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Research Papers

# Design of a Compact Gravity Equilibrator With an Unlimited Range of MotionOPEN ACCESS

[+] Author and Article Information
Bob G. Bijlsma

Faculty of Mechanical, Maritime and
Materials Engineering,
Delft University of Technology,
Delft 2628 CA, The Netherlands;
InteSpring B.V.,
Delft 2629 JD, The Netherlands

Faculty of Mechanical, Maritime and
Materials Engineering,
Delft University of Technology,
Delft 2628 CA, The Netherlands;
InteSpring B.V.,
Delft 2629 JD, The Netherlands
giuseppe@intespring.nl

Just L. Herder

Faculty of Mechanical, Maritime and
Materials Engineering,
Delft University of Technology,
Delft 2628 CA, The Netherlands
e-mail: j.l.herder@tudelft.nl

Manuscript received September 2, 2016; final manuscript received July 28, 2017; published online September 18, 2017. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 9(6), 061003 (Sep 18, 2017) (9 pages) Paper No: JMR-16-1260; doi: 10.1115/1.4037616 History: Received September 02, 2016; Revised July 28, 2017

## Abstract

A gravity equilibrator is a statically balanced system which is designed to counterbalance a mass such that any preferred position is eliminated and thereby the required operating effort to move the mass is greatly reduced. Current spring-to-mass gravity equilibrators are limited in their range of motion as a result of constructional limitations. An increment of the range of motion is desired to expand the field of applications. The goal of this paper is to present a compact one degree-of-freedom mechanical gravity equilibrator that can statically balance a rotating pendulum over an unlimited range of motion. Static balance over an unlimited range of motion is achieved by a coaxial gear train that uses noncircular gears. These gears convert the continuous rotation of the pendulum into a reciprocating rotation of the torsion bars. The pitch curves of the noncircular gears are specifically designed to balance a rotating pendulum. The gear train design and the method to calculate the parameters and the pitch curves of the noncircular gears are presented. A prototype is designed and built to validate that the presented method can balance a pendulum over an unlimited range of motion. Experimental results show a work reduction of 87% compared to an unbalanced pendulum and the hysteresis in the mechanism is 36%.

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## Introduction

The purpose of a gravity equilibrator is to counterbalance a weight [1]. The advantage of such a system is that the operating effort to move the mass is greatly reduced [2]. This property is based on the principle of constant potential energy. Normally, a system tends to converge to a state where the potential energy is at a relative minimum. When a system exhibits constant potential energy, there is no such state. The result is that any preferred position is eliminated. In order to achieve constant potential energy, a countermass or a spring can be added to the system. A spring is preferable over a countermass in terms of weight, dimensions, and inertia forces [3].

Commonly, extension springs are used to balance a weight. Most gravity balancers make use of zero free-length springs. This type of spring, of which the force is proportional to its length rather than to its elongation, is not commonly available as an off-the-shelf component. Alternative concepts that make use of more conventional type of extension or compression springs have been presented. Some of them accept that without zero free-length springs, the balance is in principle not perfect, e.g., see Ref. [4]. Others use cam and pulley-based solutions to be able to design the exact required torque to balance the weight [5,6]. Others achieve the balance by systems of linkages and sliders [7,8].

A common drawback of systems that store the energy in extensional or compression springs is that the volume they occupy changes according to the spring loading. Moreover, often such a spring, when loaded, expands into the working area of the mechanism or into space that could be available for storage. This is the case when, e.g., the weight to be balanced is that of a cabinet door or the wall of a foldable container [9].

The employment of torsion springs represents an opportunity to circumvent the named drawbacks. Torsion springs, and particularly torsion bars, are advantageous because they occupy the same volume when loaded and when unloaded. Moreover, they can be positioned along the hinge line of a pendulum, such that they do not invade the useful workspace. In literature, there are not many examples of static balancing using torsion springs.

Claus [10] designed a one degree-of-freedom bottom hinged gravity equilibrator where a cluster of torsion bars is placed near the pivot point of the rotating mass, such that the working space of the pendulum is free of components. Radaelli et al. [11] applies similar principles with multiple prestressed torsion bars to obtain a multilinear approximation of the moment characteristic. Osch [12] applies a nonlinear transmission between the cluster of bars and the mass. This transmission is established by a double-cam transmission, which consists of two cams with varying radii that are placed side by side. A string wrapped around both cams transmits the torque. The main disadvantage of these designs is that the rotating mass is limited in its range of motion, which is $0≤θin≤π/2$ rad. For a larger range, the radii of the cams are converging to the extrema; one radius becomes zero and one radius becomes maximal. This will result in extreme forces on the cams.

