Mechanical Vibration Bogumił Chiliński bogumil.chilinski@gmail.com Room 4.1.9 Consultations hours - Tuesday 11:15-12:00 March 4, 2020 B.Chiliński Mechanical Vibration March 4, 2020 1 / 41
Table of contents 1 General information - 1st lecture (2020.02.26) 2 Mechanical vibration - general information - 1st lecture (2020.02.26) 3 Periodicity of signal - 1st lecture (2020.02.26) 4 Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) 5 Damped free vibration - 1st test (2020.02.26) 6 Damped free vibration - 2nd lecture (2020.03.04) 7 Harmonic synthesis - 2nd lecture (2020.03.04) 8 Logarithmic decrement - 2nd lecture (2020.03.04) B.Chiliński Mechanical Vibration March 4, 2020 2 / 41
General information - 1st lecture (2020.02.26) General information - 1st lecture (2020.02.26) B.Chiliński Mechanical Vibration March 4, 2020 3 / 41
General information - 1st lecture (2020.02.26) General information Schedule: • 15 lectures; • First lecture - introduction; • 2nd-14th lecture - short test in the beginning (max. 5 min - non obligatory); • Last lecture - additional exam. Passing requirements: • 13 short tests (non obligatory) - extra points to additional exam (up to 15 points); • Exam (obligatory) - 50 points (26 points - passing limit). B.Chiliński Mechanical Vibration March 4, 2020 4 / 41
General information - 1st lecture (2020.02.26) General information Scope of the lecture: • Analysis and modelling of vibrating system. • Free vibrations of a single degree of freedom linear systems. • Harmonically excited vibrations of single degree of freedom linear systems. • Periodically excited vibrations of single degree of freedom linear systems. • Phase plane analysis of a linear vibration. • Phase plane analysis of a nonlinear vibration. • Free vibration of multi degree of freedom linear systems. • Excited vibration of multi degree of freedom linear systems. B.Chiliński Mechanical Vibration March 4, 2020 5 / 41
General information - 1st lecture (2020.02.26) General information Scope of the lecture (continued): • Vibrations of strings (lateral), rods (longitudinal) and shafts (torsional). • Lateral free vibrations of beams. • Lateral excited vibrations of beams. • Free vibrations of single degree of freedom nonlinear systems. • Harmonically excited vibrations of single degree of freedom nonlinear systems. • Parametric and self-excited vibrations. B.Chiliński Mechanical Vibration March 4, 2020 6 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Mechanical vibration - general information - 1st lecture (2020.02.26) B.Chiliński Mechanical Vibration March 4, 2020 7 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Mechanical vibration - general information Classification of vibration: • Deterministic or random; • Free or excited; • Damped or undamped; • Linear or nonlinear: • parametric oscillations, • self-excited oscillations; • Vibrations of Single degree of freedom, multi degree of freedom systems or continuous systems; B.Chiliński Mechanical Vibration March 4, 2020 8 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Phenomena caused by vibration Figure 1: Resonant vibration of Tahoma Narrows Bridge B.Chiliński Mechanical Vibration March 4, 2020 9 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Phenomena caused by vibration Figure 2: Vibration of a plane wing B.Chiliński Mechanical Vibration March 4, 2020 10 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Phenomena caused by vibration Figure 3: Static deflection a cable-stayed bridge B.Chiliński Mechanical Vibration March 4, 2020 11 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Phenomena caused by vibration Figure 4: A cable-stayed bridge torsional resonant vibration B.Chiliński Mechanical Vibration March 4, 2020 12 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Phenomena caused by vibration Figure 5: A cable-stayed bridge transverse resonant vibration B.Chiliński Mechanical Vibration March 4, 2020 13 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Mechanical vibration - general information The most relevant phenomena of a vibrational nature: • fatigue, • resonant vibration. B.Chiliński Mechanical Vibration March 4, 2020 14 / 41
