Balancing. Rotating Components Examples of rotating components in a mechanism or a machine. (a)

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1 alacig NOT COMPLETE Rotatig Compoets Examples of rotatig compoets i a mechaism or a machie. Figure 1: Examples of rotatig compoets: camshaft; crakshaft Sigle-Plae (Static) alace Cosider a rotatig shaft with three extruded masses as show i Fig.. ssume the shaft rotates about its axis by a costat agular velocity ω. The objective of the sigle-plae balace, also kow as static balace, is to make the shakig force to be zero, by addig aother poit-mass to the systef ecessary. Figure : Schematic represetatio of a rotatig shaft with three extruded masses: side view; frot view The orietatio of each mass-poit with respect to the x-axis, at a give time, ca be described by a agle θ i ad a positio vector. Each mass-poit will have a ormal P.E. Nikravesh 1

2 compoet of acceleratio equal to ω poitig toward the shaft axis. The iertia force associated with each mass is ω i the opposite directio of its correspodig acceleratio. For static equilibrium, the sum of these iertia forces must be zero to have zero shakig force: m 1 ω m R ω m 3 ω ω = 0 where subscript represets the added mass to balace the system. We ote that the coefficiet ω ca be dropped form this equatio to obtai m 1 m R m 3 = 0 ecause the force-equilibrium equatio becomes idepedet from the agular velocity of the shaft, it is referred to as the static balacig although the shaft is i rotatio. Equatio ca be solved for the balace mass either graphically or aalytically. Graphical solutio: The process is demostrated i Fig. 3. The solutio vector provides the magitude of (ifiite possibilities for ad ) ad the agle. m 1 ω m R ω m 3 m R θ 1 m 1 m 3 m R m 3 ω m 1 m Figure 3 alytical solutio: We ca write Eq. i a more geeral form as = 0 = This equatio ca be projected oto the x- ad the y-axes to obtai two algebraic equatios: These two equatios result ito = (x) = (y) = (x) = arcta (x) (y) + (y) (y) (x) (b.1) (b.) Two-Plae (Dyamic) alace The objective of the dyamic force balace is to elimiate the shakig force ad the shakig momet. For this purpose we eed to add two balacig masses. First we add oe mass to elimiate the shakig momet, ad the we add a secod mass to elimiate the shakig force. P.E. Nikravesh

3 To elimiate the shakig momet we cosider the side-view of the rotatig compoet. We decide o two correctio plaes, ad, i coveiet locatios as show i Fig. 4. The plae of each mass-poit has a distace from the correctio plae. The distace of plae from plae is. We add the first balacig mass (ukow at this poit) i plae. The iertia force of each mass, ω, causes a momet with respect to the correctio plae as ω. This momet is perpedicular to vector ad the rotatioal axis as show i Fig. 4 for oe of the masses. I order for the shakig momet to be zero, the sum of these momets, icludig the momet of the added mass i the correctio plae, must be zero: m 1 1 ω m R ω m 3 3 ω ω = 0 or, m 1 1 m R m 3 3 = 0 (c) This equatio ca be solved for either graphically or aalytically. Correctio plae Correctio plae m 1 1 ω m 1 ω Figure 4 Graphical solutio: The process is demostrated i Fig. 5. Sice every momet vector must be rotated 90 o with respect to its positio vector, solve the solutio vector, the rotate the solutio by 90 o to obtai vector. We ca simplify this process by ot rotatig ay of the vectors; i.e., we ca draw the vectors alog their correspodig positio vectors, as show i Fig. 5. Sice is kow, the solutio vector provides the magitude of (ifiite possibilities for ad ) ad the agle θ. The result for the first balacig mass is show i Fig. 6. Note: If the correctio plae is placed such that the poit masses are o both sides of the plae, the s must be assiged positive ad egative sigs. This will effect the directio of a momet whether to be cosidered alog or i the opposite directio. m R m 1 1 m 3 3 m R m 3 m P.E. Nikravesh 3

4 Figure 5 Correctio plae Correctio plae θ Figure 6 alytical solutio: We ca write Eq. (c) i a more geeral form as = 0 = This equatio ca be projected oto the x- ad the y-axes to obtai two algebraic equatios: These two equatios result ito = (x) = (y) = (x) θ = arcta (x) (y) + (y) (y) (x) I order to elimiate the shakig force, a secod balacig mass is added i plae accordig to the process of sigle-plae balace. I this process the added mass at must be icluded i Eq. like other extruded masses. The graphical solutio is show i Fig. 7. (c.) (c.1) m 3 m R m 1 P.E. Nikravesh 4

5 Correctio plae Correctio plae Figure 7 (c) Helpful Table table like the oe show ca be helpful i orgaizig your data ad calculatios. Mass Poit m R θ mr mr P.E. Nikravesh 5