Flexure oscillators are promising time bases thanks to their high quality factor and monolithic design compatible with microfabrication. In mechanical watchmaking, they could advantageously replace the traditional balance and hairspring oscillator, leading to improvements in timekeeping accuracy, autonomy and assembly. As MEMS oscillators, their performance can rival that of the well-established quartz oscillator. However, their inherent nonlinear elastic behavior can introduce a variation of their frequency with ampli- tude called isochronism defect, a major obstacle to accurate timekeeping in mechanical watches. Previous research has focused on addressing this issue by controlling the elastic properties of ﬂexure oscillators. Y et, these oscillators exhibit other amplitude-related fre- quency variations caused by changes of inertia with amplitude. In this article, we not only improve existing models by taking into account inertia effects but also present a new way of using them to adjust the isochronism defect. This results in a better understanding of ﬂexure oscillators and an alternative way of tuning isochronism by acting on inertia instead of stiffness. This also opens the door to new promising architectures such as the new Rotation-Dilation Coupled Oscillator (RDCO) whose symmetry has the advantage of min- imizing the inﬂuence of linear accelerations on its frequency (the other major limitation of ﬂexure oscillators). We derive analytical models for the isochronism of this oscillator, show a dimensioning with compensating inertia and stiffness variations and present a practical method for post-fabrication isochronism tuning. The models are validated by FEM and mock-ups serve as preliminary proof-of-concept. 1 INTRODUCTION 1.1 Flexure mechanical watch time bases The Swiss watchmaking industry, with an annual export of about 20 billion Swiss francs, is the country’s third largest exporter and accounts for approximately 1:5%of its GDP [1].1Mechanical watches constitute about 80%of these exports (in value) and rely substantially on innovation push- ing the limits of mechanical timekeepers to maintain their position. However, despite concerted efforts by the industry, the accuracy of the best mechanical watches seems to have reached a plateau of a few seconds per day. The general consensus in horology is that the quality factor of the time base, a dimensionless number that characterizes the damping of an oscillator, is the key limiting factor to accuracy [2,3,4]. Y et, the friction in the bearing of the traditional mechanical watch oscillator, the balance and hairspring depicted in Fig. 1, drastically limits its quality factor.2This fact, together with the recent spreading in the watchmaking industry of high-technology materials and techniques such as silicon components manufactured by deep reactive ion etching [6,7,8], has motivated the search for new time bases [9, 10]. Flexure oscillators are a promising replacement for the balance and hairspring since their motion is only guided by the deformation of elastic ele- ments, thus eliminating contact friction [11, 12], paving the way to a signiﬁcant increase in quality factor and a new level of timekeeping accuracy.3 1More information available on Switzerland’s ofﬁcial website: www.eda.admin.ch/aboutswitzerland/en/ home/wirtschaft/taetigkeitsgebiete/uhrenindustrie.html 2Air friction also has a signiﬁcant inﬂuence but eliminating it only increases the quality factor by about 50%: it was increased from 300 to 450 for a balance and hairspring in 99.8% vacuum [5]. 3Monolithic fabrication using materials with minimal internal friction such as monocrystalline silicon also contributes to increasing the quality factor [13]. Additionally, increasing the quality factor reduces power consumption, leading to greater watch autonomy [10].2 Fig. 1: Classical mechanical watch balance and hairspring oscillator [14]. The introduction of ﬂexure oscillators in mechanical watches comes with new challenges that need to be overcome before they can be implemented. First, the nonlinear elastic behavior of ﬂexures introduces a dependence of oscillation period on amplitude, i.e., an isochronism defect. Second, the deviation of ﬂexures from the motion of ideal linkages can result in changes in the mass distribution of the time base as it oscillates. This causes an isochronism defect (through amplitude-dependent inertia variation) and a contribution of gravity to the effective stiffness of the oscillator (and hence an effect on its period). These directly