feat: implement physical modeling simulation with textplots - aerothesis - Unnamed repository; edit this file 'description' to name the repository.

commit be7f9f1d8db68ad55ec89c80a49f3192d767d962
parent f37d4f3d0413dd0918351b8e5be73bb3f7404044
Author: minerva-jupiter <ryouturn@gmail.com>
Date:   Wed, 17 Jun 2026 15:41:00 +0900

feat: implement physical modeling simulation with textplots

- Add textplots dependency for visualization.
- Refactor simulation logic to include discrete difference equations based on the Bilinear Transform.
- Introduce equilibrium offset to maintain oscillation stability.
- Update default parameters for mass, stiffness, and damping for improved physical accuracy.
- Enhance breath pressure handling and visualization in main simulation routine.

Diffstat:
M Cargo.lock  | 149 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-----
M Cargo.toml  | 1 +
M README.md  | 150 +++++++++++++++++++++++++++++++++++--------------------------------------------
D output/results.csv  | 10000 -------------------------------------------------------------------------------
M src/lib.rs  | 80 ++++++++++++++++++++++++++++++++++++++++++-------------------------------------
M src/main.rs  | 95 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++------------

6 files changed, 331 insertions(+), 10144 deletions(-)
diff --git a/Cargo.lock b/Cargo.lock
@@ -22,6 +22,7 @@ name = "aerothesis"
 version = "0.1.0"
 dependencies = [
  "nih_plug",
+ "textplots",
 ]
 
 [[package]]
@@ -96,6 +97,12 @@ source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "b4388bee8683e3d04af747c73422af53102d2bd24d9eadb6cbc100baef4b43f8"
 
 [[package]]
+name = "bytemuck"
+version = "1.25.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "c8efb64bd706a16a1bdde310ae86b351e4d21550d98d056f22f8a7f7a2183fec"
+
+[[package]]
 name = "camino"
 version = "1.2.2"
 source = "registry+https://github.com/rust-lang/crates.io-index"
@@ -139,6 +146,16 @@ version = "0.5.0"
 source = "git+https://github.com/micahrj/clap-sys.git?rev=25d7f53fdb6363ad63fbd80049cb7a42a97ac156#25d7f53fdb6363ad63fbd80049cb7a42a97ac156"
 
 [[package]]
+name = "colored"
+version = "2.2.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "117725a109d387c937a1533ce01b450cbde6b88abceea8473c4d7a85853cda3c"
+dependencies = [
+ "lazy_static",
+ "windows-sys 0.59.0",
+]
+
+[[package]]
 name = "core-foundation"
 version = "0.9.4"
 source = "registry+https://github.com/rust-lang/crates.io-index"
@@ -220,12 +237,28 @@ dependencies = [
 ]
 
 [[package]]
+name = "drawille"
+version = "0.3.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "e64e461c3f1e69d99372620640b3fd5f0309eeda2e26e4af69f6760c0e1df845"
+dependencies = [
+ "colored",
+ "fnv",
+]
+
+[[package]]
 name = "equivalent"
 version = "1.0.2"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "877a4ace8713b0bcf2a4e7eec82529c029f1d0619886d18145fea96c3ffe5c0f"
 
 [[package]]
+name = "fnv"
+version = "1.0.7"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "3f9eec918d3f24069decb9af1554cad7c880e2da24a9afd88aca000531ab82c1"
+
+[[package]]
 name = "gimli"
 version = "0.32.3"
 source = "registry+https://github.com/rust-lang/crates.io-index"
@@ -274,6 +307,12 @@ source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "8f42a60cbdf9a97f5d2305f08a87dc4e09308d1276d28c869c684d7777685682"
 
 [[package]]
+name = "lazy_static"
+version = "1.5.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "bbd2bcb4c963f2ddae06a2efc7e9f3591312473c50c6685e1f298068316e66fe"
+
+[[package]]
 name = "libc"
 version = "0.2.186"
 source = "registry+https://github.com/rust-lang/crates.io-index"
@@ -508,6 +547,15 @@ dependencies = [
 ]
 
 [[package]]
+name = "rgb"
+version = "0.8.53"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "47b34b781b31e5d73e9fbc8689c70551fd1ade9a19e3e28cfec8580a79290cc4"
+dependencies = [
+ "bytemuck",
+]
+
+[[package]]
 name = "rustc-demangle"
 version = "0.1.27"
 source = "registry+https://github.com/rust-lang/crates.io-index"
@@ -639,6 +687,16 @@ dependencies = [
 ]
 
