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機(jī)械工程學(xué)院畢業(yè)設(shè)計（論文）前期工作材料
學(xué)生姓名： 陳沛東 學(xué)號： 100007102
教 科 部： 機(jī)械電子工程系
專 業(yè)： 機(jī)械設(shè)計制造及其自動化
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指導(dǎo)老師： 竺志大
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2014年04月02日
4. forces in structures and fluids?
OVERVIEW
When mechanical engineers design products, systems, and hardware, they
must apply mathematics and physical principles to model, analyze, and predict
system behavior. Successful design is supported by effective engineering
analysis; effective engineering analysis relies on an understanding of forces in
structures and machines. This is the focus of this chapter and the next element
of mechanical engineering.
This chapter introduces you to the subject of mechanics, a topic that
encompasses forces that act on structures and machines and their tendency
either to remain stationary or move. The fundamental principles that form
the basis of mechanics are Newton’s three laws of motion:
1. Every object remains in a state of rest or uniform motion of constant
velocity unless an external unbalanced force acts upon it.
2. An object of mass m, subject to a force F, experiences an acceleration in
the same direction as the force with a magnitude directly proportional to
the magnitude of the force and inversely proportional to the mass of the
object. This relationship can be expressed as F 5 ma.
3. The forces of action and reaction between two objectives are equal,
opposite, and collinear.
4.3 MOMENT OF A FORCE
When you are trying to loosen; a frozen bolt, the bolt is more easily turned when a wrench with a long handle is used. The tendency of a force.(in this case, applied to the handle's end) to make an object rotate is called a moment. The magnitude of a moment depends both on the force that is applied and on the lever arm that separates the force from a pivot point.
Perpendicular Lever Arm
The magnitude of a moment is found from its definition
(4.8)
where Mo is the moment of the force about point O , F is the magnitude of the force, and d is the perpendicular lever arm distance from the force's line of action to point O. The term torque can also be used to describe the effect of a force acting over a lever arm, but mechanical engineers generally reserve torque to describe moments that cause rotation of a shaft in a motor, engine, or gearbox. We will discuss those applications in Chapter 7.
Based on Equation 4.8, the unit for Mo is the product of force and, distance. In the USCS ， the unit for a moment is in-lb or ft-lb. In the SI, the unit N-m is used, and various prefixes are applied when the numerical value is either very large or very small. For instance; 5000N-m＝5kN-m and 0.002N-m＝2mN-m. Conversion factors between the two systems of units are shown in Table 4:2, where you can see that 1 ft-lb＝1.356 N-m. Work and energy, which are other quantities that arise in mechanical engineering, also have units that are the product of force and distance. For instance, when working in the SI, a joule (J) is defined as one newton-meter. It is the amount of work performed by a 1 newton force that moves through a distance of 1 meter. However, the physical quantities of work and energy are quite different from moments and torques, and in order to be clear when distinguishing them, the unit of N-m (and not J) should be used in the SI for moment and torque.
The expression Mo＝Fd can best be understood by applying it to a specific structure. In Figure 4.9(a), the force F is directed generally downward and to the right on the bracket. One might be interested in the moment of F about the base of the support post, which is labeled in the figure as point O. The structure could break at that location, and an engineer would examine the post to make sure that it can support F. The moment is calculated based upon both the magnitude of F and the perpendicular offset distance d between the force's line of action and point O. In fact, F could be applied to the bracket at any point along its line of action, and the moment produced about O would remain unchanged because d would likewise not change. The direction of the moment is clockwise, because F tends to cause the post to rotate that way (even though the rigid mounting would prevent the post from actually moving in this case).
In Figure 4.9(b) the direction of F has been changed. The force's line of action now passes directly through point O, and the offset distance becomes d＝0. No moment is produced, and the force tends to pull the post directly out of its base. In
short, the orientation of a force as well as its magnitude must be taken into account when calculating a moment.
