C1. Configuration of a mechanical system
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The study of a mechanical system, composed either by free particles, free rigid bodies, or assembled rigid bodies and particles (multibody system), starts with the description of its mechanical state: its configuration (position of all its points) and velocity distribution (velocity of all its points) with respect to a reference frame.
In this unit C1, we describe the configuration of mechanical systems. The description of the velocity distribution is presented in unit C2.
C1.1 Position of a particle
The position of a particle (a point) Q in a reference frame R can be described through a position vector [math]\displaystyle{ \overline{\Os_\Rs\Qs} }[/math], where [math]\displaystyle{ \Os_\Rs }[/math] is a point fixed in R (a point that belongs to the reference frame R). That vector is not univocally defined, as its origin may be any point fixed in R (Figure C1.1).
An alternative to the vectorial description of the position is the scalar description through three coordinates (Cartesian, polar...). In that case, we have to choose a coordinate origin, which may be any point fixed in R (Figure C1.2). In this course, however, we will use the vectorial description.
The main interest in mechanics is not the position of points but rather their evolution with time (their motion). A particle [math]\displaystyle{ \Qs }[/math] moves relative to a reference frame R when its position in R changes with time or, what is the same, goes through different points fixed in R with time. The set of those different points in R define the trajectory of [math]\displaystyle{ \Qs }[/math] in R (trajectory of [math]\displaystyle{ \Qs }[/math] relative to R).
C1.2 Configuration of a rigid body
When it comes to describing the configuration of a rigid body, the position of just one of its points is not enough. A first option would be giving the position of three points [math]\displaystyle{ \Ps }[/math], [math]\displaystyle{ \Qs }[/math], R not aligned. But it is evident that those vectors fulfil some restrictions: as the points of a rigid body have to be mutually fixed, the differences between pairs of points are vectors with constant value (Figure C1.4):
In a scalar description, if we provide three coordinates per point, the configuration of the rigid body is given by 9 coordinates, but as there are 3 relationships between them, only 6 coordinates are strictly necessary (Figure C1.5).
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There are many options to define the configuration of a rigid body, but in this course we will describe it through the position of one of its points and the orientation of the rigid body. Though the position of a point may be defined through a vector or three scalar coordinates, the orientation can only be described through scalar variables.
C1.3 Orientation of a rigid body with planar motion
A rigid body has a planar motion relative to a reference frame R when all its points describe trajectories contained in parallel planes. In that case, its orientation may be given through an angle defined by the intersection between a direction fixed in R (“departure” direction) and a direction fixed to the rigid body (“arrival” direction), both contained in the plane of motion. As those directions are not univocally defined, neither is the orientation angle (Figure C1.6).
The vertical arrow ([math]\displaystyle{ \Downarrow g }[/math]) shows the Earth gravitational attraction.
When the orientation angle changes its value with time, the rigid body moves according to a simple rotation about an axis perpendicular to the plane of motion, in a clockwise or counterclockwise direction, according to the orientation angle that has been chosen (Figure C1.7)
C1.4 Orientation of a rigid body moving in space
The description of the orientation of a rigid body moving in space is more complicated, and there are several ways to do it. Two options are the rotations about fixed directions and Euler rotations.
Rotations about fixed directions
They are simple rotations about three directions permanently orthogonal between them and whose orientation relative to the reference frame R is constant (“fixed” directions). A characteristic of this orientation method is that, for a same set of values of the angles and starting always from a same initial orientation, the final orientation of the rigid body depends on the sequence followed to introduce the angles. It is a sequential method.
Figure C1.8 illustrates this for a triangular object undergoing three rotations of 90[math]\displaystyle{ \deg }[/math] about three directions fixed in a reference frame R.
✏️ Example C1-4.1: the mechanical mouse of a computer
- In a mechanical mouse of a computer, the ball may rotate relative to the mouse case (R) about two orthogonal axes fixed to the case. The angle rotated about each axis is proportional to that rotated by the two small wheels that are in contact with the ball.
Euler rotations
Euler rotations are a non-sequential method to orientate rigid bodies. They are widely used in engineering because many mechanical systems include physical axes (associated with constraints between the rigid bodies) that allow this type of rotations.
