1. MODAL ANALYSIS OF DISCRETE SDOF SYSTEMS 1

2. Linear spring 400000N/m Model file1DOF.SLDASM MaterialAISI 1020 RestraintsFixed base Restraints preventing RBMs LoadsNone Objective: Modal analysis Mass 10kg 1DOF 1DOF.SLDASM 2

3. Restraints definition. These restraints are required to make the first mode correspond to the mode of vibration of SDOF 1DOF Restraints defined in local cylindrical system, only axial displacement component allowed Fixed restraint to base 3

4. Analytical solution 1DOF 4 We may assume solution as: then Equation of motion of free undamped vibration:

5. Results of modal analysis Mode 1 Numerical result: 32.36Hz Analytical result: 31.8Hz Mode 2 Numerical result: 12084Hz This is not a SDOF mode 1DOF 5

6. Model fileSWING ARM.SLDASM configuration 01 Material1060 Alloy RestraintsFixed base Fixed hinge to arm Loadsnone Objective: Modal analysis SWING ARM Linear spring 2000N/m m = 0.56kg SWING ARM.SLDASM 6

7. L 1 =0.2m k L =2000N/m L 2 =0.1m m = 0.56kg Mass of beam is negligible SWING ARM Analytical solution 7

8. SWING ARM Mode 1 Numerical result: 4.64Hz Analytical result: 4.75Hz Mode 2 Numerical result: 23.05.Hz This is not a SDOF mode 8

9. Model fileSWING ARM.SLDASM configuration 02 Material1060 Alloy RestraintsFixed base Fixed hinge to arm Loadsnone Objective: Modal analysis SWING ARM Linear spring 2000N/m m = 0.09kg SWING ARM.SLDASM 9

10. k L =2000N/m L=0.1m m = 0.09kg SWING ARM Analytical solution 10

11. 11 Mode 1 Numerical result: 43.4Hz Analytical result: 41.1Hz Mode 2 Numerical result: 2154.Hz This is not a SDOF mode SWING ARM

12. Element size 8.3mmElement size 3mm Moving beam Element size 1.5mm Moving beam Element size 1.5mm Moving beam and base

13. 13 ROLER Model fileROLER.SLDASM MaterialAISI304 RestraintsFixed base Fixed contact line Loadsnone Objective: Modal analysis k L =2000N/m m = 75.4kg Contact line

14. 14 ROLER Θ x k L =2000N/m m = 75.4kg

15. 15 ROLER Mode 1 Numerical result: 0.6Hz Analytical result: 0.66Hz Mode 2 Numerical result: 1544Hz This is not a SDOF mode

16. STABILITY 16

17. STABILITY OF A SDOF SYSTEM Non-oscillatory divergent motion is called DIVERGENT INSTABILITY Oscillatory divergent motion is called FLUTTER Over damped, critically damped, under damped motions are all well behaved. Amplitudes are finite, do not grow with time. If coefficients m, c, k are not positive, motion is not well behaved. 17

18. Divergent response Flutter response Stable response Marginally stable response Unstable response STABILITY OF A SDOF SYSTEM 18

19. PENDULUM.SLDASM Two linear springs 500N/m each m =1.0kg Model filePENDULUM.sldasm MaterialAISI 1020 RestraintsFixed base Fixed hinge to arm Loads1. Gravity down 2. Gravity up Objectives: Modal analysis Analysis of stability STABILITY OF A SDOF SYSTEM 19

20. L 1 =0.1m m =1.0kg m = 1kg k = 500N/m l = 0.1m g = 9.81m/s² Analytical solutionFEA solution Gravity down, k=500N/m Stable system STABILITY OF A SDOF SYSTEM 20

21. m =1.0kg Gravity down, k=196.2N/m Unstable system Analytical solution FEA solution L 1 =0.1m STABILITY OF A SDOF SYSTEM System becomes unstable when 21

22. L 1 =0.1m m =1.0kg Gravity up, no springsStable system Analytical solutionFEA solution STABILITY OF A SDOF SYSTEM 22