Mechanical Engineering

 

 

 

Vibration - Rigid Body

 

Undamped free vibration of a rotating body refers to a frictionless periodic motion that is maintained by gravitational or elastic restoring forces. Pendulum is an example of a rigid body that exhibits free vibration. In reality, any rotating bodies with a periodic motion without external applied force(s) is subjected to a damped free vibration because of frictional effects at the pin joint(s). Free vibrations occur upon releasing an object after applying a small displacement.

 

Like any dynamics problems, the vibration analysis of rotating bodies should start with illustrating the free-body diagram and stating the equation of motion. Figure 1 illustrates a periodic motion of a rigid body with a small angular displacement and its corresponding free-body diagram, respectively. The mass and rotational moment of inertia at the center of mass are known quantities.

 

< Figure 1: Periodic motion of a rigid body >

 

The equation of motion for the rotating body depicted in figure 1 is modelled as follows, where the sum of moment is taken at the pivot point, o:

 

 

 

As sin(θ) can be approximated to θ for a small angular displacement,

 

 

< Figure 2: Periodic motion of a plate fixed to a torsional rod. >

 

Using the known rotational frequency, the natural frequency is calculated as follows:

 

 

The period of oscillation is equal to the reciprocal of the natural frequency.

 

 

Example 1: Python for modelling the period of oscillation of a pendulum subjected to undamped free vibration.

 

Create a plot that represents the relationship between the period of oscillation and diameter of the sphere fixed to the end of a rod. The rod has a diameter of 10 mm and a length of 200 mm. The diameter of the sphere ranges from 20 mm to 100 mm. Include both scatter and line plots. Figure 3 depicts the pendulum motion and its corresponding free-body diagram. The density of the material for both the rod and sphere is 0.00271 kg/mm^3.

 

< Figure 3: Periodic motion of a pendulum. >

 

 

Mathematical Modeling:

 

Step 1: Analysis of mass and inertial properties.

    Mass of the rod and sphere

 

    Position of center of pendulum mass with respect to the pivot point (i.e., point O from figure 3)

    Rotational moment of inertia at the pivot point

 

Step 2: Rotational frequency analysis.

    Modeling the pendulum motion

     

    Rearranging the equation of motion into the  form,

     

 

Step 3: Period of oscillation.

 

 

Python Code:

 

    import numpy as np

    import matplotlib.pyplot as plt

     

    # Inputs

    # rho : Material density [kg/mm^3].

    # D_rod : Rod diameter [mm].

    # l_rod : Rod length [mm].

    # D_sphere : Sphere diameter [mm].

     

    # Output

    # T : Period of oscillation [s].

     

    # Define the inputs  

    rho = 0.00271

    D_rod = 10

    l_rod = 200

    D_sphere = np.linspace(20,100) # 20, 21, 22, ..., 100

    g = 9.81 # N/kg

     

    # Analysis of mass and inertial properties.

    m_rod = rho*np.pi*D_rod**2*l_rod/4

    m_sphere = rho*np.pi*D_sphere**2/3

     

    d_G = (m_rod*(0.5*l_rod) + m_sphere*(l_rod + 0.5*D_sphere))/(m_rod + m_sphere)

     

    I_O = 1/12*m_rod*l_rod**2 + 1/10*m_sphere*D_sphere**2 + (m_rod + m_sphere)*d_G**2

     

    # Rotational frequency analysis.

    omega_n = np.sqrt((m_rod + m_sphere)*g*d_G/I_O)

     

    # Period of osciallation.

    T = 2*np.pi/omega_n

    SF_rod = 90/(m_sphere*g/(np.pi*D_rod**2/4))

    SF_rod_critical = np.min(SF_rod)

     

    # Create the plot to show T vs D_sphere

    plt.figure()

    plt.plot(D_sphere,T,'bo',D_sphere,T,'k') # Blue dots and black lines

    plt.show()

 

Python Results:

< Figure 4: Relationship between the period of oscillation and sphere diameter for the pendulum. >

 

 

 

Example 2: Python for modelling the period of oscillation of a plate fixed to a rod with known stiffness subjected to undamped free vibration.

 

Create a plot that represents the relationship between the period of oscillation and a square plate fixed to the end of a rod subjected torsion with a side length that ranges from 20 mm to 40 mm. The thickness of the plate is 2 mm, the plate has a density of 1.05(10)-6 kg/m3, and the rod has a stiffness of 0.005 Nm/rad. Include both scatter and line plots. A small angular displacement of θ is applied to the rod. Figure 5 depicts the motion and the corresponding free-body diagram for the circular plate.

< Figure 5: Periodic motion of the square plate fixed to the torsional rod.>

 

Mathematical Modeling:

 

Step 1: Analysis of mass and inertial properties.

    Mass of the square plate

    Rotational moment of inertia at the pivot point

    Step 2: Rotational frequency analysis.

    Step 3: Period of oscillation.

 

Python Code:

    import numpy as np

    import matplotlib.pyplot as plt

     

    # Inputs

    # rho : Material density [kg/mm^3].

    # l : Side length of the square plate [mm].

    # t : Thickness of the square plate [mm].

    # k : Rod stiffness [N*m/rad]

     

    # Output

    # T : Period of oscillation [s].

     

    # Define the inputs  

    rho = 1.05*10**(-6)

    l = np.linspace(20,40) # 20, 21, 22, ..., 50

    t = 2

    k = 0.005

     

    # Analysis of mass and inertial properties.

    m_plate = rho*t*l**2

    I_O = 1/6*m_plate*l**2

     

    # Rotational frequency analysis.

    omega_n = np.sqrt(k/I_O)

     

    # Period of osciallation.

    T = 2*np.pi/omega_n

     

    # Create the plot to show T vs l.

    plt.figure()

    plt.plot(l,T,'bo',l,T,'k') # Blue dots and black lines

    plt.xlabel('Square Length [mm]')

    plt.ylabel('Period [s]')

    plt.title('Period vs Square Length')

    plt.show()

 

 

Python Results:

 

< Figure 6: Relationship between the square length and period of oscillation for a square plate fixed to the rod subjected to torsion.>

 

 

 

Reference :