# mx cx kx 0 + + = - Australian Maritime College ```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
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 &plusmn; c 2 − 4mk
⎛ c ⎞
=−
&plusmn; ⎜
⎟ −
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 &amp; 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
```