Mechanical vibration - general information - 1st lecture (2020.02.26) Mechanical vibration - general information Harmonic process (signal) x ( t ) = A · sin( ω · t + ϕ ) (1) where: A - amplitude, ω - frequency, t - time, ϕ - phase shift. B.Chiliński Mechanical Vibration March 4, 2020 15 / 41
Periodicity of signal - 1st lecture (2020.02.26) Periodicity of signal - 1st lecture (2020.02.26) B.Chiliński Mechanical Vibration March 4, 2020 16 / 41
Periodicity of signal - 1st lecture (2020.02.26) Periodic signal process Periodical function - definition ∀ t ∈ R : x ( t ) = x ( t − T ) (2) where: x ( t ) - signal, T - period. B.Chiliński Mechanical Vibration March 4, 2020 17 / 41
Periodicity of signal - 1st lecture (2020.02.26) Signal periodicity - example Case Determine a period T of the following signal: x ( t ) = sin( ωt ) (3) Solution - utilization of the definition ∀ t ∈ R : sin( ωt ) = sin( ω · ( t − T )) (4) ωt = ω · ( t − T k ) + k · 2 · π (5) for k ∈ Z B.Chiliński Mechanical Vibration March 4, 2020 18 / 41
Periodicity of signal - 1st lecture (2020.02.26) Signal periodicity - example Solution - the base period T k · ω = k · 2 · π (6) for k ∈ Z T k = k · 2 · π (7) for k ∈ Z ω The base period is for k equals k = 1 : T o = 2 · π (8) ω B.Chiliński Mechanical Vibration March 4, 2020 19 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) B.Chiliński Mechanical Vibration March 4, 2020 20 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Unexcited vibration of a harmonic oscillator System scheme k c m Figure 6: A damped harmonic oscillator B.Chiliński Mechanical Vibration March 4, 2020 21 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Harmonic oscillator Governing equation m ¨ x + c ˙ x + kx = 0 (9) where: m - mass, c - damping constant, k - stiffness coefficient. B.Chiliński Mechanical Vibration March 4, 2020 22 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Unexcited vibration of a harmonic oscillator Simplified governing equation x + c x + k ¨ m ˙ mx = 0 (10) For the further computations there is assumed that: c k m = ω 2 m = 2 · h, (11) 0 B.Chiliński Mechanical Vibration March 4, 2020 23 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Solution of the governing equation Governing equation expressed in acceleration x + ω 2 ¨ x + 2 h ˙ 0 x = 0 (12) Predicted form of the solution x ( t ) = Ce rt (13) Substitution of the predicted solution (13) leads to the following equation: � 0 r 0 � r 2 + 2 hr 1 + ω 2 Ce rt = 0 (14) B.Chiliński Mechanical Vibration March 4, 2020 24 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Solution of the governing equation Characteristic polynomial r 2 + 2 h r 1 + ω 2 r 0 = 0 1 (15) 0 ���� ���� ���� A B C Roots of the characteristic polynomial are as follows: √ B 2 − 4 AC r 1 , 2 = − B ± (16) 2 A � 4 h 2 − 4 ω 2 − 2 h ± � 0 h 2 − ω 2 r 1 , 2 = = − h ± (17) 0 2 B.Chiliński Mechanical Vibration March 4, 2020 25 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Solution of the governing equation Final result x ( t ) = C 1 e r 1 t + C 2 e r 2 t (18) There are 3 cases: • overdamped vibration - if h 2 > ω 2 0 ; • critically-damped vibration - if h 2 = ω 2 0 ; • underdamped vibration - if h 2 < ω 2 0 . B.Chiliński Mechanical Vibration March 4, 2020 26 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Solution of the governing equation Overdamped oscillator h 2 > ω 2 (19) 0 The solution has a following form: x ( t ) = C 1 e − ht − √ 0 t + C 2 e − ht + √ h 2 − ω 2 h 2 − ω 2 0 t (20) B.Chiliński Mechanical Vibration March 4, 2020 27 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Solution of the governing equation Critically-damped oscillator h 2 = ω 2 (21) 0 The solution has a following form: x ( t ) = C 1 e − ht + C 2 te − ht (22) B.Chiliński Mechanical Vibration March 4, 2020 28 / 41
Damped unexcited harmonic oscillator - 1st lecture (2020.02.26) Solution of the governing equation Underdamped oscillator h 2 < ω 2 (23) 0 The solution has a following form: x ( t ) = C 1 e − ht − i √ 0 − h 2 t + C 2 e − ht +i √ ω 2 ω 2 0 − h 2 t (24) For further simplification of the calculations the following symbol is introduced: � ω 2 0 − h 2 ω h = (25) B.Chiliński Mechanical Vibration March 4, 2020 29 / 41
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