E1C12 09/14/2010
13:54:11 Page 530
Consider the stresses created in a shaft of radius R 0 subject to a torque T. The maximum shearing
stress in a circular shaft occurs on the surface and may be calculated from the torsion formula
4
t max ¼ TR 0 =J
ð12:18Þ
where
t max ¼ maximum shearing stress
T ¼ applied torque
J ¼ polar moment of inertia (pR
2
0 =2 for a solid circular shaft)
For a shaft in pure torsion, there are no normal stresses, s x ; s y ; or s z . The principal stresses lie along
a line that makes a 45-degree angle with the axis of the shaft, as illustrated in Figure 12.26, and have
a value equal to t max . Strains that occur along the curve labeled A-A
0 are opposite in sign from those
that occur along B-B
0 . These locations allow placement of four active strain gauges in a Wheatstone
bridge arrangement, and the direct measurement of torque in terms of bridge output voltage.
Mechanical Power Measurements
Almost universally, prime movers such as internal combustion (IC) engines and gas turbines convert
chemical energy in a fuel to thermodynamic work transmitted by a shaft to the end use. In
automotive applications, the pistons create a torque on the crankshaft, which is ultimately transmitted to the driving wheels. In each case, the power is transmitted through a mechanical coupling.
This section discusses the measurement of such mechanical power transmission.
Rotational Speed, Torque, and Shaft Power
Shaft power is related to rotational speed and torque as
P
*
s ¼ v
*
 T
*
ð12:19Þ
where P s is the shaft power, v is the rotational velocity vector, and T is the torque vector. In general,
the orientation of the torque and rotational velocity vectors are such that the equation may be written
in scalar form as
P s ¼ vT
ð12:20Þ
Table 12.3 provides a summary of useful equations related to shaft power, torque, and speed as
employed in mechanical measurements. Historically, a device called a Prony brake was used to
measure shaft power. A typical Prony brake arrangement is shown in Figure 12.27. Consider using
the Prony brake to measure power output for an IC engine. The Prony brake serves to provide a welldefined load for the engine, with the power output of the engine dissipated as thermal energy in the
braking material. By adjusting the load, the power output over a range of speeds and throttle settings
can be realized. The power is measured by recording the torque acting on the torque arm and the
rotational speed of the engine. Clearly, this device is limited in speed and power, but does serve to
demonstrate the operating principles of power measurement, and is historically significant as the
first technique for measuring power.
4 Coulomb developed the torsion formula in 1775 in connection with electrical instruments.
530 Chapter 12 Mechatronics: Sensors, Actuators, and Controls
13:54:11 Page 530
Consider the stresses created in a shaft of radius R 0 subject to a torque T. The maximum shearing
stress in a circular shaft occurs on the surface and may be calculated from the torsion formula
4
t max ¼ TR 0 =J
ð12:18Þ
where
t max ¼ maximum shearing stress
T ¼ applied torque
J ¼ polar moment of inertia (pR
2
0 =2 for a solid circular shaft)
For a shaft in pure torsion, there are no normal stresses, s x ; s y ; or s z . The principal stresses lie along
a line that makes a 45-degree angle with the axis of the shaft, as illustrated in Figure 12.26, and have
a value equal to t max . Strains that occur along the curve labeled A-A
0 are opposite in sign from those
that occur along B-B
0 . These locations allow placement of four active strain gauges in a Wheatstone
bridge arrangement, and the direct measurement of torque in terms of bridge output voltage.
Mechanical Power Measurements
Almost universally, prime movers such as internal combustion (IC) engines and gas turbines convert
chemical energy in a fuel to thermodynamic work transmitted by a shaft to the end use. In
automotive applications, the pistons create a torque on the crankshaft, which is ultimately transmitted to the driving wheels. In each case, the power is transmitted through a mechanical coupling.
This section discusses the measurement of such mechanical power transmission.
Rotational Speed, Torque, and Shaft Power
Shaft power is related to rotational speed and torque as
P
*
s ¼ v
*
 T
*
ð12:19Þ
where P s is the shaft power, v is the rotational velocity vector, and T is the torque vector. In general,
the orientation of the torque and rotational velocity vectors are such that the equation may be written
in scalar form as
P s ¼ vT
ð12:20Þ
Table 12.3 provides a summary of useful equations related to shaft power, torque, and speed as
employed in mechanical measurements. Historically, a device called a Prony brake was used to
measure shaft power. A typical Prony brake arrangement is shown in Figure 12.27. Consider using
the Prony brake to measure power output for an IC engine. The Prony brake serves to provide a welldefined load for the engine, with the power output of the engine dissipated as thermal energy in the
braking material. By adjusting the load, the power output over a range of speeds and throttle settings
can be realized. The power is measured by recording the torque acting on the torque arm and the
rotational speed of the engine. Clearly, this device is limited in speed and power, but does serve to
demonstrate the operating principles of power measurement, and is historically significant as the
first technique for measuring power.
4 Coulomb developed the torsion formula in 1775 in connection with electrical instruments.
530 Chapter 12 Mechatronics: Sensors, Actuators, and Controls