Calculus Kinematics
Calculus kinematics involves using calculus to analyze the motion of objects. It focuses on concepts such as position, velocity, and acceleration, and uses derivatives and integrals to study the relationships between these quantities. By applying calculus to kinematics, it becomes possible to understand and predict the behavior of moving objects with greater precision and detail.
5 Key excerpts on "Calculus Kinematics"
eBook - ePub- Ronald J. Anderson(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...Part I Modeling: Deriving Equations of Motion 1 Kinematics Kinematics is defined as the study of motion without reference to the forces that cause the motion. A proper kinematic analysis is an essential first step in any dynamics problem. This is where the analyst defines the degrees of freedom and develops expressions for the absolute velocities and accelerations of the bodies in the system that satisfy all of the physical constraints. The ability to differentiate vectors with respect to time is a critical skill in kinematic analysis. 1.1 Derivatives of Vectors Vectors have two distinct properties – magnitude and direction. Either or both of these properties may change with time and the time derivative of a vector must account for both. The rate of change of a vector with respect to time is therefore formed from, The rate of change of magnitude. The rate of change of direction. Figure 1.1 A vector changing with time. Figure 1.1 shows the vector that changes after a time increment,, to. The difference between and can be defined as the vector shown in Figure 1.1 and, by the rules of vector addition, (1.1) or, (1.2) Then, using the definition of the time derivative, (1.3) Imagine now that Figure 1.1 is compressed to show only an infinitesimally small time interval,. The components of for the interval are shown in Figure 1.1. They are, A component aligned with the vector. This is a component that is strictly due to the rate of change of magnitude of. The magnitude of is where is the rate of change of length (or magnitude) of the vector. The direction of is the same as the direction of. Let be designated 1 as. A component that is perpendicular to the vector. That is, a component due to the rate of change of direction of the vector. Terms of this type arise only when there is an angular velocity...
eBook - ePubMechanism Design
Visual and Programmable Approaches
- Kevin Russell, Qiong Shen, Raj S. Sodhi(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
...1 Introduction to Kinematics 1.1 KINEMATICS Kinematics is the study of motion without considering forces. In a kinematic analysis, positions, displacements, velocities, and accelerations are calculated for mechanical system components without regard to the loads that actually govern them. In comparison to other engineering design disciplines such as statics, where motion and governing loads are considered according to Newton’s first law (∑ F = ∑ M = 0), and dynamics, where motion and governing loads are considered according to Newton’s second law (∑ F = m a, ∑ M = I α), kinematics is the most fundamental engineering design discipline. Because it is often necessary in the design of a mechanical system to not only consider the motion of mechanical system components, but also the forces acting on them, the component material stress and strain responses to the forces (stress analysis), and the required component dimensions for the working stresses (machine design), such analyses often follow kinematic analyses. Figure 1.1 shows kinematics, statics and dynamics, stress analysis, and machine design in an ascending order of progression. This order follows the intended sequence of use of these disciplines in mechanical design. After a mechanical system has been determined to be kinematically feasible, the static and/or dynamic loads acting on the system components are considered next. After static and/or dynamic feasibility have been achieved, the stresses and strains produced in the mechanical system components are considered next. Lastly, machine design principles and methodologies are employed to ensure the material and dimensions of the mechanical system components (and subsequently the entire mechanical system) are satisfactory for the known working stresses. * As illustrated in Figure 1.1, kinematics is the most fundamental of the listed engineering design disciplines...
