The mechanisms of one degree of freedom can be dynamically analysed by setting out a single differential equation of motion which variable is the generalized coordinate selected as independent. In front of the use of a set of generalized dependent coordinates to describe the system, the method exposed in this work has the advantage of working with a single variable but leads to complex analytical expressions for the coefficients of the differential equation, even in simple mechanisms. The theoretical approach, in this paper, is developed from Eksergian's method and Lagrange's equations. The equation of motion is written by means of a set of parameters – reduced parameters – that characterize the dynamic behaviour of the system. These parameters are function of the independent coordinate chosen and its derivative and can be obtained numerically by direct calculus or by means of a kinetostatic analysis, as is proposed. Two cases of study of the method are presented. The first example shows the study of pedalling a stationary bicycle used in a rehabilitation process. The second one shows the analysis of a single dwell bar mechanism which is driven by an electric motor.

Analysis of mechanisms from the kinematic point of view is, in general, not difficult to make except in the singular configurations and their environment. There are a number of programs that facilitate it or directly carry out it in lesser or greater extent: Geogebra (Geogebra, 2016) whose main objective is the dynamic geometry, CAD programs including a kinematic analysis module or a number of specific programs (Kurtenbach et al., 2014). The kinetostatic or inverse dynamic analysis, in which the mechanism has as many actuators as degrees of freedom and all movements are imposed, requires extra effort, both conceptual and operational. This fact is particularly difficult when trying to determine the constraint actions in the kinematic pairs of the mechanism. There are also a number of programs that perform kinetostatic analysis, such as PAM – Program of Analysis of Mechanisms – of Mechanical Engineering Department of UPC (Clos and Puig-Ortiz, 2004; Cardona et al., 2006), SAM – Synthesis and Analysis of Mechanisms – by ARTAS Engineering Software (SAM, 2016), WinMecC of Mechanical Engineering Department of Malaga University (WinMecC, 2016) and others. Finally, the direct dynamic analysis, in which the mechanism has fewer actuators than degrees of freedom, represents a qualitative leap in difficulty of implementation. To the algebraic manipulation of the above analysis the solution of differential equations must be added. There are different applications that carry out direct dynamic analysis. The use of such applications is not always justified because of their complexity and because they are not trivial to use, even for the simulation of a system of one degree of freedom. Their use may be not recommended or even unfeasible in some cases, such as when the study of the mechanism must be performed in real time, as part of the simulation and control of a production process. In these cases, there should be a low-cost model both computational and in implementation.

In the world of robotics it is usual to use reduced inertial parameters to minimize the time of calculation and simulation. Several authors have proposed methods to determine and to reduce the number of parameters to be used in the inverse dynamics of manipulators (Fogarasy and Smith, 1997; Ebrahimi and Haghi, 2013; Díaz-Rodríguez et al., 2010; Ros et al., 2012; Chen and Beale, 2003; Yoshida et al., 1995) as well as to determine the minimum number of required parameters (Gautier and Khalil, 1988). It has also been investigated the possibility that the parameters vary over time to facilitate manoeuvrability of manipulators in singular configurations (Parsa et al., 2015). With the aim of describing the inertial behaviour of mechanisms, it seems a good idea to use inertial parameters that vary over time.

The dynamic analysis equations of mechanisms are obtained usually by means of Lagrangian formulation (Fogarasy and Smith, 1997), Newtonian formulation and virtual work principle (Wu et al., 2008).

In this paper, similarly to other works of the authors (Cardona et al., 2009; Jordi et al., 2008), the direct dynamic analysis of a mechanism of one degree of freedom by using its reduced inertial parameters and reduced forces and moments is proposed. Both types of parameters are function of the independent generalized coordinate, taken for kinematic description of the mechanism, and may be obtained, for example, using applications that perform the kinetostatic analysis. With these parameters, the direct dynamic analysis leads to a single second order differential equation, equation of motion, which is easily integrated to obtain the time evolution of the generalized coordinate employed. Each constraint action, force or moment, is given by an algebraic expression that includes the reduced parameters associated with such action and the first two derivatives of the coordinate, obtained independently in the process of integration of the equation of motion. In some of the mentioned works, all the reduced inertial parameters are obtained from the Lagrangian formulation. In this study, these parameters, as well as the reduced forces and moments present in the equation of motion, are introduced from the energy theorem or Eksergian's method (Eksergian, 1930; Doughty, 1988). This approach allows the incorporation, in a simple way conceptually and operationally, of motors and passive resistances described by means of the dissipated energy.

