00036___c5acd3236334652171987c0fd330d852 - playing a...

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Sec. 1.5 VIBRATIONS 15 Deflection (a) Ideal plasticity ..il. Deflection (b) Linear elastic - ideal plastic material Fig. 1.4:l. Structural steel behaves pretty much like an ideal plastic material, ex- cept that when the load (measured in terms of the maximum shear stress) is smaller than the critical load (called the yield shear stress), the load- deflection curve is an inclined straight line (Hooke’s Law). Upon unloading, there is a small rebound. There are some details at the yield point that were ignored in the statement above. Other metals, such as copper, aluminum, lead, stainless steel, etc., behave in a somewhat similar, but more complex manner. Metals at a sufficiently high temperature may behave more like a fluid. Theories that deal with these features of materials are called the theories of plasticity, which are presented in Chapter 6. 1.5. VIBRATIONS We know vibrations by experience while driving a car, flying an airplane,
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Unformatted text preview: playing a musical instrument. The trees sway in the wind. A building shakes in an earthquake. Sometimes we want to know if a structure is safe in vibration. Sometimes we want to design a cushion that isolates an instrument from vibrations. A prototype of this kind of problem is shown in Fig. 1.5:1(a). A body with mass M is attached to an initially vertical massless spring, which has a spring constant k, and a damping constant c, and is “built-in” to a “ground” which moves horizontally with a displacement history s(t). Let x ( t ) denote the horizontal displacement of the mass and a dot over x or s denote a differentiation with respect to time. Then 2 is the acceleration of the body, k ( x - s) is the spring force acting on the body, and c(k - S) is the viscous damping force acting on the body. Newton’s second law requires that (1) MX + ~ ( 2 - S) + k ( ~ - S) = 0. If we let (2) y = x - s ,...
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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