A NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM

Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496 A NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM Mojtaba Azadi * Depatment of Mechanical Engineeing mojtaba.azadi@ualbeta.ca Univesity of Albeta, Canada Saeed Behzadipou Depatment of Mechanical Engineeing saeed.behzadipou@ualbeta.ca Univesity of Albeta, Canada Simon Guest Depatment of Engineeing sdg@eng.cam.ac.uk Univesity of Cambidge, UK ABSTRACT A novel design of a semi-active vaiable stiffness element is poposed, with possible applications in vibation isolation. Semi-active vibation isolatos usually use vaiable dampes. Howeve, it is known fom the fundamental vibation theoy that a vaiable sping can be fa moe effective in shifting the fequencies of the system and poviding isolation. Geomety change is a common technique fo building vaiable spings, but has disadvantages due to the complexity of the equied mechanism, and slow esponse due to the inetia of moving pats. In the vaiable sping intoduced hee (VS), the stiffness is changed by foce contol in the links which coesponds to infinitesimal movements of the links, and does not need a change of geomety to povide a change of stiffness. This facilitates a fast esponse. The poposed VS is a simple pestessed cable mechanism with an infinitesimal mechanism. Theoetically the level of the pestess in the cables can be used to contol the stiffness fom zeo to a maximum value that is only limited by the stength of the links. In this wok, the statics, kinematics and stability of the VS ae studied, the stiffness is fomulated, and possible configuations of the VS ae found. INTRODUCTION Stiffness elements (spings) and dampes ae the main components in any vibation isolato o absobe. Damping and stiffness ae two essential complements in passive vibation contol solutions. When passive isolatos do not povide the desied vibation contol, semi active isolatos might be consideed. Most of the semi-active isolation solutions ae based on vaiable damping (e.g. magneto heological mateials). Howeve it is known fom fundamental vibation theoy that stiffness change can be moe effective in shifting natual fequencies and hence poviding isolation. Vaiable optimum opeation of an isolato vaiable damping ae consideed in [1,2]. In [1], Jalili explains that the unpopulaity of the available vaiable stiffness sping is mostly due to thei high enegy equiements. He also mentions that low powe designs of vaiable stiffness elements suffe fom limited fequency ange, complex implementation and high cost. The main eason fo these disadvantages is that the stiffness in most of these vaiable stiffness spings changes by changing the geomety [1,3]. Geomety change equies finite motions of mechanical links which limits, the fequency shift that can be achieved, and make these systems slow. An altenative appoach fo stiffness change was intoduced by the authos in [3]. This method is based on foce contol in a pestessed mechanism. Since the foce change coesponds to infinitesimal motions, the esponse is much faste than the above mentioned geomety based solutions.. The esulting sping is a semi-active vaiable stiffness sping in which the stiffness is contolled and changed by the level of the pestess. The concept was fist pesented in [3] using pestessable cable-diven mechanisms in thei singula configuations. In [4], tensegity pism mechanisms wee investigated as the basis fo a vaiable stiffness sping. The pesent wok intoduces a new pin jointed mechanism as a vaiable stiffness sping. This sping (called the VS hee) is a pestessable mechanism whee the mechanism is infinitesimal. This design has advantages ove the tensegity pism due to its simple design and application. * Addess all coespondence to this autho. 1 Copyight 2010 by ASME