The goal of this paper is to present a new gravity equilibrator that is able to statically balance a rotating mass over an unlimited range of motion, which can be made more compact than the state of the art. The equilibrator stores the energy efficiently in torsional springs positioned along the hinge line of the pendulum. A reciprocating loading and unloading of the springs corresponding to a continuous rotation of the pendulum is obtained by a planetary set of nonlinear gears. The conceptual solution is validated by measurements on a rapid-prototype construction. The paper is based on the graduation work of the first author. More details can be found in Ref. [16].

## Method

The gravity equilibrator from Fig. 1 contains a mass and an energy storing element, i.e., a spring. The mass m rotates at a distance L from the pivot point and is connected to the input axis. The position of the mass is defined by the angle θin. The height h of the mass is measured from the datum plane h = 0. This reference plane can be chosen arbitrarily. The spring has a stiffness k and its deformation is defined by the angle θout.

The principle of constant potential energy is used to design the gravity equilibrator. It states that the sum of the potential energies in the system must be constant for all configurations Display Formula

(1)$Vg(θin)+Ve(θout)=Vpot(θin,θout)=C$

where Vg is the potential energy due to gravitational forces and Ve is the potential energy due to elastic forces.

The potential energy of the mass is calculated by Display Formula

(2)$Vg(θin)=mgh(θin)$

The potential energy that the spring stores depends on the stiffness k, the angle θout, and the initial angle θ0 and is derived by Display Formula

(3)$Ve(θout)=12k(θout+θ0)2$

The elastic energy is assumed to be zero when the spring is not rotated. Internal tensions in the material are not taken into account as they do not contribute. The angle θ0 is, therefore, equal to 0. Equations (2) and (3) are substituted in Eq. (1), which yields Display Formula

(4)$mgh(θin)+12kθout2=C$

The parameter h is chosen such that the height of the mass is equal to zero when the pendulum is at π rad. Therefore, h equals $L(1+cos(θin))$. Rearranging Eq. (4) gives θout as a function of θinDisplay Formula

(5)$θout=2mgLk·1−cos(θin)$

Equation (5) gives the relation between θin and θout such that the rotating mass is balanced for every angle, see Fig. 2.

For increasing θin, θout first increases to a maximum and after that decreases to its initial value. It can be stated that in order to achieve a perfectly balanced system, the input axis has to make a continuous rotation while the output axis has to make a reciprocating rotation.

###### Design Requirements and Limitations.

Design requirements and limitations are determined in order to specify the research.

Requirements:

• The pendulum has an unlimited range of motion. This means that the angle θin can take on any real value.

• The transmission mechanism converts a continuous rotation of the input into a reciprocating rotation of the output as depicted in Fig. 2.

• The pendulum is statically balanced over its entire range of motion.

Boundary conditions:

• The position of the pendulum is defined by a single degree-of-freedom, angle θin. This is visualized in Fig. 1.

• The system is analyzed in a quasi-static state. The system is moving with negligible velocities. The change in kinetic energy and dynamic forces are omitted.

• Preliminary research have resulted that only mechanisms with a mechanical transmission using gears are investigated.

• The mechanism is separated into three subsystems; the load element, the transmission element, and the energy storage element. It is assumed that none of these subsystems will be combined. Energy storing elements with a specific shape that replace the transmission element are outside the scope of this research.

