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In recent years, nonlinear dynamic models have been developed for flapping-type energy harvesting systems with a rigid wing, but not for those with a flexible wing. Thus, in this study, flexible wing designs of NACA0012 section are proposed and measurements of the forces of rigid cambered wings, which are used to estimate the performance of the designed wings, are conducted. Polar curves from the measured lift and drag coefficients show that JavaFoil estimation is much closer to the measured values than Eppler over the entire given range of angles of attack. As the camber of the rigid cambered wings is increased, both the lift and drag coefficients increase, in turn increasing the resultant forces. Moreover, the maximum resultant forces for all rigid cambered wings are achieved at the same angle of attack as the maximum lift coefficient, meaning that the lift coefficient is dominant in representations of the wing characteristics. Keywords: Flapping-type energy harvesting system, Flexible wing, Cambered rigid wing, Wing characteristics, NACA0012

This paper discusses design optimization of an electromagnetic soft actuator composes of two antagonistic solenoids that share a permanent magnet core. First, calculation of the magnetic field and applied force of a solenoid with a permanent magnet plunger is presented as the principal component of this electromagnetic actuator. Design optimization of the coil is discussed considering the geometrical parameters of the coil, including its length, inner and average diameters, number of turns, and packing density while the power consumption is bounded. The impact of the actuator size on the resultant force is presented and scaling limitations are discussed. Then, due to the soft nature of the actuator’s component, the impact of the cross section, that is, lateral deformation of the actuator on the magnetic field at the center of section, is investigated as well. The deformation might happen to the actuator due to the load in the transverse direction, especially when the actuator is made of flexible materials.

Operating forces and magnitude of loads from gas-liquid slug flows exerted on a horizontally orientated 90 o bend are investigated. The distributed forces are either Newtonian, associated with the fluids motion or Configurational, inherent to the internal distributions of the phases. The forces are derived through the conventional balances of mass and linear momentum arising from the volume of fluid (VOF) description of gas-liquid flows. The study uses the integral form of the momentum balance to estimate the operating forces budget. Invoking dynamical time scales separation discloses the connection of the Lamb vector (vortex-force) to the local time rate of momentum. An interesting outcome being an explicit expression for Favre-Reynolds stress that reveals the contribution of void fraction fluctuations in the redistribution of the stress across the interface. Numerical simulations are performed to determine the magnitude of Newtonian loads on bend using a segmented domain technique to represent the fully established slug flow regime. The time-dependent traces of the relevant flow variables such as liquid hold-up, flow rates and resultant forces on the bend are recorded and analysed. Compared to the isotropic component, the deviatoric stresses are shown to have a marginal contribution to the total forces. It is also shown that loading cycles on bends are much higher than slugging cycles; this is an important feature for the structural integrity assessment of pipelines with bends.

The pelvic floor has been poorly studied from a biomechanical perspective probably because of its complex structure comprised of many muscle layers and multiple muscle insertions with concave format in a three-dimensional architecture. An objective assessment of the pelvic floor muscles' (PFM) physical and mechanical capabilities is highly recommended to document changes in PFM function throughout rehabilitation protocols. Besides the magnitude of PFM force-generating capacity, other force/pressure-related parameters are important for the continence mechanism and sexual function, which are also correlated to different urogynecological dysfunctions. The existing devices for PFM assessment can be classified as manometers, dynamometers, electromyographers, or imaging techniques. So far, there is no perfect instrument or gold standard; however, each method has its advantages and disadvantages, and some of them will be discussed in this chapter. Recently, a new objective and reliable mechanical tool has been developed, being capable of measuring the magnitude and spatiotemporal distribution of pressures along the vaginal canal, taking into account the PFM structure, the directions of PFM contractions, and its ventro-cephalic resultant force. In this chapter, this device will be described and some of its clinical applications will be shown.

Traction calculation of train work studies the practical problems related to the train operation under various external forces, including the calculation or resolving of running speed and time, traction mass, locomotive or motor energy consumption, braking distance, etc. The first section of this chapter analyzes the formation of locomotive or EMU tractive effort, train running resistance, and braking pressure. The second section discusses the mathematical relations between acceleration (deceleration) and resultant force while three different forces are acting on a train; this is the train motion equation. We then employ the motion equation to solve some practical operating problems, such as traction mass, running speed, running time, braking, etc. Tractive tonnage is then discussed. The last section explains how to draw unit resultant force curves; then, based on this, the methods of calculating speed and running time are introduced.

