B264

OTHER: Mechanics - Definitions & Algorithms

• Definitions and Algorithms related to the power train.

Symbols are American National Standard Letter Symbols where possible. [See ~ MH p.115]

Note: The algorithms are in blue.

Outside Web Sites:

There is, currently, duplication on Dynamics, Aerodynamics & Mechanics.

Acceleration:

Acceleration - Linear: [a] (m/s2)

a = F * m The time-rate change of velocity and is expressed as velocity divided by time or as distance divided by time squared, that is, in feet per second per second or feet per second squared. The metric SI unit is the meter per second squared, where force is in Newtons and mass is in kilograms.

Acceleration - Angular: [α] [alpha] (rad/s2)

• The rate of change of angular velocity over time. In SI units, it is measured in radians per second squared (rad/s2). It is also rad/s2 in English units.
• Constant acceleration: = torque / moment of inertia
• Motion uniformly accelerated from initial velocity. See [MH p.151]
• The angular acceleration [α] = (ωf - ωo)/t . Knowing: initial angular velocity [ωo] (radians per second), final angular velocity [ωf] (radians per second) and the time [t] (in seconds).
• The final angular velocity [ωf] (radians per second) = ωo + α * t . Knowing: initial angular velocity [ωo] (radians per second), the angular acceleration [α] (radians per second, per second) and the time [t] ( in seconds).
• Many other algorithms are shown in the book.

Acceleration of Gravity: [g]

The acceleration of a free falling body. 32.16 ft / sec2.

Angle of Repose:

The angle at which a body will be just able to remain stationary, the frictional resistance being very nearly overcome by the tendency of the body to slide down.

The tangent of this angle equals the coefficient of friction. Tangent = opposite / adjacent.

See above; Acceleration - Angular

Angular Distance: [θ]

Angular Speed ~ Angular Velocity: [ω] | [ωR], [ωZ], [ωX] & [ωY]. Wr, Wx, Wy, Wz. [OmegaR], [OmegaZ], [OmegaX] & [OmegaY], (rad/s)

The change in angle per time unit, which is measured in radians per second in the SI system.

[Angular velocity ) (radians per second) = Angle of rotation ) (radians) / time (t) (seconds)].

Conversion; Ω = RPM * 0.1047197, RPM = Ω * 9.549304.

Axial Pitch:

See DR Gears

Angular Momentum: [L]

A measure of an object's rotation about a particular axis; more specifically, the product of its rotational inertia and rotational velocity. For an object that is small compared to the radial distance, it is the product of mass speed, and radial distance of rotation. Angular Momentum = rotational inertia x rotational velocity = JMω or mvr or

The derived SI units of angular momentum are newton·metre·seconds; N·m·s (kgm2s-1). 1 N = kilogram * meter / second2.

θ * r

The length of a circular arc of a circle with radius r and subtending an angle θ (measured in radians)

Brake Horsepower:

Brake horsepower (bhp), sometimes called shaft horsepower, is the amount of power available for useful work. Bhp is less than ihp because of the various power losses which occur during engine operation.

To determine the brake or shaft horsepower that is delivered as useful work by an engine, the sum total of all mechanical losses must be deducted from the total ihp.

Center of Gravity:

The center of gravity of a body, volume, area or line is that point at which if the body, volume, area or line were suspended it would be perfectly balanced in all conditions. For symmetrical bodies of uniform material, it is at the geometric center.

Center of Mass:

The location within a body at which its entire mass is considered to be located.

Center of Oscillation:

The .

An fictitious outward force that is due to rotation. In an inertial frame of reference, it is fictitious in the sense that it doesn't act on the rotating object but on whatever supplied the centripetal force; it is the reaction to the centripetal force. In a rotating frame of reference, it does act on the rotating body and is fictitious in the sense that it is not an interaction with an agent or entity such as mass or charge but is a force in itself that is solely a product of rotation; it has no reaction-force counterpart.

