Translational Stiffness Research Articles

Effects of Spring-like Magnetic Energy from Embedded Permanent Magnets on Elastic Actuation

Series elastic actuators (SEAs) are increasingly used in robotic applications where tasks require safe human-robot interaction. Motivated by the growing need to develop SEAs with nonlinear stiffness to tradeoff between force tracking bandwidth and low output impedance, this paper presents an analytical approach to model the magnetic force/torque of a magnetic (Mag) SEA using the distributed current source method to provide a basis to derive the noncontact magnetic compliance; both translational and rotational stiffnesses are considered. The Mag-SEA model, which relaxes several common assumptions in the published literature that treat the magnetic lead screw as an ideal transformer, is numerically illustrated. The findings demonstrate that the simplified (ideal transformer) model fails to characterize the displacement-dependent magnetic force/torque and underestimates the mechanical advantage that can be capitalized from the spring-like magnetic energy of the embedded permanent magnets.

Muscle Activity in Primary Shoulder Movers is Associated with Increases in Translational Glenohumeral Stiffness

<h3>Research Objectives</h3> Rotator cuff muscles are commonly regarded as the main stabilizers of the glenohumeral joint, but recent modeling work from our group suggests that muscles which primarily move the shoulder also contribute substantially to glenohumeral stiffness. The objective of this study was to evaluate this possibility in experiments designed to assess the relationship between muscle activity in the primary shoulder movers and glenohumeral stiffness. <h3>Design</h3> Cross-sectional study. <h3>Setting</h3> Biomechanics research laboratory. <h3>Participants</h3> 15 healthy adults (8F/7M, 26±4 years) with no shoulder pain or injury. <h3>Interventions</h3> N/A. <h3>Main Outcome Measures</h3> We measured translational glenohumeral stiffness (resistance to humeral head translation to quantify glenohumeral stability) using a linear motor that applied 15 mm stochastic anterior-posterior perturbations to the proximal humerus. Measurements were made as participants generated submaximal isometric torques. We used system identification to estimate stiffness from the applied forces and measured displacements. Active stiffness was defined as the increase in glenohumeral stiffness during torque generation compared to during no torque generation. Electromyography measurements (EMGs) were recorded from 3 rotator cuff muscles (intramuscular electrodes: supraspinatus, infraspinatus, subscapularis) and 6 primary shoulder movers (surface electrodes: anterior/middle/posterior deltoid, pectoralis major, latissimus dorsi, teres major). We used linear mixed effects models to characterize the relationship between EMGs and active stiffness. <h3>Results</h3> EMGs in rotator cuff muscles and primary shoulder movers were positively correlated with active stiffness. EMGs recorded from only the primary shoulder movers better predicted stiffness during active contractions than EMGs recorded from only the rotator cuff muscles (R2=0.80 vs. R2=0.56; P< 0.001). Including EMGs recorded from all muscles improved the fit compared to the model with only primary shoulder movers (R2=0.84 vs. R2=0.80; P< 0.001), but this difference was small. <h3>Conclusions</h3> Our results provide additional evidence that the primary shoulder movers contribute to glenohumeral stability. The stabilizing potential of these muscles should be considered when designing and implementing rehabilitation protocols used to treat individuals with reduced glenohumeral stability, such as following shoulder dislocations. <h3>Author(s) Disclosures</h3> Funded by the NIH (F31AR074288, T32GM008152, UL1TR001422), the American Society of Biomechanics, and Northwestern University.