UAVs navigating in unstructured environments need to be capable of avoiding obstacles present in their trajectory. One of the most popular approaches used for this task is to include the Artificial Potential Field (APF) Method in the navigation algorithm. In this chapter the obstacle avoidance problem is solved using the APF methodology and a backstepping nonlinear control algorithm. The APF method is based on the interaction of attractive and repulsion fields to determine a resultant force which is used to avoid obstacles. The control technique is separated in two prioritized levels: the first level controls the attitude of a quadcopter vehicle and the second deals with its position; in this level obstacle avoidance is one of the most important parts. Simulation and experimental results introduce the performance of the closed-loop system in the presence of obstacles in the trajectory.

This chapter describes automotive friction brake types, and methods of analysis for disc and drum brakes to predict the relationship between the actuation force and the brake torque generated. The brakes studied, with examples, include leading/trailing shoe (simplex), twin leading shoe (duplex), duo-servo and S-cam drum brakes, and fixed (opposed piston) and sliding caliper designs of disc brake. The basic analysis represents the friction between the friction material and the rotor as a single resultant force, and the analysis is then extended to consider the effects of contact and pressure distribution at the friction interface. These include bulk deformation arising from component flexure, localised deformation and distortion of the rubbing surfaces, and asperity contact and interaction. Classical methods of brake analysis are described, and the role of computer analysis methods, especially the finite element (FE) method, is explained. The design parameters brake factor, specific torque and ηC ∗ , and their sensitivity to changes in the coefficient of friction, are explained and discussed.

Publisher Summary This chapter provides an overview of riveted joints. In the majority of built-up structures the several members are united by riveted joints, and the strength of the joints is just as important as the strength of the members themselves. Unfortunately, the strength of riveted joints cannot be calculated with any degree of certainty, and practical design usually depends on empirical formulae based on experience. It is evident that, if the rivets are carrying any load, they must be acting as stops to check the relative movements of the plates. But, as long as the friction forces are great enough, there will be no relative movement, and the rivets will not bear any load; as soon as slip occurs, the rivets are acted on by shearing forces in the planes separating the cover plates from the main plates. It is obvious that a uniform distribution of load among the rivets cannot be attained unless the line of action of the resultant force acting on the joint passes through the centroid of the rivet-holes, and, at the same time, the rivets are symmetrically disposed with regard to the resultant force.

This chapter discusses general analysis of stress and strain. Many problems in stress calculation require for their solution. Examples of such problems are afforded by thick tubes subjected to internal or external pressure, rotating discs, the flexure of flat plates, the more exact calculation of bending stresses which is desirable in a few particular cases, the torsion of shafts of non-circular cross section, etc. In general, the method consists in considering the equilibrium and deformation of an element of the body and so forming differential equations, the solution of which enables calculating the stresses and strains at any point in the body. To specify completely the resultant force on anyone plane one must know its components in three directions. This, however, is not true, for consider the equilibrium of a small rectangular block of matter. With regard to the notation adopted for shear stresses it will be noticed that the second letter of the suffix denotes which plane the stress is acting on, whilst the first letter of the suffix shows the axis to which the direction of the stress is parallel.

This chapter focuses on the stresses that the ship experience while floating in still water and at sea. A ship floating in still water has an unevenly distributed weight owing to both cargo distribution and structural distribution. The buoyancy distribution is also non-uniform because the underwater sectional area is not constant along the length. Total weight and total buoyancy are of course balanced, but at each section there will be a resultant force or load, either an excess of buoyancy, or an excess of load. As the vessel remains intact, there are vertical upward and downward forces tending to distort the vessel. These forces are referred to as vertical shearing forces because they tend to shear the vertical material in the hull. When a ship is in a seaway, the waves with their troughs and crests produce a greater variation in the buoyant forces; therefore, the waves can increase the bending moment, vertical shear force, and stresses. Classically, the extreme effects can be illustrated with the vessel balanced on a wave of a length which is equal to that of the ship. If the crest of the wave is amidships, the buoyancy forces will tend to “hog” the vessel, and if the trough is amidships, the buoyancy forces will tend to ‘sag’ the ship.