Centrifugal force is "inertia" or momentum expressing itself against a centripetal force pulling a moving object toward a center of rotation.

• A center-seeking force that causes an object to follow a circular path.
• For blade element method, the centripetal force F acting on a body of mass M is given by the equation F = (Mv2)/r, where v is its velocity and r is the radius of its path. ????? Look into some time.
• In English; [F is in pounds], [M is in slugs (1 slug = 32.175 lbs)], [v2 is in ft/sec.], [r is in feet].
• Alternative in English F = (Wv2)/(32.16 * r); W is in pounds
• For a uniform mass per unit length of blade; F = (M * Ω2 * R) / 2. Where;
• In English; [F is in pounds], [M is in slugs], [Ω is in radians per sec.], [R is radius of blade, in feet].
• In Metric; [F is in newtons], [M is in kilograms], [Ω is in radians per sec.], [R is radius blade, in meters].
• See Blade Generated Forces - Preliminary

Centripetal Force and Centrifugal Effect:

Action-reaction force pair associated with circular motion. Centripetal (center-seeking) force is the constant inward force necessary to maintain circular motion. The centripetal force is supported by a reaction force, the centrifugal (center-fleeing) force, which acts on the circling object and on the source of centripetal force, usually located at the circle's center. The two forces are equal in magnitude and opposite in direction.

Centroid:

Center of mass.

Centroidal Distance: [y]

The distance from the base (baseline) to the center of the mass.

A Constant Velocity Joint is one that provides consistent drive shaft speeds regardless of the operating angle of the joint

Single-gimbal CMGs

The most effective CMGs include only a single gimbal. When the gimbal of such a CMG rotates, the change in direction of the rotor's angular momentum represents a torque that reacts onto the body to which the CMG is mounted, e.g. a spacecraft. Except for effects due to the motion of the spacecraft, this torque is due to a constraint, so it does no mechanical work (i.e. requires no energy). Single-Gimbal CMGs exchange angular momentum in a way that requires very little power, with the result that they can apply very large torques for minimal electrical input.

Coriolis Effect: Should this be in Flight Dynamics?

Conservation of energy. The change in rotor blade velocity to compensate for a change in the distance between the center of mass of the rotor blade and the axis of rotation of the blade as the blade flaps or cones in flight. This change in distance during flapping is the result of the flapping hinge being offset. A rotor with teetering hinge will experience the Coriolis effect during a change in the coning angle but not during the act of teetering.

For additional information see: Coriolis and Hooke's Joint

Explanation and algorithms relating to Coriolis Effect. For mathematical explanation see; [Source ~ RW, section 2.7]

Correlator: (synchronizer, anticipator)

A collective-pitch - throttle synchronization device that automatically adjusts the amount of throttle in a relationship with changes to the collective setting.

Critical Component:

A component that if it were to fail, would result in the loss of the aircraft.

xx

Displacement: [u]

The difference between an initial position and a final one. Displacement is a vector quantity.

Elastomeric Bearing:

Endurance Limit:

The endurance limit is the highest stress or range of stress that can be repeated indefinitely without failure of the metal. Graph

Energy: [Ek], [K] & [T]

The capability of a body to do work.

Kinetic Energy:

Due to translation:

EKT = (1/2) * M * V2 = (W * V2) / (2 * g); [EKT in ft-lbs], [W in pounds], [V in ft/sec.], [g in 32.16 ft/sec/sec.]

Due to rotation:

EKR = (1/2) * JMO * ω2; [EKR in ft-lbs], [JMO ~ moment of inertia of the body about a fixed axis 0, in lbs-ft-sec2]

Potential Energy:

The most common example is a body having potential energy because of its position in a field of force is that of a body elevated to some height above the earth.

EPF = W * S weight * elevation [EPF in ft -lbs.]

When both dynamic radial loads and dynamic axial loads act on a bearing at the same time, the hypothetical load acting on the center of the bearing which gives the bearings the same life as if they had only a radial or only an axial load.

