Torsional Values Research Articles

Numerical analysis of hydroelasticity problems by coupling WCSPH with multibody dynamics

This paper aims to extend the implementation of DualSPHysics-Project Chrono coupling to hydroelasticity problems involving free-surface flows and deformable elastic boundaries. In the present model, the fluid sub-domain of the phenomenon is based on the WCSPH model of DualSPHysics. The solid sub-domain, simulated by the Project Chrono engine, is modeled using the set of rigid bodies attached by hinges with torsional and damping stiffness. An analytical solution of Euler-Bernoulli beam theory is employed to adapt the rigid-multibody system to the mechanical characteristics of the elastic sub-domain via the torsional stiffness values of hinges. The stability and accuracy of the present model are tested by three cases with different flow characteristics and mechanical properties for solid sub-domain, involving a newly-designed experimental setup and two benchmark cases in the literature. Obtained results show reasonable accuracy with experimental measurements and other numerical model computations.

On the torsion values for sections of an elliptic scheme

Abstract We shall consider sections of a complex elliptic scheme ℰ {{{\mathcal{E}}}} over an affine base curve B, and study the points of B where the section takes a torsion value. In particular, we shall relate the distribution in B of these points with the canonical height of the section, proving an integral formula involving a measure on B coming from the so-called Betti map of the section. We shall show that this measure is the same one which appears in dynamical issues related to the section. This analysis will also involve the multiplicity with which a torsion value is attained, which is an independent problem. We shall prove finiteness theorems for the points where the multiplicity is higher than expected. Such multiplicity has also a relation with Diophantine Approximation and quasi-integral points on ℰ {{{\mathcal{E}}}} (over the affine ring of B), and in Sections 5 and 6 of the paper we shall exploit this viewpoint, proving an effective result in the spirit of Siegel’s theorem on integral points.

Minimizing the Negative Effects of Coolant Channels on the Torsional and Torsional-Axial Stiffness of Drills

Coolant channels allow internal coolant delivery to the cutting region and significantly improve drilling, but these channels also reduce the torsional and torsional-axial stiffness of the drills. Such a reduction in stiffness can degrade the quality of the drilled holes. The evacuation of cutting chips and the delivery of the cutting fluid put strict geometrical restrictions on the cross-section design of the drill. This necessitates careful selection and optimization of features such as the geometry of the coolant channels. This paper presents a new method that uses Prandtl’s stress function to predict the torsional and torsional-axial stiffness values. Using this method drills with one central channel are compared to those with two eccentric coolant channels, which shows that with the same cross-section area, the reduction of axial and torsional-axial stiffness is notably smaller for the design with two eccentric channels compared to a single central channel. The stress function method is further used to select the appropriate location of the eccentric coolant channels to minimize the loss of torsional and torsional-axial stiffness. These results are verified by comparison to the results of three-dimensional finite element analyses.

The torsion of the proximal femur in cementless total hip arthroplasty: a 3-dimensional evaluation.

The anatomy of the proximal femur at the time of total hip arthroplasty has been widely studied but the horizontal plane was never considered, or only limited to the torsion of the femoral neck. Using CT-scan images from a group of 178 patients scheduled for cementless total hip arthroplasty (THA), we analysed the evolution of the torsion of the proximal femoral metaphysis, in reference to the posterior bicondylar plane of the femur. The evolution of the torsion, between 20 mm below the centre of the lesser trochanter and 20 mm above, was evaluated. In cases of primary osteoarthritis, osteonecrosis, rheumatoid arthritis and epiphysiolysis capitis femoris, the mean torsion decreased from 46° to 20° without significant differences in average values between the different diagnoses, but important individual variations were found. In the groups of dysplasia and congenital hip dislocation, the torsion values were significantly higher, decreasing in mean from 59° to 25° and 63° to 34° respectively, and with important individual variations. These data are important when using cementless femoral stems, since an ideal fit-and-fill in the proximal femur zone has been shown to positively influence bone ingrowth of the stem. However, a strict adaptation of the stem to the medullary canal, without considering its torsion, can lead to an increased or decreased torsion of the prosthesis neck and thus to an instability of the arthroplasty. For these reasons, if a perfect adaptation of the stem to the intramedullary anatomy and an optimal reconstruction of the extramedullary anatomy are to be achieved, 3-dimensional planning should ideally be obtained for every patient. This will allow the best stem choice adapted to every single patient and every kind of anatomy.

