Dynamics of Machine
CONTENTS CONTENTS
972
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Theory of Machines
24 eatur tures Features
1. Introduction. 2. Natural Frequency of Free Torsional Vibrations. 3. Effect of Inertia of the Constraint on Torsional Vibrations. 4. Free Torsional Vibrations of a Single Rotor System. 5. Free Torsional Vibrations of a Two Rotor System. 6. Free Torsional Vibrations of a Three Rotor System. 7. Torsionally Equivalent Shaft. 8. Free Torsional Vibrations of a Geared System.
Torsional Vibrations
24.1. Introduction
We have already discussed in the previous chapter that when the particles of a shaft or disc move in a circle about the axis of a shaft, then the vibrations are known as torsional vibrations. In this case, the shaft is twisted and untwisted alternately and torsional shear stresses are induced in the shaft. In this chapter, we shall now discuss the frequency of torsional vibrations of various systems.
24.2. Natural Frequency of Free Torsional Vibrations
Consider a shaft of negligible mass whose one end is fixed and the other end carrying a disc as shown in Fig. 24.1. Let
θ
=
m I k q
= = = =
Angular displacement of the shaft from mean position after time t in radians, Mass of disc in kg, Mass moment of inertia of disc in kg-m2 = m.k2, Radius of gyration in metres, Torsional stiffness of the shaft in N-m.
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CONTENTS CONTENTS
Chapter 24 : Torsional Vibrations
∴
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973
Restoring force
= q.θ
=I× d 2θ
... (i)
... (ii) dt 2 Equating equations (i) and (ii), the equation of motion is and accelerating force
I× I× d 2θ dt 2 d 2θ dt 2 = − q .θ
or
+ q .θ = 0
Fig 24.1. Natural frequency of free torsional vibrations.
q + ×θ = 0 . . . (iii) ∴ 2 I dt The fundamental equation of the simple harmonic motion is + ω2 .x = 0 dt 2 Comparing equations (iii) and (iv), d 2θ
d 2θ
. . . (iv)
ω=
∴
q I 2π I = 2π ω q
1 1 q = t p 2π I
Time period,
tp = fn =
and natural frequency ,
Note : This picture is given as additional
972
l
Theory of Machines
24 eatur tures Features
1. Introduction. 2. Natural Frequency of Free Torsional Vibrations. 3. Effect of Inertia of the Constraint on Torsional Vibrations. 4. Free Torsional Vibrations of a Single Rotor System. 5. Free Torsional Vibrations of a Two Rotor System. 6. Free Torsional Vibrations of a Three Rotor System. 7. Torsionally Equivalent Shaft. 8. Free Torsional Vibrations of a Geared System.
Torsional Vibrations
24.1. Introduction
We have already discussed in the previous chapter that when the particles of a shaft or disc move in a circle about the axis of a shaft, then the vibrations are known as torsional vibrations. In this case, the shaft is twisted and untwisted alternately and torsional shear stresses are induced in the shaft. In this chapter, we shall now discuss the frequency of torsional vibrations of various systems.
24.2. Natural Frequency of Free Torsional Vibrations
Consider a shaft of negligible mass whose one end is fixed and the other end carrying a disc as shown in Fig. 24.1. Let
θ
=
m I k q
= = = =
Angular displacement of the shaft from mean position after time t in radians, Mass of disc in kg, Mass moment of inertia of disc in kg-m2 = m.k2, Radius of gyration in metres, Torsional stiffness of the shaft in N-m.
972
CONTENTS CONTENTS
Chapter 24 : Torsional Vibrations
∴
l
973
Restoring force
= q.θ
=I× d 2θ
... (i)
... (ii) dt 2 Equating equations (i) and (ii), the equation of motion is and accelerating force
I× I× d 2θ dt 2 d 2θ dt 2 = − q .θ
or
+ q .θ = 0
Fig 24.1. Natural frequency of free torsional vibrations.
q + ×θ = 0 . . . (iii) ∴ 2 I dt The fundamental equation of the simple harmonic motion is + ω2 .x = 0 dt 2 Comparing equations (iii) and (iv), d 2θ
d 2θ
. . . (iv)
ω=
∴
q I 2π I = 2π ω q
1 1 q = t p 2π I
Time period,
tp = fn =
and natural frequency ,
Note : This picture is given as additional