The static equivalent load is a load which would cause the same total permanent at the most heavily stressed contact point between the rolling elements and the raceway as under actual load conditions; that is when both static radial loads and static axial loads are simultaneous applied to the bearing.

The SKF New Life Theory introduces the concept of a fatigue load limit Pu analogous to that used when calculating endurance for other machine components. This fatigue load limit is the load below which fatigue will not occur in the bearing under ideal conditions.

Force: [F] (newton or lbf or ozf)

A push or a pull. F = m * a; F is force, M is mass, a is linear acceleration.

Fretting:

The rapid corrosion that occurs at the interface between contacting, highly loaded metal surfaces when subjected to slight vibratory motions. http://corrosion.ksc.nasa.gov/html/fretcor.htm

Splined couplings are one of the commonest methods of connecting elements in rotating machinery. Their performance is limited not only by torsional fatigue but also by fretting wear arising from angular misalignment. The wear problem is especially common in mechanical transmissions, which have lightweight and flexible casings, such as those in helicopters.

Galling:

A severe form of adhesive wear which occurs during sliding contact of one surface relative to another. Clumps of one part stick to the mating part and break away from the surface. (Can frequently occur when both a nut and bolt are zinc coated.)

Gyroscopic Inertia:

All so known as Rotational Inertia. When any object rotates, it would like to stay in that same state, and it takes a force (or torque) to change this state.

Applied Gyrodynamics

Gyroscopic Precession:

The characteristic that causes the gyroscope to react to an applied force at a point 90-degress away from the point of application, in the direction of its rotation.

Helix Angle:

The angle between the axis of a helical gear and an imaginary line that is tangent to the gear tooth. Helix angles can vary in size from 0° to 90 degrees.

Wikipedia

Hooke's Joint Effect:

This appears to be valid explanation for the flapping and teetering hinges. If only the teetering hinge is being considered then the explanation of a Knuckle Joint will be appropriate. (see: Coriolis and Hooke's Joint).

Horsepower:

• HP = Volts * Amps / 746
• HP = Watts / 746
• 1 HP = 33,000 foot-pounds per minute ~or~ 550 foot-pounds per second.

Hour-life:

Not sure of proper definition. Has to do with lifetime of a part re hours of operation. There is also the phrase Design-life.

Inertia:

That property of matter which causes it to resist any change in its motion or state of rest.

Inertial Force:

A force which must be added to the equations of motion when Newton's laws are used in a rotating or otherwise accelerating frame of reference. Some call it a "fictional force" because when the same motion is solved in the frame of the "outside world," these forces do not appear.

Joule: [J] Metric (N·m)
• The SI unit of work or energy, defined to be the work done by a force of one Newton acting to move an object through a distance of one meter in the direction in which the force is applied. Metric (N·m) Equivalently, since kinetic energy is one half the mass times the square of the velocity, one joule is the kinetic energy of a mass of two kilograms moving at a velocity of 1 m/s.

Kinetic Energy: [K]

The kinetic energy of an object is the extra energy, which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity

Linear (metric): EK = 1/2 * m * v2

• Ek is the kinetic energy in joules
• m is the mass in kilograms, and
• v is the speed in meters per second.

Rotational (metric): K = 1/2 * I * ω2

Linear (english): KE = (m * v2)/( 2 * gc)

• KE is kinetic energy in ft-lbf
• m is mass in lbm
• v is velocity in ft/sec
• gc is (32.17 lbm-ft)/(lbf-sec2)

Rotation (english): KE = (m * v2)/( 2 * gc) This is my merging of rotational Metric and linear English.

• K = 1/2 * I * ω2 ~or~ K = 1/2 * JM * ω2
• K = 1/2 * (m * r2) *((RPM / 60)* 6.283184)2
• K is the Kinetic Energy in lb-ft-RPM2
• I is the body's moment of inertia. So is JM
• m is mass in lbs
• r is the radius of any mass from the point of rotation in feet.
• RPM is the body's angular velocity in rotations per minute

Kip: []

A unit of weight equal to 1000 pounds.

