6. Lawrance, A., Modern Inertial Technology-Navigation, Guidance, and Control, Springer-Verlag, New York, 1993. 7. McConnell, K. G., Vibration Testing: Theory and Practice, New York: Wiley, 1995. 8. Machine Vibration: Dynamics and Control, London: Springler, 1992–1996. 9. Measuring Vibration, Bruel & Kjaer, 1982. 10. Sydenham, P. H., Hancock, N. H., and Thorn, R., Introduction to Measurement Science and Engi- neering, New York: Wiley, 1989. 11. Tompkins, W. J. and Webster, J. G., Interfacing Sensors to the IBM PC, Englewood Cliffs, NJ: Prentice- Hall, 1988. 19.3 Force Measurement M. A. Elbestawi Force, which is a vector quantity, can be defined as an action that will cause an acceleration or a certain reaction of a body. This chapter will outline the methods that can be employed to determine the magnitude of these forces. General Considerations The determination or measurement of forces must yield to the following considerations: if the forces acting on a body do not produce any acceleration, they must form a system of forces in equilibrium. The system is then considered to be in static equilibrium. The forces experienced by a body can be classified into two categories: internal, where the individual particles of a body act on each other, and external otherwise. If a body is supported by other bodies while subject to the action of forces, deformations and/or displacements will be produced at the points of support or contact. The internal forces will be distributed throughout the body until equilibrium is established, and then the body is said to be in a state of tension, compression, or shear. In considering a body at a definite section, it is evident that all the internal forces act in pairs, the two forces being equal and opposite, whereas the external forces act singly. Hooke’s Law The basis for force measurement results from the physical behavior of a body under external forces. Therefore, it is useful to review briefly the mechanical behavior of materials. When a metal is loaded in uniaxial tension, uniaxial compression, or simple shear (Fig. 19.29), it will behave elastically until a critical value of normal stress (S) or shear stress ( τ ) is reached, and then it will deform plastically [1]. In the FIGURE 19.29 When a metal is loaded in uniaxial tension (a), uniaxial compression (b), or simple shear(c), it will behave elastically until a critical value of normal stress or shear stress is reached. p p p F F p (a) (b) (c) 0066_frame_C19 Page 34 Wednesday, January 9, 2002 5:17 PM ©2002 CRC Press LLC 6. Lawrance, A., Modern Inertial Technology-Navigation, Guidance, and Control, Springer-Verlag, New York, 1993. 7. McConnell, K. G., Vibration Testing: Theory and Practice, New York: Wiley, 1995. 8. Machine Vibration: Dynamics and Control, London: Springler, 1992–1996. 9. Measuring Vibration, Bruel & Kjaer, 1982. 10. Sydenham, P. H., Hancock, N. H., and Thorn, R., Introduction to Measurement Science and Engi- neering, New York: Wiley, 1989. 11. Tompkins, W. J. and Webster, J. G., Interfacing Sensors to the IBM PC, Englewood Cliffs, NJ: Prentice- Hall, 1988. 19.3 Force Measurement M. A. Elbestawi Force, which is a vector quantity, can be defined as an action that will cause an acceleration or a certain reaction of a body. This chapter will outline the methods that can be employed to determine the magnitude of these forces. General Considerations The determination or measurement of forces must yield to the following considerations: if the forces acting on a body do not produce any acceleration, they must form a system of forces in equilibrium. The system is then considered to be in static equilibrium. The forces experienced by a body can be classified into two categories: internal, where the individual particles of a body act on each other, and external otherwise. If a body is supported by other bodies while subject to the action of forces, deformations and/or displacements will be produced at the points of support or contact. The internal forces will be distributed throughout the body until equilibrium is established, and then the body is said to be in a state of tension, compression, or shear. In considering a body at a definite section, it is evident that all the internal forces act in pairs, the two forces being equal and opposite, whereas the external forces