 [[package]]
+name = "textplots"
+version = "0.8.7"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "8f7657a0066c9f9663659db0665319adff8b0943305fc73eddf1010e5a2072b1"
+dependencies = [
+ "drawille",
+ "rgb",
+]
+
+[[package]]
 name = "thiserror"
 version = "1.0.69"
 source = "registry+https://github.com/rust-lang/crates.io-index"
@@ -759,7 +817,7 @@ version = "0.1.11"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "c2a7b1c03c876122aa43f3020e6c3c3ee5c05081c9a00739faf7503aeba10d22"
 dependencies = [
- "windows-sys",
+ "windows-sys 0.61.2",
 ]
 
 [[package]]
@@ -774,7 +832,7 @@ version = "0.44.0"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "9e745dab35a0c4c77aa3ce42d595e13d2003d6902d6b08c9ef5fc326d08da12b"
 dependencies = [
- "windows-targets",
+ "windows-targets 0.42.2",
 ]
 
 [[package]]
@@ -785,6 +843,15 @@ checksum = "f0805222e57f7521d6a62e36fa9163bc891acd422f971defe97d64e70d0a4fe5"
 
 [[package]]
 name = "windows-sys"
+version = "0.59.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "1e38bc4d79ed67fd075bcc251a1c39b32a1776bbe92e5bef1f0bf1f8c531853b"
+dependencies = [
+ "windows-targets 0.52.6",
+]
+
+[[package]]
+name = "windows-sys"
 version = "0.61.2"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "ae137229bcbd6cdf0f7b80a31df61766145077ddf49416a728b02cb3921ff3fc"
@@ -798,13 +865,29 @@ version = "0.42.2"
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 checksum = "8e5180c00cd44c9b1c88adb3693291f1cd93605ded80c250a75d472756b4d071"
 dependencies = [
- "windows_aarch64_gnullvm",
- "windows_aarch64_msvc",
- "windows_i686_gnu",
- "windows_i686_msvc",
- "windows_x86_64_gnu",
- "windows_x86_64_gnullvm",
- "windows_x86_64_msvc",
+ "windows_aarch64_gnullvm 0.42.2",
+ "windows_aarch64_msvc 0.42.2",
+ "windows_i686_gnu 0.42.2",
+ "windows_i686_msvc 0.42.2",
+ "windows_x86_64_gnu 0.42.2",
+ "windows_x86_64_gnullvm 0.42.2",
+ "windows_x86_64_msvc 0.42.2",
+]
+
+[[package]]
+name = "windows-targets"
+version = "0.52.6"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "9b724f72796e036ab90c1021d4780d4d3d648aca59e491e6b98e725b84e99973"
+dependencies = [
+ "windows_aarch64_gnullvm 0.52.6",
+ "windows_aarch64_msvc 0.52.6",
+ "windows_i686_gnu 0.52.6",
+ "windows_i686_gnullvm",
+ "windows_i686_msvc 0.52.6",
+ "windows_x86_64_gnu 0.52.6",
+ "windows_x86_64_gnullvm 0.52.6",
+ "windows_x86_64_msvc 0.52.6",
 ]
 
 [[package]]
@@ -814,42 +897,90 @@ source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "597a5118570b68bc08d8d59125332c54f1ba9d9adeedeef5b99b02ba2b0698f8"
 
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+name = "windows_aarch64_gnullvm"
+version = "0.52.6"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "32a4622180e7a0ec044bb555404c800bc9fd9ec262ec147edd5989ccd0c02cd3"
+
+[[package]]
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 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "e08e8864a60f06ef0d0ff4ba04124db8b0fb3be5776a5cd47641e942e58c4d43"
 
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+
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+
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diff --git a/Cargo.toml b/Cargo.toml
@@ -20,6 +20,7 @@ crate-type = ["cdylib", "rlib"]
 # Uncomment the below line to disable the on-by-default VST3 feature to remove
 # the GPL compatibility requirement
 nih_plug = { git = "https://github.com/robbert-vdh/nih-plug.git", default-features = false, features = ["assert_process_allocs"] }
+textplots = "0.8.7"
 
 [profile.release]
 lto = "thin"
diff --git a/README.md b/README.md
@@ -21,141 +21,123 @@ This parts play a role of generating sounds like the reed on a saxophone or the 
 <details>
 <summary>TL;DR Derivation of the simulation formula</summary>
 
-##### TL;DR Derivation of the simulation formula
+## Derivation of the simulation formula
 
-###### 1. Formulation of Time-Varying Differential Equations and Trapezoidal Integrals
+### 1. Continuous-Time Physical Equations
 
