# Principal vectors and equivalent mass descriptions for the equations of motion of planar four-bar mechanisms

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## Abstract

The use of principal points and principal vectors in the formulation of the equations of motion of a general 4R planar four-bar linkage is shown with two kinds of methods, one that opens kinematic loops and one that does not. The opened kinematic loop approach analyses the moving links as a system with a tree connectivity, introducing reaction forces for closing the loops. Compared with the conventional Newton–Euler method, this approach results in fewer equations and constraint forces, whereas the mass matrix entries remain meaningful, but there is a stronger coupling between the equations. Two equivalent mass formulations for the closed kinematic loop approach are presented, which need not open the loop and introduce loop constraint forces in the equations of motion. With the method of complex joint masses, the mass of the links closing the loops is represented by real and virtual equivalent masses, defining the principal points. The principle of virtual work with the inclusion of inertia terms is used to derive the equations of motion. As an example the dynamic balance conditions of the four-bar linkage are derived. With the method of the equivalent mass matrix it is shown how a constant mass matrix can be used to describe the dynamics of binary links with an arbitrary mass distribution. A four-bar linkage could be modelled with only three truss elements instead of the conventional result with three or more beam elements, which gives a significant reduction of the computational complexity.

## Keywords

Linkage Principal point Principal vector Virtual work Equivalent mass Complex joint mass Truss element## 1 Introduction

This article presents a sequence of three ways to derive the equations of motion of planar linkages. The planar 4R four-bar mechanism is used as an example. It is intended to find methods which present the equations of motion in a meaningful and insightful way such that the influence of varying the individual design parameters on the dynamics can be grasped. Such methods are useful for the synthesis of mechanisms with desired dynamic properties as a starting point in the design process. As a step towards this ultimate goal, the use of principal points and principal vectors in the derivation of the equations of motion and also of two equivalent mass descriptions are considered.

Principal points and principal vectors were introduced by Fischer [1, 2] for determining the equations of motion of articulated bodies. In particular, they were applied for the inverse dynamics for human gait analysis [3]. Later on, the principal points were also found as in the Roberson–Wittenburg formulation [4], where they were named barycentres of so-called augmented bodies, equivalent to the reduced systems introduced by Fischer. It was shown by the second author [5, 6] how the theory of principal vectors is generalized and extended to mechanisms with multiple closed loops. For determining the principal points and principal vectors, a method for equivalent mass modelling was presented, using real and virtual equivalent masses. The theory was applied to the shaking force and shaking moment balancing of mechanisms [7], with the benefit that dynamic balance properties could be set as a starting point in the design process by the synthesis of balanced mechanism solutions from inherently balanced linkage architectures. The presented method for equivalent mass modelling can be considered as a graphical interpretation of the method of complex joint masses by Freudenstein [8, 9].

Another approach for modelling the mass distribution is by the constant mass matrix for planar rigid bodies as derived by García de Jalón in the context of the natural coordinate formulation [10]. This description also includes the rotary inertia and initially removes the velocity-dependent inertia terms from the equations. It is shown that the same mass matrix can be used to model the dynamics of a truss element with a general mass distribution in the program Spacar [11].

The article starts with a kinematic analysis of the four-bar mechanism. Then the traditional derivation of the equations of motion with an opening of the closed kinematic loop is given, but principal points and principal vectors are applied. Next, equivalent complex masses are used for the description of the linear momentum of the links, and it is shown that the procedure for obtaining principal points and principal vectors with these complex masses remains unchanged. Then a description with mass matrices is shown. The equivalent mass matrix as implemented in the multibody program Spacar is applied to a numerical example. This article builds on two previously published conference papers [12, 13], which deal with the same subject.

## 2 Kinematic relations for a four-bar linkage

### 2.1 Configuration analysis

Some notation is introduced as follows. The four links are numbered 1, 2, 3 and 4. The length of link \(k\), that is, the distance between its two joints, is denoted by \(l_{k}\). The locations of the revolute joints are at the points \(\mathrm{A}_{0}\), \(\mathrm{A}_{1}\), \(\mathrm{A}_{2}\), and \(\mathrm{A}_{3}\). Link 1 between \(\mathrm{A}_{0}\) and \(\mathrm{A} _{1}\) is the crank. In the use of terminology, we assume that the crank can make a continuous rotation, so the Grashof condition is satisfied, but, except in the singular positions, this condition is not necessary for the equations to be valid. Link 2 between \(\mathrm{A}_{1}\) and \(\mathrm{A}_{2}\) is the coupler, link 3 between \(\mathrm{A}_{2}\) and \(\mathrm{A}_{3}\) is the rocker, and link 4 between \(\mathrm{A}_{3}\) and \(\mathrm{A}_{0}\) is the frame which is held fixed to the ground. A fixed coordinate system \(\mathrm{A}_{0} xy\) has its origin at \(\mathrm{A} _{0}\), its \(x\)-direction is along \(\mathrm{A}_{0} \mathrm{A}_{3}\), and its \(y\)-direction is obtained by rotating the \(x\)-direction by a right angle in the positive direction.

