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§6 － 1 Purposes and Methods of Balancing §6 － 2 Balancing of Rigid Rotors Chapter 6 Balancing of Machinery

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The center of mass of some machine elements may not coincide with their rotating centers because of the asymmetry of the structure. Even for symmetrical machine elements, the center of mass may still be eccentric because of uneven distribution of materials, errors in machining and also in casting and forging. The centrifugal force exerted on the frame is time varying, and it will therefore impart vibration to the frame. This vibration can adversely affect the structural integrity of the machine foundation. 一、 Purposes of Balancing Purposes ： We should try to eliminate the unwanted centrifugal forces in machines, especially in high-speed machinery and precision machinery. 二、 Methods 1. Balancing of rotors Rotors ——Parts constrained to rotate about a fixed axis are called rotors. § 6 － 1 § 6 － 1 Purposes and Methods of Balancing

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（ 1 ） Rigid rotors Rigid rotors——If the rotating frequency of the rotor is less than ( ) n Cl (where: n Cl is the first resonant frequency of the rotor), then the rotor is supposed to have no deformation during rotation and is called a rigid rotor. Flexible rotors——If the working rotating frequency of the rotor is larger than ( ) n Cl, then the rotor will have large elastic deformation due to imbalance during rotation. The elastic deformation makes the eccentricity larger than the original one so that a new imbalance factor is added and the balancing problem becomes more complicated. Such a rotor is called a flexible rotor. （ 2 ） Flexible rotors

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2. Balancing of mechanisms The coupler of a linkage has a complex motion. The acceleration of its mass center and its angular acceleration vary throughout the motion cycle. The coupler will therefore create a varying inertia force and inertia moment of force for any mass distribution. So the balance of linkages must be considered as a whole. The resultant inertia force of all moving parts is equal to the net unbalanced force acting on the frame of a machine, which is referred to as the shaking force. Likewise, a resultant unbalanced moment acting on the frame, caused by the inertia forces and inertia moments of all moving parts, is called the shaking moment. The shaking force and the shaking moment will cause the frame to vibrate.

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Rotors whose axial dimensions B are small compared to their diameters D (usually B /D < 0.2), the masses of such rotors are assumed practically to lie in a common transverse plane. 一、 Calculation for the Balancing of a Rigid Rotor ω D B e G G FIFI All centrifugal forces in this disk-like rotor are planar and concurrent. If the vector sum of these forces is zero, then the mass center of the system coincides with the shaft center and the rotor is balanced. Otherwise, it is called imbalance. Since the imbalance can be shown statically, such imbalance is called static imbalance. § 6 － 2 § 6 － 2 Balancing of Rigid Rotors

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O x y r1r1 m1m1 11 r2r2 m2m2 22 rbrb mbmb bb Unbalanced masses are depicted as point masses m i at radial distances r i. In this case, there are two masses, m 1 and m 2. When the rotor rotates with constant angular velocity ω, each of the unbalanced masses produces a centrifugal force F Ii F Ii = m i ω 2 r i In this case, a third mass m b with rotating radius of r b is added to the system so that the vector sum of the three centrifugal forces is zero and balance is achieved. m 1 r 1 + m 2 r 2 + m b r b = 0 ∑ F = ∑F Ii ＋ F b = 0 (m b r b ) x = - ∑ m i r i cos i (m b r b ) y = - ∑ m i r i sin i m b r b = (m b r b ) x 2 +(m b r b ) y 2 b = acrtan[(m b r b ) y / (m b r b ) x ] r’br’b m’bm’b

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If b / D >0.2 ， although the resultant of the two centrifugal forces is zero, the forces are not collinear and a resultant couple will exist. The direction of the resultant couple changes during rotation. The resultant couple will act on the frame and tend to produce rotational vibration of the frame. Such an imbalance can only be detected by means of a dynamic test in which the rotor is spinning. Therefore, this is referred to as dynamic imbalance. The rotor in Fig. is therefore statically balanced and dynamically unbalanced. 二、 Dynamic Balancing of Rigid Rotors From the above, we can see that the conditions for the balancing of a non-disk rigid rotor are: Both the vector sum of all inertia forces and the vector sum of all moments of inertia forces about any point must be zero.

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Calculation for the Dynamic Balancing L I II F 2I F 1I F 3I F 2II F 3II F 1II m2m2 m3m3 m1m1 r1r1 F2F2 r2r2 F3F3 r3r3 F1F1 l1l1 l2l2 l3l3 From Theoretical Mechanics, we know that the centrifugal force F can be replaced dynamically by a pair of forces F Ⅰ and F Ⅱ parallel to F and acting in two arbitrarily chosen transverse planes Ⅰ and Ⅱ. In this way, the complicated spatial force system has been converted into two simpler planar concurrent force systems on two planes.

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The above methods can be extended to any rotor with any number of imbalances on any number of transverse planes. The conclusion is that any number of masses on any number of transverse planes of a non- disk rigid rotor can be balanced dynamically by a minimum of two masses placed in any two arbitrarily selected transverse planes. The selected transverse planes are called balance planes. In practice, those planes on which counterweights can be mounted easily,or mass can be removed easily, may be chosen as the balance planes.

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