advertisement

12/7/2009 The Australian Maritime College is an institute of the University of Tasmania CRICOS Provider Code: 00586B Websites: AMC: www.amc.edu.au, UTAS: www.utas.edu.au Chapter 2 Free vibration x + ω2n x = 0 • Learning Outcomes + cx + kx = 0 mx • Contents: – Pp. 2-3 SHIP VIBRATION Lecture 2 Free Vibration By Dr. Hung Nguyen National Centre for Maritime Engineering and Hydrodynamics Australian Maritime College/University of Tasmania Launceston, Tasmania, Australia Chapter 2 Free vibration – Procedure to derive equations of motion for free vibration systems – Translational vibration systems – Torsional vibration systems – Compound pendulum – Free vibration with damping – Energy dissipation in viscous damping – Equivalent springs, masses and dampers Chapter 2 Free vibration • 1.2 Case study examples • 1. Free vibration of S-DOF systems • 1.2.1 Translational system: spring-mass system – Procedure: • Select suitable coordinates • Determine the static equilibrium configuration • Apply the Newton’s second law of motion – Linear motion G G G d d2 G G F = ma = ( mv ) = 2 ( mx ) = mx dt dt – Rotational motion G G G d G d2 G M = Jα = ( Jω) = 2 Jθ = J θ dt dt ( ) Chapter 2 Free vibration • 1.2 Case study examples • Activity 1: spring-mass system in horizontal position Chapter 2 Free vibration • 1.2 Case study examples • Example 2.1: Harmonic response of a water tank • Example 2.2: Free vibration response due to impact • Example 2.3: Young’s modulus from natural frequency measurement 1 12/7/2009 Chapter 2 Free vibration • 1.2 Case study examples – 1.2.3 Torsional system Chapter 2 Free vibration • 1.2 Case study examples – 1.2.4 Natural frequency of compound pendulum – Example 2.6 Chapter 2 Free vibration • 2. Damped Free Vibration • 2.1 Mass-spring-damper (dashpot) system + cx + kx = 0 mx Chapter 2 Free vibration • 1.2 Case study examples – Activity 2: Horizontal disc – Diameter: 6 m – Length: 2 m – E = 80 GN/m2 – 10 oscillations per 35 seconds – Natural frequency? – Polar moment of inertia? Chapter 2 Free vibration • 1.2 Case study examples – Activity 3: A uniform bar Chapter 2 Free vibration • 2. Damped Free Vibration – Critical damping constant and the damping ratio: – Three cases: underdamped, critically damped and overdamped ms 2 + cs + k = 0 c k −c ± c 2 − 4mk ⎛ c ⎞ =− ± ⎜ ⎟ − 2m 2m ⎝ 2m ⎠ m 2 s1,2 = C1e 2 ⎧ c ⎫ ⎪ ⎛ c ⎞ k⎪ + ⎜ ⎨− ⎟ − ⎬t 2m ⎝ 2m ⎠ m ⎪ ⎩⎪ ⎭ + C2 e 2 ⎧ c ⎫ ⎪ ⎛ c ⎞ k⎪ + ⎜ ⎨− ⎟ − ⎬t 2m ⎝ 2m ⎠ m ⎪ ⎩⎪ ⎭ 2 12/7/2009 Chapter 2 Free vibration Chapter 2 Free vibration • 2. Damped vibration Chapter 2 Free vibration • 2. Damped vibration – 2.2 Logarithmic decrement Chapter 2 Free vibration • 2. Damped vibration – 2.2 Logarithmic decrement – Procedure to find damping ratio and natural frequency by the logarithmic decrement • Step 1 • Step 2 • Step 3 – Example (pp. 2-18) – Example (pp. 2-19) Chapter 2 Free vibration • 2. Damped vibration – 2.2 Logarithmic decrement – Activity 3: It is observed that the vibration amplitude of a damped single DOF system has fallen by 50% after 5 complete cycles. The period between the two measured amplitudes is found to be 1.15 seconds. Find the damping ratio and the natural frequency of the system. Chapter 2 Free vibration • 2. Damped vibration – 2.3 Energy dissipated in viscous damping 3 12/7/2009 Chapter 2 Free vibration • 3. Equivalent springs, masses & dampers – Springs in parallel (Fig. 7) – Springs in series (Fig. 8) Any Questions? Chapter 2 Free vibration • Summary of Chapter 2 – Procedure to derive equations of motion for free vibration systems – Translational vibration systems – Torsional vibration systems – Compound pendulum – Free vibration with damping – Energy dissipation in viscous damping – Equivalent springs, masses and dampers Chapter 2 Free Vibration • Tutorial 2 References • http://en.wikipedia.org/wiki/Vibration 4