The relative and absolute positions of points will be represented by two-dimensional vectors, which are indicated by an overbar. They will also be considered complex numbers, where the absolute value is the length of the vector, denoted by the same symbol without an overbar, and the argument is the angle of the positive direction of the vector with the \(x\)-axis. For instance, the link lengths and their orientations are denoted by \(\bar{l}_{k} = l_{k} \exp (i\theta _{k}) = l_{k} \cos \theta _{k} + i l_{k} \sin \theta _{k}\), where \(\theta _{k}\) is the angle of link \(k\) with the global \(x\)-direction as shown in Fig. 1. For the fixed link 4, \(\theta _{4}=\pi \).

The scalar product is defined as the usual inner product of two vectors, for which a notation with angle brackets is chosen. If \(\bar{x}\) and \(\bar{y}\) are two complex numbers, the inner product of the corresponding vectors can be expressed as \(\langle \bar{x}, \bar{y} \rangle = \frac{1}{2}( \bar{x}^{*} \bar{y} + \bar{y}^{*} \bar{x}) = \mathrm{Re}(\bar{x}^{*} \bar{y})\), where a superscript asterisk denotes a complex conjugate and \(\mathrm{Re}\) indicates the real part.

### 2.2 Velocity and acceleration analysis

## 3 Opened kinematic loop approach

In this section, the dynamics of the four-bar linkage are analysed by the method of principal vectors in the way Fischer used them [2]. While Fischer used them only for deriving Lagrange’s equations of motion of open trees of links, here the method of principal vectors is applied in the analysis of a closed-loop linkage, where the principle of virtual work is used to obtain the equations of motion. The advantage of using the principle of virtual work is that the equations of motion are often found in a simpler way and it leads to more insight in how parameters contribute to the equations of motion.

In the opened kinematic loop approach, the closed kinematic loop of the mechanism is opened at one or more joints to form a system with a tree connectivity. In particular, the mechanism is removed from its supports to form a free system. The constraint forces are included as additional variables. The number of variables to describe the system and hence the number of resulting equations can be reduced by reimposing one of the supports. The free system for the four-bar mechanism is shown in Fig. 3. At the joints \(\mathrm{A}_{0}\) and \(\mathrm{A}_{3}\) the constraint forces \(\bar{R}_{\mathrm{A0}}\) and \(\bar{R}_{\mathrm{A3}}\) are included as additional variables. The positions of the link centres of mass \(\mathrm{S} _{k}\) with respect to the link coordinate system defined by the vector \(\bar{l}_{k}\) and the first joint as the origin is given by the vector \(\bar{s}_{k}=s_{k} \exp (i\sigma _{k})\)\((k=1,2,3)\), as shown in Fig. 1. This means that the absolute vector pointing from the first joint to the centre of mass is expressed as a complex number as \(\bar{s}_{k}\bar{l}_{k}/l_{k}\).

For this free system of planar bodies interconnected by revolute joints in a tree, the generalized coordinates can be chosen as the position coordinates of the point S and the three rotation angles of the links about the principal points. The individual rotations leave the point S invariant and owing to the definition of the principal points, leave the centre of mass of the system invariant, which is the point S. This will be confirmed by explicitly calculating the position of the centre of mass.

Alternatively, a single equation of motion can be obtained by multiplying the second equation of (22) by \(\varTheta _{2}\) and the third equation by \(\varTheta _{3}\) and adding these to the first equation, which eliminates the constraint forces and yields the equation of motion of the mechanism in \(\theta _{1}\) if the expressions for the dependent quantities are substituted. The reaction forces can then be found by substituting the accelerations in the original differential-algebraic equations.

## 4 Closed kinematic loop approach with equivalent mass formulations

In this section, a different way of using the principal vectors is shown and two methods to represent the mass distribution are presented. In the first method, the linear momentum of the centre of mass of a link is replaced by two linear momenta at the joints by means of a complex mass representation, which do not contribute to the moment of inertia. In the second method, the mass is represented by a mass matrix, which is only related to the translations of the joints. The methods have in common that the kinematic loop is not opened and no explicit constraint forces appear in the equations.

