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<div id="header">
<h1>Functional Mock-up Interface Specification</h1>
<div class="details">
<span id="revnumber">version aa3ab40,</span>
<span id="revnumber">version 906e06c,</span>
<span id="revdate">2024-11-12</span>
</div>
<div id="toc" class="toc2">
Expand Down Expand Up @@ -5730,7 +5730,12 @@ <h4 id="_physical_units"><a class="link" href="#_physical_units">2.4.3. Physical
</div>
<div class="stemblock">
<div class="content">
\[v_{\mathit{base}} = \texttt{factor} * v_{\mathit{unit}} + \texttt{if relativeQuantity then 0 else offset}\]
\[v_{\mathit{base}} = \texttt{factor} \cdot v_{\mathit{unit}} + \left\{
\begin{array}{ll}
0 &amp; \texttt{if relativeQuantity = true} \\
\texttt{offset} &amp; \texttt{if relativeQuantity = false}
\end{array}
\right.\]
</div>
</div>
<div class="paragraph">
Expand All @@ -5741,15 +5746,15 @@ <h4 id="_physical_units"><a class="link" href="#_physical_units">2.4.3. Physical
</div>
<div class="stemblock">
<div class="content">
\[{p_{\mathit{Pa}} = 10^5 p_{\mathit{bar}}}\]
\[{p_{\mathit{Pa}} = 10^5 \cdot p_{\mathit{bar}}}\]
</div>
</div>
<div class="paragraph">
<p><em>and therefore, <code>factor = 1.0e5</code> and <code>offset = 0.0</code>.</em></p>
</div>
<div class="paragraph">
<p><em>In the following table several unit examples are given.</em>
<em>Note that if in column <code>exponents</code> the definition</em> \(\frac{kg \cdot m^2}{s^2}\) <em>is present, then the attributes of <code>&lt;BaseUnit&gt;</code> are <code>kg=1, m=2, s=-2</code>.</em></p>
<em>Note that if in column <code>exponents</code> the definition</em> \(\mathrm{kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-2}\) <em>is present, then the attributes of <code>&lt;BaseUnit&gt;</code> are <code>kg=1, m=2, s=-2</code>.</em></p>
</div>
<table id="table-unit-examples" class="tableblock frame-all grid-all stretch">
<caption class="title">Table 14. Unit examples.</caption>
Expand All @@ -5774,86 +5779,86 @@ <h4 id="_physical_units"><a class="link" href="#_physical_units">2.4.3. Physical
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Torque</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>N.m</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\({kg \cdot m^2 / s^2}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>1.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-2}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(1.0\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Energy</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>J</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\({kg \cdot m^2 / s^2}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>1.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-2}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(1.0\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Pressure</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>bar</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\({\frac{kg}{m \cdot s^2}}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>1.0e5</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{kg} \cdot \mathrm{m}^{-1} \cdot \mathrm{s}^{-2}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(1.0 \cdot 10^5\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Angle</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>deg</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>rad</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.01745329251994330 (= pi/180)</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{rad}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.01745329251994330 \ \left(= \frac{\pi}{180}\right)\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Angular velocity</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>rad/s</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>rad/s</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>1.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{rad} \cdot \mathrm{s}^{-1}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(1.0\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Angular velocity</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>rpm</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>rad/s</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.1047197551196598 (= 2*pi/60)</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{rad} \cdot \mathrm{s}^{-1}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.1047197551196598 \ \left(= 2 \cdot \frac{\pi}{60}\right)\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Frequency</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>Hz</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>rad/s</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>6.283185307179586 (= 2*pi)</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{rad} \cdot \mathrm{s}^{-1}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(6.283185307179586 \ \left(= 2 \cdot \pi\right)\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Temperature</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>&#176;F</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>K</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.5555555555555556 (= 5/9)</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>255.3722222222222 (= 273.15-32*5/9)</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{K}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.5555555555555556 \ \left(= \frac{5}{9}\right)\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(255.3722222222222 \ \left(= 273.15 - 32 \cdot \frac{5}{9}\right)\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Percent by length</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>%/m</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>1/m</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.01</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{m}^{-1}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.01\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Parts per million</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>ppm</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>1</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>1.0e-6</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(1\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(1.0 \cdot 10^{-6}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Length</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>km</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>m</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>1000</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{m}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(1000\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>Length</em></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>yd</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>m</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.9144</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><code>0.0</code></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(\mathrm{m}\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.9144\)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">\(0.0\)</p></td>
</tr>
</tbody>
</table>
Expand All @@ -5875,22 +5880,44 @@ <h4 id="_physical_units"><a class="link" href="#_physical_units">2.4.3. Physical
<em>If <code>factor</code> and <code>offset</code> are also identical, again the connection equation <code>v2 = v1</code> holds.</em>
<em>If <code>factor</code> and <code>offset</code> are not identical, the tool may either trigger an error or, if supported, perform a conversion; in other words, use the connection equation (in this case the <code>relativeQuantity</code> of the <code>&lt;TypeDefinition&gt;</code>, see below, has to be taken into account in order to determine whether <code>offset</code> shall or shall not be utilized):</em></p>
</div>
</dd>
</dl>
</div>
<div class="stemblock">
<div class="content">
\[\texttt{factor(v1)} \cdot \texttt{v1} + \left\{
\begin{array}{ll}
0 &amp; \texttt{if relativeQuantity(v1) = true} \\
\texttt{offset(v1)} &amp; \texttt{if relativeQuantity(v1) = false}
\end{array}
\right.