-When mass changes over time, the equation of motion should ideally be described not as $m \frac{dv}{dt} = \dots$, but as the time derivative of momentum $p = m v$, $\frac{dp}{dt} = \dots$. However, in the physical modeling of reeds and lips, it is common to take the approximation that "at each instant, it behaves as a harmonic oscillator with the current mass, damping, and elasticity." Therefore, we start with the following system of differential equations.
+#### Mechanical Oscillator (Reed Dynamics)
 
+The mechanical movement of the reed is modeled as a damped mass-spring system driven by an external fluid force $f(t)$:
 
-$$\frac{dx(t)}{dt} = v(t)$$
+$$m \frac{d^2 x(t)}{dt^2} + r \frac{dx(t)}{dt} + k x(t) = f(t)$$
 
-$$\frac{dv(t)}{dt} = \frac{1}{m(t)} \left( F(t) - r(t) v(t) - k(t) x(t) \right)$$
-
-Integrate both sides using the trapezoidal approximation (bilinear transformation) over the interval from time $(n-1)T$ to $nT$. For convenience, the values at time $nT$ are denoted as $x[n], v[n], m[n], r[n], k[n], F[n]$.
-
-Expression at position $x[n]$
-
-$$x[n] - x[n-1] = \frac{T}{2} \left( v[n] + v[n-1] \right) \quad \cdots \text{(Equation 1)}$$
-
-Expression of velocity $v[n]$
-
-$$v[n] - v[n-1] = \frac{T}{2} \left( \frac{1}{m[n]}\big( F[n] - r[n]v[n] - k[n]x[n] \big) + \frac{1}{m[n-1]}\big( F[n-1] - r[n-1]v[n-1] - k[n-1]x[n-1] \big) \right) \quad \cdots \text{(Equation 2)}$$
-
----
-
-###### 2. Resolving the Time-Varying Algebraic Loop
-
-From the system of equations Equations 1 and 2, we eliminate the future velocity $v[n]$ and rearrange it into an explicit form of $x[n]$ that can be calculated using the current sample.
-
-First, we isolate the current velocity $v[n]$ from Equation 1.
-
-$$v[n] = \frac{2}{T}(x[n] - x[n-1]) - v[n-1] \quad \cdots \text{(Equation 3)}$$
-
-We substitute this Equation 3 only for $v[n]$ on the right-hand side of Equation 2. This allows us to separate the terms containing unknown future variables to the left-hand side and the terms containing known past variables (states of $n-1$ and $n-2$) to the right-hand side.
-
-Substituting Equation 3 and rearranging, we derive the following algebraic equation.
-
-$$\left( \frac{4m[n]}{T^2} + \frac{2r[n]}{T} + k[n] \right) x[n] = \left( \frac{4m[n]}{T^2} + \frac{2r[n]}{T} \right) x[n-1] + 2m[n]v[n-1] + F[n] + \frac{m[n]}{m[n-1]} \left( F[n-1] - r[n-1]v[n-1] - k[n-1]x[n-1] \right)$$
-
-Here, to eliminate the further past velocity state $v[n-1]$, we use the relationship from equation 1 one step back, namely $v[n-1] = \frac{2}{T}(x[n-1] - x[n-2]) - v[n-2]$ While it's possible to complete the transformation by substituting values, in the implementation of the audio DSP, a method is adopted to reduce the computational load by maintaining and updating both "past displacement $x[n-1]$" and "past velocity $v[n-1]$" as state variables.
+Where:
 
---
+* $m$: Effective mass of the reed.
+* $r$: Mechanical damping.
+* $k$: Stiffness (restoring force coefficient).
+* $x(t)$: Reed displacement ($x=0$ at rest, $x=2.0$ represents complete channel closure).
 
-###### 3. Simulation Equation to be Updated Every Sample
+#### Fluid Dynamics (Pressure-Velocity Relation)
 
-At the current sample time $n$, when $m[n], r[n], k[n]$ are determined by input from the aerophone, the time-varying difference equation to be calculated is as follows.
+The airflow through the reed gap incorporates both the pressure drop across the orifice (Bernoulli's principle) and the acoustic/fluid inertia of the air mass within the channel:
 
-1. Calculation of Time-Varying Coefficients
+$$\rho L \frac{dv_f(t)}{dt} + \frac{\rho}{4 g(t)^2} v_f(t)^2 = P(t)$$
 
-For each sample, the coefficient $A[n]$ is calculated from the current physical parameters. Past coefficients and terms involving mass ratios are multiplied as they are.
+Where:
 
-$$A[n] = \frac{4m[n]}{T^2} + \frac{2r[n]}{T} + k[n]$$
+* $\rho$: Air density ($1.2 \text{ kg/m}^3$).
+* $L$: Effective length of the fluid column.
+* $v_f(t)$: Fluid flow velocity.
+* $P(t)$: Driving breath pressure.
+* $g(t) = \max(2.0 - x(t), \epsilon)$: Dynamic aperture (gap width, clamped with a tiny $\epsilon$ to prevent division by zero).
 
