The force-deflection equation for a nonlinear spring fixed at one end is , where is the force, expressed in pounds, and is the deflection, expressed in feet. Determine the static deflection if a block is suspended from the spring. Assuming that the slope of the force-deflection curve at the point corresponding to this loading can be used as an equivalent spring constant, determine the frequency of vibration of the block if it is given a very small downward displacement from its equilibrium position and released.
(a) Given the nonlinear relationship;
(b) The stiffness; , taken as the instantaneous rate of change in the force at; ;
The natural frequency of motion (Eq.19.14);
note 1. acceleration due to gravity; .
A deflection of the second floor of a building is measured directly under a newly installed piece of rotating machinery that has a slightly unbalanced rotor. Assuming that the deflection of the floor is proportional to the load it supports, determine the equivalent spring constant of the floor system, the speed in rpm of the rotating machinery that should be avoided if it is not to coincide with the natural frequency of the floor-machinery system.
From mechanics of materials it is known that when a static load is applied at the end of a uniform metal rod fixed at end , the length of the rod will increase by an amount , where is the length of the undeformed rod, is its cross-sectional area, and is the modulus of elasticity of the metal. Knowing that and and that the diameter of the rod is , and neglecting the mass of the rod, determine the equivalent spring constant of the rod, the frequency of the vertical vibrations of a block of mass attached to end of the same rod.