2–1. The mechanism shown in Figure C2.1 is a top view of a fixture in a machining operation. Carefully examine the configuration of the components in the mechanism. Then answer the following leading questions to gain insight into the operation of the mechanism.
1. As the handle A is turned, moving the threaded rod B to the left, describe the motion of grip C.
2. As the handle A is turned, moving the threaded rod B to the left, describe the motion of grip D.
3. What is the purpose of this mechanism?
4. What action would cause link D to move upward?
5. What is the purpose of spring G?
6. Discuss the reason for the odd shape to links E and F.
7. What would you call such a device?
8. Describe the rationale behind using a rounded end for the threaded rod B.
2–10. Figure P2.10 shows another transfer mechanism that pushes crates from one conveyor to another.
Use the Working Model software to generate a model of this linkage. The motor operates clockwise at a constant rate of 40 rpm.
2–11. Figure P2.11 shows yet another transfer mechanism that lowers crates from one conveyor to another. Use the Working Model software to generate a model of
amount of donuts in the box, the amount is a scalar quantity. The following are some more examples of scalar quantities: a board is 8 ft long, a class meets for 50 min, or the temperature is 78°F—length, time, and temperature are all scalar quantities.
In contrast, a vector is not fully defined by stating only a magnitude. Indicating the direction of the quantity is also required. Stating that a golf ball traveled 200 yards does not fully describe its path. Neglecting to express the direction of travel hides the fact that the ball has landed in a lake. Thus, the direction must be included to fully describe such a quan-tity. Examples of properly stated vectors include “the crate is being pulled to the right with 5 lb” or “the train is traveling at a speed of 50 mph in a northerly direction.” Displacement, force, and velocity are vector quantities.
Vectors are distinguished from scalar quantities through the use of boldface type (v). The common nota-tion used to graphically represent a vector is a line segment having an arrowhead placed at one end. With a graphical approach to analysis, the length of the line segment is drawn proportional to the magnitude of the quantity that the vector describes. The direction is defined by the arrowhead and the incline of the line with respect to some reference axis. The direction is always measured at its root, not at its head. Figure 3.1 shows a fully defined velocity vector.
O B J E C T I V E S
Upon completion of this chapter, the student will be able to:
1. Differentiate between a scalar quantity and a vector.
2. Apply the appropriate trigonometry principles to a right triangle.
3. Apply the appropriate trigonometry principles to a general triangle.
4. Determine the resultant of two vectors, using both graphical and analytical methods.
5. Resolve vector quantities into components in the horizontal and vertical directions.
6. Subtract two vectors, using both graphical and analytical methods.
7. Manipulate vector equations.
8. Utilize a vector equation to determine the magnitude of two vectors.
C H A P T E R
T H R E E
VECTORS
3.1 INTRODUCTION
Mechanism analysis involves manipulating vector quantities.
Displacement, velocity, acceleration, and force are the primary performance characteristics of a mechanism, and are all vectors. Prior to working with mechanisms, a thorough introduction to vectors and vector manipulation is in order. In this chapter, both graphical and analytical solution techniques are presented. Students who have completed a mechanics course may omit this chapter or use it as a reference to review vector manipulation.
3.2 SCALARS AND VECTORS
In the analysis of mechanisms, two types of quantities need to be distinguished. A scalar is a quantity that is sufficiently defined by simply stating a magnitude. For example, by saying “a dozen donuts,” one describes the quantity of donuts in a box. Because the number “12” fully defines the
43
FIGURE 3.1 A 45 mph velocity vector.
3.3 GRAPHICAL VECTOR ANALYSIS
Much of the work involved in the study of mechanisms and analysis of vectors involves geometry. Often, graphical methods are employed in such analyses because the motion of a mechanism can be clearly visualized. For more complex mechanisms, analytical calculations involving vectors also become laborious.
44 CHAPTER THREE
FIGURE 3.2 The right triangle.
A graphical approach to analysis involves drawing scaled lines at specific angles. To achieve results that are consistent with analytical techniques, accuracy must be a major objective. For several decades, accuracy in mecha-nism analyses was obtained with attention to precision and proper drafting equipment. Even though they were popular, many scorned graphical techniques as being imprecise.
However, the development of computer-aided design (CAD) with its accurate geometric constructions has allowed graphical techniques to be applied with precision.
3.4 DRAFTING TECHNIQUES REQUIRED IN GRAPHICAL VECTOR ANALYSIS
The methods of graphical mechanism and vector analysis are identical, whether using drafting equipment or a CAD package. Although it may be an outdated mode in industrial analyses, drafting can be successfully employed to learn and understand the techniques.