Kinematic Modeling and Stiffness Analysis of a 3-DOF 3SPS + 3PRS Parallel Manipulator

The driving mechanism of an axisymmetric vectoring exhaust nozzle (AVEN) is a 3-degrees-of-freedom (3-DOF) parallel manipulator (PM) with a centering device. According to the characteristics of the vectoring nozzle’s movement mechanism, the 3-DOF kinematics model of the 3SPS + 3PRS PM is established. The forward and inverse positional posture models are built based on the mechanism architecture of PM and the closed-loop vector method. The Jacobian matrix and Hessian matrix are derived using screw theory, and velocity and acceleration models are established. Based on the principle of virtual work and combined with the kinematics model of PM, a stiffness model was established to analyze stiffness performance. Finally, the effectiveness of the kinematics model of the PM was verified by a simulation, and the variation of stiffness performances with the positional posture of the PM was quantitatively analyzed. The results show that the translational stiffness decreases with forward axial translation of the moving platform and the rotational stiffness is mainly determined by the posture parameters. This study is expected to offer ideas for the kinematic modeling of 3-DOF PM and provide references for application and optimizing of the AVEN’s driving mechanism.

Simplified Models to Capture the Effects of Restraints in Glass Balustrades under Quasi-Static Lateral Load or Soft-Body Impact

Structural glass balustrades are usually composed of simple glass panels which are designed under various restraint solutions to minimize large out-of-plane deflections and prematurely high tensile/compressive stress peaks under lateral loads due to crowd. Linear supports, point-fixing systems, and others can be used to create geometrical schemes based on the repetition of simple modular units. Among others, linear restraints that are introduced at the base of glass panels are mechanically described in the form of ideal linear clamps for glass, in which the actual geometrical and mechanical details of real fixing components are reduced to rigid nodal boundaries. This means that, from a modelling point of view, strong simplifications are introduced for design. In real systems, however, these multiple components are used to ensure appropriate local flexibility and adequately minimize the risk of premature stress peaks in glass. The present study draws attention to one of these linear restraint solutions working as a clamp at the base of glass panels in bending. The accuracy and potential of simplified mechanical models in characterizing the effective translational and rotational stiffness contributions of its components are addressed, with the support of efficient and accurate Finite Element (FE) numerical models and experimental data from the literature for balustrades under double twin-tyre impact. Intrinsic limits are also emphasized based on parametric calculations in quasi-static and dynamic regimes.

Limited-damage 3D-printed interlocking connection for timber volumetric structures: Experimental validation and computational modelling

Cross laminated timber volumetric construction (CLTVC) is an innovative construction technology that combines the sustainability of timber and the efficiency of modular construction, as opposed to conventional construction. However, the connection installing methods of CLTVC, such as fastening, are laborious with limited accessibility for connection installations, thus hindering the application of CLTVC in mid- and high-rise structures. Therefore, a new way of connecting CLT modules by sliding and stacking is explored herein with a newly proposed damage-control interlocking connection system, aiming to provide a more efficient assembly solution to CLTVC that does not require onsite screwing. Quasi-static monotonic and cyclic test, and numerical analyses were conducted to assess the mechanical performance of the proposed connections, which possessed adequate translational stiffness and strength of the proposed connections. The connections' ability to control deformation, while damage is moved away from timber and the embedded fasteners, was also well demonstrated in the test, as both screws and timber remained mostly intact after testing. This novel connection design showcases a new concept of modules’ assembly in volumetric construction with higher efficiency and flexibility; meanwhile demonstrates the potential in reducing the permanent damage to structural materials during service life and enabling reuse.

A three-axis modular horizontal vibration isolation system with adjustable stiffness: Design, analysis and experimental validation

In this study, a modular vibration isolator design is introduced in which the horizontal stiffness of the system can be adjusted in three axes (translational stiffness in the horizontal x and y axes and torsional stiffness about the vertical z-axis) by changing the compressive forces on the leg modules, which are composed of elastic stepped columns and a string. A novel multi-string tensioning mechanism is developed to change the axial compressive forces on all the elastic stepped columns simultaneously, which in turn vary the translational and torsional stiffness of the system. Consequently, the translational and torsional natural frequencies of the system can be tuned to a targeted value by increasing/decreasing the tension forces in the strings for light/heavy payloads. With the use of multiple leg modules in the isolator, the torsional natural frequency can be kept close to the translational natural frequencies at any value of the tension forces in the strings even if the payload masses vary over a wide range. To investigate the relationship between the tension forces and natural frequencies, a leg module is produced. Analytical and finite element models of the leg module are formed, and the results of these models are found to be in good agreement with the experimental results. It is shown that the translational and torsional natural frequencies of the isolator with multiple leg modules can be adjusted to 0.5 Hz and 0.6 Hz, respectively, keeping the lower limits of the isolation bandwidth in the translational and torsional axes less than 1 Hz. Moreover, the upper limit of the isolation bandwidths can be kept above 135 Hz. Therefore, the proposed isolation system has a very wide isolation bandwidth at low frequencies.

