EBOOK - Vibration of Mechanical Systems (Alok Sinha)

This bookis intended for a vibration course in an undergraduate Mechanical Engineering curriculum. It is based on my lecture notes of a course (ME370) that I have been teaching for many years at The Pennsylvania State University (PSU), University Park. This vibration course is a required core course in the PSU mechanical engineering curriculum and is taken by junior-level or third-year students. Textbooks that have been used at PSU are as follows: Hutton (1981) and Rao (1995, First Edition 1986). In addition, I have used the book by Thomson and Dahleh (1993, First Edition 1972) as an important reference book while teaching this course.

It will be a valid question if one asks why I am writing another book when there are already a large number of excellent textbooks on vibration since Den Hartog wrote
the classic book in 1956.


1 EquivalentSingle-Degree-of-Freedom System and Free
Vibration...................................................... 1
1.1 Degrees of Freedom 3
1.2 Elements of a Vibratory System 5
1.2.1 Mass and/or Mass-Moment of Inertia 5
Pure Translational Motion 5
Pure Rotational Motion 6
Planar Motion (Combined Rotation
and Translation) of a Rigid Body 6
Special Case: Pure Rotation about a Fixed Point 8
1.2.2 Spring 8
Pure Translational Motion 8
Pure Rotational Motion 9
1.2.3 Damper 10
Pure Translational Motion 10
Pure Rotational Motion 11
1.3 Equivalent Mass, Equivalent Stiffness, and Equivalent
Damping Constant for an SDOF System 12
1.3.1 A Rotor–Shaft System 13
1.3.2 Equivalent Mass of a Spring 14
1.3.3 Springs in Series and Parallel 16
Springs in Series 16
Springs in Parallel 17
1.3.4 An SDOF System with Two Springs and Combined
Rotational and Translational Motion 19
1.3.5 Viscous Dampers in Series and Parallel 22
Dampers inSeries 22
Dampers in Parallel 23
1.4 Free Vibration of an Undamped SDOF System 25
1.4.1 Differential Equation of Motion 25
Energy Approach 27
1.4.2 Solution of the Differential Equation of Motion
Governing Free Vibration of an Undamped
Spring–Mass System 34
1.5 Free Vibration of a Viscously Damped SDOF System 40
1.5.1 Differential Equation of Motion 40
1.5.2 Solution of the Differential Equation of Motion
Governing Free Vibration of a Damped
Spring–Mass System 41
Case I: Underdamped (0<ξ <1or0<ceq <cc) 42
Case II:Critically Damped (ξ=1orceq =cc) 45
CaseIII: Overdamped (ξ>1orceq >cc) 46
1.5.3 Logarithmic Decrement: Identification of Damping
Ratio from Free Response of an Underdamped
System(0<ξ <1) 51
Solution 55
1.6 Stability of an SDOF Spring–Mass–Damper System 58
Exercise Problems 63
2 Vibration of a Single-Degree-of-Freedom System Under
ConstantandPurelyHarmonicExcitation...................... 72
2.1 Responses of Undamped and Damped SDOF Systems
to a Constant Force 72
Case I: Undamped (ξ=0) and Underdamped
(0<ξ <1) 74
Case II: Critically Damped (ξ=1orceq =cc) 75
CaseIII: Overdamped (ξ>1orceq >cc) 76
2.2 Responseof an Undamped SDOF System
to a Harmonic Excitation 82
Case I:ω = ωn 83
Case II:ω=ωn(Resonance) 84
Case I:ω = ωn 87
Case II:ω=ωn 87


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