EP3722888A1

EP3722888A1 - Mechanical oscillator with tunable isochronism defect

Info

Publication number: EP3722888A1
Application number: EP19168095.8A
Authority: EP
Inventors: Mohammad Hussein Kahrobaiyan; Mohamed Zanaty; Simon Henein
Original assignee: Ecole Polytechnique Federale de Lausanne EPFL
Current assignee: ETA SA Manufacture Horlogere Suisse
Priority date: 2019-04-09
Filing date: 2019-04-09
Publication date: 2020-10-14
Anticipated expiration: 2039-04-09
Also published as: EP3722888B1

Abstract

Mechanical oscillator (1) for a timepiece comprising an inertial mass (3) arranged to oscillate about a neutral position under the effect of a restoring force provided by at least one elastic element (5),
characterised in that said mechanical oscillator (1) further comprises a compensation spring (11) pivotably linked to said inertial mass (3) by means of a substantially rigid first connecting bar (15), said compensation spring (11) and said first connecting bar (15) being arranged so as to apply a compensating force to said inertial mass (3) which varies in function of displacement of said inertial mass (3) from its neutral position.

This arrangement permits tuning of the isochronism and thus reduction and/or elimination of the isochronism defect of the oscillator typically caused by nonliearity of the restoring force provided by the elastic element (55)

Description

Technical Field

The present invention relates to the technical field of mechanical oscillators. More particularly, it relates to one degree of freedom mechanical oscillators with tuneable isochronism.

State of the art

The most common horological mechanical oscillators comprise an inertial mass, such as a balance, arranged to oscillate about a neutral position under the effect of a restoring force provided by an elastic element such as a hairspring. The oscillations are typically maintained by means of a conventional escapement mechanism powered by a mainspring kinematically connected to an escapement wheel by means of a suitable gear train.
Recently, various types of both rotational and translational oscillators based on elastic flexure pivots have been developed, such as those described in EP1736838 , WO2018100122 , US2017/269551 and others. These modern oscillators typically oscillate at a lower amplitude and a higher frequency than conventional balance-hairspring oscillators, and are very promising from a developmental standpoint in terms of timekeeping precision.
However, all such oscillators suffer from non-linearity of the coefficient of stiffness of the spring providing the restoring force, whether it be a conventional hairspring or a flexure pivot system. In simple terms, this is due to the fact that as the amplitude of oscillation increases, the elastic element deforms to a greater degree, which affects its geometry and hence its stiffness. This, in turn, affects the frequency of the oscillator and produces a defect in isochronism, which is defined as the aptitude of a phenomenon to reproduce itself in equal times, whatever the amplitude. An isochronous oscillator hence has a constant frequency which is independent of the amplitude of the oscillations.
The classical solution to isochronism defect is the so-called "fusée" mechanism (see e.g. EP2735919 ), in which variations of the torque of the mainspring are compensated by a chain mechanism arranged such that its output torque is maintained constant as the mainspring unwinds. Since the excitation forces to which the oscillator is subjected do not vary, the amplitude of oscillation remains substantially constant. However, fusée mechanisms are bulky and complex, and have hence found limited application in wristwatches.
Another well-known solution is the so-called constant force device (such as that disclosed in CH296060 ), in which an intermediate spring is maintained at a substantially constant state of charge. Since the escapement is driven by the intermediate spring rather than by the mainspring, as the latter unwinds, the torque to which the escape wheel is exposed remains substantially constant, and the excitation forces applied to the oscillator likewise remain substantially constant. Although significantly more compact than a fusée, such mechanisms are complex and difficult to adjust in wristwatches.
In the context of the so-called Genequand System disclosed in EP1736838 (mentioned above), document EP2290476 proposes an isochronism corrector, which compensates for the nonlinearity of the coefficient of elasticity of the restoring force by means of a pre-loaded blade spring which acts on the oscillator. This blade spring is adjusted such that it applies a force which varies in function of the amplitude of oscillation in the opposite direction to the variations of the restoring spring force. However, this device is not in permanent contact with the oscillator. As a result of this latter point, discontinuities appear in the restoring force curves which can cause rebounds when the oscillator enters into contact or loses contact with the isochronism corrector. In addition, friction or adhesion effects might alter the correcting function.
An aim of the present invention is hence to propose oscillators for which the isochronism can be tuned, and thereby to at least partially overcome the drawbacks of the prior art.