The design of the poposed semi-active vaiable stiffness sping (VS) Figue 1 a. VS b. Half of the VS (HVS) Figue 1.a shows the simplest fom of the poposed semiactive Vaiable stiffness Sping (VS) with a minimum numbe of components. It consists of two bas, two cables, uppe and lowe bases, a guide and a platfom in the middle. The lowe base is fixed to the guide, while the uppe base and the platfom have cylindical joints with the guide. The joints between the ods and the bases, and between the ods and the platfom, ae all spheical joints. Cable ends ae assumed to be connected to the cente of the spheical joints. The bas of VS ae consideed as igid links with constant lengths but two cables ae elastic with vaiable lengths. As a esult of consideing one degee of feedom (DoF) fo each cable, the VS finds two DoF. Theefoe, the uppe base can move along the guide and can also otate about the guide espect to the lowe base. Howeve, it will be clea late that because of the esistance of the cables to extension and also because of the effect of pestess level in the VS, tanslational motion and otational motion of the uppe base equie axial load and otational moment, espectively. Fo this eason. the VS acts as a tanslational sping unde axial load along the axis of the guide (Figue 6). Desiably, thoughout the tanslational deflection, the two bases move along the axis of the guide and do not otate. The middle platfom otates and moves along the axial diection. The VS also acts as otational sping unde pue otational moment applied on the bases (Figue 7). In this case, while the uppe base is otating, the height of the VS emains constant. It will be shown that the VS is a pestessed mechanism, and the pestess level in the cables and ods can povide significant contol of the total stiffness. As a esult the VS is a semi active vaiable stiffness sping, with pestess as the contol vaiable. Befoe fomulating the stiffness of the VS, its kinematics and the statics ae fist studied. Because the geomety of the uppe pat of the VS is the mio image of the lowe pat shown in Figue 1.b, the uppe pat and lowe pat have simila statics and kinematics. Fo the sake of claity and simplicity, a geneal configuation of one half of the VS (Figue 2), descibed hee as the HVS, is studied fist and the esults will be then extended to the entie VS. Paametes, and h ae the length of the ba, the length of the cable and the height of HVS, espectively. The Catesian coodinate system is set such that the axis is along the guide and the axis connect the Oigin to the lowe end of the ba ( ). Point ( ) and the lowe end of the cable ( ) ae in the same distance ( ) fom the axis. The uppe ends of the ba and cable ae connected to the platfom at point. The distance between the and the axis is. Angles and ae the angles between the ams of points and and the plane. and ae the unit vectos along the ba () and the cable ( ), espectively. is the unit vecto along the am of the platfom ()., and ae expessed in tems of the paametes as: 1 1 0 cos sin 0 1 cos sin 1 1 cos cos sin sin 0 1 cos cos sin sin (1) (2) cos sin (3) 0 Statics of HVS: Hee, equilibium of the platfom, shown as a fee body in Figue 3, is consideed. and ae the extenal loads applied on the platfom. The intenal foces in the ba and cable ae shown by and, espectively.,, and ae the foces and moments applied on the platfom fom the guide. 2 Copyight 2010 by ASME

2 Z O 2 B 1 Pestessability of HVS: The HVS is patly pestessable if the intenal foces of the links such as and exist while the extenal loads (F and M ) ae zeo. The existence of pestess in cable and ba can be found by checking Eqn.(8). It is undestood fom Eqn.(8) that in a specific configuations, whee h ϕ 1 O 1 Figue 2 a. Right hand HVS b. Top view of ight hand HVS Z B 1 sin sin 0 (9) the 2-by-2 matix becomes ank deficient and hence has a nonzeo null-space thee is a non zeo pestess vecto in equilibium with a zeo extenal load vecto. Thus, the HVS shown in Figue 2 is pestessable when sin sin 0. The null space in these configuations give the elationship between the pestess in the ba ( ) and pestess in the cable (,). The pestess vecto (null space) is found to be M F τ b Figue 3 Fee body diagam of Platfom The static equilibium equations of the platfom become: Σ 0, F τ c F Z M Z 0 Σ 0, M 0 Equations (4) and (5) gives six algebaic equations. By, eliminating the intenal foces and moments acting between the platfom and guide (,, and ), two algebaic equations (6),(7) will be obtained which can be expessed as Eqn.(8). These equations elate the intenal foces of the ba and cable ( and ) to extenal loads ( and ): sin sin (8) (4) (5) (6) (7), (10) Mathematically can be any eal numbe. Howeve, It will be shown that to have stable HVS, the foce in link should be positive (tension). As a esult, is limited to positive values, and since caies only tension, it can be a cable. Based on Eqn.