1. MODULE 08 MULTIDEGREE OF FREEDOM SYSTEMS

2. 2 Structure vibrating in a given mode can be considered as the Single Degree of Freedom (SDOF) system. Structure can be considered a series of SDOF. For linear systems the response can be found in terms of the behavior in each mode and these summed for the total response. This is the Modal Superposition Method used in linear dynamics analyses. A linear multi-DOF system can be viewed as a combination of many single DOF systems, as can be seen from the equations of motion written in modal, rather than physical, coordinates. The dynamic response at any given time is thus a linear combination of all the modes. There are two factors which determine how much each mode contributes to the response: the frequency content of the forcing function and the spatial shape of the forcing function. Frequency content close to the frequency of a mode will increase the contribution of that mode. However, a spatial shape which is nearly orthogonal to the mode shape will reduce the contribution of that mode. MODAL SUPERPOSITION METHOD

3. 3 The response of a system to excitation can be found by summing up the response of multiple Single Degree of Freedom Oscillators (SODFs). Each SDOF represents the system vibrating in a mode of vibration deemed important for the vibration response. MODAL SUPERPOSITION METHOD

4. 4 Cost of modal solution vs. Step-by-step solution Number of time steps Cost Modal solution Step-by-step solution Results of modal analysis are required as a pre- requisite for modal solution MODAL SUPERPOSITION METHOD VS DIRECT INTEGRATION

5. 5 Model fileELBOW.SLDPART MaterialAl2014 SupportsFixed to the back face LoadsHarmonic force excitation Damping 2% modal Objectives: Time Response analysis Frequency Response analysis Modal mass participation Comparison between Static and Dynamic stress results Comparison between Time Response and Frequency Response results Harmonic load Constant amplitude 25000N Frequency range 0-500Hz Fixed support to the back ELBOW

6. 6 Mode 1 96HzMode 2 103Hz Mode 3 247Hz Results of modal analysis Mode 4 380Hz ELBOW

7. 7 Modal mass participation

8. 8 Finite element mesh; use default element size and apply mesh control 5mm to the round fillet Mesh control ELBOW

9. 9 Results of static analysis Maximum static stress 18.4MPa ELBOW

10. 10 Location of sensors Sensor to monitor displacements Sensor to monitor stresses ELBOW

11. 11 UZ displacement amplitude frequency response. Modes 2, 3, 4 show Mode 1 does not show because it has 0 mass participation in Z direction mm Hz UX displacement amplitude frequency response. Modes 1, 2, 4 show Mode 3 does no show because it has almost) zero mass participation in X direction mm ELBOW

12. 12 Von Mises stress frequency response Mode1 and mode 2 are indistinguishable MPa Hz MPa Von Mises stress frequency response In he range 90-115Hz shows the effect of mode 1 ELBOW