OVERCONSTRAINED (STATICALLY INDETERMINATE) SYSTEMS

The mounting systems discussed in the previous section are typical statically indeterminate systems if more than three mounting points are used. Statically indeterminate (overconstrained) systems are frequently used, both intentionally and unintentionally, in mechanical design. Static indeterminacy has a very strong influence on stiffness. Excessive connections, if properly applied, may serve as powerful stiffness enhancers, and may significantly improve both accuracy and load carrying capacity. However, they also may play a very detrimental role and lead to a fast deterioration of a structure or a mechanism.

Effects of overconstraining depend on the design architecture, geometric di-mensions of the structures, and performance regimes. For example, overcon-straining of guideways 1 and 2 of a heavy machine tool (Fig. 5.19a)[8] plays a very positive role in increasing stiffness of the guideways. This is due to the fact that the structural stiffness of the guided part 3 is relatively low because of its large dimensions and its local deformations accommodate uneven load distribu-tion between the multiple guiding areas. On the other hand, guideways 1 and 2 for a lighter and relatively rigid carriage 3 in Fig. 5.19b are characterized by some uncertainty of load distribution that cannot be compensated by local defor-mations of carriage 3 due to its high rigidity.

Excessive connections, if judiciously designed, may significantly improve

Figure 5.19 Overconstrained guideways for (a) a heavy carriage and (b) a medium size carriage.

dynamic behavior of the structure/mechanisms, such as to enhance chatter resis-tance of a machine tool. In the spindle unit with three bearings (Fig. 5.20a), presence of the intermediate bearing (the rear bearing is not shown) may increase chatter resistance as much as 50%. However, this beneficial effect would develop only if this bearing has a looser fit with the spindle and/or housing than the front and rear bearings. In such a case, the ‘‘third bearing’’ would not generate high extra loads in static conditions due to uncertainty caused by static indeterminacy, but would enhance stiffness and, especially, damping for vibratory motions due to presence of lubricating oil in the clearance. While the spindle is able to self-align in the clearance both under static loads and at relatively slow rpm-related variations, viscosity of the oil being squeezed from the clearance under high frequency chatter vibrations would effectively stiffen the connection and generate substantial friction losses (damping action).

A similar effect can be achieved by using an intermediate plain bushing that is fit on the spindle with a significant clearance (0.2–0.4 mm per diameter) and filled with oil (Fig. 5.20b) [8]. High viscous damping provided by the bushing dramatically reduced vibration amplitudes of the part being machined (Fig.

5.20c), which represents an enhancement of dynamic stiffness of the spindle. The price for the higher stiffness and damping in both cases is higher frictional losses and more heat generation in the spindle.

A similar effect can be achieved also in linear guideways. Carriage 1 inFig.

5.21is supported by antifriction guideways 2. The (intermediate) supports 3 and 4 are flat plates having clearanceδ⫽ 0.20–0.03 mm with the base plate 5. While the intermediate supports do not contribute to the static stiffness of the system, they enhance its dynamic stiffness and damping due to resistance of oil filling the clearance to ‘‘squeezing out’’ under relative vibrations between carriage 1 and base 3.

Overconstraining (and underconstraining) of mechanical structures and mechanisms can be caused by thermal deformations, by inadequate precision of parts and assembly, and by design errors.Figure 5.22 shows round table 1 of vertical boring mill supported by hydrostatic circular guideway 2 and by central antifriction thrust bearing 3. At low rpm of the table the system works adequately, but at high rpm (linear speed in the guideway 8–10 m/s) heat generation in the hydrostatic guideway 2 causes thermal distortion of the round table as shown by broken line in Fig. 5.22. This effect distorts the shape of the gap in the hydrostatic support and the system fails to provide the required stiffness. The performance (overall stiffness) would improve if thrust bearing 3 is eliminated or if the system were initially distorted in order to create an inclined clearance in the guideway with the inclination directed opposite to the distortion caused by the thermal deformation of the round table.

Although thermal deformations in the system shown in Fig. 5.22 caused loss of linkage and resulted in an underconstrained system, frequently thermal

Figure 5.20 (a) Use of intermediate bearing 1 to enhance damping (to improve chatter resistance); (b) Influence of plain bushing having 0.2–0.4 mm clearance on the fundamen-tal mode of vibration of spindle (dashed line, without damping bushing; solid line, with damping bushing).

Figure 5.21 Intermediate support in antifriction guideways for enhancing dynamic sta-bility.

deformations cause overconstraining. For example, angular contact ball bearings 2 and 3 for shaft 1 in Fig. 5.23may get jammed due to thermal expansion of shaft 1, especially if its length lⱖ (8–12) d. Similar conditions can develop for tapered roller bearings. Jamming in Fig. 5.23 can be prevented by replacing spacer 4 with a spring, which would relieve the overconstrained condition and the associated overloads.

A similar technique is used in the design of a ball screw drive for a heavy table inFig. 5.24.The supporting bracket for nut 3 is engaged with lead screw 2, which propells table 1 supported by hydrostatic guideways 5 and 6. Bracket 4 contains membrane 7 connecting nut 3 with table 1. Such an intentional reduc-tion of stiffness prevents overconstraining of the system, which may develop

Figure 5.22 Loss of linkage (and stiffness) due to thermal deformations.

Figure 5.23 Shaft supported by two oppositely directed angular contact bearings.

due to uneven clearances in different pockets of hydrostatic guideways or to misalignment caused by imprecise assembly of the screw mechanism, among other things.

Frequently, the overconstrained condition develops due to design mistakes.

Figure 5.25shows a bearing unit in which roller bearing 1 accommodates the radial load on shaft 2 while angular contact ball bearing 3 accommodates the thrust load. In the original design, both bearings are subjected to the radial load-ing, thus creating overconstraining. It would be better to remove the excessive

Figure 5.24 Intentional reduction of stiffness (elastic membrane 7) to enhance perfor-mance of screw drive.

Figure 5.25 Overconstrained bearing unit.

constraints by dividing functions between the bearings as shown in Fig. 5.25 by a broken line. In the latter design, radial bearing 1 does not prevent small axial movements of shaft 2 thus making accommodation of the thrust load less uncer-tain.

Figure 5.26 shows a spindle unit in which the bearing system is undercon-strained. Position of the inner race of the rear bearing 1 is not determined since the position of the inner race is not restrained in the axial direction. Due to the tapered fit of the inner race it can move, thus detrimentally affecting stiffness of the spindle. The design can be improved by placing an adapter ring between inner race 2 and pulley 3. The adapter ring can be precisely machined or, even better, be deformable in order to accommodate thermal expansion of the spindle.

Figure 5.26 Spindle unit with underconstrained rear bearing 3.

5.4 INFLUENCE OF FOUNDATION ON STRUCTURAL