By Alan Palazzolo
Based on decades of analysis and instructing, this publication brings jointly all of the vital subject matters in linear vibration thought, together with failure types, kinematics and modeling, risky vibrating structures, rotordynamics, version aid equipment, and finite aspect tools using truss, beam, membrane and sturdy parts. It additionally explores intimately lively vibration keep watch over, instability and modal research. The publication offers the modeling abilities and data required for contemporary engineering perform, plus the instruments had to establish, formulate and clear up engineering difficulties effectively.
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Extra info for Vibration theory and applications with finite elements and active vibration control
So bells, traffic lights, pendulums, car antennas, turbine blades, and so on sway and ring at certain frequencies when displaced and released. Resonance occurs when an excitation or disturbance acts on an object with nearly the same frequency as the object’s natural frequency. The result may be large, damaging, and sometimes catastrophic vibrations. 1 Small-Scale Vibrations The potential energy that is created when atoms are collected in a lattice produces forces that act like spring connecting the atoms.
Piping and tubing systems are also subjected to vibration-induced cyclic stress due to pressure pulsation forces generated by attached machinery, such as reciprocating compressors, or from internally generated vortices. These systems are also subjected to static stress from internal pressure and partially constrained thermal expansion. Experienced chemical plant personnel know that an HCF-induced crack in a high-pressure gas line will sometimes emit a high-pitch whistle as the gas escapes through the crack, signaling all nearby workers to shut down the machinery and flee.
All of the remaining stress components are zero. , DI = 3 0 in. , θm = 0 002 rad 5 where X is the axial direction along the pipe axis and Y is the vertical direction. In general, KX will be different for axial and bending loads, in which case the component stresses are amplified (multiplied) by their respective KX values prior to forming σ X. For the sake of simplicity, the KX values are the same in this problem: (a) Determine the amplitude of the alternating and the mean (steady) transverse forces that the motor/fan exerts on the free end of the pipe: Alternating Fatip = Mean 3EI T tip 3EI T ∗ δa F m = 3 ∗ δm L3 L Fatip = 95 3lb 6 Fmtip = 381lb (b) Determine the mean (steady) axial force and torque exerted by the wall on the pipe: Faxial = EA A GJ A ∗ δm = 957 2lb, Γ = θ = 4068 in lb L L m 7 26 Vibration Theory and Applications with Finite Elements and Active Vibration Control (c) Determine the maximum x component of mean pipe stress including both bending and axial load contributions: σ axial m = Faxial bend Mm ∗ D0 2 x bend x , σm = , σ m = σ axial m + σ m , σ m = 8250psi A I 8 (d) Determine the maximum x component of alternating pipe stress: Ma ∗ D0 2 x bend x , σ a = σ a , σ a = 1969psi I = σ bend a 9 (e) Determine the pipe’s mean shear stress at the wall: τm = Γ ∗D0 2 , τm = 1050psi J 10 (f) Determine the maximum von Mises, mean pipe stress: σm = 1 2 x 2 KX ∗ σ m 2 + 6 τm ∗ KXY 2 , σ m = 25016psi 11 (g) Determine the maximum von Mises, alternating pipe stress: σ a = σ ax ∗ KX , σ a = 5906psi 12 (h) Determine the effective endurance limit “Se,eff” at the wall where the above von Mises stresses occur: Se, eff = − Se σ + Se , Se, eff = 18746 Sut m 13 Note that this is lower than the endurance limit (Se = 25000psi) in the absence of a mean stress.
Vibration theory and applications with finite elements and active vibration control by Alan Palazzolo