Disclosure of the invention

More specifically, the invention relates to a mechanical oscillator for a timepiece (such as a wristwatch), as defined in claim 1. This oscillator comprises an inertial mass arranged to oscillate about a neutral position under the effect of a restoring force provided by at least one elastic element, which may be e.g. a conventional hairspring, a flexure pivot system which also serves to support the inertial mass, or any other known configuration. This oscillation can be rectilinear along an axis of translation, or rotational around an axis of rotation.
According to the invention, the oscillator further comprises a compensation spring pivotably linked to said inertial mass by means of a substantially rigid first connecting bar so as to apply a compensating force to said inertial mass in such a manner that said compensating force varies in function of displacement of said inertial mass from its neutral position. The first connecting bar is hence directly or indirectly attached pivotably at one end to the compensation spring and at its other end to the inertial mass.
The restoring force can be provided by any convenient type of elastic element such as a spring, or a flexure pivot also designed to support and guide the inertial mass. All pivot points can either be conventional pivots or defined by suitable flexures.
At least a part of the force applied by the compensating spring acts on the second order term of spring stiffness (i.e. the x² or θ ² term, depending on whether the oscillator is translational or rotational) without influencing the first order term (i.e. the x or θ term). As will be explained in detail below, this means that the linear stiffness of the elastic element can be left unchanged and hence the nominal frequency of oscillation remains unchanged, whereas the nonlinear regime can be modified so as to be able to compensate for nonideal changes in the stiffness of the spring as the inertial mass moves away from its neutral position. Indeed, by carefully adjusting the force applied by the compensation spring and the geometry of the system, the x ² or θ ² term can be manipulated such that the effective stiffness of the overall restoring force applied to the inertial mass (i.e. that applied as the sum of forces acting thereupon as restoring forces, irrespective of whether they come from the elastic element or the compensation spring) is substantially linear over its full working range. As a result, the isochronism of the oscillator can be tuned, and the isochronism defect which is usually caused by nonlinearity of the elastic restoring force of the elastic element can be reduced and even eliminated.
In the case in which the inertial mass is translational, the first connecting bar is ideally arranged such that its line of action is substantially perpendicular to said axis of translation when said inertial mass is in its neutral position. Since the bar can take any shape between its pivot points without affecting its function, its line of action is referred to here, which is defined as an axis passing through its two pivot points. In the case of a rotational inertial mass, the first connecting bar is arranged such that its line of action is aligned with the axis of rotation when the inertial mass is in its neutral position. This ensures that the action of the compensation spring is symmetrical about the neutral position of the inertial mass.
Depending on whether the stiffness of the elastic element gets "harder" (stiffer) or "softer" (less stiff) with increasing displacement of the inertial mass from its neutral position, the compensating force applied by the compensation spring can be arranged to either decrease or increase respectively with increasing displacement of the inertial mass. Here, the overall compensating force is considered, i.e. the force parallel to the line of action of the first connecting bar. If the compensation spring is in tension when the inertial mass is in its neutral position, the force will increase with increasing displacement, compensating for an elastic element which softens, and if the compensation spring is in compression, the force will decrease with increasing displacement, compensating for an elastic element which hardens. As the inertial mass displaces, the component of the compensating force acting in the same direction as the restoring force hence changes in function of the geometry of the system and the arrangement of the compensation spring.
Advantageously, said compensation spring is attached to a slider arranged to translate along a respective axis, said slider being pivotally linked to said substantially rigid bar. This axis is ideally substantially parallel to the line of action of the first connecting bar when the inertial mass is in its neutral position.
Advantageously, said compensation spring is arranged such that its pre-stress (i.e. its tension or compression when the inertial mass is in its neutral position) is adjustable, which eliminates any need to precisely determine the stiffness of the compensation spring at the point of manufacture since the effective zero-order (linear) stiffness of the oscillator (and hence the nominal frequency of oscillation) can be adjusted. This can be achieved in a particularly simple way by attaching the compensation spring between the aforementioned slider and a further slider arranged such that it can be moved so as to stretch or compress the compensation spring, e.g. with a screw or cam system or similar.
Advantageously, the mechanical oscillator can comprise a further compensation spring pivotally linked to said slider by a substantially rigid second connecting bar pivotally attached to said slider such that its line of action is substantially perpendicular to that of said first connecting bar when the inertial mass is in its neutral position (and hence the oscillator is in its neutral state), said further compensation spring being arranged to apply a further compensating force to said slider which varies in function of the displacement of said slider from its neutral position. This allows to further act on the isochronism without influencing the nominal frequency, and the pre-stress of the further compensation spring may also be adjustable in the same manner as that of the first compensation spring in order to tune the isochronism of the oscillator. In such a two-bar arrangement, the pre-stress of the compensation spring affects both the nominal frequency and the isochronism (i.e. the second-order stiffness), whereas the pre-stress of the further compensation spring affects only the second-order stiffness and hence the isochronism.
Advantageously, said elastic element may be constructed as a flexure pivot arranged to support and to guide said inertial mass in translation or in rotation, as the case may be. Likewise, the compensation spring (and further compensation spring, if present) can also be formed as a flexure pivot. Indeed, the entire oscillator can be constructed as a flexure pivot system.
Any of the mechanical oscillators as defined above can be incorporated into a timepiece movement further comprising a crank arrangement or an escapement arranged to case the inertial mass to oscillate under the effect of energy supplied by a motor such as a mainspring.

