Torque

The torque and torque ( from Latin momentum moving force ) is a physical quantity in classical mechanics. It plays the same role as the force for straight movements for rotary motion. A torque can speed up or slow down the rotation of a body and twist or bend the body. The unit of measure used internationally for torque is the newton meter. The symbol is.

Two equal but exactly acting in the opposite direction forces on different lines of action of forces is called a couple. A couple of forces produces a torque of the amount of power distance lines of action.

If a force acts vertically on a lever arm length, the result is the amount of torque from the length of the lever arm multiplied by the magnitude of the force:

In general, the torque is calculated by the vector product of distance vector and force vector:

The distance vector is the vector from the reference point of the torque is the point of application of force. If several forces () at different points of a, then the total torque is the vector sum of the individual torques:

• 3.1 Dormant body
• 3.2 Rotating body

• 4.1 Example: electric motor
• 4.2 Example: Torque and power of an internal combustion engine
• 4.3 Example: power and torque of a hydraulic motor

Special denominations in the art

In the technique discussed here is the physical quantity is usually called moment.

Depending on use, the following moments are conceptually distinguished:

The torque as a directed quantity

When a force acts on the point, relative to the point it causes a torque which can be calculated as follows:

The direction of the torque vector also results from the three-finger rule that generally applies to vector products: If you point the thumb of the right hand in the direction of the distance vector and with the index finger in the direction of the force, then is the middle finger the direction of the torque vector at. Therefore, one can read the rotation of the direction of the torque vector of the corkscrew rule.

The torque vector is a pseudovector (also called " axial vector" called ). This means that, unlike the distance and the force vector it does not reverse its direction in space reflection.

For graphic representation: Like all vectors, the torque vector in the drawings are shown as an arrow ( see figure in the Introduction section). It is the length of the arrow for its amount. The direction is - as I said - the direction of rotation of the torque on. This can be indicated by an additional curved arrow about the pivot axis. Since the arrowhead is not a linear, but a rotational direction symbolizes the torque vector is sometimes drawn with a double peak.

Special case: Two dimensions

If all forces and distance vectors are orthogonal to the axis of rotation in a plane, then all greatly simplify the calculations discussed in this article, because all torques can be treated as scalars. The indication of direction is reduced in this case the sign of the torque: In accordance with the general vectorial definition torques acting when viewed from above the plane counterclockwise are (in " mathematically positive direction " ) counted as positive, corresponding torques clockwise negative. For many technical applications where the position of the axis of rotation is determined by the bearing, this two-dimensional special case of the rule.

Unit of measurement

The unit of torque in SI is the newton meter (N m). With the base units kilogram, meter and second applies:

The unit of mechanical work is also the Newton meters. However, torque and work are different physical sizes, which can not be converted into each other, so one may call the unit of work as Joule (), that of the torque but not! Work is being done when in a movement along a path a force ( component) acts parallel to the motion. In contrast, the torque force acts perpendicular to the line formed by the lever arm. The work is a scalar quantity. The torque, however, is a pseudo- vector.

The phrase " work = force times distance " here corresponds to " work = torque times angle". To illustrate this connection, can be used for the torque as energy per unit angle and the

Are used, then the direction of the vector pointing in the direction of the axis of rotation. The unit is radian for plane angle.

In technical documents and nameplates on the torque in Nm unit is specified. Other units used are, for example oz. · In (1 oz = 7.06 · in mNm ) or combinations of different (by weight) of force and length units.

Statics

Droop is the portion of the mechanics, which deals with equilibrium states. When a body is in equilibrium of forces, then it changes its speed does not (see first Newton 's law). Corresponding to this is that a body that is in torque equilibrium

Is, its rotation speed does not change. If the sum of all torques for any reference point equal to zero, so that also applies to any other reference point. Therefore, you are completely free in the choice of the reference point. Wherein pivoting bodies but provides a point on the axis of rotation. Namely, since causing no torque around the rotation axis, the coercive forces of the bearings, this facilitates the calculation of the torque balance significantly.