The determination of reduced inertial parameters and reduced forces is performed by using the mentioned programs that allow static, kinematic and kinetostatic analysis of planar mechanisms controlled by actuators, angular or linear.

To obtain the equation of motion of a mechanism of one degree of freedom,
the use of energy theorem in differential version or Eksergian's method
(Eksergian, 1930) is proposed:

For a system of one degree of freedom, kinetic energy

The total power

By developing Eq. (3), for a system of one degree of freedom, an identical equation to Eq. (2) is obtained. This fact shows that the reduced force and generalized force coincide.

When it is possible to use Eksergian's method, the inclusion of not mechanical phenomena, as motors and passives resistances, in the motion equation is conceptually simple. It can be done by means of the power that these phenomena exchange with the system, and that is described with mechanical state variables. In the Lagrange's formulation, these phenomena must be introduced by means of generalized forces associated to the non-conservative forces.

Original mechanism and with a constraint condition replaced by a constraint actuator.

If in the dynamic study of the mechanism is desired to determine a constraint
action, the kinematic condition imposed by the constraint is substitutable,
conceptually, by an actuator that ensures it – constraint actuator. Figure 1
shows a four linkage bar mechanism activated by an angular actuator

For the new system of two degrees of freedom, the kinetic energy is (Jordi
et al., 2008; Eksergian, 1930; Cardona and Clos, 2000):

The constraint force obtained with Eq. (3) is:

The coefficients of the Eqs. (5) and (6), which are reduced inertial parameters, can be determined from the calculation of the kinetic energy or, as it will be discussed below, using kinetostatic simulation programs as PAM, SAM or WinMecC.

The kinetostatic analysis allows to determine

The necessary conditions of the second analysis (

Proposed process for dynamic analysis of a system of one degree of freedom.

SAM has a utility that facilitates the realization of the two analyses
evaluated at the same instants of time. You can perform a kinetostatic
analysis by loading a file in which the temporary values, the value of the
coordinate

The process presented for the determination of the constraint action is extensible to find simultaneously as many actions as desired. In this case, as many constraint actuators as constraint actions to determine must be defined to obtain the corresponding equations. In order not to increase the complexity of the system, as many independent systems of two degrees of freedom as constraint actions to determine must be defined. Each system provides an equation for the constraint action and an equation of motion, that is obviously the same for all systems. Thus, two reduced parameters for the equation of motion and two parameters for each constraint action to determine are obtained. Figure 2 shows a schematic of the process to be followed for determining the constraint actions desired.

Model for the stationary bicycle and the inferior extremities of the cyclist.

The procedure described in the previous section is used to study two cases: the pedalling a stationary bicycle used in a rehabilitation process and the analysis of a single dwell bar mechanism which is driven by an electric motor.

The dynamics of the pedalling in a stationary bicycle in a rehabilitation process (Cardona et al., 2009; Jordi et al., 2008; Curià, 2010) is studied. The mechanism of Fig. 3 represents the model for this study. It is assumed that the ankle is fixed to the pedal, so the model has only one degree of freedom. The geometric characteristics and inertial parameters, obtained experimentally by Curià (2010), are shown in Fig. 3a. The translation kinetic energy of the bicycle and the cyclist plus the rotation kinetic energy of the wheels of a conventional bicycle are substituted by the kinetic energy of a flywheel fixed to the pinion. Some aspects as the action of several muscles, that can be modelled as simultaneous actuators, are not taken into account.

The most reasonable kinetostatic analysis to determine reduced parameters is
the one that uses an actuator controlling the rotation angle of the pedals

The motion equation of the system of Fig. 3a can be obtained by means of
Eksergian's method. The kinetic energy of the system is:

Inertial reduced parameters for pedalling a stationary bicycle.

Angular velocity of the pedal and force in the knee in leg direction.

The objectives of this analysis are to determine the motion equation and the
force in the knee in the leg direction

The expressions of generalized forces are obtained, for example, by means of
the following virtual movements:

In this movement, the mechanism has two degrees of freedom because the
restriction

In order to finish the analysis, the equations have to be solved; so it is
necessary to obtain the values of

Sketch of the single dwell bar mechanism used in the case of study.

