Natural Frequency of Free Transverse Vibrations

          Consider a shaft of negligable mass whose one end is fixed and the other end carries a body of weoght W, as shown in fig. 

               Let           s = Stiffness of shaft,

                            δ = Static deflection due to weight of the body,

                                 x = Displacement of body from mean position after time t.

                                m = Mass of body = W/g As discussed in the previous article, 

Restoring force  =     – s.x          . . .                       (i)

and accelerating force  =   m   d² x               . . . (ii)

                                                        dt²

Equating equations (i) and (ii), the equation of motion becomes

Hence, the time period and the natural frequntly of the transverse vibrations are same as that of longitudinal vibrations. Therefore 

Note : 

The shape of the curve, into which the vibrating shaft deflects, is identical with the static deflection curve of a cantiliver beam loaded at the end. It has been proved in the text book on strength of Materials that the static deflection of a cantiliver beam loaded at the free end is 

                           δ  =      Wl³                    (in metres)

                                       3EI

Where        W = Load at the free end, in newtons,

                     l = Length of the shaft or beam in metres,

                  E = Young’s modulus for the material of the shaft or beam in N/m², and

                    I = Moment of inertia of the shaft or beam in m⁴.

Effect of Inertia of the Constraint in Longitudinal and Transverse Vibrations

          In deriving the expressions for natural frequency of longitudinal and transverse vibrations, we have neglected the inertia of the constraint i.e. shaft. We shall now discuss the effect of the inertia of the constraint, as below:

1. Longitudinal vibration

          Consider the constraint whose one end is fixed and other end is free as shown in Fig.

                Let         m₁ = Mass of the constraint per unit length,

                                  l = Length of the constraint,

                                  m_C = Total mass of the constraint = m1. l, and

                                  v = Longitudinal velocity of the free end.

          Consider a small element of the constraint at a distance x from the fixed end and of length δx.  

               ∴     Velocity of the small element

          If a mass of m_C/3 is placed at the free end and the constraint is assumed to be negligible mass, then

          Total kinetic energy possesed by the constraint

          Hence the two systems are dynamically same. Therefore, inertia of the constraint may be allowed for by ading one-third of its mass to the disc at the free end.

         From the above discussion, we find that when the mass of the constraint m_C and the mass of the dosc 'm' at the end is given, then natural frequency of vibration,

2. Transverse vibration

          Consider a constraint whose one end is fixed and the other end is free as shown in Fig..

               Let             m₁ = Mass of the constraint per unit length,

                                  l = Length of the constraint,

                                  m_C = Total mass of the constraint = m1. l, and

                                  v = Longitudinal velocity of the free end.

          Consider a small element of the constraint at a distance x from the fixed end and of length δx . The velocity of this element is given by

          If a mass of 33m_C/140 is placed at the free end and the constraint is assumed to be negligible mass, then

          Total kinetic energy possesed by the constraint

          Hence the two systems are dynamically same. Therefore, inertia of the constraint may be allowed for by adding 33/140 of its mass to the disc at the free end.

         From the above discussion, we find that when the mass of the constraint m_C and the mass of the dosc 'm' at the end is given, then natural frequency of vibration,

Notes:

1. If both the ends of the constraint are fixed, and the disc is situated in the middle of it, then proceeding in the similar way as discussed above, we may prove that the inertia of the constraint may be allowed for by adding 13/35 of its mass to the disc.

2. If the constraint is like a simply supported beam, then 17/35 of its mass may be added to the mass of the disc.

Natural Frequency of Free Transverse Vibrations Due to a Point Load Acting Over a Simply Supported Shaft

          Consider a light shaft AB of length 'l' between supports, carrying a point load W at C which is at a distance of l₁ from A and l₂ from B, as shown in Fig., A little consideration will show that when the shaft is deflected and suddenly released, it will make (or undergo) transverse vibrations. The deflection of the shaft is proportional to the load W which is acting on it and if the shaft is deflected beyond the static equilibrium position under load W, then the load along with the beam will vibrate with simple harmonic motion (S.H.M) (as by helical spring). If 'δ' is the static deflection due to load 'W', then the natural frequency of the transverse vibration is 

ω_n = √g/δ

static deflection for the various types of beams under various load conditions

Some of the values of the static deflection for the various types of beams under various load conditions are given in the following table.

Values of static deflection (δ) for the various types of beams and under various load conditions