Matteucci Effect and Single Domain Wall Propagation in Bistable Microwire under Applied Torsion

Systematic experimental results on the Matteucci effect in torsioned magnetostrictive FeSiB microwire are introduced, observed during the propagation of a single domain wall (DW) under the action of axial driving magnetic field, Hdr. These data are discussed, reviewing current bibliography models and proposing new perspectives. The unidirectional DW velocity is quantified by voltage induced in tiny pickup coils along the microwire. The Matteucci electromotive force (emf) induced between the microwire's ends is associated with depinning and annihilation of the DW, introducing a new method to measure DW velocity. The emf amplitude is proportional to the DW velocity and the time derivative of the azimuthal magnetization, tailored by applied torsion. The data confirm the existence of net values of azimuthal magnetization and spontaneous torsion induced during fabrication. The clockwise/counterclockwise applied torsion dependence of Matteucci emf and DW velocity are experimentally determined for parallel and antiparallel Hdr. Asymmetric behaviors are observed for both, the sense of applied torsion and the direction of Hdr. The experimental data are discussed in terms of the magnetoelastic anisotropy introduced by torsion. Through analysis of the DW dynamics, such asymmetric behaviors are interpreted as a magnetochiral effect derived from the change of chirality of the propagating wall.

Self-contact in closed and open Kirchhoff rods

We consider the problem of an impenetrable Kirchhoff rod in self-contact. The contact is assumed to be frictionless and of point-contact type. The self-contacted rod is modeled as a set of contact-free segments each having unknown length and connected in series with each other through frictionless hard-contact condition. The Landau–Lifshitz approach is used to express analytically the centerline of each contact-free segment in terms of unknown parameters which are obtained using the hard-contact condition at each contact point and boundary condition on the full rod. The presented formulation is applicable to both closed and open rods. For the open rod case, its two end cross-sections need not be parallel either. Numerical solution of following three example problems are presented both before and after self-contact: (i) twisting of a closed rod (ii) compression and twisting of a straight rod (iii) compression and twisting of a rod with opening angle. For the case of twisting of a closed rod in the pre-self-contact regime, we also derive a minimal set of three integral equations to obtain the ring’s buckled spatial configuration. We also point to a difficulty in numerical integration of the rod’s centerline torsion over the rod length whenever the torsion value becomes large locally: a finer discretization of the rod’s length is necessitated in such regions to obtain an accurate integral value. An analytical approximation of the torsion integral is derived in such regions to make the full numerical integration computationally efficient.

A comparison of the passive fit of a 3-unit implant-supported fixed partial denture fabricated by lost-wax casting, milling soft metal blocks, or direct metal laser sintering: An in vitro study

Statement of problemStresses induced by implant-supported prostheses are related to mechanical complications. However, how the manufacturing method affects the generated stresses is unclear. PurposeThe purpose of this in vitro study was to compare the passive fit of 3-unit cobalt-chromium partial prostheses fabricated by using 3 different methods: conventional casting (milled resin with lost-wax method), machined (milling soft metal blocks), or sintered (direct metal laser sintering). Material and methodsThirty 3-unit implant-supported cobalt-chromium fixed prostheses were fabricated, 10 with each method. Four strain gauges were placed over an in vitro resin model with 2 implants in it; 2 gauges were placed above and 2 on the sides to measure the microdeformations caused by placing the structures above the implants in longitudinal flexion, vertical flexion, or torsion, before and after tightening the screws. An inferential analysis was carried out by using a statistical software program, and the Wilcoxon and Kruskal-Wallis tests were applied and then complemented with the Mann-Whitney test for multiple comparisons with the Bonferroni correction (α=.01). ResultsThe mean longitudinal deformation, vertical flexion, and torsional deformation values increased after tightening the screws with each of the 3 fabrication methods. Significant differences (P<.01) were found for all 3 methods before and after screw tightening. After screw tightening, the machined structures, followed by sintered structures, and the cast structures (P<.05) had smaller deformations in the X, Y, and Z dimensions. The longitudinal deformation was less in machined structures, and vertical flexion deformation was less in machined and sintered structures than in cast structures (P<.05). ConclusionsThe deformations registered in the in vitro model increase considerably after screw tightening, which indicates the lack of passive fit. Machined structures provide a better passive fit, followed by sintered structures, and the cast structures had the poorest fit.

