2016F Prelab2.pdf

ENME 599 – VIBRATIONS AND MACHINE DYNAMICS

TEAM #: ...............

DATE: ...............

2016F Pre-Lab. 2 SingleSingl e-DegreeDegree-Of-F Of-Freedom reedom (S (SDOF DOF)) Mechanical Oscillator Oscil lator Pre-lab assignment (one per group) DUE BEFORE THE LAB REQUIRED FORMAT :

PREFERABLY TYPED (Make sure you put appropriate UNITS)

Consider mass M at the end of a cantilever beam (shown below) of negligible mass with rectangular cross-section Π The team:

1. ............. 2. ............. 3. ............. 4. .............

1.

Find an analytical expression for the stiffness of this beam k =

P = ..... ; where δ  is  is a deflection due to the load P δ 

Hint: Use the expression (mechanics) (mechanics) for the deflection of a beam under the concentrated forceP force P = m g (it g (it involves P, L, E and I). Example dimensions of the beam are as follows: L = 0.15 m W = 0.02 m T = 2 mm

- length - width - thickness

Example mass: M = 30 g Beam material (steel): E = 200 GPa 2.

Assume that the damping factor ζ  =  = 0.01. Compute the coefficient of viscous damping, c.

3.

Derive the Equation of Motion (EOM) of this beam in symbolic symbolic form. Assume the mass of the beam is negligible. negligible.

4.

Derive a symbolic expression for the free response of this beam.

5.

Compute the angular natural natural frequency frequency of this system, ω n, (rad/s) and the corresponding natural frequency f n (Hz)

6.

Analytically solve the free response. Assume the following initial conditions:x conditions: xo = 10 mm ; vo = 0 m/s.

7.

Utilizing computer software, plot the free response of of the considered system for approximately approximately 10 oscillations (note that it will correspond to the time t = 10/f  10/f n) (assume the same initial conditions and show appropriate axis names and units). Append the necessary computer codes.

p. 1