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MuedeHydra 2026-03-16 21:14:38 +01:00
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7
photonics.typ
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@ -73,9 +73,14 @@ Diese Dokumentation ist primär für eine Linux-Umgebung ausgelegt. Windows-spez
#include "src/bildverarbeitung_3.typ"
#pagebreak()
= Bildverarbeitung 3D
#include "src/bildverarbeitung_3D.typ"
#pagebreak()
= Physik
#include "src/physik_1.typ"
#include "src/physik_2.typ"
#include "src/physik_2_elmag.typ"
#include "src/physik_2_schwing.typ"
#include "src/physik_3.typ"
#pagebreak()

14
src/bachelorarbeit.typ
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@ -183,7 +183,7 @@ Falls in einer gui umgebung gearbeitet wird gibt es dafür schaltflächen, aber
#pagebreak()
== Symbole
#grid(columns: (1fr, 1fr), gutter: 10pt, [
#table(columns: (1fr, 1fr),
#table(columns: (0.4fr, 1fr),
[$plus.minus$], [```typ $plus.minus$ ```],
[$eq$], [```typ $eq$ ```],
[$eq.triple$], [```typ $eq.triple$ ```],
@ -216,7 +216,7 @@ Falls in einer gui umgebung gearbeitet wird gibt es dafür schaltflächen, aber
[$bb(1)$], [```typ $bb(1)$ ```],
)
], [
#table(columns: (1fr, 1fr),
#table(columns: (0.4fr, 1fr),
[$alpha$], [```typ $alpha$ ```],
[$beta$], [```typ $beta$ ```],
[$gamma$], [```typ $gamma$ ```],
@ -241,6 +241,16 @@ Falls in einer gui umgebung gearbeitet wird gibt es dafür schaltflächen, aber
[$psi$], [```typ $psi$ ```],
[$omega$], [```typ $omega$ ```],
)
#table(columns: (0.4fr, 1fr),
[$A gt B$], [```typ A gt B```],
[$A gt.eq B$], [```typ A gt.eq B```],
[$A lt B$], [```typ A lt B```],
[$A lt.eq B$], [```typ A lt.eq B```],
[$A and B$], [```typ A and B```],
[$A or B$], [```typ A or B```],
[$overline(A underline(or) B)$], [```typ overline(A underline(or) B)```],
)
])
== Farben

159
src/bildverarbeitung_1.typ
View File

@ -75,12 +75,29 @@ cv::circle(img, cv::Point(10, 10), 20, cv::Scalar(0, 0, 128), 30);
[30], [Linienbreite (-1 für geffülter Kreis],
)
=== Text
#table(columns: 1fr, [```cpp
cv::putText(img, str, cv::Point(150, 500), cv::FONT_HERSHEY_DUPLEX, 1, cv::Scalar(255, 0, 0), 2);
```])
#table(columns: (0.5fr, 1fr),
[img], [Zeichen fläche (`cv::Mat`)],
[cv::Point(10, 10)], [Position vom Text (Punkt unten Links vom Text)],
[1], [Schriftgrösse],
[cv::Scalar], [Farbe in BRG],
[2], [Linienbreite],
)
=== Bild einlesen
#table(columns: 1fr, [```cpp
std::string filename = "mond.png";
cv::Mat img = cv::imread(filename, cv::IMREAD_ANYCOLOR);
```])
=== Bild speichern
#table(columns: 1fr, [```cpp
cv::imwrite("gray_img.tif", imgGray);
```])
=== Bild zu einem Graubild konvertieren
#table(columns: 1fr, [```cpp
cv::Mat img_grayray;
@ -110,5 +127,147 @@ imgGray.col(maxLoc.x).setTo(cv::Scalar(32));
imgGray.row(maxLoc.y).setTo(cv::Scalar(32));
```])
=== Bild Clamping oder Clipping
#table(columns: 1fr, [```cpp
// alle Pixel über 100 werden auf 100 Gesetzt
uint8_t maxGrayLevel = 100;
imgGray = cv::min(imgGray, maxGrayLevel);
// alle Pixel unter 50 werden auf 50 Gesetzt
uint8_t minGrayLevel = 50;
imgGray = cv::max(imgGray, minGrayLevel);
```])
=== Bild kopieren
#table(columns: 1fr, [```cpp
imgGray = img; // nur ein Pointer auf img
imgGray = img.clone(); // Effektive kopie von img
```])
=== Bild Spiegeln
#table(columns: 1fr, [```cpp
cv::flip(img, flip_img, 0); // 0 -> Flip Mode
```])
#table(columns: (1fr, 1fr),
[-1], [Flip Vertikal und Horizontal],
[0], [Flip Vertikal],
[1], [Flip Horizontal],
)
=== Durch ein Bild Iteriren
#table(columns: 1fr, [```cpp
for (auto it = img.begin<uint8_t>(); it != img.end<uint8_t>(); ++it) {
std::cout << (uint8_t)*it << "\t";
}
std::cout << std::endl;
```])
=== CPP get max value