A laboratory study of the scouting ability of two reciprocating glide path instruments in mesial root canals of extracted mandibular molars.

To evaluate in a laboratory setting the performance of two reciprocating glide path systems, WaveOne Gold Glider (WO) and R-Pilot (RP), to create a glide path in mesial root canals of mandibular molars and to assess the torsional resistance of instruments after performing the glide path. A total of 60 mesial root canals of extracted human mandibular molars were divided into two groups (n=30) according to the glide path system to be used. The data from the volume of each canal, acquired by microcomputed tomography (micro-CT), were validated statistically for the anatomical pairing of the groups. Preparation time, frequency in gaining apical patency, plastic deformation rate of instruments, and canal transportation and centring ability were recorded and compared statistically. The torsional fatigue of the instruments after use was also evaluated. Data were analysed using Fisher's exact test and Mann-Whitney U-test with a 5% significance level. No significant differences were found between groups regarding the time required to perform the glide path, the frequency distributions of the canals classified as patent and the instruments with plastic deformation after use (P>0.05). No significant differences were found between groups regarding the degree of canal transportation and centring ability at the cervical, middle and apical thirds (P>0.05). The RP groups had significantly greater maximum torsional strength values compared with the WO groups (P<0.05). The used WO group had greater angular deflection to fracture when compared to the new WO group (P<0.05). A significant difference was also found in the percentage of loss of angular deflection in a comparison of the WO group with the RP group (P<0.05). The WO and RP instruments performed the same in terms of preparation time, plastic deformation, gaining apical patency, degree of canal transportation and centring ability. The RP instruments had greater torsional strength, less angular deflection and lower percentage of loss in angular deflection than the WO. The used WO group had the greatest angular deflection values.

Myocardial mechanics in dilated cardiomyopathy: prognostic value of left ventricular torsion and strain

BackgroundData on the prognostic value of left ventricular (LV) morphological and functional parameters including LV rotation in patients with dilated cardiomyopathy (DCM) using cardiovascular magnetic resonance (CMR) are currently scarce. In this study, we assessed the prognostic value of global longitudinal strain (GLS), global circumferential strain (GCS), global radial strain (GRS) and LV torsion using CMR feature tracking (FT).MethodsCMR was performed in 350 DCM patients and 70 healthy subjects across 5 different European CMR Centers. Myocardial strain parameters were retrospectively assessed from conventional balanced steady-state free precession cine images applying FT. A combined primary endpoint (cardiac death, heart transplantation, aborted sudden cardiac death) was defined for the assessment of clinical outcome.ResultsGLS, GCS, GRS and LV torsion were significantly lower in DCM patients than in healthy subjects (all p < 0.001). The primary endpoint occurred in 59 (18.7%) patients [median follow-up 4.2 (2.0–5.6) years]. In the univariate analyses all strain parameters showed a significant prognostic value (p < 0.05). In the multivariate model, LV strain parameters, particularly GLS provided an incremental prognostic value compared to established CMR parameters like LV ejection fraction and late gadolinium enhancement. A scoring model including six categorical variables of standard CMR and strain parameters differentiated further risk subgroups.ConclusionLV strain assessed with CMR FT has a high prognostic value in patients with DCM, surpassing routine and dedicated functional parameters. Thus, CMR strain imaging may contribute to the improvement of risk stratification in DCM.

Comparison of the Torsional Resistance of 4 Different Glide Path Instruments

IntroductionThe present study aimed to compare the torsional strength of the initial files of the Mtwo (VDW, Munich, Germany) and novel Rotate systems (VDW, Munich, Germany) with the ProGlider (Dentsply Sirona, Ballaigues, Switzerland) and R-Pilot (VDW) glide path instruments. MethodsThe Mtwo (10/.04), ProGlider (16/.02), R-Pilot (12.5/.04), and Rotate (15/.04) glide path files were compared regarding their torsional strength, which was tested using a specially designed test device (N = 20). The data obtained were statistically analyzed at 5% significance level using 1-way analysis of variance and the post hoc Tukey test. The fragment surfaces and separated instruments were examined with ×50, ×100, and ×1000 magnification under a scanning electron microscope. ResultsThe R-Pilot group showed the highest torsional strength value among all groups (P < .05), whereas there was no significant difference between the torsional strength values of the Mtwo and ProGlider groups (P > .05). The Rotate group had the lowest torsional strength among all groups (P < .05). The Mtwo group showed the lowest angle of rotation among all groups (P < .05). ConclusionsAlthough the R-Pilot glide path file exhibited the highest torsional strength in all groups, Rotate showed the highest angle of rotation. Differences in torsional resistance of the instruments may be associated with their manufacturing methods and design features.