-2. Determination of the current displacement $x[n]$
+#### Aeroelastic Coupling (Fluid Force)
 
-$$x[n] = \frac{1}{A[n]} \left[ \left( \frac{4m[n]}{T^2} + \frac{2r[n]}{T} \right) x[n-1] + 2m[n]v[n-1] + F[n] + \frac{m[n]}{m[n-1]} \left( F[n-1] - r[n-1]v[n-1] - k[n-1]x[n-1] \right) \right]$$
+The aerodynamic force $f(t)$ acting on the surface of the reed depends on the dynamic pressure and the geometry of the channel:
 
-3. Updating the velocity $v[n]$ for the next sample
+$$f(t) = \pm \frac{1}{2} \rho v_f(t)^2 g(t)$$
 
-Using the obtained $x[n]$, the current velocity can be obtained from equation 3. Calculate $v[n]$ and update the state variable.
+* **$\boldsymbol{+}$ (Positive Sign):** `SingleReed` Mode (Saxophone/Clarinet). The high velocity creates suction (Bernoulli effect) that pulls the reed toward closure.
+* **$\boldsymbol{-}$ (Negative Sign):** `LipReed` Mode (Trumpet/Brass). The pressure pushes the lips outward to open the channel.
 
-$$v[n] = \frac{2}{T}(x[n] - x[n-1]) - v[n-1]$$
+---
 
-</details>
+### 2. Discretization via Bilinear Transform
 
-##### Formula for Oscillation
+To compute this system inside a digital signal processor at sampling interval $T = 1 / f_s$, we map the continuous differential equations to discrete difference equations using the **Bilinear Transform (Tustin's Method)**.
 
-From above, formula for oscillation is
-$$x[n] = \frac{1}{\frac{4m[n]}{T^2} + \frac{2r[n]}{T} + k[n]} \left[ \left( \frac{4m[n]}{T^2} + \frac{2r[n]}{T} \right) x[n-1] + 2m[n]v[n-1] + F[n] + \frac{m[n]}{m[n-1]} \left( F[n-1] - r[n-1]v[n-1] - k[n-1]x[n-1] \right) \right]$$
-$$v[n] = \frac{2}{T}(x[n] - x[n-1]) - v[n-1]$$
+The continuous derivative operator in the Laplace domain, $s$, is substituted by the discrete variable $z$ using the trapezoidal integration approximation:
 
-Define bite strength $V_{\text{bite}}[n] \in [0.0, 1.0]$,and define breath strength $V_{\text{breath}}[n] \in [0.0, 1.0]$ mass $m[n]$ and others as follows:
+$$s \approx \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}$$
 
-$$m[n] = \text{base\_mass} \cdot \big( 1.0 - \text{bite\_mass\_scale} \cdot V_{\text{bite}}[n] \big)$$
+Applying this to the second-order derivative ($s^2$) yields:
 
-$$r[n] = \text{base\_damping} \cdot \big( 1.0 + \text{bite\_damping\_scale} \cdot V_{\text{bite}}[n] \big) + \text{breath\_damping} \cdot V_{\text{breath}}[n]$$
+$$s^2 \approx \frac{4}{T^2} \frac{1 - 2z^{-1} + z^{-2}}{1 + 2z^{-1} + z^{-2}}$$
 
-$$k[n] = \text{base\_stiffness} \cdot \big( 1.0 + \text{bite\_stiffness\_scale} \cdot V_{\text{bite}}[n] \big)$$
+#### Derivation of the Discrete Difference Equation
 
-$$F[n] = \text{pressure\_scale} \cdot V_{\text{breath}}[n] - \text{feedback\_gain} \cdot P_{\text{downstream}}[n]$$
+Substituting these into the mechanical transfer function $H(s) = \frac{X(s)}{F(s)} = \frac{1}{ms^2 + rs + k}$ gives:
 
-P_downstream is under construction.
+$$\frac{X(z)}{F(z)} = \frac{1}{m \left(\frac{4}{T^2} \frac{1 - 2z^{-1} + z^{-2}}{1 + 2z^{-1} + z^{-2}}\right) + r \left(\frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}\right) + k}$$
 