#table(columns: 1fr, [```cpp
uint16_t max = *std::max_element(img_light_data.begin(), img_light_data.end());
```])
=== Normalize
*MAL TESTEN*
#table(columns: 1fr, [```cpp
cv::normalize
```])
== Beleuchtungen
#table(columns: (100pt, 100pt, 1fr, 1fr),
[Beleuchtungstyp], [Bild], [Vorteile], [Nachteile],
[Ringlicht], [#image("../img/bildverarbeitung_1/Beleuchtung_Ring.png")], [Kompakt, einfache Montage, sehr hell auf kurzer Distanz.], [Erzeugt oft punktförmige Reflexionen auf glänzenden Oberflächen.],
[Balken (Bar)], [#image("../img/bildverarbeitung_1/Beleuchtung_Balken.png")], [Sehr flexibel (Winkel/Abstand), kostengünstig, modular erweiterbar.], [Ungleichmäßige Ausleuchtung bei falscher Positionierung.],
[Dome (Kuppel)], [#image("../img/bildverarbeitung_1/Beleuchtung_Dome.png")], [Eliminiert Schatten und Glanzreflexe fast vollständig.], [Gross, teuer, benötigt geringen Arbeitsabstand zum Objekt.],
[Tunnel], [#image("../img/bildverarbeitung_1/Beleuchtung_Tunnel.png")], [Gleichmäßige Ausleuchtung langer, zylindrischer Objekte.], [Sperrig, nur für spezifische Objektformen geeignet.],
[Backlight], [#image("../img/bildverarbeitung_1/Beleuchtung_Backlight.png")], [Erzeugt perfekten Kontrast (Schattenriss), extrem präzise Kanten.], [Prüfobjekt muss zwischen Licht und Kamera passen; keine Oberflächeninfos.],
)
#table(columns: (1fr, 1fr, 1fr),
[Eigenschaft], [Kontinuierliches Licht (CW)], [Gepulstes Licht (Strobe)],
[Funktion], [Die LEDs leuchten dauerhaft mit konstanter Intensität.], [Licht blitzt synchron zur Kamera-Belichtungszeit auf.],
[Lichtausbeute], [Begrenzt durch die Wärmeentwicklung der LEDs.], [Extrem hoch (Übersteuerung bis zu 1000% möglich).],
[Bewegungsunschärfe], [Tritt bei schnellen Prozessen leicht auf.], [Wird "eingefroren", da der Blitz extrem kurz ist.],
[Lebensdauer], [LEDs altern schneller durch konstante Wärme.], [Sehr hoch, da die LEDs nur Millisekunden aktiv sind.],
[Komplexität], [Einfach: Netzteil anschließen, fertig.], [Höher: Benötigt Trigger-Signal und Blitzcontroller.],
[Fremdlicht], [Anfällig für Umgebungslicht (Hallenbeleuchtung).], [Unempfindlich, da der Blitz das Umgebungslicht "überstrahlt".],
)
== Linsenmacher gleichung
#image("../img/bildverarbeitung_1/2026-03-06-140440_hyprshot.png")
== Kamera-Schnittstellen
#table(columns: (80pt, 70pt, 60pt, 50pt, 70pt, 1fr),
[Schnittstelle], [Image data rate (MB/s)], [Cable length (m)], ["Kosten (1 low, 5 high)"], [Kameras (Netzwerk-fähig?)], [Bemerkungen],
[USB3], [~350 - 400], [< 5m], [1], [Begrenzt (Nein)], ["Kostengünstig, "Plug & Play", CPU-Last bei hohen Raten möglich."],
[GigE], [~100 (1Gbps)], [bis 100m], [2], [Viele (Ja)], [Standard in der Industrie; sehr lange Kabel; einfache Integration.],
[MIPI CSI-2], [~1000+], [< 0.3m], [1], [Wenige (Nein)], [Ideal für Embedded Vision (Raspberry Pi/Jetson); extrem kurze Wege.],
[CameraLinkHS], [~2100+], [bis 15m (Kupfer)], [5], [Wenige (Nein)], [High-End; für extrem hohe Datenraten; benötigt Framegrabber.],
[CXP-12], [~1250 pro Lane], [bis 100m], [4], [Mehrere (Nein)], [CoaXPress; hohe Bandbreite bei langen Kabeln; Framegrabber nötig.],
[FireWire], [~32 - 80], [4.5m], [2], [Begrenzt (Nein)], [Veraltet. Wird in modernen Systemen kaum noch neu verbaut.],
)
== Geschwindigkeit
=== GaussBlur (cv2.GaussianBlur)
OpenCV ist extrem hoch optimiert (C++ Backend, SIMD-Befehle). Zudem ist ein Gauß-Filter "separabel" – man kann ihn in zwei 1D-Operationen aufteilen, was mathematisch viel effizienter ist.o
=== filter2D (cv2.filter2D)
Ebenfalls OpenCV (sehr schnell), aber hier wird eine allgemeine 2D-Faltung durchgeführt, die nicht zwingend mathematisch optimiert werden kann wie der Gauß-Filter.