• Fifteen standard lead angles have been established to provide adequate coverage: 0.5,1, 1.5, 2, 3, 4, 5, 7, 9, 11, 14, 17, 21, 25, and 30 degrees. ~ from Machinery's Handbook 25.

Life Rating:

I am having a problem with this; for bearings, gears, etc. For information see; http://www.dbsmfg.com/rightspec.html

For now see; Safety Factor:

The point where structural damage occurs (objectionable permanent deformation). Limit load factors are about 2 to 3 G for observation and utility aircraft and as high as 5 G for modern attack helicopter.

NL = Maximum normal static load / Design gross weight

See OTHER: Flight Dynamics - Definitions ~ Limit Load Factor and see below.

Linear Momentum: [?]

The product of the mass [M]of the body and the velocity [V] of the center of gravity of the body . Linear Momentum = M * V, or, since M = W / g ~ Linear Momentum = W * V / g.

SI unit is kg·m/s, or, equivalently, N·s. ~ English units in ft-lb/s.

It should be noted that linear momentum is a vector quantity, the momentum being in the same direction as V.

Load Factors: The following are not in any specific order, yet. The comments may not be correct. The Load Factors should be put on Mechanics page or Flight Dynamics page only.

• Limit Load Factor (Limit Load Rating) ~ That load which can be applied without deforming the part.
• Ultimate Load Factor (Ultimate or Static Fracture Load Rating) (Aircraft Static Capacity) ~ That load which can be applied without braking the part. It is not less than 1.5 times the Limit Load Rating.

For some additional information see http://www.unicopter.com/0473.html#Transmission

L10 life:

For most bearing calculations life is expressed as the number of hours that 90% of a group of identical bearings will exceed under a give set of conditions, and is referred to as the L10 life.

Mass: [m] or [M] = W / g

A measurement of the inertia of a body.

Inertial mass is a measure of an object's resistance to changing its state of motion when a force is applied. An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily.

In the SI system of units, mass is measured in kilograms (kg).

More to come.

Mass = Weight [lb] / 32.17 [ft/sec2]

A slug has units of lbf·s²/ft. It is a mass that accelerates by 1 ft/s2 when a force of one pound-force (lbf) is exerted on it. Therefore a slug has a mass of or 32.17405 pound-mass or 14.5939 kg.

In the SI system of units, mass is measured in kilograms (kg). Many other units of mass are also employed, such as: grams (g), tonnes, pounds, ounces, long and short tons, quintals, slugs.

Mechanical Properties of Materials:

• Specific Weight
• Ultimate Tension
• Ultimate Compression
• Ultimate Shear
• Yield - Tension
• Yield - Shear
• Elastic Modulus
• Shear Modulus
• Thermal Expansion
• Ductility

See FORM: PT - Material Properties for values.

Mechanics:

The branch of physics that deals with the motion of material bodies and the phenomena of the action of forces on bodies.

In this web site ~ Information related to the mechanical aspects of helicopters. Mainly the power train.

Statics:

The branch of mechanics dealing with bodies, masses or forces at rest or in equlibrium.

Dynamics:

Aerodynamics:

Kinematics:

The description of movable mechanical structures consisting of joints and links including the number, location, and orientation of the joints.

Meters per second:

SI symbolic abbreviation is m·s−1, or equivalently, m/s;

Modulus of Rigidity [G]

Is also known as the Modulus of elasticity in Shear. It is the reatio of shear stress [τ], to shear strain [γ], in radians.

A measure of the tendency of a force to rotate the body upon which it acts about its axis. M = F * r. The SI unit is N·m.

