喜歡就充值下載吧，，資源目錄下展示的全都有，，下載后全都有，dwg格式的為CAD圖紙，有疑問咨詢QQ：414951605 或1304139763 Liand Hao Chin. J. Mech. Eng. (2019) 32:54 https:/doi.org/10.1186/s10033-019-0369-zORIGINAL ARTICLEOn Generating Expected Kinetostatic Nonlinear Stiffness Characteristics bytheKinematic Limb-Singularity ofaCrank-Slider Linkage withSpringsBaokun Li1 and Guangbo Hao2*Abstract Being different from avoidance of singularity of closed-loop linkages, this paper employs the kinematic singularity to construct compliant mechanisms with expected nonlinear stiffness characteristics to enrich the methods of compli-ant mechanisms synthesis. The theory for generating kinetostatic nonlinear stiffness characteristic by the kinematic limb-singularity of a crank-slider linkage is developed. Based on the principle of virtual work, the kinetostatic model of the crank-linkage with springs is established. The influences of spring stiffness on the toque-position angle relation are analyzed. It indicates that corresponding spring stiffness may generate one of four types of nonlinear stiffness characteristics including the bi-stable, local negative-stiffness, zero-stiffness or positive-stiffness when the mechanism works around the kinematic limb-singularity position. Thus the compliant mechanism with an expected stiffness characteristic can be constructed by employing the pseudo rigid-body model of the mechanism whose joints or links are replaced by corresponding flexures. Finally, a tri-symmetrical constant-torque compliant mechanism is fabricated, where the curve of torque-position angle is obtained by an experimental testing. The measurement indicates that the compliant mechanism can generate a nearly constant-torque zone.Keywords: Kinematic singularity, Mechanism with springs, Kinetostatic model, Nonlinear stiffness The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:/creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.1 IntroductionA mechanism with springs is defined as a rigid-body linkage whose joints are placed springs. For this type of mechanisms, the kinetostatic driving force/torque of this type of mechanisms is nonlinear with respect to the position parameter. The nonlinear relation between the driving force/torque and the position parameter is called kinetostatic nonlinear stiffness characteristic. The mech-anism with springs possessing this characteristic can be applied in constant force mechanism 1, vibration isola-tor 2 and gravity balancer 3. The mechanism attached springs is often used in the type synthesis of compli-ant mechanisms based on the rigid-body replacement method and the compliant mechanisms analysis based on the pseudo-rigid-body model 46. Compliant mecha-nisms can be fabricated in monolithic and are applied in many applications needing high precision because of absence of backlash and friction 7, such as energy har-vester based on buckled beam 8, 9, micro-switch 10 and high accurate driver 11. However, the buckled beam only generates bi-stability but other nonlinear stiff-ness characteristics. Moreover, the mechanical model of bi-stable buckled beam is very complicated 12, 13. The four-bar linkage with placed springs can be used to design compliant mechanisms with bi-stable behavior by employing pseudo-rigid-body replacement 14, which develops the configuration of the bi-stable mechanism.When the rigid-body replacement method is use to synthesize compliant mechanisms processing corre-sponding performance, designers should grasp series of performances of the rigid-body linkage. Thus one should have much experience on linkage design and Open AccessChinese Journal of Mechanical Engineering*Correspondence: G.Haoucc.ie 2 School of Engineering, University College Cork, Cork T12K8AF, IrelandFull list of author information is available at the end of the articlePage 2 of 16Liand Hao Chin. J. Mech. Eng. (2019) 32:54 performance analysis. Therefore, it is meaningful that some common attributes are used to construct compliant mechanisms with nonlinear stiffness characteristic.Kinematic singularity which is a basic property of link-ages affects the performance of linkages seriously, so many scholars pay much attention on singularity distri-bution, singularity property identification and singular-ity avoidance 15, 16. However, kinematic singularity has two sides, and can be used to construct new types of devices. Kinematic singularity of the spatial parallel link-age whose links are connected by universal joints are used to construct several types of reconfigurable parallel mech-anisms 17. When parallel mechanisms work near the singularity, they are sensitive to external load. This prop-erty is applied to design the force sensors 18, 19. A new compliant mechanism with negative-stiffness characteris-tic is synthesized by using kinematic singularity of a four-bar linkage 20. The planar parallelogram linkage when the two cranks are collinear is used to construct a type of reconfigurable