Bandgap mechanism analysis of elastically restrained periodic cylindrical shells with arbitrary periodic thickness

A unified model is established to investigate the wave propagation characteristics of cylindrical shells with arbitrary periodic variable thickness distribution and general boundary conditions. The finite periodic shell including any number of unit cells is considered in a unified pattern using energy principle and Rayleigh-Ritz procedure. System governing equation is formulated based on Love's shell theory, and an improved Fourier series expansion is employed to construct longitudinal, circumferential and radial vibration displacements, in which the unity auxiliary terms are used to overcome differential discontinuity of vibration displacements at elastic boundaries. Arbitrary periodic thickness variation information is condensed into Fourier expansion coefficients. Validity of the proposed model is verified through the comparison with results calculated from WFEM and FEM. High precision and excellent convergence can be observed with only several truncated terms of Fourier series. Results show that the bandgap characteristics is greatly dominated by the variation of shell thickness distributions, which is closely associated with the resonant frequency. Moreover, the translational spring stiffness has a more obvious influence on the bandgap characteristics comparing with the rotational spring stiffness. The mechanism of bandgap characteristics is explained, and the influences of the number of unit cells and boundary reflection on the bandgaps are also investigated.

Influence lines-based model updating of suspension bridges considering boundary conditions

The boundary conditions of the main girder are complicated for suspension bridges. And they have a significant impact on the static and dynamic response of the structure. This study proposes a model updating method based on influence lines considering boundary conditions. First, rotational and translational spring elements are used to simulate the boundary conditions of the main girder. Then, rotational spring stiffness is estimated by constructing a sensitive objective function based on the influence lines. Translational spring stiffness is estimated by constructing a sensitive objective function based on the temperature-induced displacements at both ends of the girder. To improve calculation efficiency, a Gaussian process model is used instead of the finite element model (FEM). The estimation of boundary conditions is expressed as an optimization whose main goal is to iteratively estimate the boundary conditions to minimize the objective functions. After the boundary conditions are determined, the influence lines-based model updating is carried out by using the adaptive metamodel global optimization method. To validate the proposed method, a long-span suspension bridge is taken as a case study. The results show that the boundary conditions of the main girder have been successfully estimated. The accuracy of the model updating is improved after the boundary conditions are considered. The updated FEM with accurate boundary conditions can be a benchmark model for bridge damage identification or condition assessment.

Sensitivity of the Cervical Disc Loads, Translations, Intradiscal Pressure, and Muscle Activity Due to Segmental Mass, Disc Stiffness, and Muscle Strength in an Upright Neutral Posture.