Euler rotations are 3 chained simple rotations (in series), so that the rotation about the first axis causes a rotation of the other two, and the rotation about the second axis causes the rotation of the third one. In this course, the variables ([math]\displaystyle{ \psi }[/math], [math]\displaystyle{ \theta }[/math], [math]\displaystyle{ \varphi }[/math]) are associated with the three rotations:
- 1st rotation [math]\displaystyle{ (\psi) }[/math]: about an axis with constant orientation relative to R (axis fixed in R).
- 2nd rotation [math]\displaystyle{ (\theta) }[/math]: about an axis rotating because of [math]\displaystyle{ \psi }[/math] relative to R.
- 3rd rotation [math]\displaystyle{ (\varphi) }[/math]: about an axis with constant orientation relative to the rigid body (axis rotating because of [math]\displaystyle{ \psi }[/math] and [math]\displaystyle{ \theta }[/math] relative to R).
Figure C1.9 shows the Euler axes in a gyroscope, which consists of a support fixed to the ground (R), a fork linked to the support through a revolute joint, an arm linked to the fork through a revolute joint, and a disk linked to the support arm through a revolute joint. The rotations of these elements relative to the ground are:
- Fork: rotation [math]\displaystyle{ \psi }[/math] about the vertical axis fixed to R; the angle [math]\displaystyle{ \psi }[/math] is defined on the horizontal plane.
- Arm: rotation [math]\displaystyle{ \psi }[/math], and rotation [math]\displaystyle{ \theta }[/math] about an axis rotating with [math]\displaystyle{ \psi }[/math]; the [math]\displaystyle{ \theta }[/math] is defined on the vertical plane containing the arm.
- Disk: rotation [math]\displaystyle{ \psi }[/math], rotation [math]\displaystyle{ \theta }[/math], and rotation [math]\displaystyle{ \varphi }[/math] about an axis rotating with [math]\displaystyle{ \psi }[/math] and [math]\displaystyle{ \theta }[/math] (and whose orientation relative to the disk is constant); the angle [math]\displaystyle{ \varphi }[/math] is defined on the disk plane.
A characteristic of Euler rotations is that the angle between the first and the second ([math]\displaystyle{ \beta_{\psi\theta} }[/math]), and that between the second and the third axes ([math]\displaystyle{ \beta_{\theta\varphi} }[/math]), are constant. In a gyroscope, those angles are [math]\displaystyle{ \beta_{\psi\theta} = \beta_{\theta\varphi } = }[/math] 90[math]\displaystyle{ \deg. }[/math]. However, the angle between the first and the third axes is not constant. In the gyroscope in Figure C1.9 it may vary approximately between 30[math]\displaystyle{ \deg }[/math] and 150[math]\displaystyle{ \deg }[/math].
If the angles [math]\displaystyle{ \beta_{\psi\theta} }[/math] and [math]\displaystyle{ \beta_{\theta\varphi} }[/math] are not equal to 90[math]\displaystyle{ \deg }[/math], the rigid body cannot have any orientation in space (there would be some unattainable configurations). For that reason, the first and second Euler axes are usually orthogonal, and the second and third axes as well.
The description of the orientation of a rigid body that does not belong to a multibody system (for instance, an object floating in water, or a ball in the air) is more complicated to visualize as the rotation axes are not physically associated with any links. In that case, we proceed in different ways according to whether the object has a characteristic motion (as a spinning top) or does not have that characteristic motion (as a dice in gaming).
In the first case, the axes may correspond to characteristic rotations of the object. In a spinning top, the fast rotation about its symmetry axis (introduced as an initial condition when playing with the spinning top) suggests choosing that axis as third Euler axis. If that initial rotation is fast enough, the spinning tops takes a long time before falling to the ground, and the symmetry axis precesses slowly about a vertical axis. That vertical axis can be chosen as first Euler axis. The second rotation corresponds to the approaching motion of the symmetry axis towards the ground (Figure C1.10). Actually, the motion of a spinning top is identical to that of the disk in a gyroscope.