affect timekeeping accuracy. Other external factors can also affect the period such as temperature, magnetic ﬁelds and shocks; these effects which can be addressed with existing materials and techniques [15, 16] are not treated in the present article. Flexure oscillators also have applications as MEMS time bases for electronic devices, where their frequency stability and potential for miniaturization and batch production can compete with the well-established quartz oscillator [17, 18]. The main challenge for MEMS time bases is, as for mechanical watch oscillators, to have a stable frequency regardless of external perturbations, except that the main sources of perturbation are different. For instance, isochronism defect is not a signiﬁcant issue for MEMS oscillators due to their small amplitudes whereas temperature is [19,20]. For this reason, this paper focuses on mechanical watch applications. Note that linear accelerations (such as gravity) are important sources of perturbation for both applications, hence the symmetry of the architectures presented here also has potential for MEMS time bases.4 1.2 State of the art In our previous research on ﬂexure oscillators, we developed design principles that allow to minimize gravity effects [15, 22] and introduced the concept of isochronism tuning by changing second order stiffness properties of the oscillator [16]. We showed that we could compensate isochronism defects by adjusting the stiffness of ﬂexures whose motion is of second order of the main oscillation, that is, the same order as the isochronism defect, see Eq. (6). The fact that this is is done by changing oscillator properties without affecting any other crucial characteristic of the pivot (such as such as nominal stiffness, gravity sensitivity and parasitic center shift) is called intrinsic isochronism tuning [16] and is superior to existing solutions. Indeed, the ﬁrst ﬂexure pivot oscillator to be introduced in a watch, the 2014 “Genequand System” [9, 10] used an external mechanism to tune its isochronism [23]; other oscillators based on the crossed ﬂexure pivot [24] use special values for the point at which the leaf springs cross and the angle between them to reach isochronism [22,25,26,27,28] which, however, does not qualify as isochronism tuning since changing these parameters affects other crucial properties of the pivot [16,22,29,30]. 4For instance, Ref. [20] presents a kHz MEMS ﬂexure time base using a symmetrized tuning fork which is known to minimize the effect of linear accelerations [21]. The MHz MEMS oscillator in Ref. [19] also presents a similar symmetry. 3 1.3 New concept of inertial isochronism tuning In previous research, we noticed that the deviation of ﬂexures from the motion of ideal linkages resulted in changes in the mass distribution of our time bases as they oscillate, thus affecting their isochronism. For instance, the “GIFP” oscillator [15, 22] has a parasitic tilt of its rotation axis and the “co-RCC” [15, 16, 31] has an inward motion of its intermediate bodies. These effects were neglected, assuming that they were small and could be compensated by acting on the restoring torque nonlinearity of the ﬂexures [16]. Similar inertia variations are also present in other existing ﬂexure time bases, for instance in Refs. [32,33,34]. In the present article, we show how to take into account the effect of inertia variations in our previous isochronism model. This has several advantages. First, this completes the analysis of the isochronism of ﬂexure time bases and allows to take into account inertia effects into the design instead of compensating them a posteriori. Second, this widens the spectrum of potential new ﬂexure time bases by including architectures with signiﬁcant inertia variations. For instance, we introduce the new Rotation-Dilation Coupled Oscillator (RDCO) family [35] whose symmetry has the advantage of inherently minimizing the inﬂuence of linear accelerations on oscillation frequency: the other major obstacle to the implementation of ﬂexure time bases [21, 22]. Third, we present a new way of tuning isochronism, by acting on the inertia variation of the oscillator instead of its stiffness variation, as was previously done [9, 16]. This concept is new and is the main contribution of this article. As a result, new practical methods of tuning isochronism post- fabrication emerge that are potentially simpler than the ones based on the stiffness. For instance, displacing masses or removing mass by laser ablation [36] is a much less sensitive