Sliding doors

Portas corrediças

ARTIGOS GERAIS

Sliding doors

Portas corrediças

Roberto De Luca¹

Università degli Studi di Salerno, Fisciano, Itália

ABSTRACT

Looking closely to the mechanism which make sliding doors move, one sees that the motion of the commonly used objects can be understood by solving a problem related to the motion of blocks and pulleys. The system is by itself interesting to be modelled as a problem in Newtonian mechanics, and the solution of the equation of the motion can lead us to estimate the time it takes for a sliding door to open or close.

Keywords: newtonian mechanics, physics education, problem solving.

RESUMO

Examinando em detalhes o mecanismo que produz o movimento das portas corrediças, observa-se que o movimento desses objetos usualmente usados podem ser entendidos resolvendo um problema modelado por blocos e polias. O sistema é per si interessante para ser modelado como um problema na mecânica newtoniana e a solução da equação do movimento pode nos levar a estimar o tempo de abertura e fechamento de uma porta corrediça.

Palavras-chave: mecânica newtoniana, ensino de física, solução de problemas.

1. Introducing the problem

It is a common experience to go through a sliding door when walking in a store and to observe its rather smooth motion. By looking closely to the mechanism responsible for the motion of the door, we notice that it can be schematized by a rather simple system, made of two pulleys and two material points. In order to adopt a simple model for the motion of sliding doors, let us consider a rather common way of mounting them, as shown in Fig. 1.

In Fig. 1 we notice that the two pulleys of radius r, free to rotate about a vertical axis, are connected by a string of negligible mass. Two doors, of mass M_D, follow the horizontal motion of points A and B of this string by means of connecting rods. In order for the string to transmit more easily horizontal motion to the doors in the two opposite senses, for a given angular velocity ω(t) of the pulleys, the weight of both doors is counterbalanced by the normal reactions given by a horizontal guide, on which a rather low friction force is present. In the case we neglected friction in the guides and the moment of inertia of the pulleys with respect to the quantity M_Dr², the relation between the applied moment M(t) and the acceleration a(t) of the right door would be given by the rather simple relation

However, still neglecting the mass of the pulleys, in the case viscous friction between the horizontal guides and the door supports is considered, the above relation is modified as follows

where β is the coefficient of viscous friction and V(t) is the velocity of the right door. We shall derive the above dynamical equation and shall see, in what follows, how to deal with the friction term, which may be useful to calibrate the time of closure (or opening) of the sliding doors.

2. Finding the dynamical equation

In the present section we shall find a schematic representation of the system in Fig. 1, describing the apparatus by which sliding doors are set in motion. Successively, by means of elementary mechanics, we shall derive the equation of the motion of the system.

We start by schematizing the system in Fig. 1 taking the two sliding doors as point particles of mass M_D attached to a weightless string connected to two cylindrical pulleys of negligible mass and radius r, as shown in Fig. 2. When the moment M(t) is applied to one of the pulleys, let us say the right one, the string is made to circulate and the attached masses follow the horizontal motion of the rectilinear portions of the string itself, their abscissa being, respectively, x(t) and -x(t) for the right and left door as in Fig. 2. We further assume that the two point particles move on a plane horizontal surface in such a way that the viscous friction force on the right particle is f_t = -βV(t), being the unit vector in the horizontal direction, while on the left particle, although the analytic expression of the viscous force is formally the same, it has opposite direction, since the velocity of the left particle is opposite to the velocity of the right particle.

By applying Newton's second law to the point particles and by equating to zero all moments on the right pulley, referring to Fig. 3 we have

By dividing by r both members of the first equality in Eq. (3) and by summing all three equations, we get

In this way, we can finally write

which is equivalent to Eq. (2). Notice that, for β = 0, Eq. (5) reduces to Eq. (1). Furthermore, one can easily argue that the above equation is similar to the dynamical equation for a particle moving in a viscous fluid [1]. In Eq. [5], however, we take the forcing term to depend on time. Indeed, by considering M(t) as a decreasing linear function, i.e., setting M(t) = M₀ - At, where M₀ is the applied moment at t = 0 and A is the absolute value of the slope of M(t), we write

so that a solution to Eq. (5) can be found in a similar way as in Ref. [1]. We assume that the applied moment acts up to time t = τ, at which the doors reach their full aperture d with zero velocity, or

Furthermore, in order to define the initial conditions for the differential equation (5), we take

3. Solving the dynamical equation

We are now ready to solve the dynamical equation (5). Indeed, by substituting Eq. (6) into Eq. (5) we first write

so that the general solution for V(t) can be found [2]

where the constant c can be evaluated by imposing the initial condition (8), giving c = 0. By now evaluating the integral on the right hand side of Eq. (10), we have

Let us now impose the condition V(τ) = 0 in Eq. (7), so that

We now derive the position x(t) of the point particle with respect to its initial position x(0), by further integrating Eq. (11) with respect to time, so that

4. Calculating characteristic properties of the system

In the previous section we have completely solved the dynamical problem of opening (or closing) sliding doors, assuming that the friction force on them had a viscous character. In the present section we shall calculate the time in which this process is accomplished.