=== scipy fft-based
Hier wird das Faltungstheorem genutzt: Multiplikation im Frequenzraum (FFT). Bei großen Kerneln ist das oft schneller als die direkte Faltung, aber der Overhead für die Transformation (FFT und iFFT) schlägt hier zu Buche.
=== numpy fft-based
Ähnlich wie Scipy, aber Scipy nutzt oft noch stärker optimierte Bibliotheken im Hintergrund.
=== convolve2d (scipy.signal)
Das ist die "naive" direkte Implementierung. Sie ist flexibel, aber für Echtzeit-Bildverarbeitung viel zu langsam, da jedes Pixel einzeln mit dem gesamten Kernel verrechnet werden muss.
=== GPU-Support
Ganz unten steht GPU Array not supported or not installed. Wäre die GPU (z.B. via CUDA) aktiv, lägen die Zeiten wahrscheinlich im Bereich von 0.001s oder darunter, besonders bei sehr großen Bildern.
Macht erst bei 4K+ sinn sonst ist der Vorsprung zu anderen ferfahren gleich.
=== Beispiel
Beispiel bei einem 10MP graubild:
#table(columns: (1fr, 1fr),
[GaussBlur (cv2.GaussianBlur): ], [0.024s],
[filter2D (cv2.filter2D): ], [0.184s],
[scipy fft-based: ], [0.441s],
[numpy fft-based: ], [1.521s],
[convolve2d (scipy.signal): ], [4.248s],
)
== Resampling / Aliasing
#grid(columns: (2fr, 1fr), gutter: 10pt,
[
Beim einfachen Sub-sampling wird das Bild verkleinert, indem man einfach nur jeden 8. Pixel auswählt und den Rest wegwirft.
Im Ergebnisbild siehst man seltsame, wellenförmige Muster (Moiré-Effekte), die im Originalbild der Ziegelwand gar nicht vorhanden sind. Diese künstlichen Muster entstehen durch Aliasing.
*Warum passiert das?* \
Stell dir vor, die feinen Linien der Ziegelmörtel sind sehr dicht beieinander. Wenn wir zu selten "nachschauen" (also zu wenig Pixel abgreifen), verpasst das System den eigentlichen Rhythmus der Linien. Das System "denkt" dann fälschlicherweise, es gäbe größere, gröbere Wellen.
*Die Lösung: Blurring (Tiefpassfilterung)* \
Im zweiten Versuch wird das Bild vor dem Verkleinern mit einem Gauss-Filter weichgezeichnet. Das entfernt die extrem feinen Details (die hohen Frequenzen).
- Vorteil: Die störenden Wellenmuster verschwinden.
- Nachteil: Das Bild wirkt insgesamt etwas unscharfer.
*Aliasing* \
- Die Ursache: Aliasing entsteht, wenn ein Bild oder Signal so feine Details enthält, dass die Kamera (oder der Computer) sie beim Speichern nicht schnell genug erfassen kann.
- Der Fehler: Da die Abstände der Messpunkte zu groß sind, werden die feinen Strukturen falsch interpretiert und "verfälscht" dargestellt.
- Das Ergebnis: Dadurch entstehen im fertigen Bild störende Muster, wie zum Beispiel Wellen in einer Ziegelwand oder Speichen an einem Rad, die sich scheinbar rückwärts drehen.
