CHAPTER 1 INTRODUCTION TO VIBRATIONS DEGREE OF FREEDOMS The system motion can be described completely by considering displacement x or angle θ. This system can be described completely analyzed using angle θ. The system motion can be described completely considering displacement x. 1 Degree-of-Freedom: The minimum number of independent coordinates required to determine the motion of the body at a give instant is simply one. The system motion can be described completely by two displacements x1 and x2. Both displacements are independent because spring constants may be different. X and x describe the system and they are independent of each other. The system motion can be described completely considering both angles. They are independent because of different constants. 2 Degree-of-Freedom: The minimum number of independent coordinates required to determine the motion of the body at a give instant are two. The three displacements describe the system motion. The system motion can be described completely considering either the xdisplacements, ydisplacements or the three angles. The system motion can be described completely considering the angles. 3 Degree-of-Freedom: The minimum number of independent coordinates required to determine the motion of the body at a give instant are three. IMPORTANT DEFINITIONS VIBRATION ANALYSIS PROCEDURE 1. Mathematical Modelling (DOF diagrams or any diagrams for mathematical working) 2. Deriving governing equations (Newton’s law, etc) 3. Solution (Matrix method, Laplace transformation, etc) 4. Interpretation (Results) MATHEMATICAL MODEL This is a forging hammer. Tup is a falling weight. Anvil is a steel block upon which the material is forged into the desired shape. Elastic pad is used to control vibrations and foundation block holds the system. EQUIVALENT SPRING STIFFNESS (K) 1 1 1 This is the basic formula: π = 2 πππ π₯ 2 = 2 π1 π₯ 2 + 2 π2 π₯ 2 … . Series springs: Parallel springs: Load will be divided but displacement will remain the same. Load will remain the same on both springs, though displacement could be different. FORMULAE FOR K πΊπ½ π= (shaft) π π΄πΈ π= (simple rod) π (cantilever beam) EQUIVALENT MASS EQUIVALENT DAMPING HARMONIC MOTION Displacement Velocity Acceleration A=Amplitude REPRESENTATION OF HARMONIC MOTION Vector Representation: They can be simply represented in vector form. See the figure on next page. We may define the point OP as vector. We get: π = ππ = π₯ + π¦ = π΄πππ ππ + π΄π ππππ Complex Representation: This can be represented more conveniently in complex number form. We will have: π = π + ππ Where: π = π₯ = π΄πππ π π = π¦ = π΄π πππ COMPLEX ALGEBRA REVISION MORE ON COMPLEX REPRESENTATION Using the rules of complex algebra, VECTORAL ADDITION OF HARMONIC FUNCTIONS DEFINITIONS BEATS DECIBEL