EXAMPLE 4.3
The open-end wrench shown in Figure 4.10 tightens a hexagonal head nut and bolt. Calculate the moments produced by the 35-lb force about the center of the nut when the force is applied to the wrench in the two orientations shown. The overall length of the handle, which is inclined slightly upward, is in. long between centers of the open and closed ends.
SOLUTION
(a) In Figure 4.10(a), the 35-lb force acts vertically downward. The perpendicular distance from the center of the nut to the force's line of action is d ＝ 6 in. The incline and length of the wrench's handle are immaterial insofar as calculating d is concerned; the handle's length is not necessarily the same as the perpendicular lever arm distance. The moment has magnitude
(35 1b) (6 in.)＝210 in. lb＝17.5 ft.lb
which is directed clockwise（CW）.
(b) In Figure 4.10(b); the force has shifted to an inclined angle, and its line of action has changed so that d is measured to be. in., The moment is reduced to
(35 lb) (5.375 in.)＝188 in.lb＝15.7ft.lb
We report the answer as Mo = 15.7 ft-lb (CW).
Moment Components
Just as we can break a force into rectangular components; it is sometimes useful to calculate a moment as the sum of its components. The moment is determined as the sum of portions that are associated with the two components of the force, rather than the full resultant value of the force. A motivation for calculating, the moment in this manner is, that it is often easier to find the lever arms for individual components than for the resultant force. When applying this technique, it is necessary to use a sign convention and keep track of whether the contribution made by each force component is clockwise or counterclockwise:
To illustrate this method, we calculate the moments about point O of the forces shown in Figure 4.11. We first choose the following sign convention: A moment that is directed clockwise is positive, and a counterclockwise moment is negative. The sign convention is just a bookkeeping tool for combining the various clockwise and counterclockwise moment components. Any contribution to the moment about O that acts clockwise is given a positive sign; and any contribution
in the other direction is negative. This choice of positive and negative directions is arbitrary; and we could just as easily have selected the counterclockwise direction as being positive. However, once the sign convention is chosen, we stick with it and apply it consistently.
In Figure 4.11(a). the force is broken down into the components Fx and Fy. Rather than determine the distance from point O to the line of action of F, which ought involve a geometrical construction that we want to avoid, we instead calculate the individual lever arm distances for Fx and Fy, which are more straightforward. Keeping track of the sign convention, the moment about O becomes . Each contribution to Mo is positive because Fx and Fy each tend to cause clockwise rotation. Their effects combine constructively.
The orientation of F has been changed in Figure 4.11(b). While the component Fx continues to exert a positive moment, Fy now tends to cause counterclockwise rotation about O. It therefore makes a negative contribution, and the net moment becomes .Here the two components combine in a deconstructive manner. For the particular orientation in which, the two terms precisely cancel. The moment in that case is zero because the line of action for F passes directly through O, as in Figure 4.9(b). In the general case of the moment components method, we write
(4.9)
and the positive and negative signs are selected depending on whether the component tends to cause clockwise or counterclockwise rotation:
Regardless of which method you use to calculate a moment, when reporting an answer you should state (1) the numerical magnitude of the moment, (2) the units, and (3) the direction. You can indicate the direction by using ± notation provided that you have also shown the sign convention on your drawing. The notation CW or CCW is also useful for denoting whether the moment acts clockwise or counterclockwise.
EXAMPLE 4.4
Determine the moment about the center of the nut as the 250-N force is applied to the handle of the adjustable wrench in Figure 4.12. Use (a) the perpendicular lever arm method, and (b) the moment components method.
SOLUTION
(a) As shown in Figure 4.13(a), we denote the center of the nut as point A, and the point of application of the force as point B. Using the dimensions given, distance AB is calculated as Although this is the distance from A to the location at which the force is applied, it is not the perpendicular lever arm distance d. For this reason, we need to calculate the length of segment AC. Because the applied force is tilted 35o from vertical, a line perpendicular to the force is oriented 35o from horizontal. As depicted in Figure 4.13(a); line AB lies at the angle tan-1(75/200)＝20.6o below horizontal, and so it is offset by from AC. The correct lever arm distance therefore becomes .The wrench's moment is . directed clockwise (CW).
