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Mechanics 1J

# Mechanics 1J - 2.001 MECHANICS AND MATERIALS I Lecture#7...

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2.001 - MECHANICS AND MATERIALS I Lecture #7 10/2/2006 Prof. Carol Livermore Recall: 3 Basic Ingredients 1. Forces, Moments, and Equilibrium 2. Displacements, Deformations, and Compatibility 3. Forces-Deformation Relationships Linear Elastic Springs Linear: k is a constant not a function of P or δ . If it were non-linear: Elastic: Loading and unloading are along the same curve. 1

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± ± EXAMPLE: Springs in series Q: What are the reactions at the supports? Q: How are P and δ related? FBD F x =0 R A x + P R A x = P F y R y 2
± ± FBD of Spring 1 F x =0 F 1 = P FBD of Spring 2 F x F 2 = P Look at compatibility: Undeformed 3

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±² Deformed Compatibility: δ 1 + δ 2 = δ DeFne: δ - How much things stretch u -Howmuchth ingsmove So: u x = δ ±orce-Deformation: F 1 = k 1 δ 1 F 2 = k 2 δ 2 Put it all together: δ 1 = F 1 /k 1 δ 2 = F 2 /k 2 F 1 F 2 ± 11 ² δ = δ 1 + δ 2 =+= P + k 1 k 2 k 1 k 2 So: k 2 + k 1 δ = P k 2 k 1 4
k 1 k 2 P = δ k 1 + k 2 k 1 k 2 k eff = k 1 + k 2 Plot Sanity Check 1. k 1 = k 2 = k k = k 2 = k , P 2 × δ 2 k 2 2. k 1 >> k 2 k = k 2 k 2 k 2 All ﬂexibility is due to weaker spring.

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Mechanics 1J - 2.001 MECHANICS AND MATERIALS I Lecture#7...

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