A physical property of a member that helps define rigidity or stiffness and is expressed in inches raised to the fourth power. It is a measure of the resistance to rotation offered by a section's geometry and size. m is its mass, and r is its perpendicular distance from the axis of rotation.

Inertia = mass times square of distance from axis, I = ò r2dm

SI units kg m², Former British units slug ft2 or ft-lb-sec2 (1 slug = 1 pound second2 per foot)

Inertia can be linear as well.

• Moment of inertia of plane area. [I]
• in which the axis is in the plane of the area, are found in formulas for calculating deflections and stresses in beams. To calculate values for standard shapes run program: Engineering Power Tools ~ Mechanical ~ Moment of Inertia
• Polar moment of inertia of plane area. [J]
• in which the axis is at right angles to the plane, occurs in formulas for the torsional strength of shafting. JZ = IX + IY
• Is the sum of any two moments of inertia about axes at right angles to each other. It is taken about an axis that is perpendicular to the plane of the other two axes. Occurs in formulas for the torsional strength of shafting
• Polar moment of inertia of mass. [JM] sometime denoted by [I]. In metric SI system the units are kilogram-meter2. In foot-pound-second system the units are ft-lbs-sec2 and slug-ft2. (1 slug = 1 pound second2 per foot). In inch-pound-second system the units are inch-lbs-sec2.
• Appears in dynamics equations involving rotational motion.
• The resistance to change of a body in circular motion. Also known as Gyroscopic Inertia.
• JM bears the same relationship to angular acceleration as mass does to linear acceleration.
• Value for cylinder: JM = 1/2mr2 ~ r is the radius of the cylinder.
• Value for hollow cylinder: JM = 1/2m(R2 + r2) ~ R is the outside radius of the cylinder. r is the inside radius.
• Value for disk with mass (m) considered as being concentrated at the radius (r). JM = m * r2

The quantity of motion of a moving object, equal to the product of its mass times its velocity.

SI unit is kg·m/s, or, equivalently, N·s

A more sophisticated definition of "inertia" is one that includes values of mass, speed, and direction.

Newton: [N]

A newton is the amount of force required to accelerate a body with a mass of one kilogram at a rate of one meter per second squared. [Newton's 2nd law.]

1 N = kilogram * meter / second2.

Origin of Coordinates for Total Helicopter:

The SynchroLite's origin is at the intersection of the centerlines of the two mast's.

The UniCopter's origin is at the mid-point between the centers of the two rigid hubs; and the center of the hub is the point where the centerlines of the pitch axis of the three blades intersect.

The largest diameter of the worm gear teeth. It is equal to the diameter of the outside circle.

Polar Moment of Inertia of Plane Area [J]

See up above.

Polar Moment of Inertia of Mass [JM] or [I]

See up above.

Power: [P] [W watt]

• The product of Force times Distance divided by Time.
• Power [P] = Work [W] / Time [s]
• Power [P] = Torque [Q] * v * n * 2 π = Q * ω = U * I
• Horsepower: Lift 550 pounds at the rate of 1 foot per second.
• Watt: 1 Joule per second. Metric (1 N·m/s).
• 1HP = 735 W (or about 745 if using SI horsepower or imperial horsepower)

Precession:

A change in the direction of the axis of a rotating object. See Gyroscopic Precession above.

Precessional Frequency [ωp]

The angular frequency of the axis of rotation.

Is the distance from the neutral axis of a section to an imaginary point at which the whole area of the section could be concentrated and still have the same moment of inertia. Formula: The square root of (the moment of inertia in inches^4 divided by the area of the section in inches^2) expressed in inches.

Value for circular disk w/ axis through center: k = 0.7071 r

To calculate values for standard shapes run program: Engineering Power Tools ~ Mechanical ~ Moment of Inertia.

Radius of Center of Mass: [Rcm]

xxx

Rcp = Rg2/Rcm + Rcm 'g' in this instance is acceleration due to centrifugal force. This is an assumption at this point. For best description and algorithm see [Source ~ MH p.149]

Relative Wind:

Direction of the airflow produced by an object moving through the air.

