PHYS 422 Lecture 12 Free Vibrations I

PHYS 422 Lecture 12 Free Vibrations I - force proportional...

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3/24/11
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3/24/11 Free Vibrations of Physical Systems I
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3/24/11 Mass-Spring System Vast majority of deformations of physical systems (stretching, compressing, bending, twisting) result in restoring forces displacement F Simple harmonic motion Prototype system Inertial component (KE) Elastic component (PE)
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3/24/11 Newton II (F = ma) Conservation of Mechanical Energy (E) Mass-Spring System restoring force r displacement
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3/24/11 Mass-Spring System Any equation having the form or has a solution
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3/24/11 Mass-Spring System Solve the equation: Use complex exponential Rewrite: Second derivative is a multiple of itself
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3/24/11 Mass-Spring System Assume: Substitute:
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3/24/11 Mass-Spring System ϖ t C 1 ϖ t C 2 C C
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3/24/11 Mass-Spring System F A SHM
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3/24/11 Physical Properties Governing SHM A F l0 /3 2l0 / 3 l 0 l0 / 3 2 l0 / 3 l 0 The extension l under a force F is proportional to original length l0 l/l0 ≡ strain The same strain ( l/l0 ) is caused by applying a
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Unformatted text preview: force proportional to the cross-sectional area A ∆ F/A ≡ stress For small strain ( < 0.1% l0) stress is proportional to strain stress/strain = constant (Young’s modulus of elasticity) 3/24/11 Physical Properties Governing SHM F k (spring constant) Material Density (kg/m3) Young's Modulus 109 N/m2 Ultimate Strength Su 106 N/m2 Steela 7860 200 400 Aluminu m 2710 70 110 Glass 2190 65 50b Concrete 2320 30 40b 3/24/11 Physical Properties Governing SHM m l0 + ∆ l Steel wire of length (l0) 1 m and diameter 1 mm with a 1 kg mass attached (m) 3/24/11 Floating Objects Hydrometer ρ Α recall, restoring force equal to the weight of the fluid displaced if depressed or raised from equilibrium restoring force displacement from Newton II 3/24/11 Floating Objects applies to a ship h 3/24/11...
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This note was uploaded on 03/23/2011 for the course PHYS 422 taught by Professor Staff during the Spring '08 term at Purdue.

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PHYS 422 Lecture 12 Free Vibrations I - force proportional...

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