6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant

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1 DE2-EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to Lecture resources may be found on Blackboard and at

2 Contents Three-Dimensional Kinematics and Dynamics of Rigid Bodies... 2 Kinematics... 3 Velocities and Accelerations... 3 Moving Reference Frames... 4 Euler s Equations... 7 Three-Dimensional Kinematics and Dynamics of Rigid Bodies For many engineering applications of complex kinematics and dynamics we must consider threedimensional motion. In this chapter we will explain how three-dimensional motion of rigid body is described. Then we will derive the equations of motion and use them to analyse simple motion. Design Engineering Example: The ability to model and understand the motion of the human body enables better design of garments, wearable tech, and other products that interface with the consumer during motion. It is also vital in the medical field as well; to reach the optimal design of implants and prosthetics we must first develop methods for analysing gait and biodynamics. In addition, this enables clinicians to diagnosing joint conditions, monitor progression and evaluate treatment outcomes. Design Engineering Example: In the near future it is envisaged that aerial robots (Drones) will have the capability to autonomously construct structures in many applications: building temporary shelters following natural disasters, deploying scaffolding and support structures on conventional construction sites, or assembling ramps across gaps and difficult terrain to enable access to terrestrial vehicles ( In order for such tasks to be carried out multiple robots will need to be employed simultaneously. This team of robots will need to communicate and understand were each other are, and how they are moving relative to each other.

3 Kinematics If a motor bike rides in a straight line, the wheels undergo planar motion. But, if the bike is turning the motion of the wheels is three dimensional (Figure 6.1). The same can be said for an aeroplane; planar motion when in level flight, 3D when banking or turning. A spinning top will be in planar motion at first, rotating about a fixed vertical axis. But, will eventually begin to tilt and rotate under 3D motion. Figure 6.1. Example of planar and three-dimensional motion Velocities and Accelerations We have already looked at the basic concepts needed to describe the 3D motion of rigid bodies relative to a given reference frame. In the earlier chapters on Kinematics we showed that a rigid body undergoing any motion other than translation has an instantaneous axis of rotation. The direction of this axis at a particular instant and the rate at which the rigid body rotates about the axis are specified by the angular velocity vector ω. We have also shown that a rigid body s velocity is completely specified by its angular velocity vector and the velocity of a single point of the body. For the rigid body and reference frame in Figure 6.2, suppose we know the angular velocity vector ω and the velocity v B of a point B. Then the velocity of any other point A of the body is given by; v A = v B + ω r A/B (6.1)

4 A rigid body s acceleration is completely specified by its angular acceleration vector α = dω dt, its angular velocity vector, and the acceleration of a single point of the body. If we know α and ω and the acceleration a B of the point B, the acceleration of any other point A of the rigid body is given by; a A = a B + α r A/B + ω (ω r A/B ) (6.2) Figure 6.2. Points A and B of a rigid body. The velocity of A can be determined if the velocity of B and the rigid body s angular velocity vector ω are known. The acceleration of A can be determined if the acceleration of B, the angular velocity vector, and the angular acceleration vector are known. Moving Reference Frames The velocities and accelerations in Equations (6.1) and (6.2) are measured relative to the reference frame indicated in Figure 6.2, which we will refer to as the primary reference frame (normally fixed relative to the earth). We also use a secondary reference frame that moves relative to the primary reference frame. The secondary reference frame and its motion are chosen for convenience in describing the motion of a particular rigid body. In some situation, the secondary reference frame is defined to be fixed with respect to the rigid body. In other cases, it is advantageous to use a secondary reference frame that moves relative to the primary reference frame, but is not fixed with respect to the rigid body. Figure 6.3 shows a primary reference frame, a secondary reference frame xyz, and a rigid body. The angular velocity of the secondary reference frame is specified by the vector Ω, and the angular velocity of the rigid body relative to the primary reference frame is specified by the vector ω. A third

5 component is ω rel, which is the angular velocity vector of the rigid body relative to the secondary reference frame. When the secondary reference frame is not fixed to the rigid body, we can express the body s angular velocity vector ω as the sum of the angular velocity vector Ω of the secondary reference frame and the angular velocity vector ω rel of the rigid body relative to the secondary reference frame; ω = Ω + ω rel (6.3) Figure 6.3. The primary and secondary reference frames. The vector Ω is the angular velocity of the secondary reference frame relative to the primary reference frame. The vector ω is the angular velocity of the rigid body relative to primary reference frame. The rigid body s angular acceleration vector relative to the primary reference frame can then be expressed as; α = dω = dω + dω rel = ( dω rel + Ω ω dt dt dt dt rel ) + ( dω rel + Ω Ω ) (6.4) dt The right hand side of this equation becomes zeros, leaving; α = dω rel dt + Ω ω rel (6.5)

6 If the secondary reference frame was fixed with respect to the rigid body, so that Ω = ω, the ω rel = 0. We can see from equation (6.4) that; α = dω dt Design Engineering Example: Lets consider a rotating disc, with angular velocity ω d, that is mounted perpendicular to an L-shaped shaft. The shaft rotates relative to an earth fixed reference frame with an angular velocity ω O. We want to determine the angular velocity and angular acceleration vectors of the disk relative to the earth fixed reference frame. The disc s motion relative to the earth fixed reference frame is rather complicated. However, relative to a reference frame that is fixed with respect to the shaft, the disk simply rotates about a fixed axis with constant angular velocity. If we introduce a secondary coordinate system that is fixed with respect to the shaft; the angular velocity vector we seek is the sum of the angular velocity vector of the secondary coordinate system and the disc s angular velocity vector relative to the secondary coordinate system. We introduce the secondary coordinate system, shown below, which is fixed with respect to the shaft. The angular velocity vector of the secondary coordinate system relative to the earth fixed reference frame is Ω = ω O j. The disc s angular velocity vector relative to the secondary coordinate system is ω rel = ω d i. Therefore, the angular velocity vector of the disc relative to the earth-fixed reference frame is; ω = Ω + ω rel = ω d i + ω O j Because ω d and ω O are constants, we can find from Equation (6.4) that the disc s angular acceleration vector relative to the earth fixed reference frame is; α = Ω ω rel = ω d ω O k

