Vibration Engineering Notes PDF

Vibration Engineering E N G R. R AY H , M L A O N J A O Terminologies 1. Inertia element stores and releases kinetic energy, 2. Stiffness element stores and releases potential energy, 3. Dissipation or damping element is used to express energy loss in a system. 4. Translational motion of...

Vibration Engineering E N G R. R AY H , M L A O N J A O Terminologies 1. Inertia element stores and releases kinetic energy, 2. Stiffness element stores and releases potential energy, 3. Dissipation or damping element is used to express energy loss in a system. 4. Translational motion of a mass is described as motion along the path followed by the center of mass. The associated inertia property depends only on the total mass of the system and is independent of the geometry of the mass distribution of the system. 5. Rotational motion is described as motions in which the inertia property of a mass is a function of the mass distribution, specifically the mass moment of inertia, which is usually defined about its center of mass or a fixed point O. Units of Components Comprising a Vibrating Mechanical System and Their Customary Symbols Quantity Units Translational motion Mass, m 𝑘𝑔 Stiffness, k 𝑁/𝑚 Damping, c 𝑁 ∙ 𝑠/𝑚 External force, F 𝑁 Units of Components Comprising a Vibrating Mechanical System and Their Customary Symbols Quantity Units Rotational motion Mass moment of inertia, J 𝑘𝑔 ∙ 𝑚2 Stiffness, k 𝑡 𝑁 ∙ 𝑚/𝑟𝑎𝑑 Damping, c𝑡 𝑁 ∙ 𝑚 ∙ 𝑠/𝑟𝑎𝑑 External Moment, M 𝑁∙𝑚 Mass Moments of Inertia about Axis z- Normal to the x-y Plane and Passing Through the Center of Mass 1. Slender Bar 2. Circular Disk Mass Moments of Inertia about Axis z- Normal to the x-y Plane and Passing Through the Center of Mass 3. Sphere 4. Circular Cylinder  Governing Equations of Inertia Elements When the mass oscillates about a fixed point O or a pivot point O, the rotary inertia 𝐽𝑂 is given by For a rigid body, the kinetic energy is written in the following convenient form, which is due to König,11  Examples 1. Determine the mass moment of inertia of a uniform disk hinged at a point on its perimeter.  Examples 2. Determine the mass moment of inertia of a bar of uniform mass hinged at one end.