Lectures Part#1 - Department of Mechanical Engineering

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- Phase Relationship between x, v, a x Displacement = A cos( ω t + φ ) n Displacement x (t) = A sin(v n t + f) A O t –A x& = −ω Asin( ω t + φ ) n Velocity Velocity • x (t) = v n A cos(v n t + f) n v n A O t && x Acceleration Acceleration •• x (t) = –v 2 n A sin(v n t + f) 2 = −ω A cos( ω t + φ ) n n –v n A v n 2 A O –v 2 nA t 12
- Example For m= 300 kg and ω n =10 rad/s compute the stiffness: ω n = k m ⇒ k = mω 2 n = (300)10 2 kg/s 2 = 3 ×10 4 N/m 14
Page 1 and 2: Dr. Mostafa Ranjbar Fundamentals of
Page 3 and 4: - Why to study vibration • Vibrat
Page 5 and 6: - Stiffness • From strength of ma
Page 7 and 8: - 2nd Order Ordinary Differential E
Page 9 and 10: - Periodic Motion x(t) x(t) [mm] A
Page 11: - Amplitude & Phase from the initia
Page 15 and 16: - Some useful quantities A = peak v
Page 17 and 18: - Example Hardware store spring, bo
Page 19 and 20: 21 - Derivation of the solution jt
Page 21 and 22: Viscous Damping - 24 Shear Stress (
Page 23 and 24: - Linear Viscous Damping f c = cx(t
Page 25 and 26: 28 - Derivation of the solution t t
Page 27 and 28: - Let x(t) = ae λt & subsitute int
Page 29 and 30: - Critical damping continued • No
Page 31 and 32: - The overdamped response Displacem
Page 33 and 34: - Underdamping x(t) = e − ζω n
Page 35 and 36: - Example consider the spring in Ex
Page 37 and 38: - Solution: V 0 =0.6, X 0 =0, ζ =
Page 39 and 40: - Plot of the response: Displacemen
Page 41 and 42: - MODELING AND ENERGY METHODS An al
Page 43 and 44: - Energy Methods ∫ Fdx work done
Page 45 and 46: - Example The effect of including t
Page 47 and 48: - What about gravity? kΔ mg − k
Page 49 and 50: - More on springs and stiffness l x
Page 51 and 52: - Example compute the frequency of
Page 53 and 54: - Samples of Vibrating Systems •
Page 55 and 56: - Mass of pod 10 kg empty 1000 kg f
Page 57 and 58: - Static Deflection δ , Δ = dista
Page 59 and 60: - Use these to design from availabl
Page 61 and 62: - Case a) parallel connection : k 3
Page 63 and 64:
- Free Vibration with Coulomb Dampi
Page 65 and 66:
- Free Vibration with Coulomb Dampi
Page 67 and 68:
- 86 Free Vibration with Hysteretic
Page 69 and 70:
- 2.6.4 Energy dissipated in Viscou
Page 71:
- Energy dissipation and Loss Coeff
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Lectures Part#1 - Department of Mechanical Engineering

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