This chapter deals with the angle of list. It considers a ship floating upright, where the centre of gravity and buoyancy are on the centerline. The resultant force acting on the ship is zero, and the resultant moment about the centre of gravity is also zero. It is assumed that weight already on board the ship is shifted transversely such that G moves to new centre of gravity (G 1 ). This will produce a listing moment, and the ship will list until G 1 and the centre of buoyancy are in the same vertical line. In this position, G 1 will also lie vertically under metacentre M so long as the angle of list is small. Therefore, if the final positions of the metacentre and the centre of gravity are known, the final list can be found. The final position of the centre of gravity is found by taking moments about the keel and about the centerline.

This chapter presents a brief examination of the basic principles of force and moments, because the solution of many of the problems concerned with ship stability involves an understanding of the resolution of forces and moments. A force can be defined as any push or pull exerted on a body. When two or more forces are acting at a point, their combined effect can be represented by one force that has the same effect as the component forces. Such a force is referred to as the resultant force, and the process of finding it is called the resolution of the component forces. The chapter explains resolution of forces with respect to three cases: when two forces act in the same straight line, when two forces do not act in the same straight line, and when two forces act in parallel direction. The moment of a force is a measure of the turning effect of the force about a point. The turning effect depends upon the magnitude of the force and the length of the lever upon which the force acts—the lever being the perpendicular distance between the line of action of the force and the point about which the moment is being taken. When two or more forces are acting about a point, their combined effect can be represented by one imaginary moment called the resultant moment. The process of finding the resultant moment is referred to as the resolution of the component moments.

Publisher Summary This chapter discusses the principles of statics that are essential to structural and stress analysis. A force is a vector that may be represented graphically, where the force F is considered to be acting on an infinitesimally small particle at the point A and in a direction from left to right. The magnitude of F is represented, to a suitable scale, by the length of the line AB and its direction by the direction of the arrow. In vector notation the force F is written as F. The resultant of two concurrent and coplanar forces, whose lines of action pass through a single point and lie in the same plane, may be found using the theorem of the parallelogram of forces. The law of the triangle of forces may be used in the analysis of a plane, pin-jointed truss in which, one of the three concurrent forces is known in magnitude, and direction. The law enables finding the magnitudes of the other two forces and also the direction of their lines of action.

For a kinematic chain, it is important to know how forces and moments are transmitted from the input to the output so that the links can be properly designated. The friction effects are assumed to be negligible in the force analysis. A particle is an object whose shape and geometrical dimensions are not significant to the investigation of its motion. An arbitrary collection of matter with total mass m can be divided into N particles, the i th particle having the mass, m i . A rigid body can be considered as a collection of particles in which the number of particles approaches infinity and the distance between any two points remains constant. The acceleration of the mass center can be related to the external forces acting on the system. This relationship is obtained by applying Newton's laws to each of the individual particles in the system. Any such particle is acted on by two types of forces. One type is exerted by other particles that are also part of the system. Such forces are called internal forces (internal to the system). Additionally, a particle can be acted on by a force that is exerted by a particle or object not included in the system. Such a force is known as an external force (external to the system).

Publisher Summary This chapter discusses the conventional type of tail rotor in a helicopter, along with a number of alternatives having various advantages and drawbacks. Helicopter designers have explored various ways of countering the main rotor torque in a way that reduces or eliminates one or more of these problems. These techniques give a noise reduction and a safety advantage, but currently at an increased cost. The tail rotor only requires collective pitch control and so it will have a swash plate, which moves along the shaft axis but without tilting. There are various ways of moving the swash plate via the foot pedals. The torque reaction of the main rotor is a pure couple, which is to say that it has only a turning effect and no resultant force in any direction. The tail rotor is mounted pointing sideways at the tail, and so the thrust it produces results in a moment about the centre of gravity of the machine. In any tail rotor operation that increases the inflow, the blade reaction will tilt back and the tail rotor will require more torque to drive it. This torque is provided by the engine, and delivered by the long tail drive shaft.