Tibial torsion analysis in computed tomography: development and validation of a real 3D measurement technique

PurposePathological tibial torsion is known to negatively influence the functionality of the lower extremity, and therefore, its assessment might play an important role. While 3D imaging is used for many examinations of the musculoskeletal system, for the determination of tibial torsion no 3D measurement technique has been available so far. We developed a 3D measurement method and assess its interobserver reliability as well as its correlation with standard 2D measurement methods.MethodsCT scans of 82 tibiae in 79 patients with a mean age of 41 years were included. A novel 3D measurement technique was developed and applied. Measurements were compared with two frequently used 2D measurement methods. ICC (intraclass correlation coefficient) for the new technique was determined and compared to the 2D measurement method. Furthermore, differences between left and right legs as well as between males and females were assessed.ResultsThe ICC for the 2D methods was 0.917 and 0.938, respectively. For the 3D measurements, ICCs were calculated to be 0.954 and 0.950. Agreement between 2 and 3D methods was moderate to good with ICCs between 0.715 and 0.795. Torsion values for left and right legs did not differ significantly in 2D and in 3D (26.2 vs 28.5° and 27.2 vs. 25.9°). The same is true for the differences between male and female in 2D and 3D (26.2 vs. 29.6° and 25.0 vs. 31.2°).ConclusionThe newly developed 3D measurement technique shows a high intraclass agreement and offers an applicable opportunity to assess the tibial torsion three-dimensionally.

A Validated, Automated, 3-Dimensional Method to Reliably Measure Tibial Torsion

Background: Tibial torsion is a twist in the tibia measured as an angle between a proximal axis line and a distal axis line. Abnormal torsion has been associated with a variety of painful clinical syndromes of the lower limb. Measurements of normal tibial torsion reported by different authors vary by 100% (ranging from 20° to 42°), making it impossible to determine normal and pathological levels. Purpose: To address the problem of unreliable measurements, this study was conducted to define an automated, validated computer method to calculate tibial torsion. Reliability was compared with current clinical methods. The difference between measurements of torsion generated from computed tomography (CT) and magnetic resonance imaging (MRI) scans of the same bone, and between males and females, was assessed. Study Design: Controlled laboratory study. Methods: Previous methods of analyzing tibial torsion were reviewed, and limitations were identified. An automated measurement method to address these limitations was defined. A total of 56 cadaveric and patient tibiae (mean ± SD age, 37 ± 15 years; range, 17-71 years; 28 female) underwent CT scanning, and 3 blinded assessors made torsion measurements by applying 2 current clinical methods and the automated method defined in the present article. Intraclass correlation coefficient (ICC) values were calculated. Further, 12 cadaveric tibiae were scanned by MRI, stripped of tissue, and measured using a structured light (SL) scanner. Differences between torsion values obtained from CT, SL, and MRI scans, and between males and females, were compared using t tests. SPSS was used for all statistical analysis. Results: When the automated method was used, the tibiae had a mean external torsion of 29°± 11° (range, 9°-65). Automated torsion assessment had excellent reliability (ICC, 1), whereas current methods had good reliability (ICC, 0.78-0.81). No significant difference was found between the torsion values calculated from SL and CT (P = .802), SL and MRI (P = .708), or MRI and CT scans (P = .826). Conclusion: The use of software to automatically perform measurements ensures consistency, time efficiency, validity, and accuracy not possible with manual measurements, which are dependent on assessor experience. Clinical Relevance: We recommend that this method be adopted in clinical practice to establish databases of normal and pathological tibial torsion reference values and ultimately guide management of related conditions.

Evaluation of left and right ventricular functions with three-dimensional speckle tracking in patients with mitral stenosis.