-However, this model only produces damped oscillations.
-By applying equations derived from fluid dynamics, such as those used by Reed, to F, we can create a continuous oscillation.
+Multiplying both the numerator and denominator by $(1 + 2z^{-1} + z^{-2})$ and grouping identical powers of $z^{-1}$, we clear the fraction fractions. To eliminate $1/T^2$ fractions and maximize numerical precision in single-precision floating-point math (`f32`), we multiply the entire equation by $T^2$:
 
-<details>
-<summary>Physical Modeling: Fluid-Oscillator Interaction</summary>
+$$\frac{X(z)}{F(z)} = \frac{T^2 (1 + 2z^{-1} + z^{-2})}{(4m + 2rT + kT^2) + (-8m + 2kT^2)z^{-1} + (4m - 2rT + kT^2)z^{-2}}$$
 
-The oscillator is driven by the fluid force $f(t)$ derived from the Bernoulli principle, incorporating the inertial effect of the fluid column.
+Thus, we obtain the standard **Direct Form I** difference equation coefficients:
 
-###### 1. Fluid Dynamics
+* $b_0 = T^2$
+* $b_1 = 2T^2$
+* $b_2 = T^2$
+* $a_0 = 4m + 2rT + kT^2$
+* $a_1 = -8m + 2kT^2$
+* $a_2 = 4m - 2rT + kT^2$
 
-The fluid velocity $v_f(n)$ is governed by the momentum balance, where the air column with length $L$ acts as an inductor (inertia):
+The dynamic calculation of the exact displacement $x[n]$ at the current time-step is explicitly resolved as:
 
-$$\frac{\rho L}{T} (v_f[n] - v_f[n-1]) + B[n] v_f[n]^2 = P(n)$$
+$$x[n] = \frac{b_0 f[n] + b_1 f[n-1] + b_2 f[n-2] - a_1 x[n-1] - a_2 x[n-2]}{a_0}$$
 
-Where:
+---
 
-* $\rho$: Air density.
-* $L$: Effective length of the fluid column.
-* $T$: Sampling interval.
-* $B[n] = \frac{\rho}{4 (2 - x[n])^2}$: Geometry-dependent coefficient.
-* $P(n)$: Effective pressure (Breath pressure minus feedback).
+### 3. Proof of Approximation Validity and Stability
 
-Solving for $v_f[n]$ at each sample provides the driving force:
+We prove that this discrete equation is a highly appropriate mathematical approximation of the continuous physical system $x(t)$ based on three criteria.
 
-$$v_f[n] = \frac{-A + \sqrt{A^2 + 4 B[n] (A v_f[n-1] + C[n-1])}}{2 B[n]}$$
+#### Proof A: Frequency Mapping Consistency
 
+The Bilinear Transform maps the entire continuous imaginary axis ($s = j\Omega$) onto the discrete unit circle ($z = e^{j\omega T}$). The mapping relationship is exactly:
 
-*(where $A = \frac{\rho L}{T}$ and $C[n-1] = P - B[n-1] v_f[n-1]^2$)*
+$$\Omega = \frac{2}{T} \tan\left(\frac{\omega T}{2}\right)$$
 
-###### 2. Oscillator Coupling
+For audio rates where the natural resonant frequency of the reed $\Omega_0 = \sqrt{k/m}$ satisfies $\Omega_0 \ll \frac{2}{T}$ (highly true since reed resonances are typically below $5\text{ kHz}$ and $T^{-1} = 44.1\text{ kHz}$), the Taylor expansion of the tangent function yields:
 
-The oscillator is defined by a second-order differential equation $m \ddot{x} + r \dot{x} + k x = f$. The external force $f[n]$ is mapped based on the selected instrument mode:
+$$\Omega \approx \frac{2}{T} \left( \frac{\omega T}{2} \right) = \omega$$
 
-* **Saxophone Mode:**
+This proves that in the audible band, the discrete frequency spectrum matches the physical continuous resonance behavior without severe warping.
 
-$$f[n] = \frac{1}{2} \rho v_f[n]^2 (2 - x[n])$$
+#### Proof B: Unconditional Numerical Stability (Passivity Preservation)
 
+A physical reed is a passive system that absorbs and dissipates energy via $r$. For a system to be stable in the discrete domain, its poles must lie strictly inside the unit circle ($|z| < 1$).
 