eBook - ePubInstant Notes in Sport and Exercise Biomechanics
Second Edition
- Paul Grimshaw, Michael Cole, Adrian Burden, Neil Fowler(Authors)
- 2019(Publication Date)
- Garland Science(Publisher)
...SECTION A Kinematics of linear motion A1 LINEAR MOTION Paul Grimshaw Biomechanics can be defined as the study of forces and the effects of these forces on living things. In mechanics, further subdivision includes kinematics and kinetics. Kinematics is the study of displacement, velocity and acceleration (spatial and temporal) while kinetics examine forces, moments and torques. Biomechanics and mechanics are used extensively to study human motion. Figure A1.1 defines biomechanics and kinematics in more detail. Human movement can often be classified into components of linear and angular motion. This leads to a description that is termed general motion. Linear motion (or translation) is movement along a line which may be either straight or curved and where all the body parts are moving in the same direction at the same speed. This can be classified as either rectilinear motion (motion along a straight line) or curvilinear motion (motion along a curved line). Angular motion (discussed elsewhere in this text) involves movement around an axis (either imaginary or real) with body parts (or individual body parts) moving through angles (the same or different angles) in a certain time frame (Figure A1.2). Linear kinematics deal with quantities that describe the motion of bodies such as distance, displacement, speed, velocity and acceleration. These can be classified as either scalar or vector quantities. Scalar quantities are represented by magnitude (size) only, whereas vector quantities possess both magnitude and direction. Vector quantities are represented mathematically, as a symbol with a direction sign or graphically by scaled straight lines or arrows. For example, speed is defined as the distance travelled per unit of time (i.e...
eBook - ePub- Richard Gentle, Peter Edwards, William Bolton(Authors)
- 2001(Publication Date)
- Newnes(Publisher)
...4 Dynamics Summary This chapter deals with movement. In the first part the movement is considered without taking into account any forces. This is a subject called kinematics and it is important for analysing the motion of vehicles, missiles and engineering components which move backwards and forwards, by dealing with displacement, speed, velocity and acceleration. These quantities are defined when we look at uniform motion in a straight line. This subject is extended to look at the particular case of motion under the action of gravity, including trajectories. This chapter also looks at how the equations of motion in a straight line can be adapted to angular motion. Finally in the first half the subject of relative velocity is covered as this is very useful in understanding the movement of the individual components in rotating machinery. In the second part of this chapter we consider the situation where there is a resultant force or moment on a body and so it starts to move or rotate. This topic is known as dynamics and the situation is described by Newton’s laws of motion. Once moving forces are involved, we need to look at the mechanical work that is being performed and so the chapter goes on to describe work, power and efficiency. Newton’s original work in this area of dynamics was concerned with something called momentum and so this idea is also pursued here, covering the principle of conservation of momentum...
- Paul Grimshaw, Neil Fowler, Adrian Lees, Adrian Burden(Authors)
- 2007(Publication Date)
- Routledge(Publisher)
...In mechanics there is use of a further subdivision into what is known as kinematic and kinetic quantities. Biomechanics and mechanics are used to study human motion. This section is concerned with linear (i.e., translational – where all the points move in the same direction in the same time and without rotation) kinematics. Fig. A2.1 helps to illustrate the definition of biomechanics and kinematics in more detail. Human movement or motion can be classified as either linear or angular motion. Most movements within biomechanics are a combination of translation and rotation. This leads to a description that is termed general motion. Linear motion (or translation) is movement along a line which may be either straight or curved and where all the body parts are moving in the same direction at the same speed. This can be classified as either rectilinear motion (motion in a straight line) or curvilinear motion (motion in a curved line). Angular motion (which will be discussed in the next section) involves movement around an axis (either imaginary or real) with all the body parts (or individual body parts) moving through the same angle at the same time. Fig. A2.2 identifies these types of motion in more detail. Fig. A2.1. Biomechanics, kinematics and kinetics Fig. A2.2. Different types of motion Kinematics and kinetics Linear kinematics is concerned with the quantities that describe the motion of bodies such as distance, displacement, speed, velocity, and acceleration. These quantities can be classified as either scalar or vector quantities. Scalar quantities are represented by magnitude (size) only, whereas vector quantities are represented by both magnitude and direction. Hence, vector quantities can be presented mathematically or graphically on paper by scaled straight lines or arrows. For example, speed is defined as the distance traveled per unit of time and as such it is a scalar quantity (i.e., no direction is specified). Ex 1...