###### Evaluation Criteria.

The mechanism is analyzed by two different criteria. These criteria are a measure of the performance of the mechanism:

• The first criterion is the balancing quality. This is defined by the work Wbal that is required to rotate the balanced pendulum over the work Wunbal to rotate an unbalanced pendulum. The work is determined by taking the Riemann sum of the absolute values of the moment Mbal and Munbal. The criterion is expressed in the efficiency ηbal and is determined by Display Formula

(6)$ηbal=(1−WbalWunbal)×100%$

• The second criterion is the hysteresis. A quantification of this hysteresis is given by the ratio of the amplitude of the moment Mhys that is lost due to hysteresis over the amplitude of the moment Munbal that the unbalanced pendulum exerts Display Formula

(7)$Rhys=MhysMunbal×100%$

## Conceptual Design

In order to achieve perfect balance over an unlimited range of motion a transmission is introduced. This transmission converts a continuous rotation of the input axis into a reciprocating rotation of the output axis as depicted by Fig. 2. Figure 3 is a schematic representation of a transmission that satisfies the first two requirements stated in Sec. 2.1 and is adapted from Litvin et al. [17]. This transmission consists of two gear sets. Gear set 1, see Fig. 3(b), consists of two gears with constant pitch curves. Gear set 2, see Fig. 3(c), consists of two gears with changing pitch curves, called noncircular gears. They are represented by oval gears. The relation between the angular velocities ωin and ωout of the mechanism is derived in two steps, the elaboration of gear set #1 and #2.

###### Gear Set #1.

The first gear set consists of two gears which are both circular. There are three angular velocities to be distinguished, ωin, ω1, and ω2, which are the angular velocities of the rotating mass, gear 1, and gear 2, respectively. Gear 1 is rigidly connected to the world, such that ω1 is 0 rad/s. Gear 2 is connected to the arm R which rotates with angular velocity ωin, defined positive counter-clockwise. Gear 2 meshes with gear 1, and because the latter is fixed, gear 2 is forced to rotate. The angular velocity of gear 2 is Display Formula

(8)$ω2=(1+r1r2)·ωin$

where r1 and r2 are the radii of gear 1 and gear 2, respectively. These radii are both constant and positive.

###### Gear Set #2.

The connection between the first and the second gear set is established by gear 2 and 3, which are rigidly connected such that Display Formula

(9)$ω3=ω2$

Gear 4 is connected to the output axis. The angular velocity ωout of gear 4 is a weighted combination of ωin and ω3. The factor of ωin is derived in the same way as was done for the configuration of the first gear set. The factor of ω3 is derived by the ratio $r3/r4$ between the two gears. The derivation of ωout is Display Formula

(10)$ωout=(1+r3r4)·ωin−r3r4·ω3$

By substituting Eqs. (8) and (9) in Eq. (10), ωout is expressed as a function of ωinDisplay Formula

(11)$ωout=(1−r1·r3r2·r4)·ωin$

The transmission ratio is defined by the ratio between ωout and ωinDisplay Formula

(12)$ωoutωin=1−r1·r3r2·r4$

The fraction of the four radii can be chosen in such a way that $ωout/ωin$ is positive, negative, or equal to zero. When the fraction of radii is smaller than one, the transmission ratio is positive and θout increases for increasing θin. A negative transmission ratio is obtained when the fraction of radii is larger than one and θout decreases for increasing θin. When the combination of radii is equal to one, the output angle is stationary for increasing θin.

In this mechanism, the second gear set consists of two noncircular gears. The radii r3 and r4 are variable. With changing radii, the transmission ratio also changes. The radii are changed such that the output angle θout is tuned. To achieve perfect balance, θout first has to increase from zero to a certain angle and then decreases to zero as depicted in Fig. 2.

###### Concept Variations.

The presented concept transforms a continuous rotation of the input axis into a reciprocating rotation of the output axis. A coaxial system like a planetary gearbox could result in a more compact solution with respect to the state of the art. In the presented configuration, however, this advantage in not fully exploited. From Fig. 3(b), it is seen how gear 2 rotates around fixed gear 1. The enveloping diameter that the mechanism covers is determined by the radii of the gears and is equal to $2·(r1+2r2)=2r1+4r2$.

To obtain a smaller mechanism, different concept variations are shown in Fig. 4. In Fig. 4(a), only external gears are used. In Fig. 4(b), gear 1 is replaced by a gear with internal teeth. The advantage of the internal gear is that gear 2 can rotate inside gear 1, which results in a smaller diameter of the mechanism. In Fig. 4(c), this idea is used for gear 4; gear 3 can rotate inside gear 4. Figure 4(d) shows the variation where both gear 1 and gear 4 are replaced by internal gears.

