![]() ![]() The elements may be of different shapes and sizes.Ģ. In the first step, the geometry is divided into a number of small elements. Second natural frequency - Using FEM, we will find the second natural frequency of the cantilever beam (continuous system) having accelerometer mass at free end. Hence the corrected natural frequency after consideration of mass of the accelerometer would beĢ. The mass of accelerometer is 4.8 gm = 0.0048 kg, so the total mass will be Taking data from Tables 4.1 and 4.2 for steel, Example 4.2 Obtain the fundamental natural frequency of beam by considering the mass of the sensor also. The finite element method analysis is presented subsequently for the same. However, we need to obtain the effective stiffness of equivalent discrete system with additional supports at nodes (for the second natural frequency at one location and for the third natural frequency at two locations). (4.17) Following the above procedure, other natural frequencies can be improved to account for the mass of the accelerometer. (4.16) So for the discrete beam with accelerometer, the theoretical fundamental natural frequency after considering the mass of accelerometer will be Now, if we consider the mass of accelerometer, m acc, at the free end of the beam, than the total mass at free end will be (4.14) From which the effective mass at tip can be written as Hence, the natural frequency of discrete model of the beam without an accelerometer can be written as, First natural frequency - Let us consider the beam specimen as mass-less with stiffness k and has a discrete effective mass, m eff, at the free end, which produces the same frequency as a continuous beam specimen without any tip mass. In case of non-contacting sensors, there would not be any correction required since there will not be any additional tip mass on the beam.Ĥ.3.1 Consideration of the mass of accelerometer onto a continuous cantilever beamġ. Now we would explain a simpler procedure by which corrections to the natural frequency could be made so as to get closer to the measured natural frequency. By continuous approach the solution is difficult since with tip mass the boundary condition at free end is now time dependent. The above frequencies have to be modified since there is a mass in the form of an accelerometer at the free end of the continuous beam. Table 4.2 Different geometries of the beamĮxample 4.1: Obtain the undamped natural frequency of a steel beam with l = 0.45 m, d = 0.003 m, and b = 0.02 m. Table 4.1 Material properties of various beams 4.3: The first three undamped natural frequencies and mode shape of cantilever beam 4.2: Cross-section of the cantilever beamįig. (4.12) Where b and d are the breadth and width of the beam cross-section as shown in the Fig. (4.11) Where, d is the diameter of cross section and for a rectangular cross section (4.10) where I, the moment of inertia of the beam cross-section, for a circular cross-section it is given as ![]() (4.9) The natural frequency is related with the circular natural frequency as (4.4) with The mode shapes for a continuous cantilever beam is given asĪ closed form of the circular natural frequency ω nf, from above equation of motion and boundary conditions can be written as, 4.1)įor a uniform beam under free vibration from equation (4.1), we get We have following boundary conditions for a cantilever beam (Fig. ![]() 4.1(b) depicts of cantilever beam under the free vibration. 4.1(a) shows of a cantilever beam with rectangular cross section, which can be subjected to bending vibration by giving a small initial displacement at the free end and Fig. 4.1 (b): The beam under free vibrationįig. (4.1) Where, E is the modulus of rigidity of beam material, I is the moment of inertia of the beam cross-section, Y( x) is displacement in y direction at distance x from fixed end, ω is the circular natural frequency, m is the mass per unit length, m = ρA(x) , ρ is the material density, x is the distance measured from the fixed end.įig. For a cantilever beam subjected to free vibration, and the system is considered as continuous system in which the beam mass is considered as distributed along with the stiffness of the shaft, the equation of motion can be written as (Meirovitch, 1967), ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |