In the literature, authors have made contributions in the area of partially compliant slider-crank (rocker) mechanisms possessing rigid joints that may cause backlash inherently. On contrary, fully compliant mechanisms offer no backlash which is a valuable property for the cases where high precision is required. In this paper, we proposed a fully compliant slider-crank mechanism that performs large stroke. Kinematic performance of the mechanism is investigated analytically. Dimensions of the mechanism are optimized to obtain maximum translational output, while keeping deflections of flexible hinges equal to each other and as small as possible. A design table displaying stroke, axis drift of the output segment, and critical stresses of compliant segments are presented. As an example, a compliant mechanism is designed by using rigid body replacement technique. Then, via nonlinear finite element analysis technique, analytical results are verified. Finally, a prototype is built to compare output stroke and axis drift with analytical approaches. The results of experiments verified that the theoretical approaches are consistent.
Compliant mechanisms are flexible mechanisms that transfer some or all of their motion through deformation of elastic segments. They are divided into two main categories; partially or fully compliant mechanism. Partially compliant mechanisms have at least one traditional (rigid) joint that may cause backlash inherently. By definition, a fully compliant mechanism does not possess a conventional rigid joint. Thus, fully compliant mechanisms obtain all their motion from deflection of compliant segments (Howell, 2001). This property is advantageous for the cases where precision is crucial. Compliant mechanisms have further advantages such as low cost, reduced number of parts, no need for lubrication, less wear and noise. Additionally, stored elastic energy due to deformation of compliant members brings mechanism to its original position. Pseudo-rigid-body model (PRBM) is used to simplify analysis of systems that undergo large, nonlinear deflections (Howell and Midha, 1996). Yu et al. (2016) proposed a new three degree-of-freedom (DOF) model based on PRBM for large deflection beams. In Liu and Yan's (2017) study modified pseudo-rigid- body modelling approach for compliant mechanisms with fixed guided beam flexurals was examined.
Rigid joints allow specific DOF between connected parts. However, rigid joints suffer from backlash and wear because of clearance between mating parts. Trease et al. (2005) proposed a revolute and translational compliant joint. Another novel three dimensional compliant translational joint with large stiffness ratio and small axis drift was presented (Yang et al., 2016).
Four link mechanisms (e.g. four-bar and slider-crank) have significant importance in the industry. Slider-crank mechanisms have numerous applications especially when kinematic inversions are considered. Researches on compliant slider-crank (rocker) mechanisms are limited in the literature. Hao et al. (2018) proposed a multi-mode compliant gripper. There is a fully compliant slider-crank mechanism available in their gripper. In Parlaktaş and Tanık's (2014) study the “single piece” compliant spatial slider-crank mechanism was introduced. In this study, deflections of the multiple axis flexural hinges were determined separately as bending and twist. Erkaya et al. (2016) investigated dynamics of the partially compliant slider-crank mechanism. In Alqasimi et al.'s (2016) study, a new model for a bistable compliant mechanism that is based on crank-slider mechanism was presented. In this study the slider is rigid thus the mechanism is partially compliant. In Pardeshi et al.'s (2017) study, monolithic compliant slider-crank mechanism with a rigid slider for motion amplification was proposed. Dao and Huang (2014) proposed an optimal design of a partially compliant slider-crank mechanism with circular cross-section flexure hinges.
Furthermore, there are some studies based on compliant linear-motion mechanisms which can not be categorized as slider-crank mechanism: Pavlović and Pavlović (2009) introduced compliant parallel-guiding mechanism's design procedure. A new compliant mechanism which is capable to realize axial translation of the link was presented. An inherent parasitic motion of the compliant parallel four-bar mechanism was compensated by exploiting center shift of a generalized cross-spring pivot in Hongzhe et al.'s (2012) study. Zhao, et al. (2017) designed a stiffness-adjustable compliant linear-motion mechanism.
Partially compliant mechanisms possess rigid prismatic joints (slider) in their structure. Prismatic joints inherently have disadvantages due to backlash and friction problems. An approach for optimum link proportions of the rigid body equivalent is proposed. The optimization objective is to maximize the translational motion of the slider equivalent while minimizing the relative link rotations. Minimization of link rotations is essential to keep maximum stresses in an acceptable range. Input–output motion relationships of the mechanism are determined. Resultant stresses at flexural hinges are obtained analytically. A design table is prepared for generalization of the dimensions that will be beneficial for other researchers. As a design example, an optimum mechanism is analyzed via finite element analysis (FEA) method and analytical results are verified. A prototype is manufactured and experiments are conducted. Also, it is ensured that the results of experiments are consistent with the theoretical approaches.