### 4.1 Method of complex joint masses

It should be stressed that no equivalence for the angular momentum can exist if \(f_{k}\) is not zero: a velocity \(v\) of the centre of mass in the direction of the link vector gives rise to an angular momentum for the equivalent masses, \(2m_{k} f_{k} v\), which cannot be compensated for by adjusting the moment of inertia of the link, so there is no dynamical equivalence. In the special case that \(f_{k}=0\), a reduced moment of inertia \(I_{k,\mathrm{red}} = I_{k} - m_{k} e_{k}(l_{k}-e_{k})\) gives a dynamically equivalent model.

The reduced equation of motion, in which the constraint forces have been eliminated, are obtained by multiplying the second equation of (37) by \(\varTheta _{2}\), the third equation of (37) by \(\varTheta _{3}\) and adding the resulting equations. Also, the relations (11) can be used to remove the dependency on \(\theta _{2}\) in a first step of the reduction and then to combine the two remaining equations to a single equation of motion. Finally, the expressions of the dependent angles, angular velocities and angular accelerations can be substituted to obtain a single second-order differential equation in \(\theta _{1}\), the degree of freedom.

### 4.2 Method of the equivalent mass matrix

## 5 Application to multibody system dynamics

### 5.1 Truss element

Initial and final values of the link angles, angular velocities and angular accelerations

Variable | Initial value at | Final value at |
---|---|---|

\(\theta _{1}\) | 1 rad | −0.164143028498 rad |

\(\theta _{2}\) | 0.395412477125 rad | 0.803387760489 rad |

\(\theta _{3}\) | −1.625231000527 rad | −1.505437908932 rad |

\(\dot{\theta }_{1}\) | 0 rad/s | 0.282625741349 rad/s |

\(\dot{\theta }_{2}\) | 0 rad/s | −0.124005044330 rad/s |

\(\dot{\theta }_{3}\) | 0 rad/s | −0.157299276751 rad/s |

\(\ddot{\theta }_{1}\) | −0.494982843920 rad/s | 0.269991393862 rad/s |

\(\ddot{\theta }_{2}\) | 0.090460471115 rad/s | −0.134961046282 rad/s |

\(\ddot{\theta }_{3}\) | −0.156221696196 rad/s | −0.110963980093 rad/s |

### 5.2 Shaking force balance and shaking moment balance conditions

Also a sufficient condition for the shaking moment balance if the shaking force balance conditions are fulfilled can easily be derived by considering the mass matrix in Eq. (49), which becomes diagonal for the conditions that \(f_{2}=0\) and \(I_{2} = m_{2} e_{2} (l_{2}-e _{2})\). Then also \(f_{1} = f_{3} = 0\), and this results in a reduced moment of inertia of \(I_{1}+m_{1} e_{1}^{2}+m_{2} l_{1}^{2}(1-e_{2}/l _{2}) = I_{1} - m_{1}e_{1}(l_{1}-e_{1})\) on the crank and a reduced moment of inertia of \(m_{2} l_{3}^{2} e_{2}/l_{2}+I_{3}+m_{3}(e_{3}-l _{3})^{2} = I_{3} + m_{3}e_{3}(e_{3}-l_{3})\) on the rocker, of which the shaking moments can be compensated by connecting the crank and the rocker to additional counterrotating masses. The moment balance can be checked by giving the whole linkage a virtual rotation \(\updelta \theta _{0}\) about the origin, \(\mathrm{A}_{0}\). This gives virtual displacements \(\updelta r_{\mathrm{A3}y} = l_{4}\updelta \theta _{0}\), \(\updelta r _{\mathrm{A1}x} = -l_{1}\updelta \theta _{0}\sin \theta _{1}\), \(\updelta r_{\mathrm{A1}y} = l_{1}\updelta \theta _{0}\cos \theta _{1}\), \(\updelta r_{\mathrm{A2}x} = l_{3}\updelta \theta _{0}\sin \theta _{3}\), \(\updelta r_{\mathrm{A2}y} = (l_{4}-l_{3}\cos \theta _{3})\updelta \theta _{0}\). The terms with \(l_{1}\) and \(l_{3}\) give no contribution because of the counterrotating masses. The terms with \(l_{4}\) give a contribution which has a factor \(M_{44}+M_{\mathrm{r,}44}\), which is equal to zero for the considered case. So there are no forces on the supports in the \(y\)-direction and there is no shaking moment. This could be expected from the exact dynamic equivalence of the mass distribution of the coupler with two point masses at the links. These conditions were derived by Berkof [19].

## 6 Discussion

If we compare the opened kinematic loop approach in Sect. 3 with the conventional Newton–Euler method, we see that it is almost as simple, but fewer constraint forces have to be introduced. The entries in the mass matrix still have a clear meaning. However, there is stronger coupling between the equations, as the mass matrix is no longer a diagonal matrix. The centre of gravity can easily be found, and hence the linear momentum of the linkage as a whole and of the individual links. Also the sum of the reaction forces on the ground can be easily obtained, which is advantageous for shaking force balancing. A possible disadvantage is that there are still more equations than degrees of freedom, which leads to a system of differential-algebraic equations.