(b) In Figure 4.13(b), the 250-N force is broken into its components. The horizontal portion is (250N)sin35o＝143N, and the vertical component is (250N)cos35o＝205N. Those components are oriented leftward and downward, respectively, and they each exert clockwise moment contributions about A. In Figure 4.13(b), we have shown
our positive sign convention for moments as being clockwise. By summing the moment of each component, we have
Because the net result is positive, the moment is directed clockwise.
4.4 EQUILIBRIUM OF FORCES AND MOMENTS
With groundwork for the properties of forces and moments now in place, we next turn to the task of calculating (unknown) forces that act on structures and machines in response to other (known) forces that are present. This process involves applying the principle of static equilibrium to systems that are either stationary or moving at constant velocity. In either case, no acceleration is present, and in accordance with the laws of motion, the resultant force on the system is zero.
Particles and Rigid Bodies
A mechanical system can include either la single object (for instance, an engine piston) or multiple objects that are connected together (the entire engine). When the physical dimensions of the object are unimportant in calculating forces, the object is called a particle. This concept idealizes an object as being concentrated at a single point, rather than distributed over, an extended area or volume. For the purposes of problem solving, a particle can therefore be treated as having negligible dimensions. On the other hand, if the length, width, and breadth of an object are important for the problem at hand, it is called a rigid body. When looking at the motion of the Space Shuttle as it orbits the Earth, for instance, the spacecraft is regarded as a particle because its dimensions are small compared to the size of the orbit. However, when the Shuttle is landing and engineers are interested in its aerodynamics and flight characteristics, the vehicle would instead be analyzed as a rigid body. Figure.4.14 illustrates the distinction between forces that are applied to a particle and to a rigid body; you can see how a force imbalance could cause the rigid body to rotate.
A particle is in equilibrium if the forces acting on it balance with zero resultant. Because forces combine as vectors, the resultant must be zero in two perpendicular directions, which we label x and y:
and (4.10)
For a rigid body to be in equilibrium, it is necessary that (1) the resultant of all forces is zero ， and (2) the net moment is also zero. When those conditions are met, there is no tendency for the object either to move (in response to the forces) or to rotate (in response to the moments). The requirements for equilibrium of a rigid body involve Equation 4.10 and
(4.11)
The notation Moi is used, to denote the ith moment that is applied about point O. In summary, the two relationships in Equation 4.10 specify that no net force acts in
either the x or y directions, and Equation 4.11 states that no net moment tends to cause rotation.
Sign conventions are a bookkeeping method to distinguish forces acting in opposing directions and moments that are oriented clockwise or counterclockwise. The summations in the equilibrium equation are conducted overall forces and moments present, whether or not their directions and magnitudes are known in advance. Forces that are unknown at the start of the problem are always included in the summation algebraic variables are assigned to them, and the equilibrium equations are then applied to determine numerical values.
Mathematically speaking, the equilibrium equations for a rigid body comprise a system of three linear equations involving the unknown forces and moments. One implication of this characteristic is that it is possible to determine at most three unknown, quantities when Equations 4.10-4.11 are applied to a single rigid body. By contrast; when applying the equilibrium requirements to a particle, the moment equation is not used. Therefore, only two independent equations are produced, and only two unknowns can be determined. It is not possible to obtain more independent equations of equilibrium by resolving moments about an alternative point or by summing forces in different directions. The additional equations will still be valid, but they will simply be combinations of the other (already derived) ones. As such, they will provide no new information. When you solve equilibrium problems, you should always check to be sure that you do not have more unknown quantities than independent equations.
Free Body Diagrams
Free body diagrams (abbreviated FBD) are sketches used to analyze the forces and moments that act on structures and machines, and their construction is an important skill. The FBD is used to identify the mechanical system that is being examined and to represent all of the known and unknown forces that are present. Three main steps are followed when a FBD is drawn:
1. Select an object that will be analyzed by using the equilibrium equations. Imagine that a dotted fine is drawn around the object, and note how the line would cut through and expose various forces. Everything within the dotted line is isolated from the surroundings and should appear on the diagram.