compliant gripper by applying rigid-body replacement method 21. A new medical device is designed by using the property that a parallel mechanism obtains an additional freedom when it is singular 22.In this paper, by the crank-slider mechanism with springs as an example, the kinematic limb-singularity which is a common property of rigid-body linkages, is used to con-struct the kinetostatic nonlinear stiffness characteristic. The rest of the paper is organized as follows: Section2 addresses the kinetostatic model of the mechanism and Section 3 classifies nonlinear stiffness characteristics as four types. Section4 analyzes the influences of spring stiff-ness on the nonlinear stiffness characteristics generated by the mechanism when moves from nonsingular position and passes the kinematic limb-singularity position. Sec-tion5 indicates that the mechanism only produces posi-tive-stiffness characteristic when moves from the kinematic limb-singularity position to nonsingular position. Section6 describes the approach by creating an expected zero-stiff-ness (constant-torque) characteristic of the mechanism working around the kinematic limb-singularity position. In Section7, design of a nonlinear compliant mechanism is further discussed and is validated by the experimental test-ing. Finally, Section8 draws some important conclusions.2 Kinetostatic Model oftheMechanismFigure 1 shows the schematic of the crank-slider mechanism with springs. Crank AB rotates about pin joint A in anticlockwise and drives the slider to moves along the horizontal line, where link AB and slider are connected by coupler BC. Three pin joints are placed torsional springs whose stiffness is KRA, KRB and KRC, respectively. Prismatic joint C is added extension spring whose stiffness is KPC.The Cartesian coordinates system, O-xyz, is attached on the base, where origin O is fixed on point A, the pos-itive direction of x-axis points to the horizontal right, the positive direction of y-axis is vertically up, and z-axis is determined by the right-hand rule.Vectors AB and BC are defined by r1 and r2, respec-tively. Projects of vector position C on the x-axis and y-axis with respect to the frame O-xyz are defined by r3 and e, respectively. Scalars r1 and r2 are lengths of links AB and BC, respectively. Scalars r3 and e are the coor-dinates of point C on the x-axis and y-axis, respectively. Link-length, r1 and r2, and offset, e, should satisfyso as to allow the mechanism to pass through the right limiting position, which is called the kinematic limb-sin-gularity and occurs when the crank and coupler are along the same line.Here we suppose that there is no friction and clear-ance between any two links connected by a kinematic pair. Moreover, we only discuss the kinetostatic model of the mechanism during the motion rather than con-sidering any inertial force/torque and gravity caused by links quality.The driving torque applied on link AB is set aswhere vector k is the unit vector of z-axis (vectors i and j are unit vectors of x-axis and y-axis, respectively). Torque vector Td is along the z-axis, scalar |Td| is the magni-tude of driving torque Td, where Td 0 indicates Td is along the positive direction of z-axis and Td 0 corre-sponds to direction of Td pointing to negative z-axis.The angular displacement of pin joint A iswhere A is the rotation angle of x-axis to link AB and indicates the input position angle of link AB, A0 cor-responds to the initial angle. In this paper, value of A allows no spring lose efficacy.Here we consider A as the general coordinate of the mechanism. Thus the virtual angular displacement of joint A is(1)(r1+ r2) e 0,Td= dU?dA=0,dTd?dA=d2U?d2A 0.(13a)A= arcsiner1 r2,(13b)A= arcsiner1+ r2.Td /Ua bcTmaxTminUmaxUmin2Td-AU-AUmin1Stable positionUnstable positionA0Stable positionFigure2 Torque/energy versus position anglesPage 5 of 16Liand Hao Chin. J. Mech. Eng. (2019) 32:54 Equation (7a) can lead to the following expressionEquation (14) indicates that when the mechanism locates at the two limiting positions represented by Equa-tions (13a) and (13b), the following expression is truewhich indicates that the ratio between the output velocity and the input velocity is zero and is called the kinematic limb-singularity 24.Figure3 shows the motion of the mechanism which works around the right limiting position which is also one of the two kinematic limb-singularity positions. The mechanism moves from the initial non-singular position with no deflected springs (Figure3(a), passes the kin-ematic limb-singularity position (Figure3(b) and then arrives at the end non-singular position (Figure 3(c). During the motion as Figure 3 shows, the potential energy of the spring placed at joint C increases from zero to the maximum and then falls to zero. Thus if the stiffness of the torsional springs are not too large, the potential energy of the mechanism may have one local maximum and two local minimums, which correspond to the unstable position (b as shown in Figure3) and two stable positions (a and c as Figure3 shows). This kine-tostatic nonlinear stiffness characteristic is called the bi-stable characteristic.