Figure 5 shows the rotation velocity of the pedal obtained by means of the
integration of the equation of movement. Its mean value remains nearly
constant because

The procedure described in the Sect. 2 is used to study the dynamics of a
single dwell bar mechanism. The mechanism and its characteristics, both
geometric and inertial, are shown in Fig. 6 which represents the model for
this study. The prototype shown in Fig. 7 is designed, among other functions,
to check the temporal movement law of the crank AB,

Prototype of the single dwell bar mechanism.

The orientation of the rocker QR is maintained substantially constant for a certain interval of movement of the crank AB. The crank AB is considered balanced so that its centre of inertia coincides with the fixed joint A. The centre of inertia of the connecting-rod BP is at C and the centres of inertia of the rockers OC and QR are at their midpoint. The centre of inertia of the slider coincides with the articulation P.

The motion equation of the system of Fig. 6 can be obtained by means of Eksergian's method. The kinetic energy of the system is the sum of the kinetic energies of the crank AB, the connecting-rod BP, the two rockers OC and QR and the slider. As in the previous example, if some constrain action must be determined it is necessary to use the same procedure of the previous example.

Also in this case, first of all, reduced parameters are determined by
kinetostatic simulation programs mentioned and, for a set of time instants,
the kinematic and dynamic variables for calculating inertial parameters and
reduced forces are obtained. The actuators used in these analyses are an
angular actuator that controls the angle

Prototype of the single dwell bar mechanism.

Reduced parameters of the single dwell bar mechanism.

Angular velocity of the crank AB and constraint force in the prismatic pair.

From these analyses, the inertial behaviour of the mechanism is obtained.
This must be linked with the external forces acting on it, that is the torque
of the motor applied in the crank AB

The differential equation of movement for the coordinate

The motor chosen in this application is a model Dunkermotoren GR
63

The passive resistances have been considered concentrated in bar QR and are
modelled as:

Figure 9 shows the inertial reduced parameters for one revolution of the crank AB obtained by means of the exposed procedure of Sect. 2. Calculations have been made with Scilab and simulations with PAM and SAM. All the results agree among them.

Figure 10 shows the rotation velocity of the crank obtained by means of the integration of Eq. (8) and the constraint force in the prismatic pair obtained with Eq. (9). Rotation velocity and force are periodic. Although the value of motor torque is almost constant the considerable variation of the rotation velocity is consequence of the great variation of the inertial parameters.

The proposed approach for doing the dynamic analysis of one degree of freedom mechanisms (not necessary planar) based on the use of reduced parameters leads to a simple and efficient procedure for their study.

This approach enlarges the possibilities and utilities of easy use programs that cannot do direct dynamic analysis. In particular, the inertial reduced parameters have been obtained with programs PAM, SAM and WinMecC and the reduced parameters for calculating the generalized forces with PAM.

This approach is useful when the study of the mechanism must be performed in real time, as part of the simulation and control of a production process. A comparison between the time used to integrate two cycles of stationary regime of the example 1 with a commercial software and with the proposed method has been performed. The computation time on a PC with Windows7 operating system and 64-bit processor Intel Core i7 have been 658 ms with commercial software and 68 ms with the proposed method. The results support the statement that this method can be used in real-time simulations more efficiently.

The inertial reduced parameters are obtained as a table of values. So, the integration algorithm for the equation of movement must access to them by means of an interpolation function.

The use of Eksergian's method allows the incorporation, in a simple way conceptually and operationally, of motors and passive resistances described by means of the exchanged energy, instead of doing it by means of non-conservative forces.

The exposed procedure is easily extensible to conservative forces, that only depend on position. The generalized forces associated to those can be obtained by kinetostatic analysis. Forces that depend on velocity must be dealt directly in the equation of movement. This type of forces can also be reduced to the coordinate, but the associated generalized force is, then, function of position and velocity and its determination by means of a kinetostatic analysis is not operative.

All the data used in this manuscript can be obtained by requesting from the corresponding author.

The authors declare that they have no conflict of interest.Edited by: A. Müller Reviewed by: two anonymous referees