Musculoskeletal disorders of the cervical spine have increased considerably in recent times. To understand the effects of various biomechanical factors, quantifying the differences in disc loads, motion, and muscle force/activity is necessary. The kinematic, kinetic, or muscle response may vary in a neutral posture due to interindividual differences in segmental mass, cervical disc stiffness, and muscle strength. Therefore, our study aimed to develop an inverse dynamic model of the cervical spine, estimate the differences in disc loads, translations, intradiscal pressure, and muscle force/activity in a neutral posture and compare these results with data available in the literature. A head–neck complex with nine segments (head, C1–T1) was developed with joints having three rotational and three translational degrees of freedom, 517 nonlinear ligament fibers, and 258 muscle fascicles. A sensitivity analysis was performed to calculate the effect of segmental mass (5th to 95th percentile), translational disc stiffness (0.5–1.5), and muscle strength (0.5–1.5) on the cervical disc loads (C2–C3 to C7–T1), disc translations, intradiscal pressure, and muscle force/activity in a neutral posture. In addition, two axial external load conditions (0 and 40 N) were also considered on the head. The estimated intradiscal pressures (0.2–0.56 MPa) at 0 N axial load were comparable to in vivo measurements found in the literature, whereas at 40 N, the values were 0.39–0.93 MPa. With increased segmental mass (5th to 95th), the disc loads, translations, and muscle forces/activities increased to 69% at 0 N and 34% at 40 N axial load. With increased disc stiffness (0.5–1.5), the maximum differences in axial (<1%) and shear loads (4%) were trivial; however, the translations were reduced by 67%, whereas the differences in individual muscle group forces/activities varied largely. With increased muscle strength (0.5–1.5), the muscle activity decreased by 200%. For 40 vs. 0 N, the differences in disc loads, translations, and muscle forces/activities were in the range of 52–129%. Significant differences were estimated in disc loads, translations, and muscle force/activity in the normal population, which could help distinguish between normal and pathological cervical spine conditions.

Are rotational passive stiffness and translational passive stiffness correlated? A porcine in vitro study

BackgroundQualitative clinical assessments of spinal stiffness have been demonstrated to show moderate correlations with one-another. We hypothesized that these correlations would improve in an in vitro model of the functional spinal unit. If the stiffness of spinal units are different across loading regimes (e.g. flexion-extension versus shear), then it may provide one explanation as to the variability in findings from clinical assessments, since these tests tend not to discriminate rotational and translational degrees-of-freedom. Therefore, the purpose of this investigation was to quantify the relationships between rotational and translational stiffness measures in vitro. MethodsForty-eight porcine cervical spine functional units were used in this investigation (20 C3-C4, 28 C5-C6). While under constant 300 N compressive load, range-of-motion tests for both flexion-extension (± 8 Nm, 0.5 deg./s) and anteroposterior shear (± 400 N, 0.2 mm/s) were conducted, to quantify moment-angle and force-deflection curves. Representative stiffness values were then obtained for flexion, extension, anterior shear, and posterior shear using segmented regression. The correlation matrix between these four measures was then used to explore their potential relationships. FindingsOf the six correlations conducted, only the relationship between posterior shear and extension stiffness was statistically significant (p = 0.014), despite featuring a low correlation coefficient (R2 = 0.123). InterpretationThe poor correlations between stiffness metrics in this study supports the disparate findings of tissue stiffness in vivo. Results from this investigation suggest that clinicians should be cognizant of which degrees-of-freedom they are assessing in the spine, as their stiffness values vary independently.

Dynamic estimation of soft tissue stiffness for use in modeling socket, orthosis or exoskeleton interfaces with lower limb segments

Modeling the interface between the lower limb segments and a socket, orthosis or exoskeleton is crucial to the design, control, and assessment of such devices. The present study aimed to estimate translational and rotational soft tissue stiffness at the thigh and shank during daily living activities performed by six subjects.Smooth orthogonal decomposition (SOD) was used on skin marker trajectories and fluoroscopy-based knee joint kinematics to compute stiffness coefficients during squatting, sitting and rising from a chair, level walking, and stair descending. On average, for all subjects and for all activities, in the anatomical directions observed, the translational and rotational stiffness coefficients for the shank were, respectively, 1.4 ± 1.99kN/m (median and interquartile range) and 41.5 ± 34.3Nm/deg. The results for the thigh segment were 1.79 ± 2.73kN/m and 30.5 ± 50.4Nm/deg.As previously reported in the literature dealing with the soft tissue artifact – considered as soft tissue deformation in this study - the computed stiffness coefficients were dependent on tasks, subjects, segments, and anatomical directions. The main advantage of SOD over previous methods lies in enabling estimation of a task-dependent 6 × 6 stiffness matrix of the interface between segments and external devices, useful in their modeling and assessment.

