Chapter 13 Balance of Machinery,1. Balance Purposes,13.1 Introduction,A rotating shaft or rotor will experience centrifugal forces if its center of mass does not lie exactly on the rotating centerline. The centrifugal force exerted on the frame by moving machine member will be time varying and impart vibratory motion to the frame. This vibration and accompanying noise can produce human discomfort, alter the desired machine performance or may cause failure of the rotor or the support. The purpose of balance is to reduce unbalance to an acceptable level and possibly to eliminate it entirely. ,(1) Balance of rigid rotors When a rotor is rotating about its own centerline of rotation at an angular velocity, the deformation of the rotor is small and can be negligible, and it is said to be the rigid rotor, otherwise, it is a flexible rotor. Fig 13-1 shows a rigid rotor at a constant angular velocity.,Fig.13-1 Centrifugal forces of the rotor(转子的惯性力),2. Classification of Balance,(2) Balance of flexible rotors When a rotor is rotating about its own centerline of rotation at an angular velocity, the deformation of the rotor can not be negligible, it is said to be a flexible rotor. (3) Balance of linkages The rotating links of a linkage, such as crank and rockers, can be individually balanced by the rotating balance methods. The coupler is in complex motion and has no fixed pivot, thus its mass center is always in motion, and the inertia force of the link has variable magnitude and sense. We can not attach a mass to the link for balancing it. The global mass center of the linkage normally will change position as the linkage moves. So balance of a linkage is more difficult than balance of rotors. If we can somehow force this global mass center to be stationary in the frame, we will have a state of balance for the overall linkage. ,1. Static Balance of Rigid Rotors,13.2 Balance Design of Rigid Rotors,The unbalanced forces of a rigid rotor are due to the acceleration of masses in the rotor. The requirement for static balance is that the sum of all forces in the rotating rotor must be zero. Fig13-2 shows a rigid rotor rotating with a constant angular velocity of .A number of masses, such as three, are depicted by point masses at different radii in the same transverse plane.,Fig.13-2 Static balance of rigid rotor(刚性转子的静平衡),2. Dynamic Balance of Rigid Rotors,The most general case of distribution of masses on a rigid rotor is that in which the masses lie in various transverse planes as shown in Fig13-3. The rotor revolves with a uniform angular velocity , and m1, m2, m3 are the masses attached to the rotor in planes 1, 2, 3 respectively and at radii r1, r2, r3. ,Fig.13-3 Dynamic balance of rigid rotors(转子的动平衡),1. Static Balance Test,13.3 Balance Test of Rigid Rotors,If the distance b of a rotor is small, usually b/d 0.2, the inertia moment causing a bending of the shaft can be neglected. Static balance machines are used for rotors of small axial dimensions such as fans, gears, belt wheels and impellers, etc. Fig.13-4 shows a rigid rotor 1 with the shaft laid on the horizontal parallel ways 2. By gravity G, the rotor will roll until the center of the rotor gravity lies on the lowest position.,Fig.13-4 Static balance test(静平衡实验),2. Dynamic Balance Test,Fig.13-5 Dynamic balance machine in industry(工业动平衡机) 1Base(底座) 2Power box(动力箱) 3Computer system(计算机系统) 4Spindle(主轴) 5Rotor(转子) 6Carriage(支承架),For dynamic balance of a rotor, two balance of counter masses are required to be used in any two correct planes. Dynamic balance is achieved by adding or removing masses in these two planes. This requires a dynamic balance machine. Fig.13-5 shows a common type of dynamic balance machine, which is used in industry.,3. Balance Precision,After a rigid rotor has been balanced by using a balance machine, the centerline of mass of the rotor will be coincident with the centerline of rotation of the rotor theoretically, but in practice, they can not be coincident completely. An offset between the centerline of mass and the centerline of rotation of the rotor always exists. To assure the balance precision, the actual unbalance must be less or equal to the allowable unbalance. There are two types of the allowable unbalance. They are allowable mass radius product and allowable offset. The relationship between the mr and the e is as follows:,Fig.13-6 Distribution of the allowed mass-radius product (许用质径积的分配),13.4 Balance of Planar Mechanisms,We have known that all the inertia forces acting on the moving links in a mechanism can be replaced by a single total inertia force and a single total moment of inertia which act on the frame of the mechanism. If the resultant of all the inertia forces and the resultant of all the inertia couples acting on the frame are zero, there are no shaking forces and shaking couples. The mechanism runs smoothly. We will discover that in most mechanisms, by adding appropriate balance masses, the sh