Brief description of the drawings

Further details of the invention will appear more clearly upon reading the description below, in connection with the following figures which illustrate:

Figure 1: a schematic representation of a conventional translational oscillator;
Figure 2: a schematic representation of a conventional rotational oscillator;
Figure 3: a graph showing types of spring nonlinearity;
Figure 4: a schematic representation of a translational oscillator according to the invention;
Figure 5: a schematic representation of a rotational oscillator according to the invention;
Figure 6: a pair of schematic representations showing the principle of the invention in the context of the embodiment of figure 4;
Figure 7: a schematic representation showing the principle of the invention in the context of the embodiment of figure 5;
Figure 8: a schematic representation of an embodiment of the invention in which the prestress of the compensation spring can be easily modified;
Figure 9: a schematic representation of a further embodiment of the invention based on that of figure 8, in which nonlinearity of the compensation spring is itself compensated for by means of a further compensation spring;
Figure 10: a flexure-pivot-based implementation of the embodiment of figure 9;
Figure 11: a flexure-pivot-based implementation of a rotational variant of the embodiment of figure 10; and
Figures 12a-12e: Graphs of simulations and experimental results illustrating the return force of an oscillator according to figure 10 in function of displacement of the inertial mass.

Embodiments of the invention

In the present specification, the term "flexible" should be understood as meaning that an element is intended to undergo a deformation in use which has an impact on its function, this deformation being in at least one particular direction. "Rigid" should be understood as meaning that an element does not undergo a deformation which affects its function. "Flexible" elements may be flexible in one direction and "rigid" in another, as is the case with blade and col flexures. Of course, small undesired deformations of rigid elements (and of flexible elements in a direction in which they are rigid) cannot be excluded, but they are so small as to be trivial and can be ignored. As a parametrisation, "rigid" elements are at least 100 times as stiff, preferably at least 1000 times as stiff, as "flexible" elements, considered in the direction in which they are flexible. The skilled person fully understands this principle, and it need not be described at length below.
First of all, the theory behind the present invention will be explained with reference to figures 1-3.
The types of oscillators with which the present invention is concerned are one degree-of-freedom (DOF) harmonic oscillators, and can be either linear oscillators as illustrated schematically in figure 1, or rotational oscillators as illustrated in figure 2. In each of these figures, the oscillator 1 is illustrated in its neutral state
In each case, the oscillator 1 comprises an inertial mass 3 which is arranged to oscillate about a neutral position under the effect of a restoring force provided by at least one elastic element 5 attached at one of its ends to a substantially rigid framework 7, and at its other end to the inertial mass 3. Extra masses can be present out of the plane of the system as illustrated, and the inertial mass 3 defines at least 75%, preferably at least 90%, of the oscillating inertia of the oscillator 1.
In the case of the linear oscillator 1 of figure 1, the inertial mass is guided in translation by means of a suitable bearing arrangement 9 such as rollers, sliding friction bearings, or a flexure mechanism (see for instance The Art of Flexure Mechanism Design, Simon Henein, Lennart Rubbert, Florent Cosandier, Murielle Richard, EPFL Press, 2017). Such arrangements are generally considered to be equivalent, and have simply been represented schematically by rollers. However, it must be understood that the bearings illustrated have only one degree of freedom, and in the context of figure 1 the inertial mass 1 is constrained to move only in the x direction.
In the case of the rotational oscillator of figure 2, the inertial mass 3 is a balance of any convenient form, mounted so as to be able to rotate about an axis of rotation O. This latter may be defined by a physical axis (e.g. an arbor), or may be a virtual axis defined by a flexure mechanism, such as a Remote Centre Compliance (RCC) pivot or similar. In this latter case, the flexures which support the mass 3 and define its axis of rotation O may also provide the restoring force for oscillations, and hence also constitute the at least one elastic element 5.
In either case, the oscillations are maintained by means of an escapement mechanism or similar arrangement such as a crank (not illustrated on this figure) acting directly or indirectly on the inertial mass, as is generally known, and hence need not be described in detail. In respect of the less-common crank-type driving arrangements, FR1044957 describes a very simple arrangement which can be applied easily to either a rotationally-oscillating or rectilinearly-oscillating inertial mass 3, as indeed is illustrated schematically on figure 10 (see below).
As noted above, the isochronism of an oscillator is optimised when its frequency is independent of amplitude. To this end, mechanical harmonic oscillators should obey Hooke's Law, meaning that their elastic element should have as linear a restoring force or torque (as appropriate) as possible with respect to the amplitude of oscillations. In other words, the stiffness of the elastic element should be constant with respect to the amplitude of oscillations. In addition, the effective inertia also should be constant with respect to the amplitude of oscillations in order to achieve a constant frequency.
In respect of the stiffness k of the elastic element 5 considered with respect to the degree of freedom of the inertial mass 3, this can be generically expressed in its working range by the following power series, limited to the first two non-zero-order terms: $k = k_{0} + k_{2} α^{2} + O (α) (^{4})$