Evaluation on story-based stability in multistory frame under various restraining conditions

To estimate the critical buckling load, the effective length approach isolates a critical column inside a frame and analyses the rotational and translational stiffness of its end restraints. Although designers proportion leaning columns with K = 1, which is the correct K-value based on the column failure mode, buckling analysis fails to predict these values due to the influence of nearby columns and beams. In current design practise, beam-to-column and column-base connections in steel frames are typically idealised as wholly pinned or totally stiff connections. Various studies have demonstrated, however, that the semi-rigid behaviour of the systems beam-column connections affects the internal efforts in the structural parts, therefore the buckling critical loads and corresponding effective lengths of the compressed elements were also taken into account. In this study, a numerical analysis of the Effective length of column in multi-storey frames with single bay and dual storey (G + 2) is tried with variation in storey heights using rigid and semi-rigid jointed frames with rigid and semi-rigid jointed frames is undertaken. Webber's empirical equations are efficient in rigid frame systems for calculating the effects of adjoining columns and their critical buckling loading of the isolated column. The effective length K factor with the variance in critical length with different storey heights acquired a greater value when compared to other models. Moment rotations in semi-rigid jointed frames were influenced by differences in fixity factors in semi-rigid frames. The rigidity coefficient dropped as the stiffness coefficient increased, leading in an increase in the effective length factor.

Experimental Study of Earthquake Simulator for 3D Cold-Formed Steel Frame Structure

This paper aims to perform the experimental test and numerical FE analysis of the cold-formed steel frame structure to provide new proposals to improve the strength, rotation capacity, and seismic energy dissipation in these structures. A 3D full-scale three-story CFS frame structure is experimentally investigated under four sets of input data for 100% and 50% scales of Northridge and Kocaeli earthquake acceleration records using the shaking table test. In each test, the frame acceleration and displacement are observed. A FE model is employed to assert the validity of the experimental test. The results indicated that the maximum story drift ratio of the frame ranges from 1.14% to 3.70%. It is observed that the average similarity ratio of the maximum displacement results between FE analysis and shaking table test is 92%. It is shown that the translational stiffness for the system decrease as the earthquake acceleration records decrease. It is also shown that using the image processing technique in the shaking table test can offer much more acceptable and economical solutions in determining the displacement parameters.

Analysis and design of an adjustable stiffness three-axis horizontal vibration isolator using elastic columns and a string in tension

The aim of this study is to design a horizontal vibration isolation system with adjustable quasi-zero-stiffness along three axes (translational stiffness in the horizontal x and y axes and torsional stiffness about the vertical z-axis). For this purpose, a string is placed in the central axis of the system, and a string tensioning mechanism is developed to tune the horizontal (lateral) stiffness of the elastic columns that support the top platform on which a payload is placed. By increasing/decreasing the tension force in the string for smaller/larger payloads, axial compressive forces on all the elastic columns can be varied simultaneously, which in turn enables to adjust translational and torsional natural frequencies of the system. Various parametric studies are conducted by using normalized length and force quantities such as the ratio of string length to column length, and the ratio of critical buckling force due to a tension force directed towards a fixed point to critical buckling force due to an external vertical load. Robustness of the system against eccentric payload placement is quantified. Moreover, transmissibility of the system is calculated for translational and torsional excitations. Implicit and explicit analytical, and finite element models demonstrate that by selecting the parameters of the system appropriately, the lower limit of the isolation bandwidth for both translational and torsional excitations can be kept below 1 Hz, and the upper limit can be kept above 250 Hz even if the payload mass and its mass moment of inertia change in a wide range. Thus, the system is capable of achieving a very large isolation bandwidth at low frequencies.