], [
#image("../img/bildverarbeitung_1/Resampling_orgi.png")
#v(-12pt)
Orginal
#image("../img/bildverarbeitung_1/Resampling_sub.png")
#v(-12pt)
Einfaches sub-sampling
#image("../img/bildverarbeitung_1/Resampling_blur_sub.png")
#v(-12pt)
Blurring und sub-sampling
])

24
src/bildverarbeitung_3D.typ Normal file
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@ -0,0 +1,24 @@
#import "@preview/cetz:0.4.1"
== Kamera Model
#cetz.canvas({
import cetz.draw: *
rect((0, 0), (3, 1))
content((1.5, 0.5), [Camera Model])
content((-2, 1.5), [3D World])
content((5, 1.5), [2D Image])
line((-2, 0.5), (-0.2, 0.5), mark: (end: ">"), fill: blue, stroke: blue)
line((3.2, 0.5), (5, 0.5), mark: (end: ">"), fill: blue, stroke: blue)
content((-2, -0.3), anchor: "north", [Input \ P(X,Y,Z)])
content((5, -0.3), anchor: "north", [Output \ p(u,v)])
content((0, -0.3), anchor: "north-west", [
Parameter Intrinsic: \
- fx, fy, cx, cy,
- k1-k6, p1-p2
Parameter Extrinsic:
- t, R (transformation from "camera chip" to "world")
])
})

162
src/elektronik_2.typ
View File

@ -1,4 +1,5 @@
#import "@preview/zap:0.4.0"
#import "@preview/cetz:0.4.1"
== Elektronik 2
=== Vierquadrantenkennlinienfeld
@ -16,6 +17,167 @@
)
=== Logikschaltungen
==== Logikgatter
#table(columns: (120pt, 100pt, 120pt, 1fr),
[Name], [Funktion], [Symbol], [Wahrheitstabelle \ #table(columns: (1fr,)*3, [A], [B], [Y],) ],
[Und-Gatter \ (AND)], [$Y eq A and B$], [#align(center + horizon)[
#cetz.canvas({
import cetz.draw: *
rect((0, 0), (1.5, 1.5))
line((-0.5, 1), (0, 1))
line((-0.5, 0.5), (0, 0.5))
line((1.5, 0.75), (2.2, 0.75))
content((0.75, 1.2), [&])
content((-0.75, 1), [A])
content((-0.75, 0.5), [B])
content((2.5, 0.75), [Y])
})]
], [
#table(columns: (1fr,)*3,
[0], [0], [0],
[0], [1], [0],
[1], [0], [0],
[1], [1], [1],
)],
[Oder-Gatter \ (OR)], [$Y eq A or B$], [#align(center + horizon)[
#cetz.canvas({
import cetz.draw: *
rect((0, 0), (1.5, 1.5))
line((-0.5, 1), (0, 1))
line((-0.5, 0.5), (0, 0.5))
line((1.5, 0.75), (2.2, 0.75))
content((0.75, 1.2), [$gt.eq 1$])
content((-0.75, 1), [A])
content((-0.75, 0.5), [B])
content((2.5, 0.75), [Y])
})]
], [
#table(columns: (1fr,)*3,
[0], [0], [0],
[0], [1], [1],
[1], [0], [1],
[1], [1], [1],
)],
[Nicht-Gatter \ (NOT)], [$Y eq overline(A)$], [#align(center + horizon)[
#cetz.canvas({
import cetz.draw: *
rect((0, 0), (1.5, 1.5))
line((-0.5, 0.75), (0, 0.75))
line((1.8, 0.75), (2.2, 0.75))
circle((1.65, 0.75), radius: 0.15)
content((0.75, 1.2), [$1$])
content((-0.75, 0.75), [A])
content((2.5, 0.75), [Y])
})]
], [
#table(columns: (1fr,)*3,
[0], [-], [1],
[1], [-], [0],
)],
[NAND-Gatter \ (NOT AND)], [$Y eq overline(A and B)$], [#align(center + horizon)[
#cetz.canvas({
import cetz.draw: *
rect((0, 0), (1.5, 1.5))
line((-0.5, 1), (0, 1))
line((-0.5, 0.5), (0, 0.5))
line((1.8, 0.75), (2.2, 0.75))
circle((1.65, 0.75), radius: 0.15)
content((0.75, 1.2), [&])
content((-0.75, 1), [A])
content((-0.75, 0.5), [B])
content((2.5, 0.75), [Y])