operation than removing matter from slender ﬂexures to affect their stiffness, as we suggested in Ref. [16]. Some of these results were announced in the ﬁrst author’s PhD thesis [15]. OP ABSlider PivotInertial body Extra DOF Connecting rodOP(0) P() Fig. 2: Kinematic diagram of the RDCO using ideal joints in nominal position (left) and rotated (right). Remark 1.1. It is well known that inertia variations can be used to tune the nominal frequency of rotational oscillators. This is for example implemented on balance and hairspring oscillators by 4 changing the position of inertia blocks or screws placed on the balance [37,38]. Inertia variations with temperature have also been used to minimize the effect of temperature on frequency, for example with bimetallic balances [37, pp.259-262]. These concepts are however different form the one proposed in this article as they results in a modiﬁed inertia that is not function of oscillation amplitude and therefore does not affect isochronism. Remark 1.2. Our concept is analogous to the Huygens’ 1657 theoretical isochronism corrector for the pendulum depicted in Fig. 3 [39, 40]: he proposed to compensate the isochronism defect resulting from the nonlinear relation between gravity restoring torque and angular position of the pendulum by replacing the rod with a ﬂexible cord that unwinds off a cycloid. This essentially corresponds to changing the active length Lof the pendulum as it swings. The analogy with changing the inertia Jof a rotational oscillator as it rotates can be seen from the formulas for angular frequency, namelyp k=Jfor a harmonic rotational oscillator of stiffness kandp g=Lfor an ideal pendulum under gravity g. Pushing the analogy further, tuning the isochronism defect by acting on the inertia variation would correspond to tuning the curvature of Huygens’ cycloidal cheeks. Fig. 3: Huygens’ isochronism correction for the pendulum with cycloidal cheeks changing its active length as it oscillates. 1.4 Structure of the paper Section 2 presents the new RDCO rotational oscillators family and describes its kinematics. Examples of ﬂexure implementations are given and the design is validated qualitatively on mock- ups. Section 3 derives analytical models for the inertia variation and restoring torque nonlinearity (i.e., stiffness variation) of the RDCO and describes their effect on the isochronism of the oscillator. Section 4 validates the concepts of this paper by designing a ﬂexure implementation where the inertia and stiffness variations compensate each other to minimize the isochronism defect. We show that the isochronism of the RDCO can be tuned by controlling either its stiffness variation in Sec. 4.2.1 or its inertia variation in Sec. 4.2.2. The stiffness variation is controlled by adjusting the stiffness of ﬂexures whose deﬂection is of second order of the rotation angle and the inertia variation is controlled by displacing tuning masses at a constant radius from the center of the system. These results are validated by Finite Element Method (FEM). 5 2 DESIGN AND KINEMATICS Figure 2 shows a kinematic diagram of the RDCO. The mechanism consists of n>3inertial bodies linked to each other by sliders (prismatic joints) to form a loop ( n= 3in Fig. 2). Each rigid body is linked to the ground by a connecting rod with a pivot at both extremities, one connected to the ground (pivot A) and the other to the inertial body (pivot B). The joints are placed such that they have rotational symmetry of order nwith respect to the center Oof the system. An extra degree-of-freedom (DOF) is added by allowing a pivoting motion in one of the sliders, see Fig. 2. The sole purpose of this extra DOF is to avoid overconstraining the system; it is not activated during the motion of the system. Gr ¨ubler’s formula for planar linkages [41] gives a mobility M= 1 for the RDCO in Fig. 2 with N= 8bodies and j= 10 joints with each a DOF fi= 1: M=jX i=1fi3(jN+ 1) = 103(108 + 1) = 1 (1) Knowing that there are no internal degrees-of-freedom, this shows that the 1-DOF motion depicted in in Fig. 2 is obtained without overconstraints (for a planar mechanism). 