When the rigid body does not have a characteristic motion, the simplest thing to do is choose freely the first and third axes (fixed to the reference frame and to the rigid body, respectively). The second one can be determined according to the constant angles that it should define with the other two axes ([math]\displaystyle{ \beta_{\psi\theta} }[/math], [math]\displaystyle{ \beta_{\theta\varphi} }[/math]). If we choose them to be equal to 90[math]\displaystyle{ \deg }[/math], the second axis is the intersection of the plane orthogonal to the first one and the plane orthogonal to the third one.
✏️ EXAMPLE C1-4.2: orientation of a dice
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- We may prove that the Euler angles are not sequential starting from a same initial configuration and increasing each angle by 90[math]\displaystyle{ \deg }[/math] according to different sequences. When done properly, the same final orientation is reached.
- NOTE: In a dice, the addition of the numbers on parallel sides is always 7.
Euler angles present a problem whenever the configuration of the rigid body is such that the 1st and the 3rd axes are parallel (singular configuration): the 2nd axis is not univocally defined because the planes orthogonal to the other two are coincident. For the case of the disk in the gyroscope and that of the dice in the previous example, this happens when [math]\displaystyle{ \theta=\pm(\pi/2) }[/math]. In the spinning top, this is the case when [math]\displaystyle{ \theta=0-\pm\pi }[/math].
A solution to prevent that singularity is to use two different systems of Euler angles, and switch from one to the other when the configuration approaches the singularity.
✏️ EXAMPLE C1-4.3: two families of Euler angles for a ship
- We define two families A and B of Euler angles to orientate a ship with respect to a reference frame R. The figure shows the axes for the configuration [math]\displaystyle{ \psi = \theta = \varphi = }[/math] 0:
- Starting from that orientation, the configuration [math]\displaystyle{ \psi = \varphi = 0 }[/math] , [math]\displaystyle{ \theta = \pi/2 }[/math] for the A family and the configuration [math]\displaystyle{ \psi = \theta = }[/math] 0, [math]\displaystyle{ \varphi = \pi/2 }[/math] for the B family correspond to the same ship orientation. Family A goes through a singularity because the 1st and the 3rd axes are parallel, whereas in family B the 1st and 3d axes are orthogonal, thus far from the singularity.
In ships, airplanes and vehicles in general, Euler rotations have names associated with the direction of the axes in the reference (or initial) configuration (where all angles are zero). The rotation whose axis is initially parallel to the longitudinal direction of the vehicle is called roll. The rotation whose axis initially is parallel to the transverse direction of the vehicle is called pitch. That whose the axis initially is parallel to the direction perpendicular to the two previous ones (which coincides with the vertical direction if the vehicle is parked on flat ground) is called yaw.
It should be noted that, when the vehicle is in configurations other than the initial one, these names (roll, pitch and yaw) are associated with Euler rotations, whose axes no longer have to coincide with the three directions fixed to the vehicle (longitudinal, transverse and perpendicular to these two) mentioned above.
In Video C1.4 two alternative options are shown: in the first part of the video, the first rotation is roll, the second is pitch, and the third is yaw; in the second part of the video, the first is yaw, the second is pitch, and the third is roll. In most of the literature, yaw is taken as the first Euler rotation, pitch as the second, and roll as the third rotation (as in the second part of Video C1.4 and as in the "B family" of example C1-4.3).
C1.5 Independent coordinates
Though the position of a particle (a point) relative to a reference frame can be described through three coordinates, those coordinates may not be independent when the particle undergoes restrictions because in contact with other objects. In that case, the minimum set of coordinates to describe the position is the set of independent coordinates (IC) of the particle.
✏️ EXAMPLE C1-5.1: particle in a guide
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Similarly, though the configuration of a free rigid body (without any contacts with any other object) relative to a reference frame calls for 6 coordinates, the number of independent coordinates is lower when the rigid body undergoes restrictions.
✏️ EXAMPLE C1-5.2: vehicle on flat ground
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✏️ EXAMPLE C1-5.3: wheels of a vehicle
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✏️ EXAMPLE C1-5.4: Euler pendulum
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