The transmission ratio of the four variations is given in Table 1. The second and third concept variation both have a transmission ratio which cannot be negative, regardless of the chosen radii. Therefore, these variations are excluded from the feasible concept variations. Concept 4(d) has a transmission ratio which can be positive and negative. The enveloping diameter of this mechanism is $2r1$. Because the diameter of this configuration is smaller than the diameter of the representation in Fig. 3(a), this concept is chosen for further elaboration.

## Dimensional Design

With concept 4(d) chosen, the design is worked out in detail. This process is divided in the calculation of the pitch curves of gear set #1 and gear set #2, the extension from the two-dimensional plane to the three-dimensional space, the determination of the exact teeth profiles and the calculations on the energy storing element.

###### Pitch Curves of Gear Set #1.

Gear set #1 consists of two gears with constant pitch curves. Gear 1 has internal toothing and is fixed. Gear 2 has external toothing and rotates inside gear 1. The relation between ω2 and ωin is Display Formula

(13)$ω2=(1−r1r2)ωin$

In order to achieve a periodic rotation, the fraction $r1/r2$ has to be an integer and $r1>r2$. The choice of the radii is arbitrary.

###### Pitch Curves of Gear Set #2.

Gear set #2 consists of two gears with noncircular pitch curves. The radii of these two gears are determined by differentiating Eq. (5) with respect to time to obtain the angular velocity of the output axis Display Formula

(14)$ωout=dθoutdt=(2mgLk· sin(θin)21−cos(θin))·ωin$

The transmission ratio between ωout and ωin is equal to Display Formula

(15)$ωoutωin=2mgLk· sin(θin)21−cos(θin)$

The transmission ratio of the gear mechanism is already determined in Eq. (12). By combining Eqs. (12) and (15), the relation between the radius r3 and θin is derived Display Formula

(16)$1−r1·r3r2·r4=2mgLk· sin(θin)21−cos(θin)$

Because the teeth of gear 3 and 4 have to mesh, r4 can be expressed as $r4=r3+R$, where R is the length of the arm as depicted in Fig. 3(b). With θin ranging from 0 to $2π$, r3 and r4 are derived. The pitch curves of the four gears are visualized in Fig. 5. The radius of gear 1 is twice the radius of gear 2. The radius of gear 3 ranges between two extrema as well as the radius of gear 4.

Figure 6 visualizes the relative positions of the four gears for one rotation of the input axis. Six positions are shown and these correspond to the six references in Fig. 6(a). Figure 6(b) shows the initial position; θin (blue line) and θout (green line) are both 0 rad. Figures 6(c) and 6(d) show the positions where the pendulum is rotating. In Fig. 6(e), the pendulum is at its lowest point and gear 4 is rotated to its maximal angle. While θin continues to increase, θout decreases again. position where θin has made a rotation of $2π$ rad and θout has made a reciprocating rotation.

The factor $2mgL/k$ from Eq. (5) influences the magnitude of the radii of gear 3 and 4. θout is related inversely linear to the stiffness k. With a small stiffness, the torsion bars have to be rotated over a larger angle θout such that enough potential energy can be stored. To achieve this larger angle, the fraction of radii from Eq. (11) has to change over a larger range. This can only be achieved by expanding the range of radii r3 and r4.

The opposite holds for a large stiffness. A small angle θout is required to store the potential energy. The mechanism becomes more sensitive to design errors, e.g., design tolerances or backlash. The factor is chosen such that the best compromise is made. Iterative calculations result in a factor, where $k=15·mgL$. With this factor, the maximal radius of gear 3 is not too large and the angle of rotation of the torsion bars is not too small.

###### Three-Dimensional Design.

From Fig. 5(b), it is seen that the pitch curve of gear 3 makes a rotation of $4π$ rad. This is independent of the fraction of radii. If the pitch curve is designed in the two-dimensional plane, the pitch curves of gear 3 and gear 4 intersect, and the gear train cannot function. Therefore, it is necessary to build the gears in the three-dimensional space with a pitch that is equal or larger than the width of the teeth such that interference is avoided.

###### Generation of Teeth Profile.

The teeth profiles are derived by the rack-cutter method [18]. This method generates the teeth profile by using a rack that unrolls around the pitch curve. Because the pitch curves of gear 1 and 2 have a constant radius, all teeth are equal.