Combination of Surgical Techniques Restores Multidirectional Biomechanical Stability of Acromioclavicular Joint

To measure the multiaxial stability of the acromioclavicular joint before and after transection of the acromioclavicular capsule and coracoclavicular ligaments and after sequential repair of acromioclavicular and coracoclavicular ligaments. Biomechanical testing was performed on fresh-frozen human cadaveric shoulders (N= 6). Translational and rotational stability in the vertical and horizontal planes was measured in intact specimens, after transecting the acromioclavicular and coracoclavicular ligaments, and after sequentially performing the following procedures: single-bundle coracoclavicular repair (CCR), modified Weaver-Dunn procedure (WD), and acromioclavicular stabilization (ACS). Resecting the acromioclavicular and coracoclavicular significantly reduced translational stiffness in the inferior and anteroposterior directions, as well as rotational stiffness about the vertical and anteroposterior axes. All 3 surgical procedures increased inferior translational stiffness relative to the intact condition (Intact: 38 ± 9 N/mm, CCR: 54 ± 23 N/mm (P = .03), CCR+WD 52 ± 20 N/mm (P= .07), CCR ± WD+ACS 50 ± 21 N/mm (P= .17)). However, the combination of CCR, modified WD, and ACS resulted in the greatest increase in stiffness in internal rotation (Intact: 12.5 ± 7.4 cNm/deg, CCR: 1.2 ± 1.1 cNm/deg, CCR+WD 7.2 ± 3.0 N∗m/deg [P= .023], CCR+WD+ACS 11.6 ± 4.9 cNm/deg [P= .055]). The cumulative stability of CCR, WD reconstruction, and ACS appears to be additive. Our findings provide a biomechanical justification for combining all three techniques. Biomechanical studies assessing the performance of various acromioclavicular repairs and reconstructions should therefore incorporate multiaxial testing in their protocols. Multiple points of fixation that provide multidirectional stability have the potential to improve clinical outcomes and reduce failure rates of acromioclavicular joint repair or stabilization.

Translational stiffness and resistance of sandwich panel connections at elevated temperature

When sandwich panels are used as cladding structures, they contribute to the structural behavior of the building frame. In particular, sandwich panels can noticeably improve the stability of members as well as the global stability of the building. In the design calculations for stability, the properties of the connections between sandwich panels and the supporting steel members play a key role. Although guidelines exist for the resistance and stiffness of sandwich panel connections at ambient temperature, no such rules are available for the fire limit state. To bridge this gap, this study presents an investigation on the translational (shear) behavior of sandwich panel connections at elevated temperatures. A series of connection tests are reported, followed by the description of a numerical model of the analyzed connections. The numerical model is validated against the experimental results, and a parametric study is carried out to extend the scope of the considered connections and investigate the contribution of independent variables. Finally, analytical solutions for the translational resistance and stiffness of sandwich panel connections at elevated temperatures are proposed and verified against the obtained experimental and numerical results. The analytical solutions employ the existing rules for the behavior of connections at ambient temperature and consider several approaches to incorporate the effect of elevated temperature. The panels are considered with PIR and mineral wool cores, with the thickness in a range from 100 to 230 mm. The behavior of panels is analyzed at temperatures up to 600 °C.

A complex Fourier series solution for buckling of VAT composite laminates with elastically restrained edges

Variable Angle Tow (VAT) composites have general anisotropic characteristics, and their fiber orientation angle varies continuously along an arbitrary direction. Therefore, the investigation of their buckling performance is quite challenging. Different from the commonly used numerical methods, a new method called complex Fourier series solution is proposed to study the buckling of VAT laminates with elastically restrained edges. First the prebuckling analysis is achieved by introducing Airy’s stress function and expanding it into a complex Fourier series form. And the corresponding mid-plane internal forces with non-uniform distribution are determined. Based on classical laminated plate theory, the buckling analysis of VAT laminate is carried out. The transverse displacement is also expressed as a complex Fourier series. And the critical buckling load and buckling mode shape of VAT laminate are obtained. The accuracy of the presented complex Fourier series solution is verified by comparing with the existing results. The influence of fiber orientation angles, boundary stiffness and boundary conditions on the critical buckling load and buckling mode shape is investigated in numerical examples. Results show that boundary stiffness as well as fiber path have great influence on the critical buckling load and buckling mode shape of VAT laminate. By adjusting the fiber orientation angles and the translational stiffness of the elastically restrained boundary can effectively improve the critical buckling load of VAT laminate. This paper provides a new method for the analysis of general anisotropic laminates with in-plane variable stiffness and the present complex Fourier series solution can also be used as a benchmark for the verification of other numerical results.

