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A water drinking bird or simply drinking bird (DB) is discussed in terms of a thermomechanical model. A mathematical expression of motion derived from the thermomechanical model of a drinking bird and numerical solutions are explicitly shown, which is helpful in understanding physical meanings and fundamental difference between mechanical and thermomechanical periodic motion. The mathematical and physical differences between mechanical and thermomechanical motions are clearly examined, resulting in time-independent and time-dependent coupling constants of equations of motion and continuous transitions between bifurcation solutions. The thermodynamical and irreversible process of a drinking bird motion could be theoretically examined and practically applied to energy harvesting technologies by way of the current modeling. As an example of irreversible thermodynamics, the thermomechanical model of DB will help understand heat engines manifested from microscopic to macroscopic systems.

A drinking bird (DB) known as dunking bird, insatiable birdie or dipping bird is an interesting toy, which simultaneously demonstrates wonders of thermomechanical phenomena. It is reported that a physicist, A. Einstein, was so fascinated by the drinking bird that he spent for three and one-half months to figure out or enjoy the relation between evaporation-pressure and mechanical work [

The DB toy consists of the upper bulb (head), body tube and lower bulb filled with a highly volatile liquid of dichloromethane (CH_{2}Cl_{2}) shown in

Though a drinking bird is a heat engine, the motion has not been investigated through thermodynamics and theory of irreversible thermodynamics, and in addition, it is not clearly discussed which movements of a DB are respectively mechanical and thermodynamical. One reason can be stated that a physical modeling of a drinking bird has not been clearly shown, and only experimental measurements of motion have been conducted by using modern equipments, such as voltmeter, thermistor, magnetometer and a light shutter of a photovoltaic cell [

A Stirling engine is also a well-known heat engine operated by a cyclic compression and expansion of air at different temperatures, which could be an active research area in spaceship project for energy resources by using nuclear energy. In terms of thermodynamic principles, the drinking bird, Stirling engine and nuclear radiation reactors are considered to be equivalent as “heat engines” [

Macroscopic motions and changes in nature including living things are considered to be consequences of nonlinear dynamical interactions based on energy and entropy [

The DB’s mechanical nonlinear equation of motion has bifurcation solutions, for example, an ordinary periodic oscillation around θ = 0 (see,

We introduced a thermomechanical model of DB, which continuously connects bifurcation solutions and imitates DB’s drinking motion reasonably well; the thermomechanical model clearly exhibits oscillations of lovable drinking bird. We introduce a thermomechanical model of water drinking bird by assuming a constant volatile-liquid velocity, v 0 , in order to imitate DB’s approximate periodic motions. The parameter values of the DB’s nonlinear differential equation of motion are examined in the numerical analysis and solutions are explicitly exhibited. A mechanical model and numerical calculations are explained in Section 2, and a thermomechanical model with thermal driving force and numerical simulations are discussed in Section 3, which clearly shows that DB’s motion is induced by Newtonian mechanics and irreversible thermodynamics. Conclusions and perspectives are discussed in Section 4.

The DB’s drinking period (time from a drinking to next drinking) changes drastically with environmental and mechanical conditions (temperature, humidity, friction of mechanical parts, impacts on the edge of glass, amount of water on the beak, ∙∙∙). Hence, the average period of drinking bird can only be determined roughly. The periods in the beginning are rather long, which may take about 2 minutes with the initial starting angle θ = π / 6 (see, 1(b)), and it becomes roughly converging to 30 - 40 seconds when oscillations are stable, depending on environmental and thermal conditions. All the periods of a drinking bird are not the same as those determined in mechanical pendulums and springs.

The periodic motion of a drinking bird is considered to be composed of two independent motions: a simple back and forth oscillation around θ = 0 and a drinking motion converging to θ = π which is an upside-down oscillation. However, the upside-down oscillation of a drinking bird is mechanically restricted in a real toy, not to go beyond an angle θ ~ π / 2 by the shape of arm and a hole of axis of rotation (see

The equation of motion for the mechanical drinking bird is constructed from Newtonian mechanics, or Lagrangian dynamics with the rotational kinetic energy and potential energy:

L = 1 2 I 0 θ ˙ 2 + { m 2 g a − m 1 g ( b − l 2 ) − m 3 g b } cos θ . (2.1)

where the moment of inertia is given by the sum, I 0 = I 1 + I 2 + I 3 , which consists of the moment of inertia of head I 3 , glass tube I 1 , and lower bulb I 2 , respectively. Because of the center of mass assumption, the moment of inertia of head and lower bulb are given by I 2 = m 2 a 2 and I 3 = m 3 b 2 . One should note that the mass, m 2 , is the sum of masses of liquid and lower glass bulb.

