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Vordimensionierung einer Schraubenfeder

Preliminary sizing of a spring

Böge, A.: Formeln und Tabellen Maschinenbau, 2. Auflage 2009

Das Buch gibt folgende Einheiten vor:

Default units in the book

Größe

Quantity

Symbol

Einheit

Unit

Kraft

Force

F

N

Abmessungen

Dimensions

mm

Formulas

This is a physical value formula. It works for any dimensionally correct units of the specified variables.

This is a numerical value formula. It works if the symbols represent pure numerical values with respect to the units specification in the table to the right. Providing this table is mandatory for such formulas.

Here you see a glitch in the matrix. If d just represents the numerical value and not the quantity it is strictly speaking not correct to write \[REGION[633fd628156b45ff9699a6e7f7377b1b]]\.

d 5 mm * <

Physikalische Größengleichung

Physical quantity equation with physical quantity variables.

d.erf 8 F * D.m * π τ.i.zul * / 3 nthroot :

d.erf

erforderlicher Drahtdurchmesser

Required wire diameter

F

Maximale Federkraft

Maximum force

D.m

mittlerer Wicklungsdurchmesser

Mean coil diameter

τ.i.zul

zulässige ideelle Schubspannung, abhängig vom Drahtdurchmesser

Allowable ideal shear stress

We define physical quantity variables and directly use the physical quantity equation. Any units can be used upon input definition and results display.

F 1 kN * :

D.m 20 mm * :

τ.i.zul 600 MPa * :

d.erf mm 4.39

Physical quantity equation and physical quantity variables, that's what SMath is designed for.

Zugeschnittene Größengleichung mit physikalischen Einheiten

Physical value variables and numerical value equation

d 4 mm * :

Drahtdurchmesser

Initial assumption for wire diameter

D.a D.m d + : mm 24

Außendurchmesser

External coil diameter

F.2 F : 1000 N *

Maximalkraft

Maximum force

Direct application of the numerical value equation

k.1 0.15 d 5 mm * < 0.16 cases :

k.1 0.15

d k.1 F.2 D.a * 3 nthroot * :

d 0.433 kg 13 /^m 23 /^* s 23 /^/ *

The result has a strange unit because values with units are inserted in an equation that expects numerical values.

This is particularly dangerous if the resulting unit isn't suspicious

Fix the problem by converting the physical quantities to numerical values (divide by the expected unit).

d k.1 F.2 N / D.a mm / * 3 nthroot * :

This provided a numerical value. The unit follows from the unit conventions coming along with the equation.

Not nice because you are mixing two concepts of variables.

d 4.33

required wire diameter in mm d k.1 F.2 N / D.a mm / * 3 nthroot * :

To keep track of this without converting the result to a physical value, you could use the description mechanism.

d 4.33

d description required wire diameter in mm

d k.1 F.2 N / D.a mm / * 3 nthroot * (mm * :

If you want to go on with a physical value variable, then provide the unit of the result as a factor in the equation. You can display it in whatever unit you want.

A straightforward and transparent way to use numerical value equations with physical value variables.

d mm 4.33

d cm 0.433

The "ignore units" option

Was passiert, wenn "Maßeinheiten ignorieren" gesetzt wird?

SMath provides the option to "ignore units" in math regions. Let's see what we get using this. Note: The inout variables are still defined as physical quantities with value and unit.

In der Definition aktiv

Active in definition

d k.1 F.2 D.a * 3 nthroot * :

in der Anzeige inaktiv

Inactive in display

d 0.4327 kg 13 /^m 23 /^* s 23 /^/ *

In der Definition aktiv, direkte Anzeige

Active in definition, direct display

d k.1 F.2 D.a * 3 nthroot * : 0.4327

ohne Eintragung einer Einheit ins Ergebnis

No display unit specified

d k.1 F.2 D.a * 3 nthroot * : mm 0.4327

Eintragung einer Einheit ins Ergebnis hat keine Auswirkung auf den Zahlenwert

Specification of display unit doesn't affect the displayed numerical value. That's dangerous!

d k.1 F.2 D.a * 3 nthroot * : m 0.4327

in Definition inaktiv

Inactive in definition

d k.1 F.2 D.a * 3 nthroot * :

in der Anzeige aktiv

Active in display

d 0.4327 kg 13 /^m 23 /^* s 23 /^/ *

Einstellung hat keinen Einfluss

If no unit is specified, the units in the result aren't actually ignored, so what does "ignore units" mean here?

d mm 0.4327

Eintragung einer Einheit hat keine Auswirkung auf den Zahlenwert

Specification of a unit is accepted and suppresses the weird stored unit but does not affect the numerical value.

d m 0.4327

d \0020\ 0.4327

Leere Maßeinheit (Ctrl-K, Leertaste)

Here, an invisible unit is used (space character created using Ctrl-K and Spacebar)

Summary: Hands off the "ignore units" option.

Zahlenwertvariable mit zugeschnittener Größengleichung

Numerical value equation with numerical value variables.

Zahlenwertvariable mit zugeschnittener Größengleichung

First, we recall the units conventions from the book.

Größe

Quantity

Symbol

Einheit

Unit

Kraft

Force

F

N

Abmessungen

Dimensions

mm

We define the variables using the numerical value only and annotate the unit using the description.

N F 1000 :

mm D.m 20 :

MPa τ.i.zul 800 :

mm d 4 :

Drahtdurchmesser

Initial assumption for wire diameter

mm D.a D.m d + : 24

Außendurchmesser

External coil diameter

N F.2 F : 1000

Maximalkraft

Maximum force

Abschätzung des erforderlichen Drahtdurchmessers

Estimate of the required wire diameter. We have to remove the unit from the definition of \[REGION[a17fce3389354c8cb33c7c8e7bce5d19]]\

k.1

k.1 0.15 d 5 < 0.16 cases :

The definition of k is a physical quantity equation, while the definition of d is a numerical value equation. The books switches all the time between the concepts.

k.1 0.15

mm d k.1 F.2 D.a * 3 nthroot * :

d 4.33

d description mm

Compact and close to textbook appearance. Yet the unit handling mechanism of SMath is totally bypassed and can't help to reveal errors in the equations.