WEEKS 3 Dynamics of Machinery References Theory of Machines and Mechanisms, J.J. Uicker, G.R.Pennock ve J.E. Shigley, 2003 Makine Dinamiği, Prof. Dr. Eres SÖYLEMEZ, 2013 Uygulamalı Makine Dinamiği, Jeremy Hirschhorn, Çeviri: Prof.Dr. Mustafa SABUNCU, 2014 Prof.Dr.Hasan ÖZTÜRK Dr.H.ÖZTÜRK-2010

PRINCIPLE OF SUPERPOSITION Linear systems are those in which effect is proportional to cause. This means that the response or output of a linear system is directly proportional to the drive or input to the system. An example of a linear system is a spring, where the deflection (output) is directly proportional to the force (input) exerted on the spring. The principle of superposition may be used to solve problems involving linear systems by considering each of the inputs to the system separately. If the system is linear, the responses to each of these inputs can be summed or superposed on each other to determine the total response of the system. Thus, the principle of superposition states that for a linear system the individual responses to several disturbances, or driving functions, can be superposed on each other to obtain the total response of the system. Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK

SHAKING FORCES AND MOMENTS Of special interest to the designer are the forces transmitted to the frame or foundation of a machine owing to the inertia of the moving links. When these forces vary in magnitude or direction, they tend to shake or vibrate the machine (and the frame); consequently, such effects are called shaking forces and shaking moments. If we consider some machine, say a four-bar linkage for example, with links 2,3, and 4 as the moving members and link I as the frame, then taking the entire group of moving parts as a system, not including the frame, and draw a free-body diagram, we can immediately write This makes sense because if we consider a free-body diagram ofthe entiremachine including the frame, all other applied and constraint forces have equal and opposite reaction forces and these cancel within the free-body system. Only the inertia forces, having no reactions, are ultimately extemal to the system and remain unbalanced. These are not balanced by reaction forces and produce unbalanced shaking effects between the frame and whatever bench or other surface on which it is mounted. These are the forces that require that the machine be fastened down to prevent it from moving. Prof.Dr.Hasan ÖZTÜRK Dr.H.ÖZTÜRK-2010

Prof.Dr.Hasan ÖZTÜRK

Force Analysis using the method of Virtual Work If a rigid body is in equilibrium under the action of external forces, the total work done by these forces is zero for a small displacement of the body. Work: With F, x, T, q, vectors and W a scalar. To indicate that we are dealing with infinitesimal displacements (virtual displacements), use the notation: Now apply the virtual work definition: Prof.Dr.Hasan ÖZTÜRK

Virtual Work (cont.) If we divide the virtual work by a small time step, we get: These are all external torques and forces on the body, and include inertial forces and gravity. Rewrite, to clearly show this as: Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK

Prof.Dr.Hasan ÖZTÜRK