This chapter focuses on some design aspects of pumps and turbines. It is assumed that whatever design procedures apply to pump impellers, the same procedures may be applied to turbine runners. Over a period, a number of empirical dimensionless groups have been developed that have been found to be useful in the design of axial-flow pumps and turbines. All impellers and runners are subject to axial thrust. The resultant force tends to move the impeller or runner away from the suction side because of the difference in static pressure on both sides of the impeller/runner. For any rotating machine, as the speed of rotation is increased, it may be observed that at certain speeds the shaft of the machine will oscillate or vibrate. The vibration occurs because the machine is imbalanced as it rotates. Only rarely is a rotational machine perfectly balanced. Even a small mass imbalance may produce large centrifugal forces that are balanced by the spring action of the shaft, causing the system to vibrate excessively.

Publisher Summary This chapter briefly discusses pressure management; there are three categories of pressure measurement: absolute pressure, gauge pressure, and differential pressure. Absolute pressure is the difference between the pressure at a particular point in a fluid and the absolute zero of pressure, that is, a complete vacuum. A barometer is one example of an absolute pressure gauge, because the height of the column of mercury measures the difference between the atmospheric pressure and the “zero” pressure of the Torricellian vacuum that exists above the mercury column. When the pressure-measuring device measures the difference between the unknown pressure and local atmospheric pressure, the measurement is known as gauge pressure. There are three basic methods for measuring pressure. The simplest method involves balancing the unknown pressure against the pressure produced by a column of liquid of known density. The second method involves allowing the unknown pressure to act on a known area and measuring the resultant force either directly or indirectly. The third method involves allowing the unknown pressure to act on an elastic member (of known area) and measuring the resultant stress or strain.

In this chapter, we present the plate analysis equations, integrating the local equations when the three absolute values of displacements are low in comparison to the plate thickness, and when the absolute values of the components of displacement vector gradient are much lower than one. We will limit this study to the resultant forces and moments and the inertia forces in the Reissner–Mindlin and Kirchhoff–Love type analyses. First in detail, we look at the hypothesis relating to plates that is explained in great detail. The next part of the chapter deals with the Reissner–Mindlin and Kirchhoff–Love plate theories, the segment following the chapter deals with the plate equations and plate equations in Reissner–Mindlin are calculated in different conditions. Plate equations in Kirchoff–Love analysis are seen and calculations under different conditions are made.

Publisher Summary The seismic behavior of a deck-type steel arch bridge composed of parallel twin ribs with lateral members is investigated in this chapter. The restoring force model incorporating the interaction curves of stiffened short arch ribs subjected to compression by in-plane and out-of-plane bending moments is analyzed numerically using MARC. A nonlinear seismic response analysis of a deck-type steel bridge arch is carried out, by which the effects of a RC floor slab stiffness and interaction curves are examined. The influence of interaction curves on the interpretation for yield resultant forces is shown to be a significant factor in the seismic behavior of arches. The 3-dimensional seismic analysis of a deck-type arch bridge composed of parallel twin ribs with lateral members subjected to ground acceleration is theoretically analyzed. A parametric study is conducted using the finite element method (FEM) analysis and nonlinear seismic response analysis methods by considering the effect of interaction curves on yield resultant forces and the nonlinear seismic behavior of a deck-type arch bridge.

This chapter provides an account of the sign conventions for right-handed co-ordinate systems. It discusses forces and moments. A force vector has magnitude and direction and can be represented by a single straight line in space. Vectors, which can be represented as straight lines, are physical quantities, having magnitude and direction. It describes the equilibrium of a body. From Newton's second law, the following must be true if the body is to be at rest: the resultant force must be zero and the resultant moment must be zero. A generalized displacement consists of two vectors and cannot be represented by a single straight line in space. If the normal to the plane is in the positive co-ordinate axis direction, then the positive normal and shear stresses act in the direction of the corresponding positive co-ordinate axis and vice versa . The chapter discusses specific deformations (strains). Shear strains (rotations!) are functions of shear stresses and are independent of any normal stresses. When considering structural elements as discrete elements, it is of interest to find the overall element deformations and the corresponding joint displacements.

