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MODAL ANALYSIS 1 Tacoma Narrows Galloping Gertie 2 Flutter of Boeing 747 wings 3 B52 parked B52 flying Note deflection of wings 4 1DOF.SLDASM 5 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: [K]d = F 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: 6 Non-zero solutions of a free undamped vibration present the eigenvalue problem: K - ωi 2 M = 0 i ωi 2 eigenvalue (square of circular frequency) i eigenvector (shape) ω circular frequency [rad/s] ω 2π 1 f= T f= frequency [Hz] Solutions provide with eigenvalues and associated modal shapes of vibration 7 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. Inertial forces Ki = ωi Mi 2 Elastic forces 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). 8 Material density must be defined in units derived from the unit of force and the unit of length. [mm] [N] unit of mass unit of mass density for aluminum tonne tonne/mm3 2.794x10-9 [m] [N] unit of mass unit of mass density for aluminum kg kg / m3 2794 [in] [lb] unit of mass unit of mass density For aluminum lbf = slug/12 slug/12/in3 lbf s2/in4 2.614x10-4 Notice that the erroneous mass density definition (kg / m 3 instead of tonne / mm3) will results in part mass being one trillion (1e12) times higher. 9 U SHAPE BRACKET model file model type material U BRACKET shell alloy steel thickness 2mm restraints hinge load none hinge support (no translations) hinge support (no translations) objective • demonstrate modal analysis • study convergence of natural frequencies • defining supports for shell element model • properties of lower and higher modes of vibration U BRACKET SAE models 10 cantilever beam.SLDPRT 04 models modal 11 TUNNING FORK PLASTIC PART Chapter 6 Chapter 6 12 truck.SLDPRT car.SLDPRT 04 models modal 04 models modal 13 EXERCISE helicopter blade Model file ROTOR Model type solid Material 1060 Alloy Supports fixed to the I.D. sym B.C. to hub Loads centrifugal load due to 300RPM Units mm, N, s Objectives • Modal analysis without pre-stress • Modal analysis with pre-stress Analysis is conducted on one blade only. ROTOR CHAPTER 21 14 pendulum 02.SLDPRT 04 models modal 15