Note that, for the cases where crank of a slider-crank mechanism does not fully rotate, the mechanism may be called as slider-rocker mechanism. However, this makes an ambiguity for the identity of the mechanism. In this study, we prefer to name the mechanism as slider-crank disregarding the link proportions.
In Fig. 1a, a partially compliant slider-crank mechanism that possesses a prismatic joint between the slider and ground is presented. In our design, we replaced the prismatic joint by two identical and parallel compliant segments and a rigid segment as shown in Fig. 1b. By this way, if properly designed,
translational motion for the output which acts as a slider can be achieved.
Alternatively, compliant version of Roberts or Watt (Shigley and Uicker,
1980) type four bar mechanisms can be implemented for slider replacement.
Specific points on coupler link of these mechanisms trace approximate
straight lines. However, their coupler link performs rotation as well as
translation. In our case, the output link performs no rotation but only
translation (curvilinear translation). In the literature, paired double
parallelogram mechanisms are used as sliders (Trease et al., 2005; Li et
al., 2018). The advantage of this structure is straight line motion with no
rotation. However, there is a big major disadvantage; if PRBM of this type
is constructed, it can be calculated that DOF
In Fig. 2a, PRBM of the fully compliant slider-crank mechanism is displayed. This PRBM is essentially a six-link mechanism, where two of the links are connected to a parallelogram four-bar mechanism. The coupler link of this parallelogram four-bar acts as a slider, if the fixed guided segments 1 and 2 are long enough and rigid segment 4 stays parallel to its initial position (Fig. 2b). These fixed guided segments must be identical, initially straight, and parallel to each other. By this way, we obtain a compliant parallel-guiding mechanism (Howell, 2001) where rigid segment 4 performs curvilinear translation. In the case of an extreme vertical load, fixed guided segments may buckle thus, rigid segment 4 becomes nonparallel to its initial position.
Initially, kinematic analysis of the PRBM which is essentially a rigid cascade parallelogram four-bar mechanisms is performed. The PRBM (Fig. 3) is formed as follows: input (link 2) and connecting rod (link 3) of the mechanism are combined with a parallelogram four-bar mechanism whose coupler link (link 4) acts as a slider of the fully compliant slider-crank mechanism (Fig. 2b).
Structure parameters and variables for the kinematic analysis.
Kutzbach criterion (Shigley and Uicker, 1980) for DOF of a planar mechanism
is
Referring to Fig. 3, the revolute joints at
The main objective of the position analysis is to derive closed form
equations according to the position variables as a function of input angle
It is clear that link 4 performs curvilinear translation motion due to the parallelogram structure of the four-bar mechanism. Thus, any point on the coupler link moves on congruent curves. Points
PRBM with an imaginary parallelogram four-bar mechanism.
Initially, kinematic analysis of the first four-bar mechanism in
Fig. 4, composed of links 9, 2, 3, and 7 is performed.
In order to obtain relationship between input
Structure parameters and variables for the cascade four-bar mechanism.
During rigid body replacement synthesis, if the undeflected position of the
compliant segment between rigid segments 3 and 4 is set to 180
In this section, the design approach targets three main objectives. First objective is the minimization of the relative link rotations while maximizing the translational motion of coupler of the second four-bar mechanism. Because relative rotations of the links determine the deflection amount of the hinges of the compliant slider-crank mechanism. The second objective is to equate the relative rotation of links from a reference position. Because, it is well known that, equality of deflection in both directions minimizes deflection peaks in a compliant mechanism design. The third objective is to equate all relative rotations of the links to a specific value, if possible. By this way, all hinges of compliant slider-crank mechanism can be designed with same dimensions that yields a robust design. Generally, for compliant mechanisms, the dominant loading is due to bending. Bending stress is similar for the same hinges with same dimensions and deflections.
The design of the rigid body equivalence is performed considering these
three objectives as follows: Initially, the first four-bar mechanism (
Four-bar mechanism in forward and fully withdrawn positions.
A parallelogram four bar mechanism as presented in Fig. 7 is a neat solution according to the requirements. Green, black, and red lines represent the fully withdrawn, undeflected, and forward positions of the mechanism respectively. With this approach, the angle between the extreme positions of links 2 and 7 is
Four-bar mechanism in three positions.
Considering the free design parameters, an infinite set of solution is available. However, in every solution it is determined that
Four-bar mechanism in three positions (forward-undeflected-fully withdrawn) with design parameters.
Kinematic analysis of the mechanism is simple due to the symmetrical link
proportions: The stroke of the mechanism which is the horizontal motion of
Generally, if deflections of flexural hinges are large, stresses of flexural hinges are also high. Thus, in practice, deflection values should be kept in a feasible range. By using the rigid body replacement method, the fully compliant slider-crank mechanism can be dimensioned as shown in Fig. 9.
