1. 1 MODAL ANALYSIS

2. 2 Tacoma Narrows Galloping Gertie

3. 3 Flutter of Boeing 747 wings

4. 4 B52 parkedB52 flying Note deflection of wings

5. 5 1DOF.SLDASM

6. 6 Every structure has its preferred frequencies of vibration, called resonant frequencies. Each such frequency is characterized by a specific shape of vibration. When excited with a resonant frequency, a structure will vibrate in this shape, which is called a mode of vibration. Recall that structural static analysis calculates nodal displacements as the primary unknowns: where [K] is known as the stiffness matrix, d is unknown vector of nodal displacements and F is the known vector of nodal loads. In dynamic analysis we additionally have to consider damping [C] and mass [M] In a modal analysis, which is the simplest type of dynamic analysis we investigate the free vibrations in the absence of damping and in the absence of excitations forces. Therefore, the above equation reduces to:

7. 7 Non-zero solutions of a free undamped vibration present the eigenvalue problem: Solutions provide with eigenvalues and associated modal shapes of vibration

8. 8 In resonance, inertial stiffness subtracts from elastic stiffness and, in effect, the structure loses its stiffness. The only factor controlling the vibration amplitude in resonance is damping. If damping is most often low, therefore, the amplitude of may reach dangerous levels. Note that even though any real structure has infinite number of degrees of freedom it has distinct modes of vibration. This is because the cancellation of elastic forces with inertial forces requires a unique combination of vibration frequency and vibration mode (shape). Note, that the equation of free undamped vibrations can be re-written to show explicitly that in resonance inertial forces cancel out with elastic forces. Elastic forces Inertial forces

9. 9 Material density must be defined in units derived from the unit of force and the unit of length. [mm] [N] unit of mass tonne unit of mass densitytonne/mm 3 for aluminum 2.794x10 -9 [m] [N] unit of mass kg unit of mass densitykg / m 3 for aluminum 2794 [in] [lb] unit of masslbf = slug/12 unit of mass densityslug/12/in 3 lbf s 2 /in 4 For aluminum2.614x10 -4 Notice that the erroneous mass density definition (kg / m 3 instead of tonne / mm 3 ) will results in part mass being one trillion (1e 12 ) times higher.

10. 10 U SHAPE BRACKET model file U BRACKET model typeshell materialalloy steel thickness2mm restraintshinge loadnone objective demonstrate modal analysis study convergence of natural frequencies defining supports for shell element model properties of lower and higher modes of vibration hinge support (no translations) hinge support (no translations) U BRACKET SAE models

11. 11 cantilever beam.SLDPRT 04 models modal

12. 12 TUNNING FORK Chapter 6 PLASTIC PART Chapter 6

13. 13 truck.SLDPRT 04 models modal car.SLDPRT 04 models modal

14. 14 EXERCISE helicopter blade Model fileROTOR Model typesolid Material1060 Alloy Supportsfixed to the I.D. sym B.C. to hub Loadscentrifugal load due to 300RPM Unitsmm, N, s Objectives Modal analysis without pre-stress Modal analysis with pre-stress Analysis is conducted on one blade only. ROTOR CHAPTER 21

15. 15 pendulum 02.SLDPRT 04 models modal