where k ₀ is the zero-order, i.e. linear, coefficient of stiffness, k ₂ is the second-order coefficient of stiffness which varies with amplitude α (whether expressed in linear or angular displacement, according to the type of oscillator), and the O( ) term is fourth order so considered negligible.
Figure 3 illustrates qualitatively the effect of variations of the value of k ₂, which is dependent on the changes of the geometry of the elastic element 5 as the inertial mass 3 oscillates. In the case in which k ₂ = 0, the spring is ideal, and its response is entirely linear. If k ₂ > 0, the elastic element 5 "hardens" in function of its deformation, i.e. gets stiffer the further it is deformed from its neutral state. In the opposite case, i.e. k ₂ < 0, the elastic element "softens" as it is deformed from its neutral state.
On the basis of the foregoing, stiffness relative nonlinearity can be defined as: $ε = \frac{k_{2}}{k_{0}} .$
In respect of the inertia, this is expressed in the working range of a linear oscillator in its working range by the following power series, limited to the first two non-zero-order terms: $m = m_{0} + m_{2} α^{2} + O (α) (^{4})$

where m is the mass, m ₂ is the second order coefficient of the variation of inertia in function of amplitude α, and O( ) is of fourth order and is neglected, as the use of standard 0() notation indicates. The same notation is used several times below to indicate higher order terms being neglected. In the case of a rotational oscillator, the inertia / is expressed as: $J = J_{0} + J_{2} α^{2} + O (α) (^{4})$

where J ₀ is the moment of inertia, J ₂ is the coefficient of the variation of moment of inertia as a function of amplitude α, and O( ) is as before.
To avoid unnecessary repetition, only the linear case will be treated below. The rotational case can be derived simply by substituting J for m, and expressing displacement α as an angle θ rather than as linear displacement x.
In view of the foregoing, inertia relative nonlinearity for the linear case is defined as: $µ = \frac{m_{2}}{m_{0}}$
From these equations, the kinetic energy K and potential energy V for a given oscillator can be derived as: $K = \frac{1}{2} {\dot{α}}^{2} [m_{0} + m_{2} α^{2} + O (α) (^{4})], V = \frac{1}{4} α^{2} [k_{0} + \frac{1}{2} k_{2} α^{2} + O (α) (^{4})] .$
Neglecting α ⁴ and higher-order terms in kinetic and potential energies and using Lagrange's equation of motion, we arrive at the following differential equation of motion of the oscillator (Duffing's Equation): $m_{0} (1 + µ α^{2}) \ddot{α} + k_{0} (1 + ε α^{2}) α + µα {\dot{α}}^{2} = 0.$
The angular frequency (radians/sec) of nonlinear oscillations ω is given by the following formula: $ω = ω_{0} (1 + \frac{3 ε - 2 µ}{8} α^{2}),$

where $ω_{0} = \sqrt{k_{0} / m_{0}}$
is the limit of the frequency of oscillations as amplitude goes to zero, ε and µ are as defined above.
The isochronism defect can be estimated by examining how the oscillator's frequency varies as its total energy (i.e. the sum of K + V at any given moment) varies, since this is a proxy for oscillator amplitude due to conservation of energy.
In horology, the standard measure of isochronism is the so-called "daily rate" ρ, which is the oscillator's error over exactly 24 hours and is expressed in seconds-per day (s/d). Mathematically, this is defined as: $ρ = 86400 \frac{ω - ω_{n}}{ω_{n}},$