(10), if the pestess of the cable is set to a tension equal to, the pestess of the ba will be compessive with a magnitude of. Discussion on pestessability condition The pestessability condition found above (Eqn.(10)) is a tigonometic equation with seveal sets of solutions. In ode to find the possible configuations, thee possible solutions, shown in Eqn.(11-13), ae discussed hee. 0, : Abitay (11), Right Hand design (RH) (12), Left Hand design (LH) (13) The fist solution Eqn.(11) happens when points and coincide and consequently ba and cable ae identical. In this case the HVS becomes a finite mechanism and moves unde extenal load. As a esult, this configuation cannot cay load and cannot be used as a sping. The second and thid solutions given in Eqn.(12) and Eqn.(13) ae useful pestessable configuations. The second solution, simila to the configuation shown in Figue 2, is called Right Hand (RH). The thid solution, called Left Hand (LH), is shown in Figue 4. These two solutions ae the mio image of each othe with espect to a plane which contains the Z-axis, and passes though the bisecto of (see Figue 5). It will be shown late that these configuations (RH and LH) when the cables ae in tension (links in RH and in LH) have positive stiffness and consequently ae stable. 3 Copyight 2010 by ASME

Z 2 cos 2 cos (15) B 2 O 2 B 2 At, 2 sin 2 (16) 1 ϕ O 1 Figue 4 a. Left Hand HVS b. Top view of left hand HVS Z 4 sin 2 (17) In a left hand HVS the geometical paametes ae found fom the following fomulations. 2 cos 2 cos 2 cos (18) (19) at (20) 2 sin 2 Figue 5 Right hand HVS (Figue 2) is the mio image of Left hand HVS (Figue 4) Kinematics In the VS, the bas ae chosen to be much stiffe than the cables. Fo this eason the bas ae consideed as igid links with constant length ( ) and cables as elastic elements with vaiable lengths ( ). The length of the cable and the height of the HVS at the oiginal configuation ae and, espectively. The oiginal configuation is the configuation in which no extenal load is applied on the VS and the cable and ba ae pestessed. Note that the paametes that descibe the oiginal configuation (pestessed configuation) ae disciminated by subscipt p. When extenal load is applied on the platfom of the HVS, the platfom otates while it moves axially up o down. This motion of the platfom is called a scew motion hee. The scew motion changes the cable length and the height fom and to and. The paametes in a ight hand HVS, ae found fom the followings: 2 cos (14) 4 sin 2 (21) As it was expected, the height and the length of the cable in RH and LH designs ae equal (compae Equations (16),(20) and (17),(21). One of the useful diffeences (esulted fom Eqn.(14) & Eqn.(18) between RH and LH designs is that fo equal axial motion of the platfom, it otates by the same angle but in opposite diections. This useful chaacteistic is used in the design of the VS by using one ight hand HVS and one left hand HVS attached togethe at the platfom. Fo example, in the VS (Figue 6), an axial foce applied to the uppe base moves the uppe base upwads but does not otate it. Conside anothe example, when a moment is applied on the uppe base of the VS while the bottom base is fixed as shown in Figue 7. Unde this condition, the uppe base of VS otates but does not move axially. These pue tanslation o pue otational motions of the uppe base facilitate the use of VS in standad vibation applications. Infinitesimal mechanism of HVS Small motions of the platfom (diffeential motion) aound the oiginal configuation do not change the length of the cable. This is mathematically shown by noting that the deivative of with espect to (Eqn.(15) and Eqn.(19)) vanishes at the oiginal configuations of HVS (When fo RH and fo LH). This implies that in small motions, the elastic stiffness of the cable has minimal impact on the oveall stiffness. In othe wods, the oveall stiffness is highly contollable only though changing the pestess. Small motions, such as this diffeential motion, that do not change the lengths of the links ae known as an Infinitesimal Mechanism o Infinitesimal Flex in the liteatue [5]. 4 Copyight 2010 by ASME