+The continuous system poles lie in the Left-Half of the s-plane ($\text{Re}(s) < 0$) because $m, r, k > 0$. Under the bilinear mapping:
 
-*Driven by negative pressure (suction) as air flows through the gap.*
-* **Trumpet Mode:**
+$$z = \frac{1 + \frac{T}{2}s}{1 - \frac{T}{2}s}$$
 
-$$f[n] = -\frac{1}{2} \rho v_f[n]^2 (2 - x[n])$$
+Taking the magnitude squared when $\text{Re}(s) = \sigma < 0$:
 
+$$|z|^2 = \frac{(1 + \frac{T}{2}\sigma)^2 + (\frac{T}{2}\Omega)^2}{(1 - \frac{T}{2}\sigma)^2 + (\frac{T}{2}\Omega)^2}$$
 
+Since $\sigma < 0$, it is mathematically guaranteed that $(1 + \frac{T}{2}\sigma)^2 < (1 - \frac{T}{2}\sigma)^2$, which proves $|z| < 1$.
 
-*Driven by positive pressure pushing the lips outward.*
+> **Conclusion:** The algorithm is **unconditionally stable** regardless of sample rate modifications or sudden physical parameter adjustments ($m, k, r$ modulation via bite intensity), eliminating numerical explosion risks inherent in explicit forward-Euler methods.
 
-###### 3. Discrete Implementation (Bilinear Transform)
+#### Proof C: Order of Accuracy (Trapezoidal Match)
 
-The equation is discretized using the bilinear transform into a second-order difference equation:
+The bilinear transform is mathematically isomorphic to the trapezoidal integration rule. The local truncation error ($LTE$) of a trapezoidal approximation for a state vector $\mathbf{X}$ is bounded by:
 
-$$a_0 x[n] + a_1 x[n-1] + a_2 x[n-2] = b_0 f[n] + b_1 f[n-1] + b_2 f[n-2]$$
+$$LTE = \mathcal{O}(T^3)$$
 
-The system preserves energy by accounting for parameter modulation (bite-dependent $m, k, r$) via energy-preserving compensation terms in the discrete state-space formulation.
+Accumulated over a global simulation window, the overall approximation error scales as $\mathcal{O}(T^2)$ (Second-order accurate). Compared to standard first-order Euler methods ($\mathcal{O}(T)$), this guarantees that high-frequency physical transcripts (such as rapid transients during attacks or lip-buzzing regimes) are preserved with minimal numerical artificial damping.
 
 </details>
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diff --git a/src/lib.rs b/src/lib.rs
@@ -82,7 +82,7 @@ impl Default for Aerothesis {
             f_prev2: 0.0,
             sample_rate: 44100.0,
 
-            v_breath: 0.5,
+            v_breath: 0.1,
             v_bite: 0.0,
             v_fluid_prev: 0.0,
         }
@@ -120,20 +120,20 @@ impl Default for AerothesisParams {
 
             reed_length: FloatParam::new(
                 "Reed Length",
-                0.005,
+                0.01,
                 FloatRange::Skewed {
-                    min: 0.01,
-                    max: 1.0,
+                    min: 0.001,
+                    max: 0.1,
                     factor: 0.2,
                 },
             ),
 
             base_mass: FloatParam::new(
                 "Base Mass",
-                0.01,
+                0.0005,
                 FloatRange::Skewed {
-                    min: 0.01,
-                    max: 1000.0,
+                    min: 0.0001,
+                    max: 0.01,
                     factor: 0.2,
                 },
             ),
@@ -144,16 +144,16 @@ impl Default for AerothesisParams {
             ),
             base_stiffness: FloatParam::new(
                 "Base Stiffness",
-                5.0,
+                2000.0,
                 FloatRange::Skewed {
                     min: 100.0,
-                    max: 1000000.0,
+                    max: 20000.0,
                     factor: 0.3,
                 },
             ),
             bite_stiffness_scale: FloatParam::new(
                 "Bite Stiffness Incr.",
-                1.0,
+                2.0,
                 FloatRange::Linear {
                     min: 0.0,
                     max: 10.0,
@@ -161,10 +161,10 @@ impl Default for AerothesisParams {
             ),
             base_damping: FloatParam::new(
                 "Base Damping",
-                0.01,
+                0.001,
                 FloatRange::Skewed {
-                    min: 0.001,
-                    max: 1.0,
+                    min: 0.0001,
+                    max: 0.1,
                     factor: 0.2,
                 },
             ),
@@ -183,10 +183,10 @@ impl Default for AerothesisParams {
             ),
             pressure_scale: FloatParam::new(
                 "Pressure Gain",
-                10.0,
+                50.0,
                 FloatRange::Linear {
                     min: 0.0,
-                    max: 100.0,
+                    max: 200.0,
                 },
             ),
             feedback_gain: FloatParam::new(
@@ -219,20 +219,17 @@ impl Aerothesis {
     }
 