The generation of the teeth on the noncircular gears differs from the circular gears. These gears have changing pitch curves and the shape of the teeth depends on the curvature of the pitch curve. Therefore, each tooth is different [19]. In Fig. 7, the teeth are shown on the four pitch curves.

###### Energy Storing Elements.

A torsion bar is chosen as the energy storage element because torsion bars have a high energy to volume ratio. The bar has only one large dimension which is the length. A rectangular torsion bar can be clamped to create a form closed connection such that a torque can be applied without the need of additional measures to prevent slip. The stiffness is [20] Display Formula

(17)$k=n·β·G·b·ts3l$

where n is the number of torsion bars, β is the shape coefficient, G is the shear modulus, b, ts, and l are the width, thickness, and length of the torsion bar, respectively.

The parameters n, b, ts, and l are given in Table 2 and are chosen such that enough energy can be stored to balance the mass, but the rotation of the spring is within limits.

## Prototype

A prototype is built in order to validate the design of the gravity equilibrator. An overview of the prototype together with the main components is shown in Fig. 8. The design of the prototype and the evaluation of the preliminary experiments are treated.

###### Design.

The gears are made of polyamide which is used for fully functional prototypes with high mechanical qualities and are constructed with a rapid prototyping technique called selective laser sintering (SLS).

Figure 8(a) gives the full picture of the prototype. The mass connected to the arm is visible in the foreground. On the back the torsion bars are visible. Figure 8(b) shows internal gear 1, which is rigidly connected to the ground. In Fig. 8(c), gears 2 and 3 are shown which are made up from one single part. Figure 8(d) shows gear 4 with internal toothing and which is connected to the torsion bars. Figure 8(e) shows the input axis and the structure that constraints gear 2 and 3 and supports gear 4.

###### Measurement Results.

The prototype is tested on a universal testing machine M250-2.5 CT of Testometric. The testing machine can make a vertical translation and measure the pulling force. A pulley is placed at the pivot point such that the testing machine can exert a moment on the pendulum. When the pendulum is perfectly balanced, the required moment to move the pendulum fluctuates around 0 N·m. Because the cable between the pulley and the testing machine can only be subjected to a pulling force, a counterweight is added to the mechanism. This counterweight is a mass of 5 kg that is connected via a cable to the pulley, which causes a constant moment on the input axis. A schematic overview is shown in Fig. 9.

Three consecutive measurements are performed of the total system. The input axis is rotated $4π$ rad counterclockwise and $4π$ rad clockwise. The constant moment exerted by the counterweight is subtracted from the measurement values. The results of these measurements are shown in Fig. 10, where the angle θin is located on the horizontal axis and the moment is located on the vertical axis.

The behavior of the gear transmission is measured separately by removing the pendulum and the torsion bars. The results of these measurements are shown in Fig. 11. The data are smoothed in order to filter out the peaks that occur due to the high measurement frequency. The sinusoidal line (green) in Fig. 10 represents the mean value of the moment of the unbalanced pendulum, which is used to determine the efficiency. Furthermore, the average required moment of the three measurements is shown (red), which is the hysteresis loop. The mean value between the upper and lower part of this hysteresis loop is plotted within the loop (black). This line gives an estimation of the required moment if friction were absent. The assumption is made that the magnitude of the friction for the way up and way back is equal at every position.

The work done by the balanced pendulum is derived by determining the area under the curve of the absolute mean moment that is required to rotate the mass over $4π$ rad. This area is calculated with the Riemann sum which takes the sum of areas of approximating rectangles with width $Δθin$ and height Mres [21] Display Formula

(18)$Wbal=∑i=1n|Mbal,i|·Δθin$

where $Δθin$ is 0.01 rad and n is equal to $4π/0.01$.