For those using drafting equipment, fine lines and cir-cular arcs are required to produce accurate results. Precise linework is needed to accurately determine intersection points. Thus, care must be taken in maintaining sharp draw-ing equipment.
Accurate measurement is as important as line quality.
The length of the lines must be drawn to a precise scale, and linear measurements should be made as accurately as possi-ble. Therefore, using a proper engineering scale with inches divided into 50 parts is desired. Angular measurements must be equally precise.
Lastly, a wise choice of a drawing scale is also a very important factor. Typically, the larger the construction, the more accurate the measured results are. Drawing precision to 0.05 in. produces less error when the line is 10 in. long as opposed to 1 in. Limits in size do exist in that very large constructions require special equipment. However, an attempt should be made to create constructions as large as possible.
A drawing textbook should be consulted for the details of general drafting techniques and geometric constructions.
3.5 CAD KNOWLEDGE REQUIRED IN GRAPHICAL VECTOR ANALYSIS
As stated, the methods of graphical mechanism and vector analysis are identical, whether using drafting equipment or a CAD package. CAD allows for greater precision. Fortunately, only a limited level of proficiency on a CAD system is required to properly complete graphical vector analysis.
Therefore, utilization of a CAD system is preferred and should not require a large investment on a “learning curve.”
As mentioned, the graphical approach of vector analysis involves drawing lines at precise lengths and specific angles.
The following list outlines the CAD abilities required for vector analysis. The user should be able to
䊏 Draw lines at a specified length and angle;
䊏 Insert lines, perpendicular to existing lines;
䊏 Extend existing lines to the intersection of another line;
䊏 Trim lines at the intersection of another line;
䊏 Draw arcs, centered at a specified point, with a specified radius;
䊏 Locate the intersection of two arcs;
䊏 Measure the length of existing lines;
䊏 Measure the included angle between two lines.
Of course, proficiency beyond these items facilitates more efficient analysis. However, familiarity with CAD commands that accomplish these actions is sufficient to accurately complete vector analysis.
3.6 TRIGONOMETRY REQUIRED IN ANALYTICAL VECTOR ANALYSIS
In the analytical analysis of vectors, knowledge of basic trigonometry concepts is required. Trigonometry is the study of the properties of triangles. The first type of triangle examined is the right triangle.
3.6.1 Right Triangle
In performing vector analysis, the use of the basic trigonomet-ric functions is vitally important. The basic trigonomettrigonomet-ric functions apply only to right triangles. Figure 3.2 illustrates a right triangle with sides denoted as , and and interior angles as , and . Note that angle is a 90° right angle.
Therefore, the triangle is called a right triangle.
The basic trigonometric relationships are defined as:
(3.1)
(3.2)
(3.3) These definitions can also be applied to angle :
tan ∠ B = b tangent∠ A = tan∠ A = side opposite
side adjacent = a b
Vectors 45
C A
B 35
96"
FIGURE 3.3 Front loader for Example Problem 3.1.
The Pythagorean theorem gives the relationship of the three sides of a right triangle. For the triangle shown in Figure 3.2, it is defined as
(3.4) a2 + b2 = c2
Finally, the sum of all angles in a triangle is 180°.
Knowing that angle is 90°, the sum of the other two angles must be
(3.5)
∠A + ∠B = 90°
C
EXAMPLE PROBLEM 3.1
Figure 3.3 shows a front loader with cylinder BC in a vertical position. Use trigonometry to determine the required length of the cylinder to orient arm AB in the configuration shown.
EXAMPLE PROBLEM 3.2
Figure 3.4 shows a tow truck with an 8-ft boom, which is inclined at a 25° angle. Use trigonometry to determine the horizontal distance that the boom extends from the truck.
SOLUTION: 1. Determine the Horizontal Projection of the Boom
The horizontal projection of the boom can be determined from equation (3.2):
horizontal projection = (8 ft)cos 25° = 7.25 ft cos 25° = horizontal projection
(8 ft) SOLUTION: 1. Determine Length BC
Focus on the triangle formed by points , and Figure 3.3. The triangle side BC can be found using equation (3.1).
solving:
2. Determine Length AC
Although not required, notice that the distance between and can similarly be determined using equation (3.2). Thus
solving:
AC = (96 in.) cos 35° = 78.64 in.
cos 35° = AC (96 in.) cos ∠ A = adjacent side
hypotenuse C A BC = (96 in.) sin 35° = 55.06 in.
sin 35° = BC (96 in.) sin ∠ A = opposite side
hypotenuse C in A, B
46 CHAPTER THREE
FIGURE 3.4 Tow truck for Example Problem 3.2.