The range of cutting forces acting upon a cutting tool is studied, and it is necessary to evaluate the effect of these upon the machine tool and analyze the cutting process. These forces can be resolved to determine the total resultant force acting. This can be very large and can easily cause significant deformation of the work piece, thus finishing cuts are usually taken with small forces to minimize the forces acting and to provide accurate cutting conditions. For most work piece materials, higher cutting speeds mean lower cutting forces, the higher temperatures on the flow zone and reduced contact area contribute toward this effect. The decrease in force varies with the type and condition of the material, and the range of cutting speeds in use. For heat-resistant nickel-based alloy steel, the initial chip forming force can be ten times stronger than that required to cut unalloyed aluminum.

Publisher Summary This chapter provides the method for the design of orthogonal bracing connections and then generalized to non-orthogonal connections, such as occur in trusses. The method is checked against existing physical tests and found to be safe. The method produces cost effective designs. Bracing connections constitute an area in which there is disagreement concerning a proper method of design. The American Institute of Steel Construction addressed this problem with a research program at the University of Arizona. This program describes the resultant forces on the gusset edges for a wide variety of gusset edge support conditions are seen to fall within the envelopes shown. The edge resultants appear to intersect with the line of action of the brace at a point on this line on the other side of the working point (WP) from the gusset.

Publisher Summary This chapter discusses the mechanical engineering principles of materials. The study of mechanics may be divided into two distinct areas. These are static, which involves the study of bodies at rest, and dynamic, which is the study of bodies in motion. When a set of forces act on a body they give rise to a resultant force or moment or a combination of both. The situation may be determined by considering three mutually perpendicular directions on the “free body diagram” and resolving the forces and moment in these directions. When using analysis, the moment of each element of weight within the body, about a fixed axis, is equal to the moment of the complete weight about that axis. Circular motion is a special case of curvilinear motion in which the radius of rotation remains constant. In this case, there is acceleration toward the center of ω2r. This gives rise to a force toward the center known as the “centripetal force.” This force is reacted to by the centrifugal reaction. The principle of balancing is that by the addition of extra masses to the system the out-of-balance forces may be reduced or eliminated.

The WENDELSTEIN 7-X stellarator experiment is developed as a Helias (Heli cal A dvanced S tellarator) configuration. The superconducting modular coils with nonplanar shape introduce a magnetic field with a maximum net coil force of about 4 MN. The resultant force vector of a whole field period is directed towards the torus centre. In order to support the magnetic forces, a scheme of mutual support has been developed. Each coil is enclosed by an individually adapted stainless steel housing. The coils of a field period are connected to a module by a pair of reinforcements inside the torus. For reasons of accuracy calculations are made within one period. The investigated stress levels in the coils and support depend on the mechanical contact between winding pack and coil housing. The nonplanar coils lead to difficulty in defining the bearing conditions at the ends of a period. The assumptions for prescribing these boundary conditions, which are needed for the FE analysis, are presented.

In this paper we present the results of simulations of plasma disruption events in ASDEX-Upgrade performed using a 3-D eddy current code developed for the electromagnetic analysis of next step fusion experiments. Inputs for the simulations were taken both from experimental observations of plasma current quench in the machine as well as from the original specifications used for the design of the ASDEX-Upgrade vacuum vessel. The results of the calculations have been compared with the available experimental measurements: both integral parameters such as induced currents and loop voltages and local characteristics, such as the magnetic field distribution, have been considered, in order to demonstrate the adequateness of the calculation method and model set up. The results of the analysis of the design disruption have been compared with previous work and have been used to predict resultant forces on part of the ASDEX mechanical structures.

Publisher Summary This chapter focuses on coplanar forces acting at a point. When all forces are acting in the same plane, they are called coplanar whereas when forces act at the same time and at the same point, they are called concurrent/forces. For forces acting in the same direction and having the same line of action, the single force having the same effect as both of the forces, called the resultant force or just the resultant, is the arithmetic sum of the separate forces. When two forces do not have the same line of action, the magnitude and direction of the resultant force may be found by a procedure called vector addition of forces. There are two graphical methods of performing vector addition, known as the triangle of forces method and the parallelogram of forces method. For forces acting in opposite directions along the sameline of action, the resultant force is the arithmetic difference between the two forces. When three or more coplanar forces are acting at a point and the vector diagram closes, there is no resultant. The forces acting at the point are in equilibrium.