2.1 Flexure implementation For this system to act as a mechanical oscillator, spring components must be added. This happens naturally in the ﬂexure implementation. Any combination of ﬂexures presenting equivalent properties to the ideal kinematic diagram of Fig. 2 can be used. Figure 4 shows an example where truncated circular notch ﬂexure hinges are used for pivots AandB[12, Sec. 3.5.10] and parallel leaf springs are used for the sliders. Parallel leaf springs actually have a parabolic motion but it is assumed that they closely approximate a linear motion for the small deformations considered here [12, Sec. 4.1]. In this implementation, two orthogonal axes of symmetry were chosen instead of a rotational symmetry (so as to have symmetric functions with respect to equilibrium) but the design still corresponds to the ideal kinematic diagram of Fig. 2 with n= 4inertial bodies. More examples of ﬂexure implementation are given in Sec. 2.3 and Ref. [15, Sec. 3.3]. Note that the extra DOF is omitted in most designs, assuming that there is enough ﬂexibility in the system to release the overconstraint. 2.2 Rotation-dilation coupling The kinematic behavior of the system is determined by the dimensionless ratio =d=L, where dis the distance from the center Oof the system to the axis of pivot AandLis the distance from the axis of pivot Ato the axis of pivot B, see Fig. 4 and Table 1. The signs of dandLare deﬁned with respect to the direction from OtoB. We classify the RDCO architectures according to three domains of the parameter , summarized in Table 1. (a) When > 0,dandLare of same sign. The connecting rods are connected to the ground between the inertial loop and point O. (b) When1<  < 0,dandLare of opposite sign and d < L . The connecting rods cross each other inside of the inertial loop. (c) When<1,dandLare of opposite sign and d>L . The connecting rods are connected to the ground outside of the inertial loop. 6 Fixed d L Fig. 4: Example of RDCO physical implementation. Figure 2 shows the motion of the system as it is displaced from its equilibrium position. We call this motion rotation-dilation coupling , where the rotation is deﬁned by the angle swept by a vector from the center Oto a pointPon the inertial body and the dilation is deﬁned by the change of length of this vector. As shown in Table 1 and Eq. (8), the parameter inﬂuences this dilation: •When>0, !OP() < !OP(0) and the inertial bodies move towards each other. We call it anegative dilation . •When<0, !OP() > !OP(0) and the inertial bodies move away from each other. We call it apositive dilation . 2.2.1 Particular case: = 0 When= 0, the connecting rods are attached to the ground at the center of the system and the size of the system stays constant as it rotates (no dilation). 2.2.2 Particular case: =1 When=1,dandLare of opposite sign and go to inﬁnity. It is noted that circular motion becomes straight line motion when the radius goes to inﬁnity. This motion can be implemented by replacing pivots Aby sliders with axis tangential to a circle with center O[15, Fig. 3.16]. 2.3 Qualitative design validation Mock-ups were built to validate qualitatively the kinematics of the RDCO. The conﬁgurations (a) and (b) of Table 1 are shown in Fig. 5 and 6, respectively. In these implementations, rectangular notch ﬂexure hinges [42,12] have been used for the pivots and parallel leaf springs for the sliders. 7 (a)>0 (b)1<< 0 (c)<1Nominal position d>0L>0 d<0 L>0 L<0 d>0Rotated Negative dilation Positive dilation Positive dilation Table 1: Inﬂuence of parameter on the kinematics of the RDCO. The mock-ups were fabricated by laser cutting a 5mm thick Polyoxymethylene (POM) sheet. They have an outer diameter of 200mm and an admissible angular stroke of approximately 18degrees for the conﬁguration with  > 0and25degrees for the conﬁguration with 1<  < 0. This hardware showed that the system behaves qualitatively as predicted: the system has one DOF and a motion of the inertial bodies closely approximating a rotation about point Ocoupled to a dilation. The variation in diameter of the mock-ups can be observed relatively to the red circle printed in the background. The two systems conﬁrm that the dilation is negative when  >0, see Fig. 5 and Ref. [43], and positive when 1<< 0, see Fig. 6 and Ref. [44]. Remark 2.1. Due to its three-dimensional design, the mock-up of Fig. 6 had to be assembled in multiple parts. As a result, the connecting rods do not cross perfectly in the center in the nominal position and one of the parallel leaf springs was removed to prevent overconstraining the pivot. This is equivalent to the extra DOF of Fig. 2. 3 ANALYTICAL MODEL The goal of this paper is to use the inertia variation of the oscillator in addition to its stiffness variation to tune its isochronism defect. We thus start by deriving in Sec. 3.1 an expression for the isochronism defect of a general rotational oscillator with varying stiffness and varying inertia. We then compute these two effects for the RDCO in Sec. 3.2 and 3.3, respectively. 