})]
], [
#table(columns: (1fr,)*3,
[0], [0], [1],
[0], [1], [1],
[1], [0], [1],
[1], [1], [0],
)],
[NOR-Gatter \ (NOT OR)], [$Y eq overline(A or B)$], [#align(center + horizon)[
#cetz.canvas({
import cetz.draw: *
rect((0, 0), (1.5, 1.5))
line((-0.5, 1), (0, 1))
line((-0.5, 0.5), (0, 0.5))
line((1.5, 0.75), (2.2, 0.75))
content((0.75, 1.2), [$gt.eq 1$])
content((-0.75, 1), [A])
content((-0.75, 0.5), [B])
content((2.5, 0.75), [Y])
})]
], [
#table(columns: (1fr,)*3,
[0], [0], [1],
[0], [1], [0],
[1], [0], [0],
[1], [1], [0],
)],
[XOR-Gatter \ (EXCLUSIVE OR)], [$Y eq A underline(or) B$], [#align(center + horizon)[
#cetz.canvas({
import cetz.draw: *
rect((0, 0), (1.5, 1.5))
line((-0.5, 1), (0, 1))
line((-0.5, 0.5), (0, 0.5))
line((1.5, 0.75), (2.2, 0.75))
content((0.75, 1.2), [$eq 1$])
content((-0.75, 1), [A])
content((-0.75, 0.5), [B])
content((2.5, 0.75), [Y])
})]
], [
#table(columns: (1fr,)*3,
[0], [0], [0],
[0], [1], [1],
[1], [0], [1],
[1], [1], [0],
)],
[XNOR-Gatter \ (EXCLUSIVE NOT OR)], [$Y eq overline(A underline(or) B)$], [#align(center + horizon)[
#cetz.canvas({
import cetz.draw: *
rect((0, 0), (1.5, 1.5))
line((-0.5, 1), (0, 1))
line((-0.5, 0.5), (0, 0.5))
line((1.8, 0.75), (2.2, 0.75))
circle((1.65, 0.75), radius: 0.15)
content((0.75, 1.2), [$eq 1$])
content((-0.75, 1), [A])
content((-0.75, 0.5), [B])
content((2.5, 0.75), [Y])
})]
], [
#table(columns: (1fr,)*3,
[0], [0], [1],
[0], [1], [0],
[1], [0], [0],
[1], [1], [1],
)],
)
==== Totem Pole
#grid(columns: (1fr, 1fr), gutter: 10pt,
[#image("../img/elektronik_2/7400_Circuit.svg")],
[#image("../img/elektronik_2/74LS00_Circuit.svg")],
)
// Sind beide Eingege auf Hight so schaltet die Schaltung durch und der Ausgang wird gegen GND gezogen. Ist mindestens ein Ausgang auf LOW so beginnt der obere Transistor zu leiten und der Ausgang wir Hight. \
// Die Diode ist dafür da um zu verhindern das der Transistor sich öffnen kann und es zu einem Kurzschluss kommt! \
// Zudem dürfen die Ausgänge von TTL schaltungen niemals zusammen geschaltet werden sondern müssen durch eine Open Collector schaltung gehängt werden.
- *Logik*: Sind beide Eingänge auf High, schaltet der untere Transistor durch und zieht den Ausgang gegen GND (Low). Ist mindestens ein Eingang auf Low, sperrt der untere Transistor, der obere beginnt zu leiten und zieht den Ausgang auf High.
- *Schutz*: Die Diode verhindert, dass beide Transistoren gleichzeitig leiten, was einen Kurzschluss der Versorgungsspannung verursachen würde.
- *Verbindung*: Totem-Pole-Ausgänge dürfen niemals direkt parallel geschaltet werden. Für solche "Wired-Logic"-Verbindungen müssen stattdessen Open-Collector-Ausgänge verwendet werden.
Rechts ist eine Low-Power-Schottky-TTL \
Diese Schaltung ist die "High-Speed-Variante". Sie ist schneller, braucht weniger Strom (siehst du auch an den höheren Widerstandswerten wie 20k statt 4k) und nutzt den Darlington-Effekt bei V4​, um die Diode einzusparen.