Hornets build combs that are clearly oriented toward the center of gravity and they compensate for any change made in the direction of their cells, provided the gravitational vector remains unchanged. Under natural conditions the first few cells are built by the queen in the spring, following the hibernation period, and building is then continued by the workers. Recently several reports have been published concerning the fact that groups of hornets (V. orientalis workers) may build in the absence of the queen and that such combs resemble those built by the queen. In previous investigations on building orientation, hornets placed in artificial breeding boxes (ABBs) on a horizontal centrifuge were subjected to constant centrifugal and gravitational forces, the resultant of which ranged between 1-1.5 g. Under these conditions young hornets (1-2 days of age) built combs in the direction of the resultant force, while the direction of building by adult hornets (3-7 days of age) was dependent on the presence or absence of a straight roof. Adults placed in the ABBs of rectangular shape built combs the orientation of which was determined by the Earth's gravity alone (perpendicular to the roof) whereas those in spherical ABBs with a convex roof built combs in the direction of the resultant gravitational and centrifugal forces, much the same as combs built by the young hornets. We were, however, also interested in studying the building orientation of hornets placed under conditions of changing resultant forces. The present paper describes comb building by hornets under the alternating gravitational forces of a vertical centrifuge.

1. The effect of persistent acceleration on comb construction by Oriental hornet (Vespa orientalis) workers was assessed experimentally within breeding boxes of various sizes, and shapes. Groups of hornets in the building phase were subjected to a centrifugal and gravitational force whose resultant ranged between 26° and 45°. The comb construction within such boxes was compared to that within control boxes under ordinary gravitational pull. 2. It was found that: (a) juvenile hornets (1–2 days of age) placed in quasi-rectangular boxes built in the direction of the resultant force; (b) juvenile and adult hornets (3–7 days of age) placed in spherical-shaped containers also built in the direction of the resultant force; (c) adult hornets who had spent their first days of life in a stationary rectangular box, ‘tried’ to build in the direction of the gravitational force when exposed to a centrifugal force; (d) adult hornets made to spin in quasi-rectangular boxes tilted in the direction of the calculated resultant, built in the direction of the resultant. 3. Only the roof of the breeding box gives geometric cues of critical importance.

Publisher Summary This chapter provides an overview of the idea of projectiles. For a projectile motion, acceleration, being a vector, might also be expressed as the sum of two components, the directions of which would be most convenient when at right-angles. However, the acceleration of a body of certain mass is related to the resultant force on the body. The velocity of the body is changing throughout the motion. Putting together all the relevant vectors in the motion, it is concluded that (1) acceleration is constant; (2) velocity is not constant, either in magnitude or in direction; and (3) displacement is not constant. To vary the elementary experience of Newton's equations of motion, the motion of a body has been explained, in which only a component of its weight is able to contribute to the acceleration of the body.

Integral equations for the resultant forces on a piece–wise smooth crack line are formulated and coupled to the standard BEM equations for the outer boundary of a finite plane. The resulting equations are a generalization of the equations for infinite geometries (Cheung and Chen, 1987). The integrals along the crack line, with the dislocation densities as unknowns, contain only a weak logarithmic singularity. An improvement of the numerical formulation at a kink of the crack line is introduced. Two numerical experiments are presented and compared with alternative numerical calculations.

This chapter discusses parallel forces and rigid bodies. The turning effect of a force about a fulcrum is called the moment of the force about the fulcrum. The moment of a force about a fulcrum is the product of the magnitude of the force and its perpendicular distance from the fulcrum. The resultant of two parallel forces acting on a rigid body is that the single force that would produce the same effect on the body. The chapter discusses the systems of parallel forces so that the moments have been easy to calculate and the balancing of clockwise and anticlockwise moments has been easy to verify. It is a simple matter to adapt the existing apparatus to examine the case of two non-parallel forces acting on a beam. The more forces that act on a body, the more difficult it becomes to investigate its possible equilibrium. A system of two forces keeping a body in equilibrium has been illustrated by the suspension of a body from a spring balance.

