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Force-displacement relationship for a spring. we show the displacement-force relationship to be nonlinear, which is the general case. For the region where the curve is linear, the analytic expres­ sion is written, f=Kx (Newtons) (1-35) where K is the spring constant. Over this linear region, the spring satisfies Hooke's law. Physical springs satisfy this relationship for forces below the elastic limit of the material. Deviations from this linear relation may occur for a number of reasons including, temperature effects, rotation of the spring ends, hysteresis, and others.

Depends on the velocity, often in a complicated way. Several representa­ tive variations are illustrated in Fig. 1-28. Curve A is reasonably rep­ resentative of the frictional force relation for a dash-pot, an oil-actuated device, which is illustrated in rough physical design and schematic representation in Fig. 1-29. Curve B in Fig. 1-28 illustrates all three important types of friction. A region exists for which the force is inde­ pendent of velocity, a region for which the friction is substantially constant and independent of velocity, and a region of nonlinear dependence on velocity.

1-42. To describe the features of these elements, the flow laws for both laminar and turbulent conditions must be used. For turbulent flow through pipes, orifices and valves, the steady-flow energy equation (the first law of thermodynamics) for adiabatic flow of ideal gases is, w = KAYV2g(Pl-p2)lp (kg/sec) where w = gas flow rate, kg/sec, A = area of restriction, Y = rational expansion factor = y = specific heat ratio for gases, p = gas density, kg/m3, K = a flow coefficient, p = pressure, kg/m2.

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