#grid(columns: (1fr, 1fr), gutter: 10pt,
[#image("../img/elektronik_2/OpencollectorV3.png")],
[#image("../img/elektronik_2/electronic-analog-common-e.png", width: 50%)],
)
=== Sinussignale
=== Passive Zweipole
=== Aktive Zweipole (Versorgungen)

109
src/lineare_algebra_2.typ
View File

@ -153,4 +153,113 @@ dots.v, dots.v, dots.down, dots.v;
0, 0, dots.h, 1;
)$
=== Diagonal-Matrix
$D eq mat(delim: "[",
lambda_1, 0, dots.h, 0;
0, lambda_2, dots.h, 0;
dots.v, dots.v, dots.down, dots.v;
0, 0, dots.h, lambda_n;
)$
=== Inverse
$ A dot A^(-1) eq bb(1) $
- $n_V$ Anzahl Spalten
- $n_R$ (Rang) Echte Zeilen welche nicht nur aus Nullen bestehen
- $n_D eq n_V minus n_R$ (Defekt)
#table(columns: (1fr, 1fr), [
$ A eq mat(delim: "[",
1, 3;
2, 6;
) $
- $n_V eq 2$
- $n_R eq 2$
- $n_D eq n_V minus n_R eq 2 - 2 eq 0$ $arrow$ Matrit ist Invertierbar
], [
$ A eq mat(delim: "[",
1, 0, 5;
0, 1, 2;
0, 0, 0;
) $
- $n_V eq 3$
- $n_R eq 2$
- $n_D eq n_V minus n_R eq 2 - 2 eq 1$ $arrow$ Matrit ist *NICHT* Invertierbar
])
==== Reguläre Matrizen (invertierbar)
Invertierbar: $A dot A^(-1) eq bb(1)$ \
Determinante: $det(A) eq.not 0)$ \
Lösbarkeit: Genau eine Lösung
==== Singuläre Matrizen (nicht invertierbar)
Nicht invertierbar: Es gibt keine Matrix, die sie rückgängig machen kann. \
Determinante: Die Determinante ist exakt Null $det(A) eq 0$ \
Lösbarkeit: Ein Gleichungssystem hat entweder gar keine oder unendlich viele Lösungen.
=== Lineare Abbildungen
#table(columns: (1fr, 140pt),
[Identität], [
$ bb(1) eq mat(delim: "[",
1, 0;
0, 1;
) $
], [Streckung um den Faktor a in x Richtung. bzw. b in y Richtung], [
$ A eq mat(delim: "[",
a, 0;
0, b;
) $
], [Punktspiegelung am Ursprung P], [
$ P eq minus bb(1) eq mat(delim: "[",
-1, 0;
0, -1;
) $
], [Projektion auf die x-Achse: $P_x$], [
$ P_x eq mat(delim: "[",
1, 0;
0, 0;
) $
], [Projektion auf die y-Achse: $P_y$], [
$ P_y eq mat(delim: "[",
0, 0;
0, 1;
) $
], [Spiegelung an der x-Achse: $S_x$], [
$ S_x eq mat(delim: "[",
1, 0;
0, -1;
) $
], [Spiegelung an der y-Achse: $S_y$], [
$ S_y eq mat(delim: "[",
-1, 0;
0, 1;
) $
], [Spiegelung an der Geraden y], [
$ S_(x y) eq mat(delim: "[",
0, 1;
1, 0;
) $
], [Drehung um den Ursprung um $pi\/2$ $(90 degree)$], [
$ R(pi \/ 2) eq mat(delim: "[",
0, -1;
1, 0;
) $
], [Drehung um den Ursprung um $-pi\/2$ $(-90 degree)$], [
$ R(-pi \/ 2) eq mat(delim: "[",
0, 1;
-1, 0;
) $
], [Drehung um den Ursprung um $phi$], [
$ R(phi) eq mat(delim: "[",
cos(phi), -sin(phi);
sin(phi), cos(phi);
) $
])
=== Bild und Kern
*woche 4*
=== Orthogonale Matrizen
$ A^(-1) eq A^T $

117
src/physik_2.typ → src/physik_2_elmag.typ
View File

@ -93,120 +93,3 @@ $bold(F)_E eq Q dot E$ #h(20pt) $[E] eq frac(N, C)$
=== B-Feld
$bold(F)_B eq Q dot v crossmark B$ #h(20pt) $[B] eq frac(N, A dot m) eq T "(Tesla)"$
#pagebreak()
#pagebreak()
#pagebreak()
== Schwingungen und Wellen
Periodendauer $T [s]$ \
Frequenz $f eq frac(1, T)$ #h(20pt) $[f] eq "Hz"$