The function of both ventricles has been suggested to be affected in patients with mitral stenosis. In this study, it was aimed to investigate deformation properties of right (RV) and left ventricle (LV) in rheumatic mitral stenosis (MS) patients with three-dimensional speckle tracking echocardiography (3D-STE). A total of 60 patients were included in the study (20 patients with mild MS diagnosis, 20 patients with moderate MS diagnosis, and 20 healthy volunteers). Three-dimensional echocardiography datasets were obtained for both ventricles in all patients. LV global longitudinal strain (GLS), LV torsion, RV free wall (FW) LS, and interventricular septal (IVS) LS measurements were analyzed. The LV ejection fraction (EF), RV fractional area change, peak systolic velocity of the tricuspit annulus, isovolumic acceleration, and tricuspid annular plane systolic excursion values were statistically similar and in the normal range. The LV GLS measurements were significantly different among the groups by being highest in the control group and least in the moderate stenosis group. Patients with MS showed higher torsional values, correlated with MS severity. IVS LS, RVFW LS values obtained by RV analysis also differed significantly among groups. The RVFW-GLS values only showed significant difference between the control group and moderate MS group. Patients with MS showed lower LV-GLS and higher LV torsion values. RV deformation indices showed significant decrease in correlation with the severity of the mitral stenosis. In conclusion, our data suggest that subclinical LV and RV systolic dysfunction is present in mild-moderate MS patients and this dysfunction can be detected by 3D-STE.

True 3D parameters of the spinal deformity in adolescent idiopathic scoliosis.

Spinal deformities in adolescent idiopathic scoliosis (AIS)are measured on 2D radiographs. Due to the 3D nature of the curvein AIS, such 2D measurements fail to differentiate between the truecurve patterns, which in turn may adverslyimpact the clinical care and surgical planning. The use of 3D models of the spinal radiographs largely remains limited to the 3D measurements of the 2D parameters. The use of the true 3D variables of the spinal curves in describing the differences between the AIS patients is not fully explored. A cohort of 141 Lenke 1 AIS with two-view spinal stereoradiographs and 3D models of the spines were included. The 3D model of the spine was used to determine the spinal centerlines. The writhe and torsion of the 3D centerlines, which, respectively, quantify the coiling and twist of the curve, were calculated using differential geometry. Patients were clustered based on the writhe and torsion values to determine the patient groups with significantly different 3D curve characteristics. The relationship between the writhe and torsion was statistically determined. The distribution of the writhe and torsion groups between the lumbar modifier types was determined. Two writhe and two torsion clusters were determined. Lumbar orientation of plane of maximum curvature (PMC) was significantly different between the torsion clusters and thoracic and lumbar PMC and thoracic Cobb angles were significantly different between the writhe groups, p < 0.05. More than 50% of the patients had high writhe and low torsion except for Lumbar modifier C that mainly belonged to the low writhe group. Two geometrical parameters of the spinal centerline determine true 3D characteristics of the scolioticcurves. The parameters were complimentary and weakly correlated, quantifying different characteristics of the scoliotic spines.

Helicopter Shaft Heating Parameters for Polyamide-11 Coating

In this study, the heating parameters required for polyamide-11 (P11) coating of the helicopter shaft were investigated. The torsion value created by the force applied to the middle of the helicopter's shaft is measured. Heating before coating affects the torsion value of the shaft. A new heating mechanism has been designed to cover the shaft surface without changing the torsion value of the shaft. Thus, the coating surface is left inside the furnace and the torsion measurement area outside of the furnace. Taguchi method was used to determine heating parameters such as furnace type, initial temperature and heating time. The heating process was performed by connecting K type probes to the shaft surface and temperature changes were recorded. It has been determined that with the fanless furnace, the coating surface does not reach the required minimum temperature, the shaft surface reaches the sufficient temperature with the fanned furnace, and PA11 coating can be applied effectively.

Cyclic and Torsional Fatigue Resistance of Seven Rotary Systems.