A B
a C
b c
FIGURE 3.5 The oblique triangle.
2. Determine Horizontal Projection of Truck and Boom
The horizontal distance from the front end of the truck to the end of the boom is
3. Determine the Overhang
Because the overall length of the truck is 11 ft, the horizontal distance that the boom extends from the truck is
13.25 ft - 11 ft = 2.25 ft 6 ft + 7.25 ft = 13.25 ft
3.6.2 Oblique Triangle
In the previous discussion, the analysis was restricted to right triangles. An approach to general, or oblique, triangles is also important in the study of mechanisms. Figure 3.5 shows a general triangle. Again, , and denote the length of the sides and , , and represent the interior angles.
For this general case, the basic trigonometric functions described in the previous section are not applicable. To ana-lyze the general triangle, the law of sines and the law of cosines have been developed.
The law of sines can be stated as
(3.6)
The law of cosines can be stated as:
(3.7) In addition, the sum of all interior angles in a general triangle must total 180°. Stated in terms of Figure 3.4 the equation would be
Problems involving the solution of a general triangle fall into one of four cases:
Case 1: Given one side and two angles and . To solve a problem of this nature, equation (3.8) can be used to find the third angle:
The law of sines can be rewritten to find the remaining sides.
Case 2: Given two sides and and the angle opposite to one of the sides .
To solve a Case 2 problem, the law of sines can be used to find the second angle. Equation (3.6) is rewritten as
Equation (3.8) can be used to find the third angle:
The law of cosines can be used to find the third side.
Equation (3.7) is rewritten as:
c = 3{a2 + b2 - 2ab cos ∠C}
Vectors 47
A C
B 96"
25°
78"
FIGURE 3.6 Front loader for Example Problem 3.3.
Case 3 Given two sides and and the included angle .
To solve a Case 3 problem, the law of cosines can be used to find the third side:
The law of sines can be used to find a second angle.
Equation (3.6) is rewritten as
Equation (3.8) can be used to find the third angle:
∠B = 180° - ∠A - ∠C
∠A = sin -1e aa
c b sin ∠Cf c = 2a2 + b2 - 2ab cos ∠C (∠C)
b)
(a Case 4 Given three sides.
To solve a Case 4 problem, the law of cosines can be used to find an angle. Equation (3.7) is rewritten as
The law of sines can be used to find a second angle.
Equation (3.6) is rewritten as
Equation (3.8) can be used to find the third angle:
Once familiarity in solving problems involving general triangles is gained, referring to the specific cases will be unnecessary.
∠B = 180° - ∠A - ∠C
∠A = sin-1e aa
c b sin ∠C f
∠C = cos -1aa2 + b2 - c2
2ab b
EXAMPLE PROBLEM 3.3
Figure 3.6 shows a front loader. Use trigonometry to determine the required length of the cylinder to orient arm AB in the configuration shown.
EXAMPLE PROBLEM 3.4
Figure 3.7 shows the drive mechanism for an engine system. Use trigonometry to determine the crank angle as shown in the figure.
SOLUTION: 1. Determine Angle BAC
By focusing on the triangle created by points , and , it is apparent that this is a Case 4 problem.
Angle BAC can be determined by redefining the variables in the law of cosines:
= cos-1e(5.3 in.)2 + (1in.)2 - (5in.)2
2(5.3 in.)(1 in.) f = 67.3°
∠BAC = cos-1eAC2 + AB2 - BC2 2(AC )(AB) f
C A, B SOLUTION: 1. Determine Length BC
By focusing on the triangle created by points A, B, and , it is apparent that this is a Case 3 problem. The third side can be found by using the law of cosines:
Because determining the remaining angles was not required, the procedure described for Case 3 problems will not be completed.
= 41.55in.
= 3(78in.)2 + (96in.)2 - 2(78in.)(96in.) cos 25°
BC = 3AC2 + AB2 - 2(AC)(AB) cos∠BAC C
48 CHAPTER THREE
A B C 5.3"
5"
Crank angle
1"
FIGURE 3.7 Engine linkage for Example Problem 3.4.
2. Determine the Crank Angle
Angle BAC is defined between side AC (the vertical side) and leg AB. Because the crank angle is defined from a horizontal axis, the crank angle can be determined by the following:
3. Determine the Other Interior Angles
Although not required in this problem, angle ACB can be determined by
Finally, angle ABC can be found by
∠ABC = 180° - 67.3° - 10.6° = 102.1°
= sin-1e a1 in.