Rotational Inertia: [JM or I]

See above; Moment of Inertia ~ Polar moment of inertia of mass

Rotational Speed: [RPM or (Ω or ω)]

Conversion; Ω = RPM * 0.1047197, RPM = Ω * 9.549304.

Safety Factor:

The safety factors that are used on the mechanical and structural components of helicopters.

A number as opposed to a vector, function.

A physical quantity that can be completely charactirized by just one number.

The distance between one thread and its neighbor.

The distance the nut advances when the screw is rotated one complete revolution.

Slug: [M]

A unit of mass, which is equal to 1 pound second2 per foot. M = W / g in which M = mass in slugs, W = weight in pounds, and g = acceleration due to gravity commonly taken as 32.16 feet per second.

Strain:

Systems of Measurement:

English: .

Metric SI: .

Throat Diameter:

The diameter of the throat circle at the center line of the worm gear face (the lowest point on the tooth face). .

Time: [t]

xx

Torque: [τ] [tau] [Q]

• The magnitude of a torque is defined as the product of a force and the length of the lever arm (radius).
• Torque is a moment. A measure of the tendency of a force to rotate the body upon which it acts about its axis. Torque is the turning moment exerted by a tangential force acting at a distance from the axis of rotation
• The rate of change of angular momentum  τ [ft-lb] = 5252 * HP / RPM:   τ [in-lb] = 63025 * HP / RPM:   τ [oz-in] = (HP / RPM) * 106. The SI unit for torque is the newton metre (N·m).
• Note that omega = Me!RPM * 0.1047197.
• When a driver turns a steering wheel, he exerts two equal but opposite forces on it. The two forces form a couple. The turning effect of a couple is the sum of moment of the two forces. The moment of a couple is called a torque.
• Torque as a function of time: (from Wikipedia)

τ = dL / dt

where L is angular momentum

L = JM * ω

so if JM is constant,

τ = JM * dω / dt = JM α = 243 ft-lb^2 * 1.03 radians/sec. = 250.05 ft-lb ???

where α is angular acceleration, a quantity usually measured in radians per second squared.

Torque rise, or torque reserve, is a measure of lugging ability (low-end power) when the going gets tough and the engine revs down

In the linked case the rated torque of the diesel engine is 477 lb.-ft. at the maximum speed of 2400 RPM and the peak torque is 657 lb.-ft. @ 1200 RPM. The torque rise is: 657 - 477 divided by 477 = 38%. The idle speed is 1000 RPM.

Torsion:

Torsion is a mode of loading where a shaft is subjected to a torque about its longitudinal axis, resulting in a twisting deformation.

Amplitude modulation of torque measured in degrees peak-to-peak referenced to the axis of shaft rotation.

The point where structural failure occurs. The Ultimate Load Factor equals the Limit Load Factor times the Safety Factor.

Velocity: (mps or ipm)

Constant Linear Velocity:

• The time-rate change of distance and is expressed as [Velocity (V) = Distance (S) / Time (t)], that is feet per second, miles per hour, centimeters per second, meters per second, etc.
• Speed and direction taken together form a vector known as velocity.

Constant Angular Velocity:

• [Angular velocity ) (radians per second) = Angle of rotation ) (radians) / time (t) (seconds)].

Vibration:

OTHER: Mechanics - Vibration - Engine Induced

OTHER: Flight Dynamics - Definitions & Algorithims ~ Vibration:

Whole Depth:

The distance between the throat and the base of the tooth. It is equal to the addendum plus the dedendum.

Work: [Energy] [W]

• The product of Force times Distance.
• Work = Force [F] * Displacement [u]
• Joule: One Newton moving an object through a distance of one meter
• 1 Joule = 0.7376 lbf-ft (foot-pound force)
• 1 watt hour = 3600 J

Yield Point:

xx

Last Revised: June 28, 2012