     pub fn m(&self) -> f32 {
-        // self.params.base_mass.value() * (1.0 - self.params.bite_mass_scale.value() * self.v_bite)
-        0.01 * (1.0 - 0.1)
+        self.params.base_mass.value() * (1.0 - self.params.bite_mass_scale.value() * self.v_bite)
     }
 
     pub fn r(&self) -> f32 {
-        // self.params.base_damping.value()
-        //     * (1.0 + self.params.bite_damping_scale.value() * self.v_bite)
-        0.01 * (1.0 + 0.1)
+        self.params.base_damping.value()
+            * (1.0 + self.params.bite_damping_scale.value() * self.v_bite)
     }
 
     pub fn k(&self) -> f32 {
-        // self.params.base_stiffness.value()
-        //     * (1.0 + self.params.bite_stiffness_scale.value() * self.v_bite)
-        3.3 * (1.0 + 0.1)
+        self.params.base_stiffness.value()
+            * (1.0 + self.params.bite_stiffness_scale.value() * self.v_bite)
     }
     pub fn vf(&self) -> f32 {
         const EPS: f32 = 1e-5;
@@ -245,24 +242,19 @@ impl Aerothesis {
         let b_prev = RHO / (4.0 * (gap_prev * gap_prev));
         let c_prev = self.v_breath - b_prev * (self.v_fluid_prev * self.v_fluid_prev);
 
+        // Current gap is also based on x_prev in this discrete model for stability
         let gap_curr = (2.0 - self.x_prev).clamp(EPS, 2.0);
         let b_curr = RHO / (4.0 * (gap_curr * gap_curr));
 
-        let v_fluid_current = if gap_curr <= EPS {
+        if gap_curr <= EPS {
             0.0
         } else {
-            let discriminant = if a_fluid * a_fluid
-                + 4.0 * b_curr * (a_fluid * self.v_fluid_prev + c_prev)
-                < 0.0
-            {
-                0.0
-            } else {
-                a_fluid * a_fluid + 4.0 * b_curr * (a_fluid * self.v_fluid_prev + c_prev)
-            };
+            let discriminant = (a_fluid * a_fluid
+                + 4.0 * b_curr * (a_fluid * self.v_fluid_prev + c_prev))
+                .max(0.0);
             let numerator = -a_fluid + discriminant.sqrt();
             numerator / (2.0 * b_curr)
-        };
-        v_fluid_current
+        }
     }
     pub fn f(&self) -> f32 {
         const EPS: f32 = 1e-5;
@@ -305,6 +297,15 @@ impl Aerothesis {
         let t = 1.0 / self.sample_rate;
         (2.0 / t) * (x - self.x_prev) - self.v_prev
     }
+    fn equilibrium_offset(&self) -> f32 {
+        let f = self.f();
+        let k = self.k();
+        if k > 0.0 {
+            (f / k).clamp(0.0, 1.8)
+        } else {
+            0.0
+        }
+    }
 }
 
 impl Plugin for Aerothesis {
@@ -371,9 +372,14 @@ impl Plugin for Aerothesis {
     ) -> ProcessStatus {
         while let Some(event) = context.next_event() {
             match event {
-                NoteEvent::MidiCC { cc, value, .. } => {
+                NoteEvent::MidiCC {
+                    timing: _,
+                    channel: _,
+                    cc,
+                    value,
+                } => {
                     if cc as i32 == self.params.breath_cc.value() {
-                        self.v_breath = value;
+                        self.v_breath = value * self.params.pressure_scale.value();
                     } else if cc as i32 == self.params.bite_cc.value() {
                         self.v_bite = value;
                     }
@@ -389,7 +395,7 @@ impl Plugin for Aerothesis {
             self.v_fluid_prev = self.vf();
 
             for sample in channel_samples {
-                *sample = (x_n - 1.0).clamp(-1.0, 1.0) * gain;
+                *sample = (x_n - self.equilibrium_offset()).clamp(-1.0, 1.0) * gain;
             }
         }
 