(14)dr3?dA= r1sinA b?a.(15)dr3?dA= 0If and only if the pin joints are attached springs, the mechanism does not exhibit the phenomenon that the potential energy increases firstly and then decreases, which means that there is no maximal potential energy during the motion because the pint joints rotate in one direction during the motion. Thus, the mechanism only produces the positive-stiffness characteristic but does not generate the bi-stable characteristic.According to Eqs. (10) and (11), the driving torque is to resist the all of the force/torque caused by all of the springs and the total potential energy of the mecha-nism is the sum of the potential energy of each spring. In other words, the mechanism may produce four types of kinetostatic nonlinear stiffness characteristics which are determined by the stiffness of springs placed at the joints.Four nonlinear stiffness characteristics including bi-stable characteristics, local negative-stiffness char-acteristic, local zero-stiffness characteristic and posi-tive-stiffness characteristic are shown in Figure4, which describes the driving torque varies with the input posi-tion angle, A. Unlike a generic elastic spring or structure, the driving force/torque applied on the mechanism with springs does not obey the Hookes law. If the mechanism is carried out the motion as Figure3(a)3(c) shows, it may produce four types of nonlinear stiffness character-istics depicted by Figure4(a)(d), which are addressed as follows: (1) Figure 4(a) describes the bi-stable characteristic which includes three domains, where domains i and iii are positive-stiffness and domain ii is nega-tive-stiffness. As Tdmax Tdmin 0. Thus we can conclude that the mechanism is located at the local minimal energy point when A = A1 and A = A3, respectively. According to Ref.28, the mechanism is in equilibrium when A = A1 and A = A3 corresponding to a and c as Figure2 shows, respectively.Differentiating Eq. (16) with respect to A yieldsC1= 4r31cosA0sin3A0 10er21cosA0sin2A0+ 8e2r1cosA0sinA0 r31cosA0sinA0 3r1r22cosA0sinA0 2e3cosA0+ 2er21cosA0+ 2er22cosA0,C2= a0?4r21sin3A0 6er1sin2A0+ 2e2sinA03r21sinA0 r22sinA0+ 4er1?,C3= 4r31sin4A0+ 10er21sin3A0 8e2r1sin2A0+ 5r31sin2A0+ 3r1r22sin2A0+ 2e3sinA0 8er21sinA0 er22sinA0+ 4e2r1 r31 3r1r22,C4= a0?4r21sin2A0cosA0 6er1sinA0cosA0+2e2cosA0 3r21cosA0 r22cosA0?.(19)r1cosA+ a r1cosA0 a0= 0,(20)r1sinA b/a = 0.(21)dTddA=d2Ud2A= KPC(r1sinA b/a)2+ KPC(r1cosA+ a r1cosA0 a0)?r1cosA?r21sinAcosA+ er1sinA?ab2?a3?.If the mechanism is located at A = A2, which is the solution of Eq. (20), thenCombing Eqs. (5a), (22b) and (22c) obtainsAccording to Eqs. (21), (22a) and (22d), the following equation can be obtainedEquation (17) can lead toThus we can conclude that the mechanism is in unsta-ble equilibrium when located at A = A2 corresponding to b as shown in Figure2.When the geometry parameters are given as r1 = 10 cm, r2 = 50 cm and e = 3 cm, and the initial input position angle is set to A0 = 5, the driving torque and potential energy variations versus the input position angle is shown in Figure5. In this paper, the unit of translational spring and the torsional spring is N/cm and Ncm/(), respectively. It should be pointed out that the initial input position angle should satisfy(22a)(r3 r30)|A=A2= (r1cosA+ a r1cosA0 a0)|A=A2 0,(22b)?r21sinAcosA+ er1sinA?A=A2 0,(22c)cosA|A=A2 0.(22d)?r1cosA?r21sinAcosA+ er1sinA?ab2?a3?A=A2 0.(23a)dTddA?A=A2=d2Ud2A?A=A2 0.Page 8 of 16Liand Hao Chin. J. Mech. Eng. (2019) 32:54 so as to allow the mechanism to pass the right kinematic limb-singularity position with starting from a non-singu-lar position.Figure 5 indicates that when KRA = KRB = KRC = 0 and KPC 0, the kinematic limb-singularity position is in the unstable equilibrium point. Moreover, it can be shown that the increment of the translational spring stiffness increases both of the values of driving torque in positive direction and in negative direction. The potential energy is also increased by the increment of the translational spring stiffness.4.1.2 Nonlinear Stiffness Characteristics When KRB = KRC = 0, KPC = 0, andKRA 0Substitution of the springs stiffness into Eq. (10) obtains the driving torque asarcsiner1 r2 A0 0.(26)U =12KRA(A A0)2.(27)Td= KRB?A arcsinr1sinA er2+A0+ arcsinr1sinA0 er2?1 r1cosA?a?.(28)U =12KRB?A arcsinr1sinA er2+A0+ arcsinr1sinA0 er2?2.KKKKKKb Potential energy versus input position angleInput position angle ()Position angle / Input position angle A ()a Driving torque versus input position angleDriving torque Td (Ncm)Potential energy U (Ncm)Figure5 Bi-stable characteristic when KRA = KRB = KRC = 0 and KPC 0Page 9 of 16Liand Hao Chin. J. Mech. Eng. (2019) 32:54 4.1.4 Nonlinear Stiffness Characteristics When KRA = KRB = 0, KPC = 0, andKRC 0The driving force can be simplified asConsidering to Eq. (6), the physical meaning of Eq. (29) is that the driving torque is to resist the torque due to the torsional spring added at the pin joint C.Substitution the springs stiffness into Eq. (11) obtains the potential energy as follows(29)Td= KRC?arcsinr1sinA er2arcsinr1sinA0 er2? r1cosA?a.