Stiffness of the VS Stiffness of the VS (Figue 1.a) is the elation between the extenal load applied at the uppe base and the displacement of the uppe base, while the lowe base is fixed. Two types of stiffness can be consideed fo the VS: tanslational stiffness when 0 and the base moves axially; and otational stiffness when 0 and the uppe base tuns unde pue moment applied about the axis (Figue 7). The VS (Figue1.a) is a seial aangement of a ight hand and left hand HVS with equal stiffness. As a esult, the stiffness of the VS is half that of the HVS. In the followings, without loss of geneality, only the RH design (Figue 2) and its elated kinematics equations Eqn.(14),(15) ae consideed fo stiffness deivations. Tanslational Stiffness of a ight hand HVS As was shown peviously, a ight hand HVS is a pestessable mechanism. It means at the oiginal configuation of a ight hand HVS (whee, pestess foces and can exist in the ba and the cable. It will be shown that this pestesses geneate a stiffness called pestess o antagonistic stiffness. Unde a pue axial foce along the guide (Figue 6), the platfom moves down to a new equilibium configuation and the foces of the ba and cable change fom to and to and, espectively. At this new equilibium, the tanslational stiffness of the ight hand HVS ( ) is found by: F Z h ϕ 1 2 F Z (22) Figue 6. a. Tanslational VS b. Tanslational ight hand HVS Whee and ae the change in the foce applied along the axis (z) and the coesponding displacement, espectively. Using the equilibium equation, the axial foce ( ) can be expessed in tems of the geometical paametes and the tension of the cable using Eqn.(7). sin sin sin (23) The tension of the cable ( depends on the elasticity of the cable ( ), the cuent length of the cable (Eqn.(15)) and the natual length of the cable : (24) The tanslational stiffness ( ) is then found fom Eq(22) by using chain ule and using equations (14),(15),(23),(24): (25) whee and ae : 1 cos 2 cos, temed the tanslational cable foce stiffness, is a function of the cable foce ( ) and change of the geomety., temed the tanslational elastic stiffness, is a function of elasticity of the cables ( ) and the geomety. It is notable that at oiginal configuation of a ight hand HVS, whee, tanslational elastic stiffness ( ) is zeo( Eqn. (29)). This means that at this configuation the stiffness is not a function of elasticity of cable. It only depends on the tanslational cable foce stiffness (. In this case, since the cable foce is only the pestess ( ), this stiffness is a function of geomety and pestess and is called the pestess stiffness o antagonistic stiffness (Eqn.(28)). Following the same agument on the left hand HVS (Equations (18),(21)), simila esults ae found. The elastic stiffness of the left hand HVS at its oiginal configuation ( ) is zeo. In addition, the pestess stiffness of a left hand HVS is the only souce of stiffness at its oiginal configuations. It is also equal to the pestess stiffness of a ight hand HVS at its oiginal configuation (Eqn.(28)). 2 sin 2 cos 2 (28) =0 (29) (26) (27) In summay, at oiginal configuation the stiffness of a HVS comes only fom pestess stiffness ( ), and this depends on angle and the cable pestess. This impotant esult indicates that HVS can be designed as a vaiable stiffness sping in which the stiffness changes with cable pestess level. 5 Copyight 2010 by ASME

Rotational Stiffness of a ight hand HVS Unde a pue moment ( 0) about the axial axis, the platfom tuns to a new equilibium configuation and the foce of the ba and cables change fom to and to and, espectively. At this new equilibium, the otational stiffness of the ight hand HVS (Figue 7.b) is found by: M Z h ϕ 1 Figue 7. a. Rotational VS b. Rotational ight hand HVS Fist, the pue moment ( ) is found fom Eqn.(7): sin sin 2 M Z (30) (31) Then by using Equations.(15) and (24), and ae expessed in tems of vaiable. Applying Eqn.(31) to the esult gives the otational stiffness of the ight hand HVS ( ): (32) whee and ae : cos cos 2 (33) (34), temed the Rotational cable foce stiffness, is a function of the cable foce ( ) and change of the geomety., temed the Rotational elastic stiffness, is a function of elasticity of the cables ( ) and the geomety. It is notable that at oiginal configuation, whee, otational elastic stiffness ( ) is zeo and the stiffness oiginates only fom the tanslational cable foce stiffness (. Unde this condition, since the cable foce is only the pestess of the cable ( ), the stiffness is called the pestess stiffness/ antagonistic stiffness. 