Mechanical Engineering Question: undamped free vibrations ( I have some of the work 9 easy k pt)

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Date Posted: 7/24/2008 2:01:17 PM Status: Live

undamped free vibrations ( I have some of the work 9 easy k pt)

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Question Details:

NOTE: I only need help with the first part! And my work is below

The mechanism shown is a simplified version of what is used in an Impact Test. The radius of gyration about the point O is k_O = 0.95 m. However, the radius to the center of gravity of the entire object from the point O is r_G = 0.9 m. First derive the equation of motion using Newton’s Laws – do not plug in numbers for the variables yet….just keep them as letters (m, k_O, r_G, etc.). Then, if the hammer is rotated to -0.4 rad and given an initial velocity of 0.7rad/s, find:
1. Natural period. Does this vary with initial conditions? (2.01 s)
2. Phase angle. Does this vary with initial conditions? (-1.06 rad)
3. Position θ as a function of time using BOTH solutions to the differential equation of motion. (θ = 0.224sin(3.13t) – 0.4cos(3.13t) OR θ = 0.458sin(3.13t – 1.06))
4. Maximum displacement and first time which it occurs. (0.458 rad, 0.841 s)

Maximum velocity and first time which it occurs. (1.43 rad/s, 0.339 s)

given:

Ko = 0.95 m

Rg = 0.9 m

-Wrsinθ = Iα + mr^2α = -mgrθ = Ko*r*g^2 + mr^2α

0 = (Ko +m)*r^2*α +mgrθ = α + m*g*r/(Ko +m)*r^2

since τ natural period = √m*g*r/(Ko +m)*r^2 = gives me the wrong answer

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undamped free vibrations ( I have some of the work 9 easy k pt)