(30)U =12KRC?arcsinr1sinA er2arcsinr1sinA er2?2.When r1 = 10cm, r2 = 50cm, e = 3cm, and A0 = 5, Figure8 depicts the driving torque and potential energy represented by Eqs. (29) and (30), respectively.Figure 8 shows that the mechanism produces the positive-stiffness characteristic when the pin joint C is attached a torsional spring exclusively.In addition, when KRA = KRB = KRC, Figures6 through 8 indicates that the stiffness of the driving torque curve caused by KRB is the greatest, the stiffness due to KRA is the second largest and the stiffness due to KRC is the lowest.4.2 Influences ofSpring Stiffness ontheNonlinear Stiffness CharacteristicsSection4.1 illustrates that KPC makes the mechanism to generate the bi-stable characteristic including the nega-tive domain and KRA, KRB or KRC only allow the mecha-nism to exhibit the positive-stiffness characteristic. The total torque can be obtained by linear superposition of the torque due to KRA, KRB, KRC and KPC. Therefore, an expected nonlinear stiffness characteristic may be con-structed by designing different values of KRA, KRB, KRC and KPC on the condition of KPC 0.When r1 = 10cm, r2 = 50cm, e = 3cm, A0 = 5, and KPC = 1N/cm, the nonlinear stiffness characteristics of the mechanism for different values of KRA, KRB and KRC is described by Figure9, where KRA = KRB = KRA,B.Figure9 indicates that one nonlinear characteristic can transformed to another one when the torsional springs stiffness, KRA, KRB and KRC, are set to different values when the translational spring, KPC, is nonzero. For a given translational spring stiffness, when the torsional spring stiffness is small, the mechanism exhibits the bi-stable characteristic. Increment of torsional springs stiffness delays the unstable equilibrium position and advances the second stable point. The bi-stable characteristic may degenerate to the local negative-stiffness characteristic and even the positive-stiffness characteristic with large increment of torsional springs stiffness.In addition, existence of local maximum potential energy point is the precondition of the bi-stable char-acteristic. When the torque curve has local negative-stiffness domain but no maximum potential energy point, the mechanism does not exhibit the snap-through phenomenon.When r1 = 10cm, r2 = 50cm, e = 3cm, A0 = 5 and KPC = 1N/cm, Figure10 depicts the nonlinear stiffness characteristic of the mechanism when one torsional spring stiffness is zero exclusively.Figure10 shows that when KPC is given as a constant, KRB has the greatest effect, KRA has the second greatest effect, and KRC has the smallest effect on the nonlinear stiffness characteristic of the mechanism, respectively.Figure6 Stiffness characteristics for different values of KRA when KRB = KRC = 0, and KPC = 0Page 10 of 16Liand Hao Chin. J. Mech. Eng. (2019) 32:54 5 Nonlinear Stiffness Characteristic withInitial LimbSingularity PositionSection4 shows that the mechanism may generate the positive-stiffness when torsional spring stiffness is great enough. Section5 manly discusses another approach for producing the positive-stiffness characteristic by making the mechanism to move from the right kinematic limb-singularity position (Figure3(b) to the nonsingular posi-tion (Figure3(c).The torque versus position angle of the mechanism starting from the right limiting kinematic-singularity position can be derived by substitutinginto Eq. (10), and is not detailed here.Within this situation, as the translational spring placed at prismatic joint C moves in one-direction, the potential energy increases with the increment of the input rota-tion angle, and does not exist the local minimum except the initial position. Thus the bi-stable characteristic does A0= arcsiner1+ r2not exist caused by KPC. For the three torsional springs attached at the three pin joints, the potential energy only increase. Therefore, the total potential energy increases during the motion of the mechanism, and the mechanism only exhibits the positive-stiffness characteristic.When r1 = 10cm, r2 = 50cm, e = 3cm, the torque curve versus the position angle is described by Figure11.Figure 11 verifies that the torque curve exhibits the positive-stiffness characteristic caused each spring. Thus the total torque caused by all of the springs does exhibit the positive-stiffness.6 Expected Nonlinear Stiffness Characteristic CreationAccording to Sections4 and 5, the mechanism only gener-ates the positive-stiffness characteristic when the mecha-nism moves from the kinematic limb-singularity position with