Fully compliant slider-crank mechanism overlapped with its PRBM.
As shown in the PRBM, links 4–6 form a parallelogram four-bar mechanism where link 5 and 6 correspond to fixed guided beams. According to the length of the top and bottom parts of long compliant segments are equal to
Design table of the fully compliant slider-crank mechanisms.
By using Table 1, numerous fully compliant slider crank mechanisms can be designed for different dimensions. This generalized design table will be beneficial during preliminary design stage satisfying very large slider strokes with acceptable stresses.
A fully compliant slider-crank mechanism is designed by using the method
given in Sect. 6 and the equations in Table 1 as follows: let lengths of rigid segments 2, 4, and 6 be
It is verified that, average of the output stroke and axis drift values of the slider are in close agreement with the analytical results. Resultant von-Mises stresses at the flexural hinges when the input rotation is set to
its maximum value (20
Stroke analyses for fully compliant slider-crank mechanism.
Maximum stress values at compliant segments for fully compliant slider-crank mechanism.
A prototype of the mechanism is built for collecting experimental data. The
mechanism is manufactured in one piece from polypropylene with a plate
thickness of 15 mm. During the manufacturing process 3 axis CNC router is
used setting 7.5 mm s
Experimental setup.
Theoretical and experimental stroke and axis drift of the fully compliant slider-crank mechanism.
The datum point (zero axis drift position) is selected as follows: the datum axis is the mid-point of highest and lowest positions of the output. Note that, there are differences in error values for the left and right axis drifts. Because, fine manufacturing of the long compliant segments with classical machining process is very hard. The manufacturing error that deteriorates initial parallelism of the long compliant segments causes difference between theoretical and experimental results. If the prototype were produced by plastic injection molding with a metal mold as in a mass production case, we would be able to achieve smaller error values.
It is calculated that the mean absolute error of all data points for the
stroke measurements is 0.13 mm. If this value is compared with the reference
dimension
In the design of compliant mechanisms, there are always some errors due to roughness of PRBM approach. However, PRBM is not used for final dimensioning of a compliant mechanism. It is a tool which is used during the preliminary design stage. Precise dimensioning of compliant mechanism is generally finalized with finite element analysis tools. The design approach in this study is no exception to this issue.
Slider-crank is one of the most commonly used mechanisms in the industry and has numerous applications. In the literature, studies on “partially” compliant slider-crank mechanism can be found where the slider is a rigid joint. There are some fully compliant slider-crank mechanisms available however those are not optimized for large stroke and minimum link rotation. In this paper, a novel design procedure for a fully compliant slider-crank mechanism is proposed. In this design, the prismatic joint is replaced with a compliant parallel-guiding mechanism. The compliant parallel-guiding mechanism is the output of the system that performs curvilinear translation provided that the output is not loaded with large forces. The novelty of this design approach is the slider's large stroke capacity. It is achieved by maximizing the translational motion of the slider while minimizing the relative link rotations thus stresses in compliant links.
A design approach for optimum link proportions of the PRBM of the fully compliant slider-crank mechanism is proposed. Input-output motion relationship of the mechanism and analytical stress values at flexural hinges are determined. In order to generalize the design approach, a procedure is employed and finally a design table that can be used for various tasks is presented. This table will be very beneficial for compliant slider crank designers.
As an example, a mechanism is synthesized using the design table. This mechanism is analyzed with FEA method to verify analytical results. It is shown that the results of the proposed theoretical model and FEA model are consistent. Finally, a single piece prototype is manufactured from polypropylene; thus, it has the advantage of ease of manufacturing when compared with the partially compliant mechanisms. Experimental setup is employed for stroke and axis drift measurements. It is observed that the output displacement and axis drift in the mathematical and the prototype are almost the same. Thus, it is verified that the proposed methods are consistent.
As a result, validity of the design approach is ensured by two ways. It is remarkable that the maximum error between the theoretical model and prototype is less than 0.2 % (for slider stroke). This is a negligible error for a mechanism designed via PRBM approach where generally noticeable errors take place.
We believe this fully compliant slider-crank mechanism design procedure will be beneficial for many designers and may find many applications especially where backlash free design is required.
Data can be made available upon reasonable request. Please contact Engin Tanık (etanik@hacettepe.edu.tr).
The video file of the prototype is available at
ET proposed the original design and defined the methodology. ÇMT applied the methodology, performed analytical solutions, employed FEA, and wrote the paper. VP studied on the theoretical model. YY supervised ÇMT during the study.
The authors declare that they have no conflict of interest.
This paper was edited by Lionel Birglen and reviewed by three anonymous referees.