where ω is the frequency of oscillations and ω_n is the nominal frequency of oscillations, i.e., the frequency of oscillations at a nominal amplitude α _n.
The relative energy variation E_% with respect to oscillator energy at nominal amplitude E_n expressed as a percentage is: $E_{%} = 100 \frac{E - E_{n}}{E_{n}} .$
Since the oscillator's total energy (K + V) is proportional to the square of amplitude α, the relative energy variation is given by: $E_{%} = 100 \frac{α^{2} - α_{n}^{}}{α_{n}^{}} .$
The isochronism defect σ of the daily rate of an oscillator due to a change in its total energy is defined as: $σ = \frac{ρ}{E_{%}}$
Substituting Eqs. (3)-(5) into Eq. (6), the following formula describes how the above-described relative nonlinearities of the stiffness k and inertia m affect the isochronism defect: $σ = 864 \frac{\frac{3 ε - 2 µ}{8} α_{n}^{}}{1 + \frac{3 ε - 2 µ}{8} α_{n}^{}} .$
Alternatively, simpler equation for isochronism defect can be arrived at by using the Taylor series expansion of σ around $(3 ε - 2 µ) α_{n}_{2} = 0,$
which gives: $σ = 108 (3 ε - 2 µ) α_{n}_{2} + O [{(3 ε - 2 µ)}^{2} α_{n}^{}]$
As can be seen from either of these equations, if the relative nonlinearity ε or µ can be varied, the isochronism defect σ can be varied.
The principle of the present invention revolves around mechanisms which cause ε to vary, thereby acting on the value k ₂ of the term k₂α ² of equation (1). In doing so, the nonlinearity of the elastic element 5 can be compensated for without modifying the linear stiffness unless desired, and hence without modifying the nominal frequency of oscillation of the oscillator 1 unless desired. At the present time, a practical way of acting on the value of m ₂ of the term m₂α ² of equation (2) has not been achieved, hence the problem of compensating for this source of isochronism defect remains open for the moment.
Figure 4 illustrates a first embodiment of an oscillator 1 according to the invention. This oscillator 1 is translational, similar to that illustrated in figure 1, and only the differences with this latter figure will be described in the following.
As mentioned above, the inertial mass 3 is constrained by an appropriate guide arrangement 9 so as to move along a rectilinear axis, here deemed to be the x axis, under the effect of the elastic element 5, whose linear stiffness is referred to as k ₅ (i.e. the zero-order stiffness) in the following text. The oscillator 1 further comprises a compensation spring 11 with zero-order stiffness k ₁₁, attached at one end to a frame element 7, and at its other end to a slider 13 which is arranged to translate along an axis y, which is perpendicular to the axis x, guided by a suitable bearing 9, again illustrated schematically by rollers, representing a sliding bearing. As will be seen later, this bearing 9 can also be a flexure mechanism giving the slider 13 substantially one degree of freedom in translation parallel to the y direction. Slider 13 is deemed to have negligible mass. Axis y intersects the centre of mass of inertial mass 3 when this latter is in a neutral position, as illustrated, and the slider 13 is attached to the inertial mass 3 by means of a substantially rigid first connecting bar 15 with effective length L, pivoted on each of these elements by means of either a conventional pinned pivot (as illustrated) or by a flexure pivot such as a col, a blade spring or equivalent. Effective length L is measured between the respective pivot points, and its line of action is along an axis joining its pivot points, irrespective of the actual shape of the bar. When the inertial mass 3 is in its neutral position, the line of action of bar 15 extends along the y axis, i.e. perpendicular to the degree of freedom of the inertial mass 3. Other arrangements are possible, such as pairs of functionally-parallel bars arranged such that their combined line of action is equivalent to that of the single bar 15 illustrated.
Presuming that both springs 5, 11 have no pre-stress when the inertial mass 3 is in its neutral position, when inertial mass 3 undergoes a translation with a distance x, the compensation spring 11 applies a compensating force to the inertial mass 3, whose component parallel to the x axis varies as a function of the length L and the stiffness of compensation spring 11, and the displacement x of inertial mass along the x axis. By resolving forces and using a power series expansion, the effective stiffness k to which the inertial mass is subject is: $k = k_{5} + \frac{k_{11}}{2 L^{2}} x^{2} + O (x^{4}),$
As can be seen, the effective stiffness k comprises a component deriving from the stiffness of spring 5 and from the compensation spring 11, this latter only acting upon the x² term in such a situation, leaving the linear term (and hence the nominal frequency of oscillation) unchanged.
Figure 5 illustrates a second embodiment of an oscillator 1 according to the invention, which applies the same arrangement of slider 13 and compensation spring 11 as that of figure 4, but in the context of a rotational hairspring-balance oscillator. In this case, inertial mass 3 is arranged to oscillate about an axis of rotation O, and the substantially rigid first coupling bar 15 is pivotally attached to said mass 3 at a location which is eccentric with respect to the axis of rotation O. When the oscillator 1 is in its neutral position (as illustrated), the line of action of the substantially rigid bar 15 intersects the axis of rotation of the inertial mass 3; in other words, the axis of rotation O and the two pivot points of the bar 15 are aligned along the same axis.
Figure 6 illustrates an embodiment similar to that of figure 4, in which the nominal stiffness is tuneable rather than simply being set by the fixed value of k ₅. In the arrangements of figures 4 and 5, the nominal value of k ₁₁ would have to be changed in order to tune the frequency of oscillation.
In the embodiment of figure 6, a constant force p is applied to the slider 13 by any convenient means, with positive p being in a direction away from the inertial mass 3, the effective stiffness experienced by the inertial mass 3 is given as: $k = k_{5} + \frac{p}{L} + O (x^{2})$