2. The coordinate system is drawn next to indicate the positive sign conventions for forces and moments. It will be meaningless to report an answer of, say, "-25 N-m" or "+250 lb" without having defined the directions associated with the positive and negative signs.
3. In the final step, all forces and moments are drawn and labeled. These forces might represent weight or contact between the free body and other objects that were removed when the body was isolated. When a force is known, its direction and magnitude should be written on the diagram. Forces are included even if their magnitudes and directions are not known at this step in the analysis: If the direction of a force is unknown (for instance, upward/downward or leftward/rightward), you should just draw it one way or the other on the FBD, perhaps using your intuition as a guide. After applying the equilibrium equations and consistently using a sign convention, the correct direction will be determined through your calculations. If you find the quantity to be positive, then you know that the correct direction was chosen. On the other hand, if the numerical value turns out to be negative, the result simply means that the force acts opposite to the assumed direction.
EXAMPLE 4.5
During crash testing of an automobile, the lap and shoulder seat belts each become tensioned to 300 lb, as shown in Figure 4.15(a). Treating the buckle B as a particle, (a) draw a free body diagram, (b) determine the tension T in the anchor strap AB, and (c) determine the angle at which T acts.
SOLUTION
(a) The free body diagram of the buckle is shown in Figure 4.15(b). The xy coordinate system is also drawn to indicate our sign convention for the positive horizontal and vertical directions. Three forces act on the buckle: the two given 300-lb forces and the unknown force in the anchor strap. For the buckle to be in equilibrium, these three forces must balance. Although both the magnitude T and direction θ of the force in strap AB are unknown, both quantities are shown on the free body diagram for completeness.
(b) We sum the three forces by using the vector polygon approach, as shown in Figure 4.15(c). The polygon's start and end points are the same because the three forces acting together have zero resultant; that is, the distance between the polygon's start and end points is zero. The tension is determined by applying the law of cosines (equations for oblique triangles are reviewed in Appendix B) to the side-angle-side triangle in Figure 4.15(c):
from which we calculate .
(c)The anchor strap’s angle is found from the law of sines:
and
EXAMPLE4.6
A pair of wire cutters is shown in Figure 4.16. A machinist applies the 70-N gripping force to the handles. What is the magnitude of the cutting force on the electrical wire at A?
SOLUTION
The coordinate system and positive sign convention for the force and moment directions are shown in Figure 4.17. We next draw the free body diagram for one jaw/handle assembly, which is treated as a rigid body because it can rotate and the distances between the forces are significant to the problem: When a blade is pressed against the wire, the wire in turn pushes back with the (unknown) force A on the: blade. Force B is carried by the hinge pin, which connects the two jaw/handle pieces. The 70-N gripping force is given and shown on the FBD acting at the end of the handle.
With the free body diagram completed in Figure 4.17, the cutting force A is found by applying the equilibrium equations for a rigid body. The force balance requirement in the vertical direction becomes
There are two unknowns, A and B, and so an additional equation is needed. By summing moments about point B, we have
The negative sign is present because the 70-N force produces a counterclockwise moment about B. The cutting force is found to be A＝315 N, and after back substitution, B＝385 N. Because these are each positive, the directions assumed and shown on the FBD are correct.
These wire cutters operate according to the principle of a lever. Each jaw/handle assembly rotates slightly about point B. The cutting force at A is proportional to the force on the handles, and it is also related to the ratio of distances AB and BC. The mechanical advantage for a machine is defined as the ratio of the output and input forces, or in this case, (315 N)/(70 N)＝4.5. Thus, the wire cutters amplify the machinist's gripping force by some 450%.
EXAMPLE4.7
The forklift shown in Figure' 4.18(a) weighs 3500 1b, and carries an 800-lb shipping container. There are two front wheels and two rear wheels on the forklift. (a) Draw a free body diagram of the forklift. (b) Determine the contact forces between the wheels and ground. (c) How can you determine the front wheel force without having to find the force at the rear wheels? (d) How heavy a load can be carried