Publisher Summary This chapter discusses the properties of coplanar forces acting at a point. When forces are all acting in the same plane, they are called coplanar. When forces act at the same time and at the same point, they are called concurrent/forces. Force is a vector quantity and, thus, has both a magnitude and a direction. A vector can be represented graphically by a line drawn to scale in the direction of the line of action of the force. For forces acting in the same direction and having the same line of action, the single force having the same effect as both of the forces, called the resultant force or just the resultant, is the arithmetic sum of the separate forces. For forces acting in opposite directions along the same line of action, the resultant force is the arithmetic difference between the two forces. An alternative to the graphical methods of determining the resultant of two coplanar forces is by calculation. This can be achieved by trigonometry using the cosine rule and the sine rule, or by resolution of forces.

This chapter discusses vector algebra and coplanar forces. If a system of coplanar forces acts at a point, it becomes convenient to know which single force acting at the point would produce the same result as the original system. Force is a vector quantity and so the resultant force is the vector sum of all the forces in the system. The chapter explains how the resultant of a system of coplanar forces is that single force, if it exists, which produces the same effect as the original system. Vector addition is called a binary operation because it combines only two vectors at a time. The chapter discusses the associative law for addition of vectors.

This chapter discusses the effects of forces on materials. A force exerted on a body can cause a change in either the shape or the motion of the body. The unit of force is the Newton ( N ). No solid body is perfectly rigid and when forces are applied to it, changes in dimensions occur. Such changes are not always perceptible to the human eye as they are small. It is important for engineers and designers to appreciate the effects of the forces on materials, together with their mechanical properties. The three main types of mechanical force that can act on a body are tensile, compressive, and shear. Tensile force tends to stretch a material, such as the rope or cable of a crane carrying a load is in tension. It increases the length of the material on which it acts. On the other hand, compression will decrease the length of the material on which it acts. Shear is a force that tends to slide one face of the material over an adjacent face. It can cause a material to bend, slide, or twist.

This chapter describes the equilibrium of rigid bodies. In a rigid body, the distances between every pair of points in the body are constant. A rigid body can undergo translational motion, rotational motion, or a combination of both translational and rotational motions. A rigid body translates if every point in the body moves the same distance parallel to a line. A rigid body rotates if one point in the body is fixed. In addition, when the forces acting on a rigid body combine to maintain the body in a state of rest or of motion with constant velocity, the body is said to be in a state of equilibrium. There are two conditions of equilibrium which, if satisfied, guarantee the rotational and translational equilibrium of a rigid body. The chapter presents the two conditions and elaborates them through examples. The concept of center of gravity is developed and three kinds of equilibrium are classified according to their stability. The chapter also introduces the principles of force and torque to make it possible to discuss equilibrium.

This chapter discusses some types of motion of a rigid body. In mechanics, rigid body means that the relative position of the parts of a body remains unchanged during the motion. The body, thus, moves as a whole. The simplest motion of a rigid body is one in which it moves parallel to itself; this is called translation. For example, if a compass is moved smoothly in a horizontal plane, the needle will retain a steady north-south direction and will execute a translational motion. In translational motion of a rigid body, every point in it has the same velocity and describes a path of the same shape, there being merely a displacement between the paths. Another simple type of motion of a rigid body is rotation about an axis. The chapter describes the energy of a rigid body in motion, rotational angular momentum, gyroscope, and inertia forces.

This chapter discusses the conditions for static stability. An airplane will continue in steady straight flight only if the resultant force acting on it is zero and the resultant moment about the centre of gravity is zero. Under any other circumstances, the airplane will be subjected to the appropriate linear or angular accelerations. The airplane is, therefore, in a state of equilibrium or trim when these resultants are zero. The airplane is statically stable when it tends to revert to the equilibrium condition after a disturbance in pitch. This observation only indicate a tendency to revert to the equilibrium condition by considering pitching moments assessed under static conditions.

This chapter discusses the flow of limited jets around obstacles. The chapter presents a study wherein the fluid flow was reflected about the upper horizontal wall of a vessel and the coordinate origin was shifted for future convenience. The wall was replaced by a streamline that was located within the fluid. The chapter further presents some equations that give the general solution for a finite-width jet flow around a wedge. In terms of the parameters in these equations, not only the flow rates in the jets but also all the geometric and hydrodynamic characteristics of the problem can be found out. The pressure at each point in the flow is computed by an application of the Bernoulli integral; however, the resultant force acting on the wedge can be computed without integration. Because of the flow symmetry, only the lower half of the flow replacing the axis of symmetry x by a solid wall should be considered.