The work done by the unbalanced pendulum is derived in the same manner Display Formula

(19)$Wunbal=∑i=1n|Munbal,i|·Δθin$

Using Fig. 10, the results are

$Wbal=1.5 J$
$Wunbal=11 J$

Substituting the obtained values in Eq. (6), the efficiency is derived Display Formula

(20)$ηbal=(1−1.5 J11 J)×100%=87%$

The hysteresis loop in Fig. 10 shows that a certain amount of losses are present in the system. To give a quantitative measure to this hysteresis, the peak-to-peak amplitude of Mhys and the peak-to-peak amplitude of Munbal are determined. From 0 to $4π$ rad, the peak moment to move the balanced pendulum is 0.5 N·m. On the way back from $4π$ to 0 rad, the peak moment is −0.5 N·m. The hysteresis is, therefore, 1.0 N·m. The peak-to-peak amplitude of the moment from the unbalanced pendulum is equal to 2.8 N·m. With Eq. (7) the ratio of the hysteresis is derived Display Formula

(21)$Rhys=1.02.8×100%=36%$

## Discussion

During assembly, problems were encountered regarding the alignment and tolerances of several components.

The gears are manufactured by SLS. The accuracy of this manufacturing method is $±0.3 mm$. During the design of the gears, the tolerance was not taken into account. This resulted in gears that were slightly different than intended. The teeth of the gears did not mesh as was intended. When the gears were assembled, they were pressed against each other causing high friction between the gears. This problem is solved by milling a small amount from each tooth such that the gears mesh better. This had a positive effect on the friction.

A possible solution to overcome this problem in the future is to account for the accuracy of this method by scaling the gears. In this case, the gears are too large, so when the gears are scaled down, the teeth have less friction when assembled.

The accuracy of other rapid prototyping manufacturing methods is in the same order of magnitude as SLS.

The hysteresis in the system is considered to be large when compared to the maximal moment that the unbalanced pendulum exerts in the pivot point. It is assumed that this hysteresis is mainly caused by the friction in the system itself because only one extra pulley and one cable are added to make a connection between the system and the testing machine. In order to determine which components cause this friction, the components have been measured separately. The average hysteresis measured in Fig. 11 in the unloaded gear train is approximately 0.07 N·m, but peaks up to 0.2 N·m are measured. This friction is caused by the teeth that mesh with each other. The points where the friction is larger, the teeth press harder against each other.

A measurement with the gear train and the pendulum gave the same hysteresis as when the pendulum was not attached, so it is concluded that the pendulum itself does not account for a significant amount of hysteresis.

At the moment that the balanced pendulum is rotated, the hysteresis is 0.66 N·m as calculated in Eq. (21). The extra hysteresis of 0.59 N·m can probably be assigned to the teeth that are now loaded, causing the teeth to press harder against each other.

When taking the gear train apart, it is seen that the component from Fig. 8(e) slides against gear 1 and 4. This sliding causes friction that accounts for a part of the 0.59 N·m.

When the pendulum is at $π/2$ rad, gear 4 slides against the housing of gear 1. This sliding accounts also for a part of the hysteresis. This is also visible in Figs. 10 and 11, where the peak-to-peak amplitude of the hysteresis loop is larger.

The hysteresis in the system is caused not only by dimensional problems, such as the sliding of the components against each other, but also by the friction between the teeth. The solution may lie in the manufacturing method and the employed material.

A hard steel machined gear would probably suffer much less from friction but also result in higher admittable loads and longer endurance. This type of requirements is specific to the application scenario. Evaluating the real industrial applicability by means of an improved construction is left as a recommendation for future research.

During the dimensional design, the pitch curves of the noncircular gears were obtained. The pitch curves of these gears have a discontinuity at the moment that the pendulum is at 0 rad. It was not possible to smooth this jump such that the pitch curves would be continuous again. This is explained by expanding Fig. 2, see Fig. 12. Angle θout changes discontinuously when θin is $i·2π$ rad, with $i=Z$.

This discontinuity in θout is caused by $1−cos(θin)$ in Eq. (5). θout can never become negative but has the largest steepness at the moment that $1−cos(θin)$ is zero. A discontinuity of the pitch curve can result in high jerk and associated forces, vibration, and noise. This is especially true in systems with high speeds of revolution. In quasi-static motion, there were no such effects observed.

One may say that θout can also be negative, because the same energy is achieved when the torsion bars are rotated in the other direction. When this is implemented, the pitch curves of gear 3 and 4 become indeed continuous, see Fig. 13. The problem that now occurs is that there are four loops present in gear 3 and two loops in gear 4. It becomes virtually impossible to create gears that can function without intersecting.

## Conclusions

A gravity equilibrator is designed that can theoretically balance a rotating pendulum with a single degree-of-freedom over an unlimited range of motion using a geared transmission and torsion bars. A general method is presented to calculate the shape of the pitch curves of the noncircular gears such that perfect balance is achieved. The shape can be adjusted by choosing different parameters for the mass m, the length of the arm L, and the stiffness of the torsion bars k. Different masses can be balanced by changing the active length of the torsion bars or the number of torsion bars. Prelimenary experiments result in a work reduction of 87% over a rotation of $4π$ rad when compared to an unbalanced pendulum. The hysteresis in the mechanism is 36%. Other manufacturing methods and improvements on the prototype are needed to reduce the hysteresis in the system.

## Funding Data

• Ministerie van Economische Zaken (Grant No. KWR09086).

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Boresi, A. , and Schmidt, R. , 2003, Advanced Mechanics of Materials, 6th ed., Wiley, New York.
Stewart, J. , 2003, Calculus, Thomson Brooks/Cole, Pacific Grove, CA.
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## References

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## Figures

Fig. 1

An exploded view of the different components to distinguish the parameters of the gravity equilibrator. On the left, a torsion spring of which the angle of rotation is defined by θout, and on the right, a rotating mass on an arm, that is, defined by the angle θin.

Fig. 2

The relation between θin and θout to obtain a statically balanced system. For increasing θin, θout first increases and then decreases again.

Fig. 3

A schematic representation of the gear train: (a) a three-dimensional representation of the gear train to visualize the topology, (b) the first gear set which consists of two circular gears, and (c) the second gear set which consists of two noncircular gears, visualized by oval gears

Fig. 7

The pitch curves of the four gears with the teeth: (a) the pitch curve of gear 1 and the internal teeth, (b) the pitch curve of gear 2 and the external teeth, (c) the pitch curve of gear 3 and the external teeth, and (d) the pitch curve of gear 4 and the internal teeth

Fig. 5

The pitch curves of (a) gear set #1 and (b) gear set #2. (a) The outer gear (black) is gear 1, the inner gear (yellow) is gear 2. (b) The inner gear (red) is gear 3 and the outer gear (blue) is gear 4. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 4

The four possible variations based on the original concept: (a) the original concept where all four gears have external toothing, (b) the concept where gear 1 is replaced by a gear with internal toothing, (c) the concept where gear 4 is replaced by a gear with internal toothing, and (d) the concept where both gear 1 and 4 are replaced by gears with internal toothing

Fig. 10

The measurement results of three consecutive measurements from 0 to 4π rad and back. The hysteresis loop is shown (red) together with the mean value of the hysteresis loop (black). Also, the mean value of the moment exerted by the unbalanced pendulum (green) is shown for reference. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 8

Pictures of the prototype: (a) a general overview, (b) fixed internal gear 1, (c) gear 2 and gear 3 combined, (d) output gear 4, and (e) the structure that connects the multiple parts

Fig. 6

A visualization of the positions of the gears at different moments. These moments are referred to in (a): (a) the graph where the angle θin and θout are plotted, (b) the initial position where θin and θout are zero, (c) and (d) other moments where both θin and θout are increasing, (e) at this moment θout is maximal, (f) a moment where θin increases and θout decreases, and (g) the initial position which is equal to (b).

Fig. 9

A schematic overview of the measurement setup. A pulley is attached to the pendulum. Two different cables are attached to this pulley. One of these cables is connected to the testing machine and one cable is connected to a counterweight of 5 kg.

Fig. 13

The pitch curves when the angle θout can also become negative. It is seen that the pitch curves of gear 3 (red) and gear 4 (blue) are now continuous but have multiple loops. This configuration of gears is impossible without having intersecting gears. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 11

The hysteresis in the gear transmission which is not loaded by the torsion bars. Only the counterweight is added to the mechanism for the measurements. The hysteresis in the unloaded system is around 0.

Fig. 12

Angle θout changes discontinuously when the pendulum is in the upright position. This results in a jump in the pitch curves of the noncircular gears 3 and 4. This discontinuity is caused by the required energy that the spring has to absorb to achieve perfect balance.

## Tables

Table 2 The parameters of the torsion bars that are required to balance a mass
Table 1 Transmission ratio of the four concept variations given in Fig. 4, where gear 3 and 4 are noncircular gears

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