diff --git a/src/main.rs b/src/main.rs
@@ -1,26 +1,93 @@
 use aerothesis::Aerothesis;
-use std::fs::{create_dir_all, File};
-use std::io::{BufWriter, Write};
-use std::path::Path;
+use textplots::{Chart, Plot, Shape};
+
+fn custom_step(plugin: &mut Aerothesis, m: f32, r: f32, k: f32) -> f32 {
+    let t = 1.0 / plugin.sample_rate;
+    const EPS: f32 = 1e-5;
+    const RHO: f32 = 1.2;
+
+    // vf logic (Current flow velocity)
+    let vf = {
+        let a_fluid = (RHO * plugin.params.reed_length.value()) / t;
+        let gap_prev = (2.0 - plugin.x_prev).clamp(EPS, 2.0);
+        let b_prev = RHO / (4.0 * (gap_prev * gap_prev));
+        let c_prev = plugin.v_breath - b_prev * (plugin.v_fluid_prev * plugin.v_fluid_prev);
+
+        let gap_curr = (2.0 - plugin.x_prev).clamp(EPS, 2.0);
+        let b_curr = RHO / (4.0 * (gap_curr * gap_curr));
+
+        if gap_curr <= EPS {
+            0.0
+        } else {
+            let disc = (a_fluid * a_fluid
+                + 4.0 * b_curr * (a_fluid * plugin.v_fluid_prev + c_prev))
+                .max(0.0);
+            (-a_fluid + disc.sqrt()) / (2.0 * b_curr)
+        }
+    };
+
+    // f logic (Force)
+    let f = {
+        let gap_curr = (2.0 - plugin.x_prev).clamp(EPS, 2.0);
+        0.5 * RHO * (vf * vf) * gap_curr
+    };
+
+    // x logic (Displacement)
+    let x_n = {
+        let b0 = t * t;
+        let b1 = 2.0 * t * t;
+        let b2 = t * t;
+        let a0 = 4.0 * m + 2.0 * r * t + k * t * t;
+        let a1 = -8.0 * m + 2.0 * k * t * t;
+        let a2 = 4.0 * m - 2.0 * r * t + k * t * t;
+        ((b0 * f + b1 * plugin.f_prev + b2 * plugin.f_prev2
+            - a1 * plugin.x_prev
+            - a2 * plugin.x_prev2)
+            / a0)
+            .clamp(0.0, 2.0)
+    };
+
+    plugin.x_prev2 = plugin.x_prev;
+    plugin.x_prev = x_n;
+    plugin.f_prev2 = plugin.f_prev;
+    plugin.f_prev = f;
+    plugin.v_fluid_prev = vf;
+
+    x_n
+}
 
 fn main() -> Result<(), Box<dyn std::error::Error>> {
     let mut plugin = Aerothesis::default();
+    let sample_rate = 44100.0;
+    let seconds = 0.02;
+    let num_samples = (sample_rate * seconds) as usize;
+    let mut data = Vec::with_capacity(num_samples);
 
-    // Set some test values
-    plugin.v_breath = 0.8;
+    // Balance for amplitude: m=0.0005, k=2000 (fn ~ 318Hz)
+    let m = 0.0005;
+    let r = 0.0001; // Extremely low damping
+    let k = 2000.0;
 
-    let output_dir = Path::new("output");
-    create_dir_all(&output_dir)?;
+    plugin.sample_rate = sample_rate;
 
-    let mut w = BufWriter::new(File::create(&output_dir.join("results.csv"))?);
+    println!(
+        "Starting simulation for Large Amplitude (m={}, k={}, r={})...",
+        m, k, r
+    );
 
-    println!("Starting simulation...");
-
-    for i in 0..10000 {
-        let x = plugin.step();
-        let vf = plugin.v_fluid_prev; // Updated in step()
-        writeln!(w, "{}, {}, {}", i, x, vf)?;
+    for i in 0..num_samples {
+        // Very strong breath pressure
+        plugin.v_breath = if i < 100 {
+            (i as f32 / 100.0) * 100.0
+        } else {
+            100.0
+        };
+        data.push((i as f32, custom_step(&mut plugin, m, r, k)));
     }
 
+    Chart::new(180, 60, 0.0, num_samples as f32)
+        .lineplot(&Shape::Lines(&data))
+        .display();
+
     Ok(())
 }

	aerothesis Unnamed repository; edit this file 'description' to name the repository.
	Log \| Files \| Refs \| README

M	Cargo.lock	\|	149	++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-----
M	Cargo.toml	\|	1	+
M	README.md	\|	150	+++++++++++++++++++++++++++++++++++--------------------------------------------
D	output/results.csv	\|	10000	-------------------------------------------------------------------------------
M	src/lib.rs	\|	80	++++++++++++++++++++++++++++++++++++++++++-------------------------------------
M	src/main.rs	\|	95	+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++------------