5 in.b sin 67.3°f = 10.6°
∠ACB = sin-1e aAB
BCb sin∠BAC f Crank angle = 90° - 67.3° = 22.7°
3.7 VECTOR MANIPULATION
Throughout the analysis of mechanisms, vector quantities (e.g., displacement or velocity) must be manipulated in dif-ferent ways. In a similar manner to scalar quantities, vectors can be added and subtracted. However, unlike scalar quanti-ties, these are not simply algebraic operations. Because it is also required to define a vector, direction must be accounted for during mathematical operations. Vector addition and subtraction are explored separately in the following sections.
Adding vectors is equivalent to determining the com-bined, or net, effect of two quantities as they act together.
For example, in playing a round of golf, the first shot off the tee travels 200 yards but veers off to the right. A second shot then travels 120 yards but to the left of the hole. A third shot of 70 yards places the golfer on the green. As this golfer looks on the score sheet, she notices that the hole is labeled as 310 yards; however, her ball traveled 390 yards
yards).
As repeatedly stated, the direction of a vector is just as important as the magnitude. During vector addition, does not always equal 2; it depends on the direction of the individual vectors.
1 + 1 (200 + 120 + 70
3.8 GRAPHICAL VECTOR ADDITION
Graphical addition is an operation that determines the net effect of vectors. A graphical approach to vector addition involves drawing the vectors to scale and at the proper orientation. These vectors are then relocated, maintaining the scale and orientation. The tail of the first vector is designated as the origin (point . The second vector is relocated so that its tail is placed on the tip of the first vector. The process then is repeated for all remaining vectors. This technique is known as the tip-to-tail method of vector addition. This name is derived from viewing a completed vector polygon. The tip of one vector runs into the tail of the next.
The combined effect is the vector that extends from the tail of the first vector in the series to the tip of the last vector in the series. Mathematically, an equation can be written that represents the combined effect of vectors:
R = A +7 B +7 C +7 D +7 . . . O)
( + 7)
Vectors 49
FIGURE 3.8 Vectors for Example Problem 3.5.
Vector R is a common notation used to represent the resultant of a series of vectors. A resultant is a term used to describe the combined effect of vectors. Also note that the symbol is used to identify vector addition and to differentiate it from algebraic addition [Ref. 5].
It should be noted that vectors follow the commutative law of addition; that is, the order in which the vectors are added does not alter the result. Thus,
+>
The process of combining vectors can be completed graphi-cally, using either manual drawing techniques or CAD soft-ware. Whatever method is used, the underlying concepts are identical. The following example problems illustrate this concept.
1B +> A +> C2 = . . .
R = 1A +> B +> C2 = 1C +> B +> A2 =
EXAMPLE PROBLEM 3.5
Graphically determine the combined effect of velocity vectors A and B, as shown in Figure 3.8.
SOLUTION: 1. Construct Vector Diagrams
To determine the resultant, the vectors must be relocated so that the tail of B is located at the tip of A. To verify the commutative law, the vectors were redrawn so that the tail of A is placed at the tip of B. The resultant is the vector drawn from the tail of the first vector, the origin, to the tip of the second vector. Both vector diagrams are shown in Figure 3.9.
2. Measure the Resultant
The length vector R is measured as 66 in./s. The direction is also required to fully define vector R. The angle from the horizontal to vector R is measured as 57°. Therefore, the proper manner of presenting the solution is as follows:
R= 66 in./s 57°
FIGURE 3.9 The combined effect of vectors A and B for Example Problem 3.5.
0
30° 60°
25 in./s Scale:
50
A = 59 in./s B = 30 in./s
Q
50 CHAPTER THREE
A = 200 lb
B = 226 lb C = 176 lb D = 300 lb
15° 45° 20°
0 100
lb Scale:
200
FIGURE 3.10 Vectors for Example Problem 3.6.
SOLUTION: 1. Construct Vector Diagrams
To determine the resultant, the vectors must be relocated so that the tail of B is located at the tip of A. Then the tail of C is placed on the tip of B. Finally, the tail of D is placed on the tip of C. Again, the ordering of vectors is not important, and any combination could be used. As an illustration, another arbitrary combination is used in this example. The resultant is the vector drawn from the tail of the first vector, at the origin, to the tip of the fourth vector. The vector diagrams are shown in Figure 3.11.
2. Measure the Resultant
The length vector R is measured as 521 lb. The direction is also required to fully define the vector R. The angle from the horizontal to vector R is measured as 68°. Therefore, the proper manner of presenting the solution is as follows:
R = 521 in./s 68°