=== Ort-Zeit Funktion
$x(t) eq A dot cos(omega t plus delta)$ \
// Phase $omega t dot delta$ \
// Phasenwinkel $delta$ Wert von $delta$ bei $t eq 0$ \
// Kreisfrequenz $omega$ \
// $x(t) eq x(t plus T)$ wir setzen $delta eq 0$ \
// $A cos(omega t) &eq A dot cos(omega (t plus T)) \
// &eq A dot cos(omega t plus omega T)$ \
// da $cos 2 pi$ periodisch $omega T eq 2 pi$
// $omega eq frac(2 pi, T) eq frac(2 pi, frac(1, f)) eq 2 pi f$ \
// $[omega] eq frac(1, s)$
$omega eq frac(2 pi, T) eq 2 pi f$ #h(20pt) $[omega] eq frac(1, s)$
Geschwindigkeit: \
// $V_x &eq frac(d x (t), d t) \
// &eq - A omega sin(omega t plus delta)$
$V_x eq - A omega sin(omega t plus delta)$
Beschleunigung: \
// $a_x &eq frac(d v_x, d t) eq frac(d^2 x(t), d t^2) \
// &eq - A omega^2 cos(omega t plus delta) \
// &eq minus omega^2 dot x(t)$
$a_x &eq - A omega^2 cos(omega t plus delta) \
&eq minus omega^2 dot x(t)$
// Amplitude von:\
// $v_x : A omega$ \
// $a_x : A omega^2$ \
=== Newtonssches Gesetz (Bewegungsgleichung)
// $m dot a eq sum F$ \
// $m dot a_x eq minus k dot x arrow a_x eq minus frac(k, m) dot x$ \
// oder \
// $m dot a_x eq minus omega^2 dot x dot m$ \
// gleichsetzen der gleichungen: \
// $minus k dot x eq minus m dot omega^2 dot x$ \
// $k eq m dot omega^2$ \
// $omega eq sqrt(frac(k, m))$ \
// mit: \
// $omega eq frac(s pi, T)$ $arrow$ $T eq 2 pi sqrt(frac(m,k))$ \
$omega eq sqrt(frac(k, m))$ \
Dehnung der Feder: \
$k dot Delta l eq m dot g$
// = pp S 21
// $F eq minus k dot x$ \
// $E_"pot" eq minus integral_0^x F d s eq minus integral_0^x (minus k s ) d s eq frac(1, 2) k x^2$ $(c eq 0)$
//
// $E_"pot" (x eq 0) eq 0$ Bei Gleichgewichtslage \
// $E_"pot" (x = A) eq frac(1, 2) k A^2)$ Maximeirt ...
// $E_"ges" &eq E_"pot" plus E_"kin" eq frac(1, 2) k x^2 plus frac(1, 2) m v^2 \
// &eq frac(1, 2) k (A cos(omega t plus delta))^2 plus frac(1, 2) m (minus A omega sin(omega t plus delta))^2 \
// &eq frac(1, 2) k A^2 cos^2(omega t plus delta) plus frac(1, 2) m A^2 omega^2 sin^2(omega t plus delta) \
// &eq frac(1, 2) k A^2 underbrace((cos^2(omega t plus delta) plus sin^2omega t plus delta), eq 1) \
// &eq frac(1, 2) k A^2 $
=== Gleichgewichtslage
$E_"pot" (x) eq 0$ Bei Gleichgewichtslage \
// = pp S 27
// Federkraft von m: \
// $F eq minus k y$ \
// Gewichtskraft von m: \
// $F eq m dot x$ \
// Newtown 2: \
// $m dot a_y eq minus k y plus m dot g$ \
// Einführung neue Koordinate: \
// $y^' eq y minus y_0$
//
//
// $sum f eq -(y^' plus y_0) dot k plus m dot g$ \
// mit $k y_0 eq m dot g$:
// $sum F eq - y^' dot k$
//
// $ y^'(t) eq A dot cos(omega t plus delta) $
// masse m schwingt um Gleichgewichtslage $y_0$

116
src/physik_2_schwing.typ Normal file
View File

@ -0,0 +1,116 @@
#import "@preview/cetz:0.4.1"
== Schwingungen und Wellen
Periodendauer $T [s]$ \
Frequenz $f eq frac(1, T)$ #h(20pt) $[f] eq "Hz"$
=== Ort-Zeit Funktion
$x(t) eq A dot cos(omega t plus delta)$ \