This chapter discusses centers of gravity and centroids. A solid body is made of a very large number of small bodies or particles; the earth attracts each one of these particles toward its center by the force of gravity. Each gravitational force is known as the weight of the particle. Because the earth is so large in comparison with any body, on it the radial lines of force attracting each particle to the earth's center can be regarded as parallel to each other. If all these tiny gravitational forces are added together, the total represents the weight of the body itself. In other words, the resultant of all the gravitational forces acting on the tiny particles is equal to the weight of the body. The resultant force is the one that produces the same effect as all these tiny gravitational forces acting together, and its line of action can be represented by the thick vertical line.

Publisher Summary This chapter provides an overview on vectors. Vectors have two properties: length and direction. When a scalar a and a vector v combine to form a new vector av called the scalar multiple of a and v. If v is expressed in coordinates, av is express in coordinates as a(x, y) = (ax, ay). If two forces v and w are applied at the same point, the resultant force is v + w, that is, the law of parallelogram of forces. The opposite to vector v is denoted by the vector -v. The difference between vectors v − w is obtained by v − w = v + (-w). The geometric interpretation of the difference of vectors illustrates that the directed line segment from w to v has the same length and direction as the vector v − w. If v is any nonzero vector, then v/||v|| is a unit vector, length 1, in the same direction, where ||v|| is the length of v.

Publisher Summary This chapter discusses the applications of vector calculus. It describes the equivalence of force systems. Two systems of forces are said to be equivalent if when they separately act upon a rigid body they would produce the same effect in each case. If two forces F1 and F2 act along lines that intersect, they are equivalent to the force F1 + F2 acting at the point of intersection of the two fines. Thus, the resultant of two intersecting forces is given by the parallelogram of forces. The resultant of two parallel forces—which are not equal and opposite—is a third parallel force. The line of action of the resultant forces passes through the centroid of masses proportional to F1 and F2 placed at points and r1 and r2 respectively. Equal and opposite forces F and −F acting along parallel lines constitute a couple. A couple has the property that the moment about any point of the pair of forces constituting the couple does not depend on the point about which the moment is taken.

When we think of forces we usually consider the action of lifting a weight, bending a piece of wire or setting a vehicle in motion. Whenever we move, or move other objects around us, then forces are brought into play. One of the most important forces is the one that causes objects to fall downwards, the Force of Gravity. This chapter discusses the types of force, effects of forces, and the equilibrium of forces. The effect of a number of forces acting on a body depends not only on the magnitude of the forces and their directions, but also on the points at which the forces are applied to the body. This action of the force turning about an axis or fulcrum is called the turning moment or moment of the force, or torque. The chapter also presents the experiment to find the coefficient of friction between two surfaces in contact.

This chapter provides an overview of the general plane motion of a rigid body. It discusses the meaning of the general torque equation for plane motion. A system of plane forces acting on a rigid body are equivalent to a single force equal to the resultant of the system of forces and applied through the center of mass of the body, thus, giving the body a pure translation and to a couple whose magnitude is equal to the product of the resultant force and the distance of its line of action from an axis through the center of mass, thus, giving the body a rotation round this axis. In general, a system of external forces applied to a rigid body will give the center of mass the same acceleration which they would give it if their resultant passed through the center of mass and, in addition, give the body an angular acceleration round the center of mass.

This chapter discusses the forces acting on the rocket in flight. A body moving through air is subjected to the action of a system of forces distributed along its surface. The resultant of these forces is called the total aerodynamic force. The component of aerodynamic force along the tangent to the trajectory of the center of gravity of the body, or the projection of the total force onto the direction of the velocity vector, is called the frontal resistance force. This force is always opposite to the direction of motion. The component of the total aerodynamic force along the normal to the trajectory, or its projection on a normal to the direction of velocity, is called the lift force. The chapter also explores the properties of the atmosphere, taking into account that the magnitudes of aerodynamic forces acting on the rocket in flight depend on these properties.