Forced Vibration of a Cantilever Beam(Continuous System)



5.3 Mathematical analysis

Fig. 5.1 (a): A cantilever beam

 

Fig. 5.1 (b): The beam under forced vibration

 

Fig 5.1(a) is showing a cantilever beam which is fixed at one end and other end is free, having rectangular cross-section.

Fig 5.1(b) is showing a cantilever beam which is subjected to forced vibration. An exciter is used to give excitation to the system. The exciter is capable to generate different type of forcing signal e.g. sine, swept sine, rectangular, triangular etc.

In actual case, the beam is a continuous system, i.e. the mass along with the stiffness is distributed throughout the beam. The equation of motion in this case is ( Meirovitch, 1967),

                   

                                 (5.1)

 

where, E is the modulus of rigidity of beam material, I is the moment of inertia of the beam cross-section, y ( x , t ) is displacement in y direction at distance x from fixed end, m is the mass per unit length, m = ρA(x) ρ is the material density, A>x ) is the area of cross-section of the beam, f(t) is the forced applied to the system at x = L 1

 

Free Vibration Solution: For a cantilever beam (Fig. 5.1), the boundary conditions are given by,

                                  

                                                       (5.2a)

                                  

                                                (5.2b)

 

For a uniform beam under free vibration from equation (5.1), we get

                                            

                                                           (5.3)

 

with

                                                   

A closed form of the circular natural frequency ω nf , from above equation of motion for first mode can be written as

                                  

                                                  (5.4)

 

Second natural frequency

 

                                                                                                              (5.5)

 

Third natural frequency

 

                                                                                                         (5.6)

 

The natural frequency is related with the circular natural frequency as

 

                                                                                                           (5.7)

 

where I , the moment of inertia of the beam cross-section, for a circular cross-section it is given as

                                                                                                                   (5.8)

 

Where, d is the diameter of cross section and for a rectangular cross section

                                                                                                                       (5.9)

 

Where b and d are the breadth and width of the beam cross-section as shown in the Fig. 5.2.

 

Fig. 5.2: Cross-section of the cantilever beam

 

&where b and d are the breadth and width of the beam cross-section as shown in the Fig. 5.2.

Fig. 5.3: The first three undamped natural frequencies and mode shapes of cantilever beam



 

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