The glass tube is assumed to have the mass m 1 with length l = a + b , and the moment of inertia is obtained as [

I 1 = m 1 l 2 ( 1 / 3 − ( 1 − b l ) + ( 1 − b l ) 2 ) . (2.2)

The angle θ is measured from the vertical direction as in

I 0 θ ¨ + g l { m 2 a l − m 1 ( b l − 1 2 ) − m 3 b l } sin θ = 0, (2.3)

where I 0 = I 1 + m 2 a 2 + m 3 b 2 .

The velocity-dependent friction term, c θ ˙ (c is a free parameter), is included in the equation of motion for accounting an oscillation damping, and by defining the effective mass, m ∗ , for convenience as:

m ∗ = m 2 a l − m 1 ( b l − 1 2 ) − m 3 b l , (2.4)

the mechanical equation of motion for a DB is written as,

θ ¨ + c θ ˙ + g l m ∗ I 0 sin θ = 0. (2.5)

Now, one can check the nonlinear Equation (2.5) whether it can express the motion of DB reasonably well or not by adjusting parameters, a , l , c , m 1 , m 2 , m 3 ; initial conditions, θ ( 0 ) = π / 6 and θ ˙ ( 0 ) = 0 , are used in numerical simulations. The following parameters are fixed: g = 980 ( cm / s 2 ) , a = 6.0 ( cm ) , l = 13.0 ( cm ) , c = 0.04 ( 1 / s ) .

In the mechanical DB model, one obtains bifurcation solutions: a simple back and forth damping oscillation converging to θ = 0 and an upside-down damping oscillation converging to θ = π , as shown in

Based on the mechanical DB simulations, we remark:

(Remark 1) The imitation and construction of the motion of drinking bird cannot be possible based only on Newtonian mechanics, because the nonlinear equations derived from Newtonian mechanics cannot describe continuous transition between bifurcation solutions.

The shape of a mechanical DB could be more realistic, which corresponds to changing distributions of mass and the moment of inertia. Changing parameters do not produce DB’s periodic motion. Hence, different external force to keep

periodic motion should be essential, which originates from thermodynamic force. The properties of nonlinearity of DB equation make analyses of solutions difficult when thermal effects are included, but it results in a physically and mathematically interesting solution, which will be discussed and numerically shown in the following sections.

We will demonstrate a method to imitate DB’s periodic motion, which is beyond classical Newtonian mechanics as discussed in the previous section. From experimental observation, one can examine that physical parameters of equation are changing with time, because the liquid inside a glass tube (CH_{2}Cl_{2}) is moving upward with time until the drinking bird drinks water. The DB’s mechanical constituents which should become time-dependent would be, respectively, the moment of inertia, I 1 , I 2 , I 3 , masses, m 1 , m 2 , m 3 , and the center of gravity of m 2 fixed at a; the center of gravity of m 2 shifts upward because liquid in m 2 moves upward with time.

The constituents become time-dependent by way of upward moving liquid induced by external thermodynamic force. The velocity of upward moving liquid is not smooth, however, we assume the average velocity of liquid as v 0 , whose appropriate value will be determined by numerical simulations. The mass and moment of inertia, m 1 = 7.0 ( g ) and I 1 given by the Equation (2.2), are fixed with length l = 12.4 ( cm ) for simulations. The DB’s motion is not sensitive to variations of mass m 1 .

Let us suppose that at time t, the volatile liquid (density, ρ l , g/cm^{3}) in the glass tube (cross-section, s, cm^{2}) goes upward with the speed v 0 and length, v 0 t , as in

masses of a glass bulb and liquid, and the density of liquid is specifically given by ρ l ~ 1.336 g / cm 3 , and the cross-section of the inner glass tube is, s = 0.1256 cm 2 (the internal radius ~0.20 cm) [

The mass of liquid is ρ l s v 0 t and the mass at the lower bulb becomes m 2 − ρ l s v 0 t . The moment of inertia for volatile water around the axis of rotation is given by the sum of moments from a − v 0 t to a and a 2 ( m 2 − ρ l s v 0 t ) given by the remaining mass at a:

I 2 ( t ) = ∫ a − v 0 t a r 2 d m + a 2 ( m 2 − ρ l s v 0 t ) ( d m = ρ l s d r ) = ρ l s ( a 3 3 − ( a − v 0 t ) 3 3 ) + a 2 ( m 2 − ρ l s v 0 t ) = a 2 m 2 ( 1 − ρ l s v 0 2 t 2 a m 2 + ρ l s v 0 3 t 3 3 a 2 m 2 ) . (3.1)

Because of time-dependent volatile water distribution, the center of mass of m 2 is shifted from a to x 2 ( t ) with time as:

x 2 ( t ) = 1 m 2 ( ∫ a − v 0 t a r d m + a ( m 2 − ρ l s v 0 t ) ) ( d m = ρ l s d r ) = a − ρ l s ( v 0 t ) 2 2 m 2 = a ( 1 − ρ l s v 0 2 t 2 2 a m 2 ) . (3.2)

One should note that the shift, x 2 ( t ) , should be substituted for a in Equation (2.4), because a in Equation (2.4) means the position of center of mass m 2 at t = 0 , corresponding to x 2 ( 0 ) = a .