8 (a) (b) Fig. 5: Mock-up of the RDCO with  > 0andn= 4inertial bodies in nominal position (a) and rotated (b), showing a negative dilation. Video available in Ref. [43]. (a) (b) Fig. 6: Mock-up of the RDCO with 1<  < 0andn= 3inertial bodies in nominal position (a) and rotated (b), showing a positive dilation. Video available in Ref. [44]. 9 3.1 Isochronism defect of the perturbed rotational harmonic oscillator We consider the case of an oscillator whose restoring torque can be expressed by a power series having only odd terms (assuming the restoring torque to be antisymmetric with respect to equilibrium position) M=k0+k23+O 5 =k0(1 +2) +O 5 ; (2) wherek0is the nominal stiffness (i.e., the limiting stiffness as rotation angle goes to zero) and =k2 k0therelative restoring torque nonlinearity [22], and whose inertia can be expressed by a power series having only even terms (assuming the inertia variation to be symmetric with respect to equilibrium position) J=J0+J22+O 4 =J0 1 +2 +O 4 ; (3) whereJ0is the nominal inertia and=J2 J0therelative inertia variation . This oscillator satisﬁes the differential equation =k0 1 +2 J0(1 +2)+O 5 ; (4) which, when using again series expansions around = 0, becomes =k0 J0k0 J0()3+O 5 : (5) Solving this equation using standard methods of perturbation theory [45, Eq. 2.3.34] yields the frequency-amplitude relation of the oscillator !() =!0 1 +3 () 82 +O 4 ; (6) where!0=p k0=J0is the nominal frequency as amplitude approaches zero. Equation (6) gives an explicit expression for the isochronism defect. It shows that isochronism tuning can be performed either by modifying the relative restoring torque nonlinearity by acting onk2(without affecting k0) or by modifying the relative inertia variation by acting on J2(without affectingJ0), hence not affecting the nominal frequency !0. This will be done on the RDCO in Sec. 4.2.1 and 4.2.2, respectively. Remark 3.1. When the relative restoring torque nonlinearity is equal to the relative inertia variation (=), Eq. (5) becomes the differential equation of the simple rotational harmonic oscillator with constant angular frequency !0, i.e., with zero isochronism defect. 10 3.2 Stiffness variation of the RDCO The rotational stiffness of the RDCO is calculated with the following assumptions: •The ﬂexure elements of the RDCO are considered as springs and the other bodies as rigid. •The system is considered symmetric: all pivots Ahave rotational stiffness kA, all pivotsBhave rotational stiffness kBand all the sliders have translational stiffness kt. •The restoring force or restoring torque of the ﬂexure elements can be expressed by series expansion having only odd terms of displacement, see Eq. (2). •The rotations are small and terms can be expressed using series expansions around = 0. •No external force (such as gravity) is acting on the oscillator; we assume that its symmetry minimizes the inﬂuence of gravity on its restoring torque [22]. The rotational stiffness of the oscillator is derived from the strain energy of the system with the following steps: 1. Derive the motion of the two pivots of the connecting rod for a given rotation of the oscillator. 2. Derive the motion of the sliders for a given rotation of the oscillator. 3. Express the strain energy of the system for a given rotation of the oscillator. 4. Compute the rotational stiffness of the system from its total strain energy. 3.2.1 Motion of the pivots of the connecting rod The angles AandBswept by the pivots of the connecting rod when the system rotates by an angleare obtained by trigonometry, see Fig. 7b. For small displacements, they can be expressed using series expansions: B= arcsindsin L=+1 6 21 3+O 5 A=+B= (1 +)+1 6 21 3+O 5 :(7) 3.2.2 Motion of the sliders The motion of the sliders corresponds to the change in distance qbetween the pivots of the inertial bodies, see Fig. 7b. We ﬁrst derive the distance from the center of the oscillator to these pivots as the oscillator rotates r() =dcos+LcosB=R0 1 22 +O 4 ; (8) whereR0=L+d. We then get the motion of the sliders q() = 2 sin n (r()r(0)) =R0sin n 2+O 4 : (9) 11 R0 SySxmr,Jr O (a) r sdL O A Bq (b) Fig. 7: Parameters of the RDCO stiffness and inertia models in equilibrium position (a) and rotated by angle(b). 