¡Descarga Aerospace Eng. Project: 2DOF System Analysis with NASTRAN y más Ejercicios en PDF de Ingeniería Aeroespacial solo en Docsity! MAJOR IN AEROSPACE ENGINEERING AEROSPACE DESIGN II PROJECT REPORT: INTRODUCTION TO NASA STRUCTURE ANALYSIS ”NASTRAN” Project report submited by: Carlos Garcia NIA: 100383047 Diego Alonso NIA: 100382962 Pedro Gilabert NIA: 100363316 Professor: Dr. Pablo Fajardo 2 UNCOUPLED 2DOF SYSTEM 1 Introduction This report studies the behavior of a 2-DOF system with masses, springs and dampers using NASTRAN as the tool for computations. Using finite element analysis NASTRAN can obtain the normal mode frequencies and the transient responses to a excitation of the simple systems proposed here. For pre and postprocessing both no user interface was used, and all the data was directly introduced and interpreted on the text files with which the code operates. The system studied here are composed by point masses connected by linear springs and dampers, creating a coupled system whose analytical solution is hard to obtain, but that the code handles pretty easily. 2 Uncoupled 2DOF System 2.1 System Model The uncoupled 2 DOF system was defined as two independent masses attached to one spring each. Nevertheless the stiffness of each of the springs has to be computed taking into account that both of the uncoupled masses are equal and have a value of M1 = M2 = 1kg and the natural frequencies of the systems are given as fR1 = 10Hz and fR2 = 20Hz. Figure 1: Mass-spring 2DOF uncoupled system For each of the two mass-spring systems, the natural frequency is defined as: f = 1 2π √ K M (1) So as the goal is to define the model of the system, the stiffness of each spring is needed. In order to compute these parameters, the equation 1 was solved for the K as follows: K1 = M1(2πfR1) 2 = 3947.84 N m (2) K2 = M2(2πfR2) 2 = 15791.37 N m (3) 1 3 COUPLED SYSTEM 3.1 Sensitivity Analysis for the Maximum Response Until this point only normal modes of the system were studied. From now on a force with a shape similar to (1-cos) is applied in the M1 and the study of the system will be focused on the transient response of it using the feature SOL 112 in NASTRAN. For this purpose two different configurations of the system were considered. First a 1DOF spring and damper system with a stiffness of K1 = 3947.84N/m and a damping ratio of 3%. Secondly the 2DOF coupled system with K1 = KCoupling = 3947.84N/m and K2 = 15791.37N/m. A sensitivity analysis for the maximum response of M1 was made for a changing time step and the results of each of the systems were represented in the following graph. Figure 6: Maximum displacement vs Time step It can be seen that for the 1DOF damped system the response is much higher which means that for a mass attached to a spring and subjected to a oscillating force, adding a coupled mass with a spring will absorb the oscillations better than adding a damper to the similar mass-spring system. 4 4 2DOF VIBRATION ABSORBER 4 2DOF Vibration Absorber Last thing to be studied on this report was the modification of the 2 DOF system in order to obtain a vibration absorber. For that purpose, as it can be seen below, the spring that connected the floor with mass 2 has been removed. Figure 7: Vibration absorber model Now in order to study the performance of the vibration absorber defined, the particular displacement vector of the system can be computed assuming harmonic oscillation of the system (x = Aeiwdrt), leading to a first component of A defined as: a1 = Fo(Ka −m2w 2 dr) (K +Ka −mw2 dr)(Ka −maw2 dr) (4) If a reduction on the displacement is needed this constant should be zero. In order to reach this, the coupling stiffness needs to be dependent on the frequency of the external force as: Ka = maw 2 dr (5) By inserting forces with frequencies of 10 and 100 Hz, the response of the vibration absorber could be studied. A trend towards the stable response of zero displacement is observed, however the system still oscillates and the complete absorption of the force is not observed, which contradicts the analytical solution from which the kcoupling is obtained 5 4 2DOF VIBRATION ABSORBER Figure 8: Vibration absorber model The frequency of 10 Hz yields a displacement much higher than the 100 Hz one as we get much closer to resonance, but the 20 Hz frequency makes the system tend to the steady state solution of zero displacement. Figure 9: Vibration absorber model 6
