Lateral stiffness

Rotors with differing rotor lateral stiffnesses. Slotted rotors, electrical machinery, Keyway. 

Modal balaneing is one of the proven methods for flexible rotor balan-eing. Modal balaneing has also been applied to problems of dissimilar lateral stiffness, hysteretie whirl, and to eomplex shaft-bearing problems. In many diseussions on modal balaneing fluid-film damping is not ineluded. In other... 

A very practical way to infer the contact area was later developed by Carpick et al. [65] and Lantz et al. [66]. In these experiments, a small (up to nanometer) lateral modulation, djc, is applied to the sample, and torsion of the cantilever is monitored with a lock-in amplifier to detect the lateral force response, dF (Fig. 5). In this way, the lateral stiffness, [51], given by... 

The shape of AFM cantilevers (much thinner than the width) results in torsional deflection when forces push the tip laterally as in friction measurements (when the tip is sliding) or lateral stiffness measurements (when the tip is stuck). 

Fig. 6. Lateral stiffness vs. load data for a silicon nitride tip vs. mica surface in ultra-high vacuum. Solid line is fit of the JKR model to the data. Reprinted with pennission from ref. [67].

Carpick, R.W., Ogletree, D.F. and Salmeron, M., Lateral stiffness A new nanomechanical measurement for the determination of shear strengths with friction force microscopy. Appl. Phys. Lc//., 70(12), 1548-1550(1997). 

Here, P is the piezoelectric constant of the piezoelectric tube, is the lateral stiffness of the cantilever. 

Standard geotechnical test reports address typical static properties of soil such as shear strength and bearing capacity but may not provide dynamic properties unless they are specifically requested. In these situations, it is necessary to use the static properties. Dynamic soil properties which are reported may be based on low strain amplitude tests which may or may not be applicable to the situation of interest. Soils reports will generally provide vertical and lateral stiffness values for the foundation type recommended. These can be used along with ultimate bearing capacities to perform a dynamic response calculation of the foundation for the applied blast load. 

As for other materials, the soil stiffnesses, Kv, Kh, and K0 are limited by ultimate soil capacities. Furthermore, reversals of movement and uplift can generate zero resistance and must be appropriately included in the analysis. The lateral stiffness, Kh, is determined from friction, adhesion, and passive pressure as applicable with an appropriate moment arm, h. 

Fourth, in the vertical and horizontal directions, the stiffness should be very different. When the AFM is operated in the repulsive-force mode, frictional forces can cause appreciable image artifacts. Choosing an appropriate geometry for the shape of the lever can yield substantial lateral stiffness, thus minimizing the disturbing artifacts. 

In analogy to indentation experiments, measurements of the lateral contact stiffness were used for determining the contact radius [114]. For achieving this, the finite stiffness of tip and cantilever have to be taken into account, which imposes considerable calibration issues. The lateral stiffness of the tip was determined by means of a finite element simulation [143]. As noted by Dedkov [95], the agreement of the experimental friction-load curves of Carpick et al. [115] with the JKR model is rather unexpected when considering the low value of the transition parameter A(0.2Further work seems to be necessary in order to clarify the limits of validity of the particular contact mechanics models, especially with regard to nanoscale contacts. 

When some critical crack length (between 200 pm and 400 pm, i.e. about 15% of the contact diameter) is reached, a brittle propagation stage is observed which is associated with a sudden and drastic drop in the lateral stiffness, K. The measured crack width in the plane of the contact is then of the order of magnitude of the contact diameter. Post-mortem microscope observation of specimen cross sections in the contact zone (Fig. 8) indicates that the depth of the cracks is of the order of magnitude of the contact radius (i.e. about 900 pm). The two deep cracks induced at the edge of the contact may thus be viewed as some kind of half-penny cracks whose radii are approximately equal to the radius of the contact. In the subsequent part of this paper, the two deep cracks will be referred to as primary cracks . 

Figure 7. Schematic representation of the one dimensional Frenkel Kontorova Tomlinson model, a and b denote the lattice constant of the upper sohd and the substrate, respectively. The substrate is considered rigid, and its center of mass is kept fixed. In the shder, each atom is coupled with a spring of lateral stiffness to its ideal lattice site and with a spring of stiffness 2 to its neighbor. The PT model is obtained for 2 0, while the Frenkel Kontorova model corresponds to k = 0. We will drop the subscripts for these two cases since a single spring is relevant.

Hence, the optimal design of a cantilever is a compromise between different factors. Depending on the application, the appropriate dimensions and materials are chosen. Cantilevers for AFMs are usually V shaped to increase their lateral stiffness. They are typically 100 to 200 pm long [L], each arm is about 20 pm wide [w] and 0.5 pm thick [h). The spring constant of V -shaped cantilevers is often approximated by that of a rectangular bar of twice the width of each arm [104]. A more detailed analysis of the mechanical properties of cantilevers and calculations... 

The paper considers a fundamental basic aspect of suspended vertical ropes—namely that the maximum suspended length will ultimately be governed by their strength, their mass per metre and the required factor of safety for their specific application. These thoughts are applied to the guide rope whose function in a rope guided shaft system is to provide lateral stiffness to control the lateral motion of the conveyance. 

The lateral stiffness of a Koepe head rope of linear mass per metre at the conveyance when it is distance z from the top of the shaft is ... 

The combined lateral stiffness of ( p guide ropes and (n ) Koepe head ropes can be written ... 

The aerodynamic load is apphed at the mcixrmum value achieved i.e., at the mctximum winding speed. Because this force is held constcuit cuid apphed qucisi-staticcJly the mctximum lateral deflection occurs approximately at the position along the guide ropes that hcis the maximum flexibility. (Head rope lateral stiffness influences this position shghtly). Hence y due to the aerodynamic load can be written... 

To obtain an approximation for the maximum quasi-static rotational response, the maximum flexibility of the guides are again considered and because both the head rope lateral stiffness and head rope load vary slightly during the wind their values at the shaft mid-point are adopted. Hence it can be shown that... 

Hence the dilemma faced by a designer of a deep rope guided shaft can be appreciated from this Table 3. The designer must either seek guide rope factor of safety dispensation from the statutory authorities or he must find other ways of providing extra lateral stiffness (more and/or heavier rope guides) or provide greater lateral clearances for the deep shaft. 

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