3.2.3 Strain energy The restoring force of the ﬂexures can be expressed by series expansion: MA() =kA;0(1 +A2)+O 5 ; MB() =kB;0(1 +B2)+O 5 ; Ft(q) =kt;0(1 +tq2) q+O q5 :(10) The strain energy of the system for a rotation follows from the sum of the strain energies of each elastic joint U=nZA 0MA()d+ZB 0MB()d+Zq 0Ft()d : (11) 3.2.4 Rotational stiffness Substituting Eq. (7)-(10) into Eq. (11) and deriving and dividing the strain energy Ubyyields the rotational stiffness of the RDCO kR=dU d=kR;0(1 +R2) +O 4 ; (12) 12 where kR;0=n (+ 1)2kA;0+2kB;0 (13) is the nominal stiffness and R=2(+ 1)(21)kA;0+ 22(21)kB;0+ 3 (+ 1)4kA;0A+ 62kt;0R2 0sin2 n 3 ((+ 1)2kA;0+2kB;0): (14) is the relative restoring torque nonlinearity according to Eq. (2). Note that the contribution of the relative restoring torque nonlinearity Bof the external pivot is neglected assuming that it is small and that4is small. Remark 3.2. We deﬁne the rotational stiffness by the secant stiffness deﬁned as the restoring torque divided by the angular displacement ksec=M() . This quantity should not be confused with thetangent stiffness deﬁned as the derivative of the restoring torque ktan=dM d. 3.3 Inertia variation of the RDCO The inertia of the RDCO as it rotates is calculated with the following assumptions: •All the inertial bodies are the same. •All the joints are ideal. •The connecting rods are massless. •The rotations are small and terms can be expressed using series expansions around = 0. The inertia of the RDCO for a rotation is J() =n Jr+mrs2() ; (15) whereJrandmrare respectively the inertia and mass of one of ninertial bodies and sis the distance from the center Oof the oscillator to the COM of one inertial body depicted in Fig. 7. This distance can be computed as follows s2() =S2 x+ (r()(R0Sy))2=S2 0R0Sy2+O 4 ; (16) whereris given in Eq. (8), S0=s(0)is the distance from Oto the COM of one inertial body at equilibrium and Syis the projection of that distance against the axis formed by the pivots of the connecting rod, see Fig. 7a. Substituting Eq. (16) into Eq. (17) yields the inertia of the RDCO as it rotates J() =n Jr+mr S2 0R0Sy2 +O 4 (17) 13 with relative inertia variation according to Eq. (3) R=mrR0Sy Jr+mrS2 0: (18) The results obtained with this model are compared to the ones obtained by numerical simula- tion in Fig. 8 and show a good match. The FEM model is described in Sec. 4.1. Fig. 8: Inertia Jof the RDCO versus rotation angle obtained by FEM and with the analytical model. Remark 3.3. The sign of the inertia variation in Eq. (18) depends on the sign of , as was already noted through the sign of the dilation in Sec. 2.2. 4 EXAMPLE OF IMPLEMENTATION AND NUMERICAL VALIDATION 4.1 Design with compensating inertia and stiffness variation In order to validate the concepts presented in this paper, we design a ﬂexure implementation of the RDCO and show, ﬁrst, that the inertia and stiffness variation can compensate each other in order to reach isochronism and, second, that the isochronism of the RDCO can be tuned by varying either its stiffness variation or its inertia variation . We validated these results by FEM. Two conﬁgurations exist where the inertia and restoring torque defects can compensate each other: (a) <0and the positive dilation of the system (  >0) compensates for the increasing stiffness of the system ( >0). 14 (b)>0and the negative dilation of the system ( <0) compensates for the decreasing stiffness of the system ( <0). Conﬁguration (b) where  >0was chosen. This makes the implementation easier since the connecting rods do not have to cross each other as depicted in Fig. 6. The design is shown in Fig. 4. It does not target watch speciﬁcations but aims at respecting realistic proportions for a titanium prototype of diameter 80 mm and thickness 5 mm manufactured by electrical discharge machining (EDM).5A relatively large value of = 0:25was chosen in order to highlight nonlinear effects. Truncated circular notch ﬂexure hinges were used for pivots AandBwith the following dimensions: minimum thickness of 50 m and radius of 10.5 mm for pivot Aand minimum thick- ness of 150 m and radius of 2 mm for pivot B(whose deformation is much smaller). Parallel leaf springs were used for the sliders with a thickness of 130 m and a length of 32 mm. It was shown in Ref. [46, Chap. 7] that these dimensions can be reached by EDM manufacturing. Figure 9 shows that the inertia and stiffness variation of this oscillator obtained by FEM overlap, hence compensating each other and allowing to reach theoretical isochronism. These results validate the design and concepts presented in this paper. The oscillator was simulated using the commercial FEM software ANSYS [47] with a mesh of hexahedral elements of type SOLID186 that is reﬁned on the ﬂexures such that there are three elements across their thickness and ﬁve along their height. The truncated circular notch ﬂexures have 10 elements along their length and the parallel leaf springs 32, see Fig. 10. The inertia variationJvar=J() J0is obtained directly from the FEM data and the stiffness variation is obtained by calculating the secant stiffness for each data point, i.e, kvar=ksec k0=M() k0, see Remark 3.2. Fig. 9: Inertia and stiffness variation of the example RDCO versus rotation angle . 