7 You will pick up 3D kinematics again in Robotics 1 next year. THE NEXT SECTION IS OPTION AND WILL NOT BE INCLUDED IN THE EXAM! Euler s Equations To complete the kinematics and dynamics loop I have included the following optional section. In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of inertia. So in basic term we are using the technique we just employed to study the kinematics of a moving reference frame and adding mass so that it becomes a dynamics problem. But only for cases where the secondary reference frame is body fixed to the body in question and parallel to the body s principal axes of inertia. Recap: From Dynamics we know that the moment of inertia of a rigid body is a tensor* that determines the torque needed for a desired angular acceleration about a rotational axis. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation. It is an extensive (additive) property: For a point mass the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis).

8 Tensor: In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Examples of such relations include the dot product, the cross product, and linear maps. We will be exploring tensors and tensor matrices more in stress analysis. The Inertia Tensor can be expressed in a matric as; I xx I xy I xz I = [ I yx I yy I yz ] I zx I zy I zz Which can be visualised on a cube element as shown in figure 6.4. Figure 6.4. Inertia tensors acting on a cube element. With the origin of the XYZ axes at the centre of mass of the cube. Euler's equations consist of Newton s second law; F = ma. Which states that the sum of the external forces on a rigid body equals the product of its mass and the acceleration of its centre of mass, and equations of angular motion. Previously in 2D dynamics we showed how this related to equation of angular motion; M O = I O α, for an object in rotation and M = Iα for an object in planar motion. Now that we want to introduce 3D motion we need to consider additional reference systems. Here is where the Euler equations come in. They essentially combine (6.5) into the equations of angular motion. The generalised form is given as: ΣM = I dω dt + Ω (Iω ) (6.6)

9 Where M and I may have components in more than one plane. This is often written as the matrix equation; ΣM x I xx I xy I xz dω x dt 0 Ω z Ω y I xx I xy I xz ω x [ ΣM y ] = [ I yx I yy I yz ] [ dω y dt] + [ Ω z 0 Ω x ] [ I yx I yy I yz ] [ ω y ] ΣM z I zx I zy I zz dω z dt Ω y Ω x 0 I zx I zy I zz ω z (6.7) If the secondary coordinate system used to apply Equation (6.7) is body fixed, the terms dω x dt, dω y dt, and dω z dt are the components of the rigid body s angular acceleration, α. But this is not generally the case if the secondary coordinate system rotates but is not body fixed (Equation 6.5). Using the Euler equations to analyse three-dimensional motions of rigid bodies typically involved three steps: 1. Choose a coordinate system: If an object rotates about a fixed point, O, it is usually preferable to use a secondary coordinate system with its origin at O and express the equations of angular motion in terms of the total moment about a fixed point O. Otherwise, it is necessary to use a coordinate system with its origin at the centre of mass of the body and express the equations of angular motion in terms of the total moment about the centre of mass of the body. In either case it is usually preferable to choose a coordinate system that simplifies the determination of the moments and products of inertia. 2. Draw a free-body diagram: Isolate the object and identify the external forces and couples acting on it. 3. Apply the equations of motion: Use Euler s equations to relate the forces and couples acting on the object to the acceleration of its centre of mass and it angular acceleration vector. Design Engineering Example: Consider the robotic arm, below, that is used during an assembly process; the 4 kg rectangular plate is held at O by the manipulator. Point O is stationary. At the instant shown, the plate is horizontal, its angular velocity vector is ω = 4i 2j rad/s, and its angular acceleration vector is α = 10i + 6j rad/s 2. We want to determine the couple, C, exerted on the plate by the manipulator.

10 Reminder: A couple is a system of forces with a resultant moment but no resultant force. Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass. The total moment about O is the sum of the couple exerted by the manipulator and the moment about O due to the plate s weight: ΣM O = C + (0.15i + 0.3j) [ 4(9.81)k] ΣM O = C ± 11.77i j N m To obtain the unknown couple C, we can determine the total moment about O from Equation (6.7). We let the secondary coordinate system be body fixed, so its angular velocity Ω equals the plate s angular velocity ω. First determine the plate s inertia matrix using the M4DE formula sheet; So, Inputting this into Equation (6.7); I xx = 1 3 (4)(0.6)2 = 0.48 kg m 2 I yy = 1 3 (4)(0.3)2 = 0.12 kg m 2 I zz = I xx + I yy = 0.6 kg m 2 I xy = 1 (4)(0.3)(0.6) = 0.18 kg m I = [ ]

11 ΣM x [ ΣM y ] = [ ] [ 6 ] + [ 0 0 4] [ ] [ 2] ΣM z ΣM x 5.88 [ ΣM y ] = [ 2.52 ] N m ΣM z 0.72 We substitute into the above equation for; ΣM O = C ± 11.77i j = 5.88i j k And solve for C; C = 5.89i 3.37j k N m