Single Degree Undamped Vibrations
(i) In many practical situations, damping has little influence on the natural frequency and may be neglected in its calculation.
(ii) In absence of damping, the system can be considered as conservative and principle of conservation of energy offers another approach to the calculation of the natural frequency.
(iii) The effect of damping is mainly evident in diminishing of the vibration amplitude at or near the resonance.
Vibration Model:
The basic vibration model consists of a mass, spring (stiffness) and damper (damping) as shown in figure
The inertia force model is
F2 = 
Where m is the mass in kg 
is the acceleration in m/sec2 and Fi is the inertia force in N.
The linear stiffness force model is
Fs= kx
Where k is the stiffness (N/m), x is the displacement and Fs is the spring force.
The damping force model for the viscous damping is
Fd = c
̇
Where c is the damping coefficient in N/m/sec, is the velocity in m/sec and Fd is the damping force.
For simplicity at present the damping is not considered.
The direction of x in the downward direction is positive. Also velocity, ̇, acceleration, , and force, F, are positive in the downward direction
On application of Newton's second law, we have
m = mg – k (Δ + x)
We have kΔ = mg (i.e. spring force due to static deflection is equal to weight of the suspended mass), so the above equation becomes
m + kx = 0
The choice of the static equilibrium position as reference for x axis datum has eliminated the force due to the gravity. Equation can be written as
Where ωn is the natural frequency (in rads/sec)
It can be observed that
(i) The motion is harmonic (i.e. the acceleration is proportional to the displacement).
(ii) It is homogeneous, second order, linear differential equation (in the solution two arbitrary constants should be there).
(iii) The general solution of equation can be written as
x = A sin ωnt + B cos ωnt
Where A and B are two arbitrary constants, which depend upon initial conditions i.e. x (0) and x ̇(0). Equation can be differentiated to give
̇= Aωn cos ωnt – B ωn sin ωnt
On application of initial conditions in, we get
x (0) = B and
̇ = Aωn
and B = x(0)
On substituting, we get
Where X is the amplitude, ωn is the circular frequency and Ø is the phase. The undamped free vibration executes the simple harmonic motion as shown in figure
Since sine & cosine functions repeat after 2 π radians (i.e. Frequency × Time period = 2 π), we have
ωnt = 2 π
The time period (in second) can be written as
The natural frequency (in rads/sec or Hertz) can be written as
We also have
On substituting, we get
Here T , f , ωn are dependent upon mass & stiffness of the system, which are properties of the system.
Above analysis is valid for all kind of SDOF system including beam or torsional members. For torsional vibrations the mass may be replaced by the mass moment of inertia and stiffness by stiffness of torsional spring. For stepped shaft an equivalent stiffness can be taken or for distributed mass an equivalent lumped mass can be taken.
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