When liquid in the glass tube reaches the head, the mass of head, m 3 , changes with time. It is denoted by m 3 ∗ ( t ) as:

m 3 ∗ ( t ) = ( m 3 + ρ l s ( v 0 t − l ) when v 0 t ≥ l m 3 when v 0 t < l (3.3)

resulting in the moment of inertia of the head:

I 3 ( t ) = m 3 ∗ ( t ) b 2 . (3.4)

Now, the time-dependent moment of inertia is expressed as:

I ( t ) = I 0 + I 2 ( t ) + I 3 ( t ) , (3.5)

and the substitution of x 2 ( t ) and m 3 ∗ ( t ) for a and m 3 in (2.4) will yield:

m ∗ ( t ) = m 2 x 2 ( t ) l − m 1 ( b l − 1 2 ) − m 3 ∗ ( t ) b l . (3.6)

The numerical simulations are ready to be performed whether it is possible to produce periodic DB motions by adjusting parameters, m 1 , m 2 , m 3 , v 0 , a , l , and initial starting values are θ ( 0 ) = π / 6 and θ ˙ ( 0 ) = 0 .

By employing the result above, the equation of motion for the thermomechanical DB could be written as:

θ ¨ + c θ ˙ + g l m ∗ ( t ) I ( t ) sin θ = 0 , (3.7)

and the nonlinear differential Equation (3.7) should be compared with the mechanical equation of motion of (2.5). The Equations (2.5) and (3.7) are almost identical, except for time-dependent mass, m ∗ ( t ) and moment of inertia, I ( t ) .

However, one must be careful that although the Equation (2.5) can be derived from Lagrngian (2.1), the Equation (3.7) cannot be derived from Lagrngian (2.1) with m ∗ ( t ) and I ( t ) . A Lagrangian corresponding to (3.7) does not exist, and even if one could formally write down a complicated Lagrangian to obtain (3.7), it breaks the law of energy conservation fundamental for Newtonian mechanics. The energy of DB’s motion is not conserved, or in other words, irreversibly dissipated, because the motion of a DB is an irreversible thermodynamic phenomena. This is the reason why we call the Equation (3.7) as a “thermomechanical model”.

The thermomechanical model can produce DB’s one periodic motion reasonably well, and the thermomechanical DB Equation (3.7) has a continuous solution changing from a standard periodic motion (oscillations around θ = 0 ) to a drinking motion which is converging to the upside down oscillations (oscillations around θ = π ) shown in

When the angle of bird’s water dipping ( θ ≃ π / 2 ) resets volatile liquid in the glass tube, some mechanical properties of a DB are initialized (more or less initialized, because mechanical and thermodynamic state are not the same at all). The initialization in the equation of motion means that time-dependent

quantities become

( x 2 ( t ) , m ∗ ( t ) , I ( t ) ) → ( a , m ∗ , I 0 ) t 1 , (3.8)

at a drinking time t 1 . In order to initialize as (3.8), we use a step-function which is denoted as Unitstep ( t 1 − t ) :

Unitstep ( t 1 − t ) = ( 1 t 1 ≥ t 0 t 1 < t (3.9)

The time of initialization or drinking period, t 1 , is determined from the solution to produce the bird’s drinking angle θ ≃ π / 2 in numerical simulations. By inserting the step-function in the drinking bird calculation, the initialization, ( x 2 ( t ) , m ∗ ( t ) , I ( t ) ) Unitstep ( t 1 − t ) → ( a , m ∗ , I 0 ) t 1 , must be performed in the numerical calculation. The time t 1 indicates a period of bird’s drinking time, and the step-function technique is also used for m 3 ∗ ( t ) in Equation (3.3), when volatile water in the glass tube reaches the bird’s head.

A continuous transition of motion from a normal oscillation to an upside down oscillation is shown in

The period of drinking water is T = 38.8 (second) and the number of oscillation (frequency) is f = 27 in

m 1 ( g ) | m 2 ( g ) | m 3 / m 2 | v 0 ( cm / s ) | l ( cm ) | a ( cm ) |
---|---|---|---|---|---|

7.0 | 14.0 | 0.5 | 0.25 | 12.4 | 6.0 |

The motion of a real DB toy is not accurately periodic like a spring or a pendulum, because initial and starting conditions for each DB’s periodic oscillation are not completely the same; for example, the quantity of volatile water returned in the lower bulb, the effect of impacts of bird’s beak at the edge of glass of water, temperature and humidity of environment are not identical at all. These varied conditions change the speed of volatile water in the glass tube, v 0 , essential for the periodic oscillation, and hence, though all the starting conditions for the real DB’s periodic motions are a little different, sufficiently stable periodic motions are produced.