5Titanium alloy TiAl6V4 with Y oung’s modulus E= 114 GPa was chosen for its machinability and high ratio of fatigue limit to Y oung’s modulus that make it suitable for ﬂexures [46, Table B.14].15 Fig. 10: Close-up view of the mesh of the RDCO ﬁnite element model. Remark 4.1. In order to satisfy the analytical model’s assumption that stiffness and inertia are symmetric functions with respect to the equilibrium position, a ﬂexure implementation with axial symmetry was chosen instead of a rotational symmetry such as shown in Fig. 6. 4.1.1 Numerical validation of analytical stiffness model The analytical restoring torque nonlinearity of the oscillator is calculated by substituting the rotational stiffness of the pivots and translational stiffness of the sliders from [46, Eq. 4.20 and 5.9] into Eq. (14). We showed in previous research that the nonlinearity Aof pivotAcannot be cal- culated accurately using Euler-Bernoulli beam theory [15,22]. This issue is resolved by assuming thatA= 0to ﬁnd a ﬁrst set of dimensions and then ﬁne-tuning the restoring torque nonlinear- ityRof the oscillator by changing the thickness htof the parallel leaf springs, as explained in Sec. 4.2.1. For the given implementation, Fig. 9 displays similar analytical and FEM results with respective restoring torque nonlinearities R=0:225andR=0:205, showing that the analyt- ical model provides an estimation with less than 10% error despite neglecting A. Note that the FEM value of Rwas obtained by ﬁtting Eq. (2) to the data. 4.1.2 Numerical validation of analytical inertia model Since our ﬂexure implementation has two different types of inertial bodies, see Fig. 4, Eq. 18 is adapted as follows: R=R0(mr;1Sy;1+mr;2Sy;2) Jr;1+Jr;2+mr;1S2 0;1+mr;2S2 0;2(19) where indices 1 and 2 refer to the two different types of inertial bodies. With the mass parameters in Table 2, this equation returns an inertia variation R=0:217that matches the numerically computedR=0:204with less than 10% error, see Fig. 9. Note that the FEM value of Rwas obtained by ﬁtting Eq. (3) to the data in Fig. 8. 16 R0S0;1S0;2Sy;1Sy;2mr;1mr;2Jr;1 Jr;2 34.0 mm 34.2 mm 30.5 mm 34.0 mm 30.3 mm 11.3 g13.5 g2.29 kg mm22.99 kg mm2 Table 2: Mass parameters of the RDCO ﬂexure implementation used for numerical validation. 4.2 Isochronism tuning Using the models of Sec. 3.3 and 3.2, we present two ways of tuning the isochronism of the RDCO, by acting either on stiffness or inertia. 4.2.1 Stiffness isochronism tuning The isochronism of the RDCO can be tuned by varying the relative restoring torque nonlinearity Rwithout affecting any other crucial property of the oscillator. As in our previous work [16], this can be done by changing the stiffness kt;0of ﬂexures that perform a second order motion of the rotation angle and thus only contribute to the stiffness variation in Eq. (12) without affecting the nominal stiffness kR;0in Eq. (13). Although the analytical model cannot exactly predict the inertia and stiffness variation of the oscillator, this technique can be used to ﬁne-tune the dimensions obtained using the analytical model to have the two effects accurately compensate each other as shown in Fig. 9. The tuning calculated with Eq. (14) is compared to FEM results in Fig. 11. Recall that the model cannot accurately predict the absolute restoring torque nonlinearity since Ais unknown. The results are hence shown in terms of variation of restoring torque nonlinearity obtained for a relative thickness variation htof the parallel leaf springs. The difference in slope between the analytical and FEM models could be explained by the fact that ﬂexures only approximate the motion of the ideal joints used in the analytical model. It is indeed known that parallel leaf springs actually have a parabolic trajectory and that the rotation axis of notch ﬂexure hinges moves slightly during their deformation [12,42]. Remark 4.2. The thickness htof the parallel leaf springs was chosen as tuning parameter due to its low impact on the design. However, their length Ltcould also have been used, which might be more suited for post-fabrication ﬁne-tuning [16]. 