// Phase $omega t dot delta$ \
// Phasenwinkel $delta$ Wert von $delta$ bei $t eq 0$ \
// Kreisfrequenz $omega$ \
// $x(t) eq x(t plus T)$ wir setzen $delta eq 0$ \
// $A cos(omega t) &eq A dot cos(omega (t plus T)) \
// &eq A dot cos(omega t plus omega T)$ \
// da $cos 2 pi$ periodisch $omega T eq 2 pi$
// $omega eq frac(2 pi, T) eq frac(2 pi, frac(1, f)) eq 2 pi f$ \
// $[omega] eq frac(1, s)$
$omega eq frac(2 pi, T) eq 2 pi f$ #h(20pt) $[omega] eq frac(1, s)$
Geschwindigkeit: \
// $V_x &eq frac(d x (t), d t) \
// &eq - A omega sin(omega t plus delta)$
$V_x eq - A omega sin(omega t plus delta)$
Beschleunigung: \
// $a_x &eq frac(d v_x, d t) eq frac(d^2 x(t), d t^2) \
// &eq - A omega^2 cos(omega t plus delta) \
// &eq minus omega^2 dot x(t)$
$a_x &eq - A omega^2 cos(omega t plus delta) \
&eq minus omega^2 dot x(t)$
// Amplitude von:\
// $v_x : A omega$ \
// $a_x : A omega^2$ \
=== Newtonssches Gesetz (Bewegungsgleichung)
// $m dot a eq sum F$ \
// $m dot a_x eq minus k dot x arrow a_x eq minus frac(k, m) dot x$ \
// oder \
// $m dot a_x eq minus omega^2 dot x dot m$ \
// gleichsetzen der gleichungen: \
// $minus k dot x eq minus m dot omega^2 dot x$ \
// $k eq m dot omega^2$ \
// $omega eq sqrt(frac(k, m))$ \
// mit: \
// $omega eq frac(s pi, T)$ $arrow$ $T eq 2 pi sqrt(frac(m,k))$ \
$omega eq sqrt(frac(k, m))$ \
Dehnung der Feder: \
$k dot Delta l eq m dot g$
=== Gleichgewichtslage
$E_"pot" (x) eq 0$ Bei Gleichgewichtslage \
=== Energie
$E_"pot" eq frac(1, 2) k y^2$ \
=== Fadenpendel
#grid(columns: (100pt, 1fr), gutter: 10pt, [
$ T eq 2 pi dot root(, frac(l, g)) $
], [
#cetz.canvas({
import cetz.draw: *
line((0, 0), (6, 0), stroke: (thickness: 3pt))
line((3, 0), (1, -2))
circle((1, -2), radius: 0.5, fill: gray)
content((2.5, -1), [$l$])
content((4.5, -0.5), anchor: "north-west", [
- $l$ Lënge des Pendels
- $g$ Erdbeschleunigung $(9.81m\/s^2)$
])
})
])
=== Physikalisches Pendel
#grid(columns: (100pt, 1fr), gutter: 10pt, [
$ T eq 2 pi dot root(, frac(l, m dot g dot d)) $
], [
#cetz.canvas({
import cetz.draw: *
circle((0, 0), radius: 2, fill: gray)
circle((0, 1), radius: 0.2, fill: white)
circle((0, 0), radius: 0.1, fill: black)
line((0, 0), (0, 1), mark: (symbol: ">"), fill: blue, stroke: blue)
line((0, 0), (2, 0), mark: (symbol: ">"), fill: blue, stroke: blue)
line((2.4, 1.85), (0.2, 1.2), mark: (end: ">"), fill: blue, stroke: blue)
content((-0.3, 0.4), [$d$])
content((1, 0.4), [$r$])
content((0, -1), [$m$])
content((2.5, 2), anchor: "north-west", [Drehpunkt])
content((2.5, 1), anchor: "north-west", [
- d Abstand vom Drehpunkt zum Massemittelpunkt
- r Radius vom Kreis
- m Masse vom Pendel
])
})
])
=== Drehpendel (Torsionspendel)
#grid(columns: (200pt, 1fr), gutter: 10pt, [
$ T eq 2 pi dot root(, frac(I, D*)) $
- $D∗$ Direktionsmoment (Torsionskonstante des Drahtes)
- $I$ Trägheitsmoment des Körpers
], [
#image("../img/schwingungen_und_wellen/Drehpendel.png")
])