The periodic thermomechanical DB motions (up to DB’s 3-dipping motions) will be shown in the section, and they are sensitive to changes of parameter values and initialization timings, and one often obtains unexpected upside-down, two- or three-rotation solutions, which makes hard to control to obtain expected periodic solutions; one would need some time to get used to changing parameter values to obtain solutions shown in

In theoretical terms, the DB’s motion is a dissipative and irreversible process, and mechanics and thermodynamics are integrated in the motion. Therefore, the drinking bird is not only fascinating but also scientifically fundamental in order to study fundamental principles. The thermomechanical model (3.7) could produce a useful mathematical expression to imitate DB’s periodic motions, and properties of nonlinear equations with time-dependent coefficients are revealed.

Other mathematical models could be constructed [

Although mathematical properties and structure of solutions are interesting, mathematics is only useful in science as a method of taxonomy in order to describe observations rigorously. The equation of thermomechanical model imitates DB’s periodic motion reasonably well, and it would be useful for investigations of mechanism and related variables for understanding fundamental principles such as energy and entropy and possible electromechanical applications.

Based on the thermomechanical DB simulations, we would like to remark as:

(Remark 2) The construction and duplication of the thermomechanical drinking bird show essentially complementary relations between Newtonian mechanics and thermodynamics to operate natural motions by way of energy, speed of matter, heat and temperature. Whatever principles we could conjecture to explain thermomechanical phenomena, natural phenomena are more fundamental than any consistency of humane logical thought-experiment.

The more we study DB’s motions, the more we realize that it is scientifically interesting and enlightening in terms of physical principles and applications. It would help people in science reflect on physical principles and science, such as energy and entropy, status of mathematics in science, fundamental meanings of applicability and application. The drinking bird also encouraged us to consider fundamental logical structure of science, regarding reproducibility, self-consistency and testability. Reproducibility means exact recordings of experiments (mathematics as taxonomy), correct reproductions of observed data and natural phenomena. Self-consistency means no contradictions within a model as theoretical consistency and calculations. Testability includes experimental and possible theoretical tests as well as specific applications in science; applications are required for testability, which must not be taken for granted. These three criteria are essential to define science.

Although mathematical analyses combined with supercomputers and experimental technologies are helpful and powerful instruments to observe natural phenomena, they correspond to taxonomy in science: exact reproductions, expressions and calculations. One should always have in mind that physical principles and laws to maintain three criteria are fundamental and ultimate entity for science.

It should be emphasized that mathematical properties (nonlinearity, differential equations, ...) and supercomputer simulations of natural phenomena are only for the purpose of taxonomy and correct recordings of observations. Beyond taxonomy, we have to test our models to ultimately understand dynamics and principles of nature, which is the most fundamental property in science. The DB’s periodic motion can be mathematically expressed, however, it is not the end of the problem, the beginning of pursuit of fundamental physics. The end of science or physics is in mind of people who believe that the end of physics exists. Natural phenomena are beyond human imagination.

The theory of DB’s irreversible thermodynamics is fundamental and applicable to any structure considered to be a heat engine from microscopic to macroscopic scales, such as the energy flow of nuclear energy reactors, flow of electric current through semiconductors, thermocouples, chemical reactions and electron transport processes in batteries, energy conversions of biological systems, etc. Hence, it is interesting to conjecture thermomechanical principles, laws and relations to quantum electrodynamics (QED) and quantum hadrodynamics (QHD) [

It is fundamental to check the theory of irreversible thermodynamics, applications and applicability manifested in the DB analysis, and it could be a possible theoretical foundation for new applications to renewable and sustainable energy harvesting technologies [

A drinking bird is a fascinating toy to understand and appreciate possibility of the theory of irreversible thermodynamics, more fascinating than we discussed in the paper. It encourages us to study fundamental physics, mathematics and logic in science, and we realize that this is an appropriate concluding remark in this study.

We would like to acknowledge Dr. J. Denur at Department of Physics, University of North Texas, USA, for his valuable comments on the article.

The authors declare no conflicts of interest regarding the publication of this paper.

Uechi, S.T., Uechi, H. and Nishimura, A. (2019) The Analysis of Thermomechanical Periodic Motions of a Drinking Bird. World Journal of Engineering and Technology, 7, 559-571. https://doi.org/10.4236/wjet.2019.74040