A power arrow can be moved along its line of action. In the position where the distance is perpendicular to the force vector arrow perpendicular to the rotation axis, it is referred to as a lever arm. Amount of default, then: " torque equal lever arm times force ". With two attacking forces ( which are then referred to as force and load) is the above torque balance equivalent to the lever rule:

( Note that, strictly speaking, only the amounts are the same, because the two torques are in opposite directions and therefore have the opposite sign. )

Dynamics

The dynamics dealing with states that are not in equilibrium. According to Newton's 2nd law introduces a resultant force on a body to a change in velocity. Analogously, a resultant torque means a change in the angular velocity. The inertial behavior in terms of rotation depends not only on the mass of a body, but also on their spatial distribution. This is expressed by the inertia moment. In a rotation around a fixed axis of the torque in the direction of the axis:

It should be noted that the moment of inertia is not only dependent on the position of the rotation axis (see Steiner's theorem ) but also on its direction. If you want to formulate the above equation for any general direction in space, we must instead use the inertia tensor.

Can be the connection between the torque and the rotation also on the rate of change of the angular momentum to express:

In the two-dimensional special case of a torque causes only an acceleration or deceleration of a rotational movement. In general three-dimensional case it can, however, change the direction of the axis of rotation. (see for example: precession )

Correspondences between linear and rotary motion

Torque increases in classical mechanics for rotary motion a similar role as a force for linear movement:

Measuring the torque

Resting body

The rotating body is held by a static counter-torque at rest. The force acting on the stationary body and the torque to be measured is equal to the counter torque which is generated, for example using a torque wrench and its value is the product of the lever arm and the reaction force to the key handle.

When tightening a screw or a nut using a torque wrench torque that is both generated and measured, which opposes the screw tightening,. In this case, the rotatable body is not complete at rest, when the operation of the tightening is completed.

Rotating body

The rotational speed changing torque can be determined by measuring the angular acceleration when the moment of inertia is known. The evaluation is performed using the formula

In a power transmission, for example a rotating shaft, the function of the torque thereby acting interested by the rotational speed ( torque curve). For this, the state of constant speed shall be made. Are measured, the performance and the speed. The evaluation is performed using the formula

Measuring the performance is done using a so-called power brake: pendulum machine, Prony brake or Water brake.

Torques on selected machines

Example: Electric

Electric motors have a relatively high starting torque, which can be increased in three-phase motors by temporary delta operation. The picture shows the output torque of an induction motor as a function of the speed. The normal operating range to the right of the tipping points K1 and K2 on the steep curve. The area to the left of the tipping points is the starting range to be as fast as possible by driving because of the poor efficiency.

Example: Torque and power of an internal combustion engine

The term used in automobiles, the maximum torque of the engine at a particular speed refers to the maximum output torque from the motor to the crankshaft. The torque output to the crankshaft at full load is not constant over the whole speed range of the engine, but has a certain range of the useable speed range of the maximum.

The torque M for four -stroke engines is calculated from:

Here, Vh is the stroke volume and the mean effective pressure PE, the factor in the denominator of the formula is derived from the work of a torque, which is performed along the circumference. The value is multiplied for four-stroke motors with 2, as four-stroke engines do work only every second revolution. For two-stroke engines shall apply accordingly:

Sample calculation for the torque of a production vehicle with 2000 cc ( = 0.002 m³) displacement, the four-stroke engine at a speed of 2000 rpm, a central pressure of 9 bar (= 900,000 Pa, 1 Pa = 1 N / m²) is achieved, calculated in SI units:

The equation for the performance of a rotational movement is ( see above)

And for a speed-dependent power

In an internal combustion engine, a torque of 143 Nm at 2000 rpm write, the power is calculated as follows:

Example: power and torque of a hydraulic motor

The hydraulic performance of a hydraulic motor is calculated from the pressures and the motor input or output and the swallowed oil volume ( the volume per revolution ):

From the equation for the performance of a rotational movement (see above)

Follows the torque to: