2 * Y1) = -K1*Y1 - K2*(Y1-Y2) 43. M2*(-F**2 * Y2) = K2*(Y1-Y2) - K3*Y2 44. SUBSTITUTING FOR K1, K2, K3, M1, AND M2 YIELDS 45. -F**2 * Y1 = -5*Y1 + 4*Y2 46. -F**2 * Y2 = Y1 - 2*Y2 47. WRITING F**2 AS LAMBDA AND TRANSPOSING, WE HAVE 48. 5*Y1 - 4*Y2 = LAMBDA*Y1 49. -Y1 + 2*Y2 = LAMBDA*Y2 50. THE NATURAL FREQUENCIES ARE THUS SEEN 51. TO BE THE EIGENVALUES OF THE COEFFICIENT MATRIX. 52. NOTE--FOR TESTING PURPOSES, THE SOLUTION IS 53. 1 AND SQRT(6) = 2.449 54. 55. -----START POINT----- 56. 57ECHO 58DIMENSION 100 VARIABLES 59. 60. STEP 1-- 61. DEFINE THE MATRIX 62. 63READ MATRIX A 645 -4 65-1 2 66END OF DATA 67PRINT A 68. 69. STEP 2-- 70. DETERMINE THE EIGENVALUES 71. 72LET E = MATRIX EIGENVALUES A 73PRINT E 74