4.2.2 Inertia isochronism tuning The isochronism of the RDCO can be tuned by varying the inertia variation Rwithout changing any other crucial property of the oscillator. Equation (18) shows that this can be done by changing parameterSywithout changing Jr,mrorS0, for instance by moving the COM of the inertial part on a circle centred at point Oas depicted in Fig. 12. Note that the stiffness properties of the oscillator are not affected by these changes in mass distribution. Figure 13 shows a practical way of implementing this tuning by moving tuning masses placed on the inertial bodies of the RDCO by an angle on a circle centred at O. Note that the nominal position = 0is offset with respect to the axis formed by the pivots of the connecting rods such that the tuning can produce either a positive or negative isochronism defect. In order to provide numerical data for the tuning, we choose a reference amplitude 1= 10degrees and express the isochronism defect in seconds per day for a 20% amplitude decrease from the 1reference. This is computed by applying Eq. (6) in the deﬁnition for daily rate [16, Eq. 17 Fig. 11: Analytical and FEM results for the restoring torque nonlinearity tuning (i.e., isochronism tuning) of the RDCO by varying the thickness htof the parallel leaf springs. Sy;2 Sy;1 S0 Fig. 12: Tuning of the inertia variation of the RDCO. (7)]. The results are plotted in Fig. 14, showing an effective way of tuning the isochronism of the RDCO with a precision of order 1 s/day for the chosen amplitude range. Note that the predictions of the analytical model match the numerical result with less than 0.2 s/day error. Remark 4.3. The masses were chosen such that the correction is of order 1 s/day for the tuning range depicted in Fig. 13. The tuning can be adjusted by changing the mass ratio, inertia ratio and COM radius ratio between the mobile and ﬁxed parts of the inertial bodies. In this example, these 18 (a) = 0 (b) = 11 degrees (c) =20degrees Fig. 13: Inertia isochronism tuning masses for the RDCO. Fig. 14: Isochronism tuning for the RDCO by varying the angular position of tuning masses. Results obtained by FEM and with the analytical model are displayed in terms of daily rate for a 20% amplitude decrease from reference amplitude 1= 10 degrees. ratios have the respective values of 0.38, 0.007 and 1.29 for body 2, see Table 2. 5 CONCLUSION AND FUTURE WORK This article presents a new way of tuning the isochronism of ﬂexure mechanical time bases by using their inertia variation, as opposed to the previously known methods that focused on stiffness variation. A simple and practical way of implementing this tuning post-fabrication was devised by displacing masses. A new family of ﬂexure rotational oscillators was introduced that embodies both inertia and stiffness isochronism tuning concepts. Additionally, the symmetry of these architectures has the advantage of addressing another important limitation of ﬂexure time 19 bases: the inﬂuence of linear accelerations on their period [15, 16, 21, 22], making them also interesting candidates for MEMS time bases [17,18] The concept of inertia isochronism tuning introduced here is not limited to the RDCO family: it is indeed common for ﬂexure oscillators to have second order motions resulting in variations of their mass distribution that could be exploited to tune their isochronism. This is for example the case with the motion of the intermediate bodies of the “co-RCC” oscillator [16,31] or the parasitic angle of the rotation axis of the “GIFP” oscillator [15,22]. Our new concept of inertia isochronism tuning paves the way for new solutions to this historical problem and has led to a new family of ﬂexure architectures with rotation-dilation kinematics that could ﬁnd promising applications in other ﬁelds. For instance, the RDCO has potential in ﬂexure (a.k.a. compliant) gripper applications where the dilation kinematics offer self-centering gripping with radial symmetry that can be superior to the conventional 2 ﬁnger gripping [48, 49] while the rotation kinematics allow simple actuation with a conventional rotary actuator. ACKNOWLEDGEMENTS We thank Ilan Vardi for his constructive criticism and careful review of the manuscript. We would also like to thank Sim ´on Pr ˆecheur-Llarena for his assistance in fabricating the mock-ups of Fig 5 and 6. Finally, we thank Julia Bierent for her assistance in the dimensioning and simulation of the example of implementation.