2 sin 2 (35) =0 (36) Tanslational and Rotational Stiffness of a VS The tanslational stiffness and otational stiffness of the VS ae half of the tanslational stiffness ( ) and otational stiffness ( ) of the left hand side HVS, espectively. The big advantage of the VS ove the HVS is the simple motion of the uppe base in the VS unde pue axial foce o pue otational moment. In the HVS, the motion of the platfom is always a combined otation and tanslation, while in the VS, the uppe base only moves along the axis unde tanslational foce and only tuns unde pue otational moment. On the othe hand, the HVS has a moe compact size and highe stiffness, which may be moe favoable in paticula applications. In vibation contol applications, VS seves as a vaiable stiffness pat. The dead weight of the mass (if exists) can be balanced by the low damping and low stiffness passive mount (e.g. ubbe mount). Because of balancing the dead weight, the VS will wok at the oiginal configuation. (see case study of [3] as an example) As a esult, among the seveal equation deived above, only two shot equations need to be used fo design. These equations ae the tanslational and otational pestess stiffness of the VS (Eqn.(28) and Eqn.(35)). Stability The HVS and the VS ae stable as long as they have positive stiffness and cables ae in tension. At the oiginal configuation (no extenal loads), as long as the pestess of the cables ae tensile ( 0), the HVS and the VS have positive stiffness (see Equations (28),(35)) and consequently they ae stable. Conside an HVS, when extenal loads ae applied and the configuation of the HVS changes fom the oiginal configuation (e.g. in ight hand HVS ) to a new equilibium (new ) found fom Eqn.(7). Thoughout the change of configuation unde extenal loads the cable foce changes fom to. The stiffness at this new equilibium is found fom geneal stiffness equations: Eqn.(25) fo ight hand HVS and fom Eqn.(32) fo left hand HVS. If at this new equilibium, stiffness and the cable foce ae both positive, the HVS is stable. If the cable foce becomes negative, the cable becomes slack and the mechanism collapses. Inceasing the pestess of the cable postpone the slackness and incease the woking ange of the HVS and similaly the woking ange of the VS. 6 Copyight 2010 by ASME

Summay It was shown that the poposed mechanism (VS) is a foce contolled vaiable stiffness sping with simple tanslational and otational motion. The VS consists of two pestessed mechanisms (called HVS) assembled in a seial configuation. These two HVS s ae mio images of one anothe. They ae pestessed mechanisms with one scew-like fist ode infinitesimal mechanism. The stiffness of the VS can be contolled by the tension of the cables. At the oiginal configuation (unloaded configuation) of the VS, whee the VS has zeo elastic stiffness, the stiffness of VS is meely detemined by the pestess level in the cables and bas. Theefoe foce contol on these elements, implemented by foce actuatos, effectively change the stiffness. At oiginal configuation the VS is stable unde extenal loads, as long as the stiffness is positive and the cables ae in tension. Pestessing the cables inceases the stiffness, and inceases the stable woking ange of the VS. Refeences [1]. N. Jalili, 2002, A compaative study and analysis of semi-active vibation-contol systems, Jounal of Vibation and Acoustics,124(4), pp. 593 605. [2]. J. Sun, M. Jolly, M. Nois, 1995, Passive, adaptive and active tuned vibation absobes a suvey, 50th annivesay of the design engineeing division, A Special Combined Issue of the Jounal of Mechanical Design and the Jounal of Vibation and Acoustics, 117(3B), pp. 234 242. [3]. Azadi, M., Behzadipou, S., Faulkne, G., 2009, Antagonistic Vaiable Stiffness Elements, Mechanism and Machine Theoy Jounal, 44(9), pp. 1746-1758. [4]. Azadi, M., Behzadipou, S., Faulkne, G., 2009, Vaiable Stiffness Tensegity Pism Sping, Poceeding of 2009 ASME Intenational Design Engineeing Technical Confeences, DETC2009-87407. [5]. C.R. Calladine, S. Pellegino, 1991, Fist-ode Infinitesimal Mechanisms, Intenational Jounal of Solids and Stuctues, 27(4), pp. 505-515. 7 Copyight 2010 by ASME