act singly. Hooke’s Law The basis for force measurement results from the physical behavior of a body under external forces. Therefore, it is useful to review briefly the mechanical behavior of materials. When a metal is loaded in uniaxial tension, uniaxial compression, or simple shear (Fig. 19.29), it will behave elastically until a critical value of normal stress (S) or shear stress ( τ ) is reached, and then it will deform plastically [1]. In the FIGURE 19.29 When a metal is loaded in uniaxial tension (a), uniaxial compression (b), or simple shear(c), it will behave elastically until a critical value of normal stress or shear stress is reached. p p p F F p (a) (b) (c) 0066_frame_C19 Page 34 Wednesday, January 9, 2002 5:17 PM ©2002 CRC Press LLC Fundamental Concepts Angular Displacement, Velocity, and Acceleration The concept of rotational motion is readily formalized: all points within a rotating rigid body move in parallel or coincident planes while remaining at fixed distances from a line called the axis. In a perfectly rigid body, all points also remain at fixed distances from each other. Rotation is perceived as a change in the angular position of a reference point on the body, i.e., as its angular displacement , ∆ θ , over some time interval, ∆ t . The motion of that point, and therefore of the whole body, is characterized by its clockwise (CW) or counterclockwise (CCW) direction and by its angular velocity , ω = ∆ θ / ∆ t . If during a time interval ∆ t , the velocity changes by ∆ ω , the body is undergoing an angular acceleration , α = ∆ ω / ∆ t . With angles measured in radians, and time in seconds, units of ω become radians per second (rad s –1 ) and of α , radians per second per second (rad s –2 ). Angular velocity is often referred to as rotational speed and measured in numbers of complete revolutions per minute (rpm) or per second (rps). Force, Torque, and Equilibrium Rotational motion, as with motion in general, is controlled by forces in accordance with Newton’s laws. Because a force directly affects only that component of motion in its line of action, forces or components of forces acting in any plane that includes the axis produce no tendency for rotation about that axis. Rotation can be initiated, altered in velocity, or terminated only by a tangential force F t acting at a finite radial distance l from the axis. The effectiveness of such forces increases with both F t and l ; hence, their product, called a moment , is the activating quantity for rotational motion. A moment about the rotational axis constitutes a torque. Figure 19.45(a) shows a force F acting at an angle β to the tangent at a point P , distant l (the moment arm) from the axis. The torque T is found from the tangential component of F as (19.58) The combined effect, known as the resultant , of any number of torques acting at different locations along a body is found from their algebraic sum , wherein torques tending to cause rotation in CW and CCW directions are assigned opposite signs. Forces, hence torques, arise from physical contact with other solid bodies, motional interaction with fluids, or via gravitational (including inertial), electric, or magnetic force fields. The source of each such torque is subjected to an equal, but oppositely directed, reaction torque. With force measured in newtons and distance in meters, Eq. (19.58) shows the unit of torque to be a Newton meter (Nm). FIGURE 19.45 (a) The off-axis force F at P produces a torque T = ( F cos β ) l tending to rotate the body in the CW direction. (b) Transmitting torque T over length L twists the shaft through angle φ . TF t lF cos b()l== L + O d T T F P I CCW CW β F t b b' b' b a Axis (a) (b) φ 0066_Frame_C19 Page 49 Wednesday, January 9, 2002 5:27 PM ©2002 CRC Press LLC . b()l== L + O d T T F P I CCW CW β F t b b' b' b a Axis (a) (b) φ 0066_Frame_C19 Page 49 Wednesday, January 9, 20 02 5 :27 PM 20 02 CRC Press LLC . or shear stress is reached. p p p F F p (a) (b) (c) 0066_frame_C19 Page 34 Wednesday, January 9, 20 02 5:17 PM 20 02 CRC Press LLC Fundamental Concepts Angular Displacement, Velocity, and Acceleration . or shear stress is reached. p p p F F p (a) (b) (c) 0066_frame_C19 Page 34 Wednesday, January 9, 20 02 5:17 PM 20 02 CRC Press LLC 6. Lawrance, A., Modern Inertial Technology-Navigation, Guidance,