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Name: Gerion Jay C. Babuyo, ce, USC‐ERDT Date: Julyl 21, 2015 Professor: Rodolfo A. Chua, Jr. , ce, m.eng, ASEP Subject: MCE 211S ‐ Dynamic of Structures _____________________________________________________________________________________________________________________ Assignment No. 1 The weight ...

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Name: Gerion Jay C. Babuyo, ce, USC‐ERDT Professor: Rodolfo A. Chua, Jr. , ce, m.eng, ASEP Subject: MCE 211S ‐ Dynamic of Structures

Date: Julyl 21, 2015

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Assignment No. 1 The weight W of the building is 200 kips and the building is set into free vibration by releasing it (at time t=0) from a displacement of 1.20 in. If the maximum displacement on the return swing is 0.86 in. at time t=0.64 sec. determine: a.) The lateral spring stiffness b.) The damping ratio c.) The damping coefficient

W ≔ 200000 lb u0 ≔ 1.20 in u0.64 ≔ 0.86 in

The system completes one cycle in 0.64 seconds, this gives us the natural period of the undamped vibration which may is approximately equal to the natural period of the damped vibration. From the reference book pg. 50 "Damping has the effect of lowering the natural frequency and lengthening the natural period, but this effects are negligible for damping ratios below 20%". Thus a.) Spring stiffness k Tn ≔ 0.64 s TD ≔ Tn TD ＝ 2

⋅

‾‾‾‾ W ―― g.k 2

⎛2 ⎞ ⎛W⎞ lb k ≔ ―― ⋅ ―― = 49927.845 ― ⎜ T ⎟ ⎜⎝ ⎟⎠ in ⎝ D⎠

b.) Damping ratio ξ The logarithmic decrement is given by the equation; ⎛ u1 ⎞ 2 π⋅ξ ln ⎜―⎟ ＝ ――― 2 ⎝ u2 ⎠ ‾‾‾‾‾ 1−ξ

We assume that the damping ratio is too small (< 20%), thus this gives us that; 2 ‾‾‾‾‾ 1−ξ

is approximately equal to 1

⎛ u0 ⎞ ln ⎜―― ⎟＝2 π ⋅ ξ ⎝ u0.86 ⎠

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Assignment No. 1 ‐ Free Vibra on

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Name: Gerion Jay C. Babuyo, ce, USC‐ERDT Professor: Rodolfo A. Chua, Jr. , ce, m.eng, ASEP Subject: MCE 211S ‐ Dynamic of Structures

Date: Julyl 21, 2015

_____________________________________________________________________________________________________________________

⎛ u0 ⎞ ln ⎜―― ⎟ ⎝ u0.64 ⎠ ξ ≔ ―――― = 0.053 2

Notice that the damping ratio is lower than 20% which we assumed, thus we can say that the system false on the underdamped system and our assumption is correct. c.) Damping coefficient c wn ＝ wD wn ≔

‾‾‾‾‾ k⋅ 1 ――= 9.817 ― W s ⎛ W ⎞ lb ⋅ s ⋅ wn = 539.294 ―― c ≔ ξ ⋅ 2 ―― ⎜⎝ ⎟⎠ in

Assume that the mass and stiffness of the structure of figure below are given: if the system is set free into free vibration with the initial conditions y(0) = 0.7 in. and y'(0) = 5.6 in/sec, determine the displacement and velocity at t = 1.0 sec assuming: a.) Undamped system, c = 0 b.) c = 2.8 kips-sec/in.

2

s m ≔ 2000 lb ⋅ ― in y0 ≔ 0.7 in in y'0 ≔ 5.6 ― s

;

lb k ≔ 40000 ― in

a.) Undamped system, c = 0 For an undamped system, the solution to the homogeneous differential equation is; y' (0)) y (t)) ＝ y (0)) cos ⎛⎝wn ⋅ t⎞⎠ + ―― sin ⎛⎝wn ⋅ t⎞⎠ wn

we have;

wn ≔

‾‾‾ k 1 ― = 4.472 ― s m

at t ≔ 1.0 s the displacement is: y'0 sin ⎛⎝wn ⋅ t⎞⎠ = −1.383 in y1 ≔ y0 cos ⎛⎝wn ⋅ t⎞⎠ + ―― wn

at t ≔ 1.0 s the velocity is: ⎞ y'0 d ⎛ in sin ⎛⎝wn ⋅ t⎞⎠⎟ = 1.708 ― ―― ⎜y0 cos ⎛⎝wn ⋅ t⎞⎠ + ―― s w ⎠ n dt ⎝ _____________________________________________________________________________________________________________________

Assignment No. 1 ‐ Free Vibra on

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Name: Gerion Jay C. Babuyo, ce, USC‐ERDT Professor: Rodolfo A. Chua, Jr. , ce, m.eng, ASEP Subject: MCE 211S ‐ Dynamic of Structures

Date: Julyl 21, 2015

_____________________________________________________________________________________________________________________

b.) Damped system,

s c ≔ 2800 lb ⋅ ― in

For the damped system, the solution to the homogeneous differential equation is; y (t)) ＝ ⎛⎝

we have; wn ≔

−ξ ⋅ wn ⋅ t⎞

⎠ ⎛⎝A cos ⎛⎝wn ⋅ t⎞⎠ + B sin ⎛⎝wn ⋅ t⎞⎠⎞⎠

‾‾‾ k 1 ― = 4.472 ― s m

A ≔ y0 = 0.7 in

;

c ξ ≔ ―――― = 0.157 2 ⋅ m ⋅ wn

;

wD ≔ wn ⋅

2 1 ‾‾‾‾‾ 1 − ξ = 4.417 ― s

y'0 + ⎛⎝ξ ⋅ wn ⋅ y0⎞⎠ B ≔ ―――――― = 1.379 in wD

at t ≔ 1.0 s the displacement is: y1 ≔ ⎛⎝

−ξ ⋅ wn ⋅ t⎞

d ⎛⎛ ―― ⎝⎝ dt

−ξ ⋅ wn ⋅ t⎞

⎠ ⎛⎝A cos ⎛⎝wD ⋅ t⎞⎠ + B sin ⎛⎝wD ⋅ t⎞⎠⎞⎠ = −0.756 in

at t ≔ 1.0 s the velocity is: in ⎠ ⎛⎝A cos ⎛⎝wD ⋅ t⎞⎠ + B sin ⎛⎝wD ⋅ t⎞⎠⎞⎠⎞⎠ = 1.118 ― s

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