This study aims to evaluate the cyclic and torsional fatigue resistance of seven rotary systems. A total of 140 instruments were tested, 20 of each system including Genius (GN) 25/0.04, TruShape (TS) 25/0.06, Logic (LOG) 25/0.06 taper, Vortex Blue (VB) 25/0.06, ProTaper Gold (PTG) 25/0.08, Hyflex CM (HCM) 25/0.06 and Hyflex EDM (EDM) 25/0.08 taper. Cyclic fatigue resistance testing was performed using an artificial stainless steel canal with a curvature (60° angle and 5-mm radius) located at 5 mm from the tip. The files (n=10) rotated until fracture, and time was recorded in seconds. The torsional test was evaluated according to ISO 3630-1. Data were analysed with one-way ANOVA and Tukey's tests (α=5%). The fractured surface of the instruments were assessed using scanning electron microscopy to confirm the type of fracture. The cyclic fatigue resistance value of EDM was significantly higher than those of all tested instruments (P<0.05). LOG showed a higher cyclic fatigue resistance than GN or TS (P<0.05). There was no difference among the other groups (P>0.05). The torsional test showed that PTG 25/0.08 had the highest torsional strength value of all instruments tested followed by VB and EDM (P<0.05). The LOG showed significant difference only with GN (P<0.05). No difference was found among the other groups (P>0.05). In relation to angular deflection, the GN; TS; HCM, and EDM showed significantly higher values until fracture than the other groups (P<0.05). No difference was found among PTG, LOG, and VB (P>0.05). Our in vitro study EDM group had the highest cyclic fatigue resistance among all the tested instruments. For the torsional test, the PTG showed highest torsional strength and lowest angular deflection values.

Comparison of Mechanical and Structural Properties of Nickel-titanium Alloy with Titanium-molybdenum Alloy and Titanium-niobium Alloy as Potential Metals for Endodontic Files.

The objective of this study was to compare the mechanical and structural properties of the nickel-titanium (Ni-Ti) alloy already used in endodontics with titanium-molybdenum (Ti-Mo) and titanium-niobium (Ti-Nb) alloys to determine if these can be suggested in the manufacture of endodontic files. Orthodontic wires made of the different alloys were used. The previously mentioned alloys were characterized by energy-dispersive X-ray spectroscopy (EDX), X-ray diffraction (XRD) and torsion tests. Cyclic fatigue tests were performed on a simulated canal with a curvature of 86° to 375 rpm. The fractured surfaces of the wires were observed by means of scanning electron microscopy (SEM). A Kruskal-Wallis test and U Mann Whitney test were used to determine significant differences in cyclic fatigue between groups. In the mechanical tests, similar values of torsion were found for the three alloys. In XRD, the Ti-Nb showed less structural changes. In the cyclic fatigue test, Ti-Nb was found to be significantly more resistant with respect to Ni-Ti and Ti-Mo. Based on our in vitro study, Ti-Nb is suggested as a possible alloy for the manufacture of rotary files due to its impressive properties.

Harnessing distinct deformation modes of auxetic patterns for stiffness design of tubular structures

Auxetic materials with a negative Poisson's ratio show a unique lateral expansion under tension while their deformation under shear is similar to that of normal materials. Here, we present a novel method to design the stiffness of tubular structures by exploiting these load-dependent distinct deformation modes of auxetic materials. Auxetic cutting patterns are engraved on a tube whose bending and torsional stiffness values are predicted computationally for a comprehensive set of design parameters and measured experimentally for representative designs. Results demonstrate that a wide range of bending and torsional stiffness values can be effectively achieved and controlled independently. The proposed design principle is simple and powerful, and hence can be easily extended to design the mechanical properties of non-tubular structures as well.

Design and Torsional Rigidity Analysis on Frame of Formula Garuda 17 Based on Student Formula Japan 2017 Rules

The objective of this study are to: (1) Produce the design of the Formula Garuda 17 car that complies with Student Formula Japan 2017. (2) Produce the frame design of the Formula Garuda 17 car that has a torsional rigidity that can withstand bending and twisting loads. The type of research used is research and development, also known as R & D, and the development model of the Formula Garuda 17 car frame, which is a model for developing Analysis, Design, Development, Implementation, and Evaluation (ADDIE). The results showed that the frame design was in accordance with Student Formula Japan 2017 regulation, and the torsional rigidity value met the target of 1,026.7NM / degree while the Factor of safety value in the bending force was 3.3 so that this frame was categorized as safe for the race.

The total torsion of knots under second order infinitesimal bending

In this paper we consider infinitesimal bending of the second order of curves and knots. The total torsion of the knot during the second order infinitesimal bending is discussed and expressions for the first and the second variation of the total torsion are given. Some examples aimed to illustrate infinitesimal bending of knots are shown using figures. Colors are used to illustrate torsion values at different points of bent knots and the total torsion is numerically calculated.