Problem 1: Composite Beams

Lets look at a composite beam as shown in the figure below.

W

T

The overall dimensions are L = 3m,W

top and bottom layers are made of steelof copper = 200. The left endGPa_,direction. Once the beam is bent, we are letting it go making= 0.3. The beam is made of three layers of equal width. The_{, = 7960}is clamped. We are first applying a traction ofkg _m3. The middle layer is softer and made

the beam vibrate.on the entire rightGPa,x = 0

^y

(1a) bration periods and compute the frequency of oscillation.Plot the displacement profile (in meters) as a function of time for points: Choose appropriate non-dimensionalization(L,W/2,H/2) over at least 10 vi-

method. Hint

(1b) Lets make the middle layer thickerDid the frequency and overall amplitude increase or decrease?3_W/5 and outer steel layers thinner W_/5. Repeat the above analysis.

Problem 2: Vibrations of a composite plate

L = 1m,W = 1m,H = 0.01m x = 0,x = L,y = 0 y = W

Consider a thin plate (while the remaining parts are made of steel. You can use the properties in the previous question forplate is made of two different materials. The middle patchare first applying a traction ofin the downward ( ) direction. Once the plate is deformed, we are letting it go making it vibrate.^{5 2} in a small middle part) clamped at all side faces0.4Lx 0.6L, 0.4W y 0.6W is made of Copperand and . The. We

E,

^{T = 10 N/m 0.49Lx 0.51L,0.49W y 0.51W,z = H}

(2a) Plot the displacement profile (in meters) as a function of time for pointsquency of oscillation for each point.^y , and over at least 10 vibration periods and compute the fre-, ,

(L/2,W/2,H) (L/4,W/2,H)

(3L/4,W/2,H) (L/2,W/4,H) (L/2,3W/4,H)

(2b) Lets vary the middle patch to be. Repeat the above analysis. Did the frequency and overall amplitude of the points increase or(0.35Lx 0.55L,0.35W y 0.55W) and (0.3Lx 0.6L,0.3W

ydecrease? 0.6W)