where L is defined as before and x is the displacement of the mass 3 from its neutral position in the x direction.
Hence, by modifying p, the nominal stiffness k to which the inertial mass 3 is subjected can be modified without changing the zero-order stiffness k ₅ of the spring 5 itself, and hence the nominal frequency of the oscillator 1 can be modified.
When -Lk₅ < p < 0, the effective stiffness decreases with increasing p but remains positive, whereas when p < -Lk₅, the stiffness decreases and becomes negative. When p = -Lk ₅, stiffness is zero. In practice, the effective stiffness must remain positive in order to provide the restoring force for oscillations.
Figure 7 shows the same principle applied to a rotational oscillator 1. The length L ₁ is the distance from the pivot point of the inertial mass 3 (which is highly schematically represented) with respect to the frame 7 to its pivot point with respect to connecting bar 15, and L ₂ is the effective length of the connecting bar 15 between its two pivot points. The effective rotational stiffness of this mechanism is: $k = k_{5} + {pL}_{1} (1 + \frac{L_{1}}{L_{2}}) + O (θ) (^{2})$
When -k ₅/L ₁(1 + L ₁/L ₂) < p < 0, the effective stiffness decreases with increasing p but remains positive but remains positive, whereas when p < -k ₅/L ₁(1 + L ₁/L ₂ ), the effective stiffness decreases and becomes negative. When p = -k ₅/L ₁(1 + L ₁/L ₂ ), the effective stiffness is zero. Again, in order to apply a restoring force, the effective stiffness remains positive.
Figure 8 illustrates an oscillator 1 which combines the principles of both figure 4 and figure 6. In essence, the compensation spring 11 is arranged so as to supply not only a compensating force which varies by virtue of Hooke's Law, but also the constant force p mentioned above by means of pre-stressing compensation spring 11. To this end, one extremity of compression spring is attached to the slider 13 as in figure 4, and its other extremity is attached to a further slider 17 which is arranged to be displaceable by a distance d from a position in which the compensation spring 11 is not stretched or compressed when the bar 15 is parallel to the y axis, guided by a suitable bearing 9. This pre-stress generates the constant force p, and the overall force applied is the sum of the pre-stress and the variations in stress due to Hooke's Law as the spring 11 changes length. The force applied by compensation spring 11 hence varies as the slider 13 displaces. The displacement d of further slider 17 can be arranged by means of an appropriate arrangement such as a screw, a cam or similar (not illustrated).
The effective stiffness of the system is hence: $k = k_{5} + \frac{k_{11} d}{L} - \frac{3 k_{11} x^{2}}{2 L^{2}} + O (x^{4})$
where k ₁₁ is the nominal linear stiffness of compensation spring 11.
The same principle can be applied to the rotational embodiment of figure 7, which does not need to be illustrated or described at length.
Figure 9 illustrates a further embodiment of an oscillator 1 according to the invention which is based upon that of figure 8, and in which both the nominal frequency and the isochronism can be adjusted.
Compared to the variant of figure 8, this oscillator 1 also comprises a substantially rigid second connecting bar 19 pivotally connected to slider 13 perpendicular to the first bar 15 when the system is in a neutral position. This bar 19 is also pivotally connected to a further slider 21, and has an effective length L ₂ between its pivot points. Again, slider 21 is guided by an appropriate bearing 9.
A further compensation spring 23 with linear spring constant k ₂₃ links slider 21 with a yet further slider 25 in a manner analogous to the compensation spring 11. Again, slider 25 is guided by a suitable bearing 9. This spring 23 is preloaded by moving slider 25 by a distance d ₁ from a position in which the spring 23 is not stressed when the system is in the neutral position illustrated. In this figure, d ₀ corresponds to d in figure 8. As a result, further compensation spring 23 acts in respect of compensation spring 11 in the same manner that this latter acts in respect of elastic element 5. In other words, the spring stiffness component of further compensating spring 23 acts to compensate the second-order stiffness term of compensation spring 11, and its pre-stress due to displacement d ₁ acts to modify the effective zero-order stiffness to which the slider 13 is subjected.
The effective stiffness of the system is hence: $k = (k_{5} + \frac{k_{11} d_{0}}{L_{1}}) + (\frac{k_{11}}{2 L_{1}^{2}} + \frac{k_{11} d_{0}}{2 L_{1}^{3}} + \frac{k_{23} d_{1}}{2 L_{1}^{2} L_{2}}) x^{2} + O (x^{4})$
As can be seen, adjusting d ₀ acts upon both the nominal stiffness and the second-order stiffness, whereas adjusting d ₁ acts only upon the second-order stiffness and hence only upon the isochronism without affecting nominal frequency. Hence, d ₀ can be adjusted to set the nominal frequency, and then d ₁ can be set to correct the isochronism without affecting the nominal frequency.
In the case in which compensating spring 11 is not adjustable and has no pre-stress, d ₀ is simply zero in the above equation.
Again, the same principle can be applied to a rotational oscillator, mutatis mutandis, which does not need to be illustrated in detail.
As noted above, all the realisations can be constructed based around flexure pivots, and as an example, figure 10 illustrates an entirely flexure-based realisation of the oscillator 1 of figure 9. The simpler embodiments of the earlier figures can be constructed in a similar manner by eliminating the elements not present therein, such as the second connecting rod 19, sliders 21 and 25, and spring 23, and even slider 17 if the force of the compensation spring 11 is preset.
Spring 5 is integrated with the corresponding bearing 9 and is constructed as a parallelogram flexure pivot comprising a pair of functionally-parallel elongated bars 5a terminated by blade flexures or cols 5b (which are equivalent), attached at one end each to framework 7, and at the opposite end each to inertial mass 3. It should be noted that the exact shape of the bars 5a is unimportant, so long as the pivot functions as a parallelogram flexure pivot. For small displacements of the inertial mass 3 that substantially obey the small angle approximation (i.e. which cause the bars 5a to incline by no more than 15° from their neutral orientation), the movement of the inertial mass 3 is substantially rectilinear along the x direction, for which its stiffness k ₅ in the x direction can be defined as is generally known. Alternatively, the bar-and-blade flexures can be replaced with simple blade flexures.