Let us first evaluate the expression for the position of the door at t = τ and set

Because of Eq. (12), we can rewrite Eq. (14) as follows

We can now solve the algebraic Eq. (15) for τ, obtaining the following solutions

The largest solution is the one sought, since it gives a finite value of τ for very small β. In this case, indeed, which we shall consider to be real (a door with much too friction would be too difficult to open!) we can make a third order expansion of the exponential function in Eq. (12) and write

where we shall retain only terms in which β appears to second order or less. In this way

By setting to zero the term in parenthesis and by solving, to first order in βτ/M_D, the resulting algebraic equation, we obtain

We may notice that this closure time τ_C is compatible with the solution τ₊ in Eq. (16) if the first order expansion of the latter, namely

coincides with τ_C, as expressed in Eq. (20). In this way, by simply comparing Eq. (19) and Eq. (20), we have

Finally, then, by substituting the value of α in Eq. (19), we obtain

From the above analysis we may notice that the absolute value α of the slope of the applied moment is expressed, in Eq. (21), in terms of the sole parameters m₀ and d, due to the fact that the speed of the two doors is taken as null when the latter reach their final positions. As a consequence, the closure time τ_C is seen to depend on the parameters of the problem as in Eq. (22).

In order to better understand the role played by the parameter α in determining change in τ_C, let us take β/M_D = 0.1 s^-1 and d = 2.4 m to be fixed parameters. Let us then consider two different cases: the first for m₀ = 6.0 m/s², the second for m₀ = 12.0 m/s². In the second case the value of m₀ is simply doubled with respect to the first. In Fig. 4a and Fig. 4b the displacement and the velocity of the right door, as function of time, are shown for m₀ = 12.0 m/s², respectively.

In Fig. 5, on the other hand, the applied moment M, divided by 2M_Dr, is shown as a function of time for the two cases. From Fig. 4a we notice the smooth S-shaped motion of the right door, starting from rest and ending its run with zero velocity; therefore, the velocity of this door, shown in Fig. 4b, initially rises, until it attains a maximum value, and then decreases, until it reaches again the t-axis at t = τ_C. The different dynamical behavior of the door for two different values of m₀ = M(0)/2M_Dr can be resumed in Fig. 5, where the appropriate coefficient α, which determines the downward slope of the curve, is calculated as in Eq. (21). From Eq. (21) it is clear that, for higher values of the initially applied moment, the parameter α must be larger, so that, by Eq. (22), it results that the closure time τ_C must be lower. Therefore, the curve obtained for m₀ = 12.0 m/s² is truncated at time t = τ_C = 0.764 s, while the curve obtained for m₀ = 6.0 m/s², for which the parameter a is lower, has a higher value of τ_C (1.075 s).

In order to see in more details for what range of m₀ the closure time diminishes as m₀ increases, we take the derivative of τ_C with respect to m₀, assuming β/M_D and d constant, obtaining

In this way, we obtain a negative value of dτ_C/dm₀, if the following relation is satisfied

Given now that we only consider the case βτ_C/M_D < < 1, and being the desired closure times of the order of one second, we see that the inequality in Eq. (24) can well be always satisfied for the characteristic parameters of the present problem.

As a final remark, let us consider the instantaneous power P(t) provided by the electric motor to the sliding doors. By definition, we have

In Fig. 6 we show the time dependence of the instantaneous power provided to the doors by the electrical motor. In particular, in Fig. 6a the value of P(t)/2M_D vs. t is shown, which gives the instantaneous power per each kilogram of mass of the moving system necessary to get the desired motion for three different values of the normalized initially applied moment m₀. Notice that negative values of P(t) means that the motor is applying an inverse torque to the sliding doors to make them decelerate. In Fig. 6b, on the other hand, the squared value of the quantity P(t)/2M_D is shown for the same three values of the parameter m₀. The time average value of the square root of the latter quantity is indicative of the effective power dissipated by the motor (r.m.s. value). Notice, in this respect, that, as it is possible to argue from Fig. 6b simply by inspection, a lesser amount of energy is needed in decelerating the sliding doors rather than in accelerating them, for the given choice of parameters.

5. Conclusions

We have considered the problem of the closure of two sliding doors. The rather simple way of realizing the motion of both doors has led us to consider a simplified model for describing the dynamics of the system. By the adopted scheme, the system consists of two point particles attached to a string, which runs over two ideal pulleys. The problem is rather interesting, since it makes use of the commonly adopted models for initiating the student to the study of the dynamics of point particles, and yet it is related to a frequently observed phenomenon: The closing or opening of a sliding door.

The characteristic properties of these systems are analyzed in details, by assuming that a viscous friction force is present between the guides and the doors supports. The resulting first-order differential equation, describing the motion of both doors, is solved and the closure time τ_C is found by imposing smooth closing, i.e., by taking the final velocity of the door be zero at t = τ_C.

Finally, because of what has been found, the present work can be useful not only as an application of basic physics to commonly observed phenomena, but also for engineering studies relating to the development of sliding doors with optimum closure or opening times.

Acknowledgments

The author is gratefully indebted to N. Di Brizzi, C. Formisano and F. Romeo for helpful discussions on the subject. The author also wishes to thank S. Ganci for advices.

Recebido em 14/1/2008; Aceito em 31/1/2008; Publicado em 2/8/2008

• [1] F.W. Sears, M.W. Zemansky and H.D. Young, University Physics (Addison-Wesley, Reading, 1976), 5^th ed.
• [2] W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Wiley, Nova York, 1977).

Publication Dates

• Publication in this collection
  21 Aug 2008
• Date of issue
  Sept 2008

History

• Accepted
  31 Jan 2008
• Received
  14 Jan 2008