Bidirectional deep-subwavelength band gap induced by negative stiffness

Owing to its tremendous potential application in wave manipulation, metamaterials have attracted significant interest since their appearance at the start of this century. The resonant mechanism breaks the dependency of the lattice constant on the operation frequency, successfully achieving wave manipulation in the subwavelength range. However, it fails to simultaneously control two types of wave propagations in the deep-subwavelength range due to the specific configuration of the resonator. In this study, we introduce a novel negative-stiffness mechanism into the resonator to open a bidirectional deep-subwavelength band gap. Both the translational and torsional stiffness of the resonator can be neutralised by the proposed negative-stiffness mechanism. More importantly, adjusting the parameters of the negative-stiffness mechanism provides a convenient way to tune the bidirectional stiffness characteristic of the resonator, which is beneficial for manipulating the elastic wave in a large frequency range. The fundamental principle of opening the bidirectional deep-subwavelength band gap is examined through mechanical analysis, dispersion relation, and wave propagation features. By cascading the presented resonator in a host structure, this study realises the suppression of the propagations of two types of elastic waves within the deep-subwavelength frequency range as well as the filtration of one of these two types of elastic waves in a particular frequency range.

Design of flexure revolute joint based on compliance and stiffness ellipsoids

In view of the problems of low accuracy, small rotational angle, and large impact caused by flexure joints during the deployment process, an integrated flexure revolute (FR) joint for folding mechanisms was designed. The design was based on the method of compliance and stiffness ellipsoids, using a compliant dyad building block as its flexible unit. Using the single-point synthesis method, the parameterized model of the flexible unit was established to achieve a reasonable allocation of flexibility in different directions. Based on the single-parameter error analysis, two error models were established to evaluate the designed flexure joint. The rotational stiffness, the translational stiffness, and the maximum rotational angle of the joints were analyzed by nonlinear finite element analyses. The rotational angle of one joint can reach 25.5° in one direction. The rotational angle of the series FR joint can achieve 50° in one direction. Experiments on single and series flexure joints were carried out to verify the correctness of the design and analysis of the flexure joint.

Design, static analysis and development of a new three-legged shake table

In this paper, an alternative design is proposed based on a family of three-legged manipulators. Such manipulators have two actuators (one vertical and one horizontal) in each leg, unlike the standard UP̅S Stewart platform, which has one actuator in each leg. The arrangement of the two actuators is such a way that, to have vertical motion of the shake table only the Vertical Motion Actuators (VMA) are actuated and for longitudinal or lateral motion, the Horizontal Motion Actuators (HMA) alone are moved. Due to its inherent features such as simplified kinematics, control and distributed loading, a study is carried out to determine the performance of such three-legged manipulators as a shake table. Sinusoidal motion and white noise motions are given to the actuators and shown that the VMA forces have linear relationship with the platform forces. The translational stiffness and the torsional stiffness are studied separately for the manipulators. In the dynamic analysis, it is highlighted that the gravity load of the legs is borne by the Vertical actuators, irrespective of the motion being spatial or planar. Hence, this topology provides scope for lighter electromechanical actuation. The performance analysis of the 3 legged configuration is accomplished using simulation results, in comparison to a 7-UP̅S configuration of shake table. A prototype of the shake table is fabricated and tested with earthquake data of El Centro.