Substantially rigid bar 15 is also constructed as an elongated bar with a blade or col flexure at each end, its effective length L ₁ being measured between the effective pivot points, which are generally considered as being at the midpoints of the flexures. Since many of the flexure pivots used correspond to the type used for bar 15, there is no need to re-describe them exhaustively each time.
Slider 13 is constrained to move substantially only parallel to the y axis by a flexure pivot bearing 9 comprising a single flexure pivot of the same bar-and-blade/coltype as rigid bar 15. Alternatively, this bearing 9 can be a single blade flexure or a parallelogram flexure pivot of the type 5, 9 constraining the inertial mass 3 or comprising a pair of simple blade flexures. Such a parallelogram flexure pivot may be constructed of simple blade flexures or bar-and-blade/col type flexures.
Compensation spring 11 is formed as a single, simple blade flexure, and further slider 17 is guided by a pair of functionally-parallel blade flexures 9b forming its bearing 9, although only a single flexure is necessary. These flexures can each be replaced by an equivalent flexure of the bar and blades/cols type (i.e. of the types used for guiding the inertial mass 3 and the slider 13 in the construction of figure 10).
Second connecting bar 19 is likewise formed as a substantially rigid bar terminating at each end in a blade flexure or col, and has an effective length L ₂ again defined between the effective pivot points as for the connecting bar 15.
Slider 21 is again connected to slider 25 via a spring 23 comprising a pair of functionally-parallel blade flexures 23a, although again only one is required, and alternative forms are possible as described above, such as bar-and-blade/col flexures.
Finally, slider 25 is again joined to framework 7 by yet another pair of functionally-parallel blade flexures 25a forming a parallelogram flexure pivot. Again, the same comments apply.
It should be noted that the exact shape of the sliders 13, 17, 21 and 25 is unimportant and can be adapted to the needs of the geometry of any particular construction.
On this figure, a simple crank arrangement as disclosed in FR1044957 has been illustrated, in order to show how the inertial mass 3 can be induced to oscillate under the effect of torque applied by a motive source to a driving wheel 27a. Alternatively, an escapement-type system can also be used. This applies to all embodiments, but has only been illustrated here in order to avoid over-encumbering the figures.
As a result, the oscillator 1 of this embodiment functions in the same manner as that of figure 9, displacements d ₀ and d ₁ applied to sliders 17 and 25 acting as before.
Figure 11 illustrates a variant of the oscillator 1 of figure 10, incorporating a rotary inertial mass 3 similar to that of figures 2, 5 and 7 instead of a translational mass 3. Only the differences with respect to figure 10 will be described in the following.
Inertial mass 3 is formed as a balance-type rotary mass, supported by a pair of blade flexures 5c arranged at right angles to one another in the plane of the oscillator 1. These blades 5c constitute a Remote Centre Compliance (RCC) flexure pivot, and also provide a restoring force for the oscillations of inertial mass 3. As a result, they constitute both the spring 5 and bearing 9 for the mass 3, and ensure that it can oscillate about its virtual centre of oscillation O. Other forms of flexure pivots are of course possible.
Substantially rigid bar 15 is attached by one of its terminal flexures to the mass 3 at a convenient point, this flexure being oriented towards the axis of rotation O when the mass 3 is in a neutral position.
The bearing 9 joining the slider 13 to the frame 7 is illustrated as being a parallelogram flexure rather than a single flexure as in figure 10, as was described as one of the non-illustrated variants for this latter.
In the illustrated embodiment, a conventional escapement 29 has been schematically illustrated, powered by an escape wheel 29a itself driven by a source of energy such as a mainspring. However, a crank system as illustrated in the context of figure 10 can equally be applied.
Otherwise, the system is the same and functions in exactly the same manner other than the oscillations of the mass 3 being in rotation rather than in translation. Furthermore, the same variants in respect of the nature of the various bearings 9 of figure 10 apply equally to the embodiment of figure 11.
Experiments were carried out with a variant of the oscillator of figure 10, in which the bearing 9 connecting the slider 13 to the framework 10 was a parallelogram flexure pivot as in figure 11. The stiffness to which the inertial mass 3 is subject along the x axis is assumed as: $k = β_{0} + β_{1} x^{2} + β_{2} x^{4} + β_{3} x^{6}$
Displacements of d ₁ = p[mm] and d ₀ = 0 were applied by means of slider 25 and slider 17 respectively, which gave the inertial mass a relatively long stroke (approximately +/- 5 mm for the oscillator referenced below) within its substantially linear stiffness region, with the displacement at which this long linear stroke occurs being referred to as d ₁ = p*.
Figure 12 illustrates graphs plotting return force against displacement of the inertial mass for various values of p for an oscillator dimensioned as explained in [Zanaty, Mohamed, Ilan Vardi, and Simon Henein. "Programmable multistable mechanisms: Synthesis and modelling." Journal of Mechanical Design 140.4 (2018): 042301.]
From these graphs, it is clear that adjusting the value of p does not alter the slope of the substantially linear portion of the graph, which remains at 70 mN/mm, while the higher order stiffness terms can be modified, which tunes the isochronism without affecting the linear stiffness region and hence without affecting the nominal frequency. Furthermore, from these experiments it was clear that:

The term β ₀ is constant and independent of d ₁.
The term β ₁ is a function of d ₁ and is zero at d ₁ = p* where p* is a function of the mechanism dimensions
The term β ₂ is zero.
The term β ₃ is constant.

Although the invention has been described in terms of specific embodiments, variations thereto are possible without departing from the scope of the invention as defined in the appended claims.

Claims

Mechanical oscillator (1) for a timepiece comprising an inertial mass (3) arranged to oscillate about a neutral position under the effect of a restoring force provided by at least one elastic element (5),
characterised in that said mechanical oscillator (1) further comprises a compensation spring (11) pivotably linked to said inertial mass (3) by means of a substantially rigid first connecting bar (15), said compensation spring (11) and said first connecting bar (15) being arranged so as to apply a compensating force to said inertial mass (3) which varies in function of displacement of said inertial mass (3) from its neutral position.
Mechanical oscillator (1) according to the preceding claim, wherein said inertial mass (3) is arranged to oscillate in translation along an axis of translation (x).
Mechanical oscillator (1) according to the preceding claim, wherein said substantially rigid first connecting bar (15) is arranged such that its line of action is substantially perpendicular to said axis of translation (x) when said inertial mass (3) is in its neutral position.
Mechanical oscillator (1) according to claim 1, wherein said inertial mass (3) is arranged to oscillate in rotation about an axis of rotation (O).
Mechanical oscillator (1) according to the preceding claim, wherein said substantially rigid first connecting bar (15) is arranged such that its line of action is aligned with said axis of rotation (O) when said inertial mass (3) is in its neutral position.
Mechanical oscillator (1) according to any of claims 1-5, wherein said compensation spring (11) is attached to a slider (13) arranged to translate along a respective axis, said slider (13) being pivotally linked to said substantially rigid first connecting bar (15).
Mechanical oscillator (1) according to claim 6, wherein said compensation spring (11) is arranged such that its pre-stress is adjustable.
Mechanical oscillator (1) according to claim 7, wherein said compensation spring (11) is attached between said slider (13) and a further slider (17) which is arranged movably.
Mechanical oscillator (1) according to one of claims 6-10, further comprising a further compensation spring (23) pivotally linked to said slider (13) by a substantially rigid second connecting bar (19) pivotally attached to said slider (13) such that the line of action of said second connecting bar (19) is substantially perpendicular to the line of action of said first connecting bar (15) when the inertial mass (3) is in its neutral position, said further compensation spring (23) being arranged to apply a further compensating force to said slider (13) which varies in function of the displacement of said slider (13) from its neutral position.
Mechanical oscillator (1) according to claim 9, wherein said further compensation spring (23) is arranged such that its pre-stress is adjustable.
Mechanical oscillator (1) according to claim 10, wherein said further compensation spring (23) is attached between a further slider (21) and a yet further slider (25), said yet further slider (25) being arranged movably.
Mechanical oscillator (1) according to any preceding claim, wherein said elastic element (5) is a flexure pivot arranged to support and to guide said inertial mass (3).
Mechanical oscillator (1) according to any preceding claim, wherein said compensation spring (11) is formed as a flexure pivot arrangement.
Mechanical oscillator (1) according to one of claims 9-11 or to one of claims 12-13 as dependent upon one of claims 9-11, wherein said further compensation spring (23) is formed as a flexure pivot arrangement.
Timepiece movement comprising a mechanical oscillator (1) according to any preceding claim and a crank arrangement (27) or escapement (29) arranged to cause said inertial mass (3) to oscillate.