=
\texttt{factor(v2)} \cdot \texttt{v2} + \left\{
\begin{array}{ll}
0 &amp; \texttt{if relativeQuantity(v2) = true} \\
\texttt{offset(v2)} &amp; \texttt{if relativeQuantity(v2) = false}
\end{array}
\right.\]
</div>
</div>
<div class="paragraph">
<p><code>factor(v1) * v1 + (if relativeQuantity(v1) then 0 else offset(v1)) = factor(v2) * v2 + (if relativeQuantity(v2) then 0 else offset(v2))</code><br>
<em>where</em> <code>relativeQuantity(v1) = relativeQuantity(v2)</code> <em>is required</em>.</p>
<p><em>where</em> <code>relativeQuantity(v1) = relativeQuantity(v2)</code> <em>is required</em>.</p>
</div>
<div class="paragraph">
<p><em>As a result, wrong connections can be detected (for example, connecting a force with an angle-based variable would trigger an error) and conversions between, say, US and SI units can be either automatically performed or, if not supported, an error is triggered as well.</em></p>
</div>
<div class="paragraph">
<p><em>This approach is not satisfactory for variables belonging to different quantities that have, however, the same <code>&lt;BaseUnit&gt;</code>, such as quantities <code>Energy</code> and <code>Torque</code>, or <code>AngularVelocity</code> and <code>Frequency</code>.</em>
<p>+
<em>This approach is not satisfactory for variables belonging to different quantities that have, however, the same <code>&lt;BaseUnit&gt;</code>, such as quantities <code>Energy</code> and <code>Torque</code>, or <code>AngularVelocity</code> and <code>Frequency</code>.</em>
<em>To handle such cases, quantity definitions have to be taken into account (see <code>&lt;TypeDefinitions&gt;</code>) and quantity names need to be standardized.</em></p>
</div>
<div class="paragraph">
<p><em>This approach allows a general treatment of units, without being forced to standardize the grammar and allowed values for units (for example, in FMI 1.0, a unit could be defined as <code>N.m</code> in one FMU and as <code>N*m</code> in another FMU, and a tool would have to reject a connection, since the units are not identical.</em>
<p>+
<em>This approach allows a general treatment of units, without being forced to standardize the grammar and allowed values for units (for example, in FMI 1.0, a unit could be defined as <code>N.m</code> in one FMU and as <code>N*m</code> in another FMU, and a tool would have to reject a connection, since the units are not identical.</em>
<em>In FMI 2.0, the connection would be accepted, provided both elements have the same <code>&lt;BaseUnit&gt;</code> definition).</em></p>
</div>
</dd>
<div class="dlist">
<dl>
<dt class="hdlist1">Dimensional analysis of equations</dt>
<dd>
<div class="paragraph">
Expand All @@ -5901,8 +5928,8 @@ <h4 id="_physical_units"><a class="link" href="#_physical_units">2.4.3. Physical
<div class="stemblock">
<div class="content">
\[\begin{align*}
J \cdot \alpha = \tau \rightarrow [kg.m^2]*[rad/s^2] = [kg.m^2/s^2] &amp; \quad \text{// o.k. ("rad" is treated as "1")} \\
J \cdot \alpha = f \rightarrow [kg.m^2]*[rad/s^2] = [kg.m/s^2] &amp; \quad \text{// error, since dimensions do not agree}
J \cdot \alpha = \tau \rightarrow [\mathrm{kg} \cdot \mathrm{m}^2] \cdot [\mathrm{rad} \cdot \mathrm{s}^{-2}] = [\mathrm{kg} \cdot \mathrm{m}^2 \cdot \mathrm{s}^{-2}] &amp; \quad \text{// o.k. ("rad" is treated as "1")} \\
J \cdot \alpha = f \rightarrow [\mathrm{kg} \cdot \mathrm{m}^2] \cdot [\mathrm{rad} \cdot \mathrm{s}^{-2}] = [\mathrm{kg} \cdot \mathrm{m} \cdot \mathrm{s}^{-2}] &amp; \quad \text{// error, since dimensions do not agree}
\end{align*}\]
</div>
</div>
Expand All @@ -5918,12 +5945,12 @@ <h4 id="_physical_units"><a class="link" href="#_physical_units">2.4.3. Physical
<div class="ulist">
