Transitorio III.- El Poder Ejecutivo reglamentará la presente ley, después de su publicación.”
NACIONAL DE LA CONTRATACIÓN ADMINISTRATIVA
The following formula calculates the torque required for rotating motion to begin:
where:
T = acceleration torque in lb-ft
WK2= total reflected inertia (total of motor, gear reducer, and load) in lb-ft2
∆
N = change in speed required in seconds t = time to accelerate the total system loadA term above that may not be familiar, is the term reflected inertia. Basi- cally, this term is used to describe the inertia found at the motor shaft, and is a standard term found throughout the rotating machinery industry. A speed reducer (gear box or belt coupler) changes the inertia that the motor shaft actually sees. More on this subject will be discussed later in this chap- ter.
Acceleration Time
The next formula is a rearranged version of the previous one, only it allows for the calculation of the time needed to accelerate, given a speci- fied amount of torque, inertia, and change in speed.
where:
t = time to accelerate the total system load
WK2 = total reflected inertia (total of motor, gear reducer, load) in lb-ft2
∆
N = change in speed required in seconds T = acceleration torque in lb-ftThe following are examples of how inertia is calculated and how much time is needed to accelerate a machine with the specified requirements.
One note regarding calculations should be added here. When performing inertia calculations, two measurement units are commonly used: lb-ft2 and in-lb-sec2. For the most part, many calculations are defined in lb-ft2, which is the units of measure for WK2 or WR2. However, in many motion control (servo) applications, inertia is defined in terms of in-lb-sec2, which is the units of measure for the moment of inertia (J). When performing inertia calculations, be consistent with the formulas and units of measure used. For most practical motion-control applications involving inertia
t 308 N WK T 2 × ∆ × = T 308 N WK t 2 × ∆ × =
Chapter 2: Review of Basic Principles — Mechanical Principles 55
given as J, the following conversions can be used to convert lb-ft2 to in-lb- sec2 and vice versa.
To convert a calculated answer of lb-ft2 to in-lb-sec2, divide the answer by 2.68.
To convert a calculated answer of in-lb-sec2 to lb-ft2, multiply the answer by 2.68.
Solid Cylinder Inertia Calculations
To calculate the inertia of a solid cylinder, the following formula is used.
WK2 = .000681 × ρ × L × D4 where:
WK2 = inertia of a cylinder in lb-ft2
ρ = density of cylinder material in lb/in3 L = length of the cylinder in inches
D = diameter of the cylinder in inches
Note: The units of measure for WK2 are in lb-ft2. Refer to Figure 2-47.
For an example of a solid cylinder, let’s consider a solid aluminum roll that has a length of 72 inches and a diameter of 18 inches. Its inertia would be:
WK2 = .000681 × 0.0977 × 72 × 184 WK2 = .000681 × 0.0977 × 72 × 104976 WK2 = 502.62 lb-ft2
Figure 2-47. Inertia of solid cylinders
Ερ
D
L
Common Material Densities ( ) Aluminum = 0.0977
Brass = 0.311 Steel = 0.2816
Now that the inertia has been calculated (WK2), the time it would take to accelerate the system can be determined as follows:
From 0 to 1200 rpm, with an acceleration torque available of 30 lb-ft, the formula for acceleration time will be used:
Therefore:
t = 65.27 s
If the time calculated is too long, then the easiest item to control would be the amount of available acceleration torque—meaning the motor. The motor would have to be upsized to have increased available acceleration torque. Motor torque will again be discussed in the chapter on DC and AC motors.
Hollow Cylinder Inertia Calculations
To calculate the inertia of a hollow cylinder, basically the same formula can be used, but without the inertia of the hollow section. The formula would be:
WK2 = .000681 × ρ × L × (D24 – D14) Figure 2-48 shows a hollow cylinder with the formula parts.
where:
WK2 = inertia of a cylinder in lb-ft2
ρ = density of cylinder material in lb/in3 L = length of the cylinder in inches
D2 = outside diameter of the cylinder in inches D1 = inside diameter of the cylinder in inches.
If we take the previous example and only solve inertia for the hollow cyl- inder above, we would go through the following calculations. We will assume the same information regarding the cylinder:
Solid aluminum roll has a length of 72 inches, an outside diameter of 18 inches, and an inside hollow diameter of 12 inches. Therefore, we would go through the following calculations:
WK2 = .000681 × 0.0977 × 72 × (184 – 124) T 308 N WK t 2 × ∆ × = 9240 603144 30 308 1200 62 . 502 t = × × =
Chapter 2: Review of Basic Principles — Mechanical Principles 57
WK2 = .000681 × 0.0977 × 72 × (104976 – 20736) WK2 = .000681 × 0.0977 × 72 × 84240
WK2 = 403.34 lb-ft2
If we use the above inertia, we can again find the time it would take to accelerate the system. We will use the same example:
From 0 to 1200 rpm, with an acceleration torque available of 30 lb-ft, the formula for acceleration time will be used:
Therefore:
t = 52.38 s
We have determined that the amount of time to accelerate the hollow-cyl- inder system is definitely less, compared with a solid cylinder. However, it should be noted that the amount of time reduction is not as much as expected. With 66% of the cylinder removed, there was only a reduction of about 13 s of acceleration time. This indicates a characteristic of iner- tia—the largest amount of inertia is concentrated around the rim, as opposed to the center part of the object.
The same formulas would be used to determine the inertia of pulleys, sheaves, and other rotating objects. Simply break up the objects into solid and hollow cylinders and apply the formulas previously discussed. Then
Figure 2-48. Inertia of hollow cylinders
Ερ
L
Common Material Densities ( ) Aluminum = 0.0977 Brass = 0.311 Cast Iron = 0.2816 Steel = 0.2816 T 308 N WK t 2 × ∆ × = 9240 484008 30 308 00 12 403.403 t = × × =
add all the inertia components calculated and determine the time required for acceleration of the system.
Reflected Inertia
One additional note should be stated about inertia. The term reflected iner- tia is the inertia actually found at the motor shaft. This term is standard throughout the industry. A speed reducer (gear box or belt type of cou- pling device) changes the inertia that the motor actually sees. Figure 2-49 shows this phenomenon.
As seen in Figure 2-49, if an adjustable speed drive was sized for an inertia of 12 lb-ft2, it may be much larger than actually necessary. Reflected iner- tia also includes inertia of the actual motor.
Power transmission devices (gear boxes, belts, and pulleys) serve to make the motor’s job easier. In our next section, we will look at gear boxes and speed reducers in more detail and determine what relationship exists between speed, torque, and horsepower.