If the mass of one or more links is represented by equivalent complex masses at the joints as explained in Sect. 4.1, the centre of mass and hence the linear momentum of the system can easily be found; the conditions of shaking force balancing (58) for the four-bar linkage become almost trivial. The mass description remains relatively simple. A disadvantage is that the equivalent masses are not fully dynamically equivalent, so the equations of motion cannot be derived by applying standard techniques to the equivalent masses only.

Principal points and principal vectors are useful for the opened kinematic loop approach, as the off-diagonal terms of the mass matrix and the convective inertia terms are easily obtained. To a lesser extent, this applies also for the method with equivalent complex masses, although this is not so obvious in the considered simple example. In both approaches, the motion of the centre of mass of the system is easily found with the principal vectors, which yields conditions for shaking force balance.

The description with equivalent mass matrices in Sect. 4.2 includes the complete dynamics, although the mass matrix is filled and does not have a block diagonal structure. The equations of motion can initially be described in terms of the joint coordinates, without making use of the angles of the links. The independent coordinate can be chosen as one of the angles or one of the position coordinates of the joints. This mass description can be used for the finite truss element, leading to simpler models in terms of the kind of element used and the number of coordinates and constraints.

The methods presented can be applied to other linkages, with similar advantages and disadvantages, although the equations can become more complicated. General planar mechanisms with links interconnected by pin joints can be directly treated by the methods shown here. An example is the seven-body mechanism form [20], which was modelled by truss elements and simulated in another publication [21].

Principal points and principal vectors can also be defined for systems of spatial bodies interconnected with spherical joints [2, 4] and they can be used to obtain the equations of motion. Some other types of joints, such as a revolute joint, a universal joint or a homokinetic coupling, can be described by adding constraints on the relative motion in a spherical joint. The method of equivalent masses can be extended to some special spatial systems. In general, a binary link between spherical joints cannot be modelled in this way, as the orientation of this link is not determined by the positions of two its joints and even the linear momentum is generally not determined by the velocities of these two joints. Six classical spatial truss elements which interconnect four points can be used to describe the dynamic properties of a rigid body [21].

## 7 Conclusions

This article has led us from a familiar approach with opened kinematic loops via a method with equivalent complex masses to a method using a constant mass matrix, which were all applied to formulate the equations of motion of a general 4R planar four-bar mechanism. In the first two methods, the use of principal vectors has been shown. The principle of virtual work with the inclusion of inertia terms was used to derive the equations of motion.

The opened kinematic loop approach, compared with the conventional Newton–Euler method, results in fewer equations and constraint forces, while the mass matrix entries remain meaningful, but there is a stronger coupling between the equations. For the closed kinematic loop approach with equivalent complex masses, no explicit loop constraint forces are introduced in the equations. The mass of the coupler link could be modelled onto the other links of the four-bar linkage by using real and virtual equivalent masses, defining the principal points. The complex masses give a correct representation of the linear momentum, but they do not give a full dynamic equivalence if the centre of mass of the link is not on the line connecting the two joint positions. This makes this approach useful in problems mainly involving linear momentum, such as shaking force analysis.

With the method of the equivalent mass matrix, it was shown how a constant mass matrix can be used to describe the dynamics of binary links with an arbitrary mass distribution. This seems to lead to the simplest form of the equations of motion, but has as a disadvantage that the link angles appear as derived quantities and are no longer directly present in the equations of motion.

The constant mass matrix, which makes use of the joint coordinates only, can describe the mass properties of a truss element in a finite element formulation, which is fully dynamically equivalent to that of a rigid link. In addition, a body which is allowed to undergo a uniform dilatation, as is approximately the case in some auxetic metamaterials, can be described by this element. As an example, the general 4R four-bar linkage was modelled with only three truss elements instead of three or more beam elements, which is a significant reduction in the complexity of the model.

Although all three presented methods can be used to obtain the equations of motion and the reaction forces of the four-bar linkage, the equations remain complicated for all methods if an explicit form is aimed at. For different purposes, different methods to derive the equation can be useful. In particular, the methods which use principal points and principal vectors have advantages if the main interest is in the linear momentum and the resultant reaction forces, which can be used to obtain conditions for dynamic force balance.

The described methods are a step towards obtaining insight into the dynamic equations such that they can direct the synthesis process towards desired dynamic conditions.

## Notes

### Acknowledgements

This publication was financially supported by the Netherlands Organisation for Scientific Research (NWO, 15146).

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