<ul>
<li>
<p><em>a = b + c, and <code>Unit</code> of c is provided, but not <code>Unit</code> of a and b:</em><br>
<p><em>\(a = b + c\), and <code>Unit</code> of c is provided, but not <code>Unit</code> of a and b:</em><br>
<em>The Unit definition of <code>c</code> (in other words, <code>Unit.name</code>, <code>&lt;BaseUnit&gt;</code>, <code>&lt;DisplayUnit&gt;</code>) is also used for <code>a</code> and <code>b</code>.</em>
<em>For example, if BaseUnit(c) = <code>rad/s</code>, then BaseUnit(a) = BaseUnit(b) = <code>rad/s</code>.</em></p>
</li>
<li>
<p><em>a = b*c, and <code>Unit</code> of a and of c is provided, but not <code>Unit</code> of b:</em><br>
<p><em>\(a = b \cdot c\), and <code>Unit</code> of a and of c is provided, but not <code>Unit</code> of b:</em><br>
<em>If <code>rad</code> is either part of the <code>&lt;BaseUnit&gt;</code> of <code>a</code> and/or of <code>c</code>, then the <code>&lt;BaseUnit&gt;</code> of <code>b</code> cannot be deduced (otherwise it can be deduced).</em>
<em>Example: If <code>BaseUnit(a) = kg.m/s2</code> and <code>BaseUnit(c) = m/s2</code>, then the <code>BaseUnit(b)</code> can be deduced to be <code>kg</code>.</em>
<em>In such a case <code>Unit.name</code> of b cannot be deduced from the <code>Unit.name</code> of <code>a</code> and <code>c</code>, and a tool would typically construct the <code>Unit.name</code> of <code>b</code> from the deduced <code>&lt;BaseUnit&gt;</code>.]</em></p>
Expand Down Expand Up @@ -5956,8 +5983,8 @@ <h4 id="_physical_units"><a class="link" href="#_physical_units">2.4.3. Physical
<div class="content">
\[v_{\mathit{display}} =
\left\{\begin{array}{ll}
\texttt{factor} * v_{\mathit{unit}} + \texttt{offset} &amp;\text{if} \; \texttt{inverse = false} \\
\texttt{factor} * \frac{1}{v_{\mathit{unit}}} &amp;\text{if} \; \texttt{inverse = true}
\texttt{factor} \cdot v_{\mathit{unit}} + \texttt{offset} &amp;\text{if} \; \texttt{inverse = false} \\
\texttt{factor} \cdot \frac{1}{v_{\mathit{unit}}} &amp;\text{if} \; \texttt{inverse = true}
\end{array}\right.\]
</div>
</div>
Expand All @@ -5969,11 +5996,11 @@ <h4 id="_physical_units"><a class="link" href="#_physical_units">2.4.3. Physical
</div>
<div class="stemblock">
<div class="content">
\[T_F = (9/5) * (T_K - 273.15) + 32\]
\[T_F = \frac{9}{5} \cdot (T_K - 273.15) + 32\]
</div>
</div>
<div class="paragraph">
<p><em>and therefore, <code>factor = 1.8 (=9/5)</code> and <code>offset = -459.67 (= 32 - 273.15*9/5)</code>.</em></p>
<p><em>and therefore, \(\texttt{factor} = 1.8 \ \left(= \frac{9}{5}\right)\) and \(\texttt{offset} = -459.67 \ \left(= 32 - 273.15 \cdot \frac{9}{5}\right)\).</em></p>
</div>
<div class="paragraph">
<p><em>Both the <code>DisplayUnit.name</code> definitions as well as the <code>Unit.name</code> definitions are used in the variable elements.</em></p>
Expand Down Expand Up @@ -6142,7 +6169,7 @@ <h4 id="definition-of-types"><a class="link" href="#definition-of-types">2.4.4.
If not defined and no other information about the nominal value is available, then <code>nominal = 1</code> is assumed.
For array variables, this nominal value applies to all elements of the array.<br>
<em>[The nominal value of a variable can be, for example, used to determine the absolute tolerance for this variable as needed by numerical algorithms:</em><br>
<code>absoluteTolerance = nominal * tolerance * 0.01</code><br>
<code>absoluteTolerance = nominal \cdot tolerance \cdot 0.01</code><br>
<em>where <code>tolerance</code> is, for example, the relative tolerance defined in <a href="#DefaultExperiment">Section 2.4.6</a>.]</em></p></td>
</tr>
<tr>
Expand Down

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