RelativeAccelerationSensor

Ideal sensor that measures the relative angular acceleration between two rotational mechanical splines.

This component calculates the relative angular acceleration between two connected splines, spline_a and spline_b. The relative angle, ${\varphi }_{rel}$, is first determined as the difference between the absolute angles of the two splines:

$$\begin{gathered} ph{i}_{rel}= \\ ph{i}_b- \\ ph{i}_a \end{gathered}$$

The relative angular velocity, $w_{rel}$, is then calculated as the time derivative of this relative angle:

$$\begin{gathered} w_{rel}=\frac{d \\ ph{i}_{rel}}{dt} \end{gathered}$$

Finally, the output, relative angular acceleration $a_{rel}$, is obtained as the time derivative of the relative angular velocity:

$$\begin{gathered} a_{rel}=\frac{d{w}_{rel}}{dt}=\frac{d^2 \\ ph{i}_{rel}}{d{t}^2} \end{gathered}$$

The sensor is ideal, which implies it does not exert any torque back on the connected splines. This is explicitly modeled by the equation $\tau \ast a=0$ for spline*a.

This component extends from PartialRelativeSensor

Usage

RotationalComponents.RelativeAccelerationSensor()

Connectors

• spline_a - This connector represents a rotational spline with angle and torque as the potential and flow variables, respectively. (Spline)

• spline_b - This connector represents a rotational spline with angle and torque as the potential and flow variables, respectively. (Spline)

• a_rel - This connector represents a real signal as an output from a component (RealOutput)

Variables

Name	Description	Units
phi_rel	Relative angle between two splines	rad
w_rel	Relative angular velocity between two splines	rad/s

Behavior

$$\begin{array}{rl}0& =\mathtt{spline}\mathtt{\_}\mathtt{b}\mathtt{.}\mathtt{tau}\left(t\right)+\mathtt{spline}\mathtt{\_}\mathtt{a}\mathtt{.}\mathtt{tau}\left(t\right)\\ \mathtt{phi}\mathtt{\_}\mathtt{rel}\left(t\right)& =\mathtt{spline}\mathtt{\_}\mathtt{b}\mathtt{.}\mathtt{phi}\left(t\right)-\mathtt{spline}\mathtt{\_}\mathtt{a}\mathtt{.}\mathtt{phi}\left(t\right)\\ \mathtt{w}\mathtt{\_}\mathtt{rel}\left(t\right)& =\frac{\mathrm{d}\mathtt{phi}\mathtt{\_}\mathtt{rel}\left(t\right)}{\mathrm{d}t}\\ \mathtt{a}\mathtt{\_}\mathtt{rel}\left(t\right)& =\frac{\mathrm{d}\mathtt{w}\mathtt{\_}\mathtt{rel}\left(t\right)}{\mathrm{d}t}\\ 0& =\mathtt{spline}\mathtt{\_}\mathtt{a}\mathtt{.}\mathtt{tau}\left(t\right)\end{array}$$

Source

"""
Ideal sensor that measures the relative angular acceleration between two rotational mechanical splines.

This component calculates the relative angular acceleration between two connected
splines, `spline_a` and `spline_b`. The relative angle, \$\\phi_{rel}\$,
is first determined as the difference between the absolute angles of the two
splines:

math \phi_{rel} = \phi_b - \phi_a

The relative angular velocity, \$w_{rel}\$, is then calculated as the time
derivative of this relative angle:

math w_{rel} = \frac{d\phi_{rel}}

Finally, the output, relative angular acceleration \$a_{rel}\$, is obtained
as the time derivative of the relative angular velocity:

math a_{rel} = \frac{dw_{rel}}{dt} = \frac{d^2\phi_{rel}}

The sensor is ideal, which implies it does not exert any torque back on the connected splines.
This is explicitly modeled by the equation \$\tau_a = 0\$ for `spline_a`.
"""
component RelativeAccelerationSensor
  extends PartialRelativeSensor
  "Relative angular acceleration between two splines as output signal"
  a_rel = RealOutput() {
    "Dyad": {
      "placement": {"icon": {"x1": 450, "y1": 950, "x2": 550, "y2": 1050, "rot": 90}}
    }
  }
  "Relative angle between two splines"
  variable phi_rel::Angle
  "Relative angular velocity between two splines"
  variable w_rel::AngularVelocity
relations
  phi_rel = spline_b.phi - spline_a.phi
  w_rel = der(phi_rel)
  a_rel = der(w_rel)
  0 = spline_a.tau
metadata {
  "Dyad": {
    "icons": {"default": "dyad://RotationalComponents/RelSensor-Angle-Vel-Acc.svg"}
  }
}
end

Flattened Source

"""
Ideal sensor that measures the relative angular acceleration between two rotational mechanical splines.

This component calculates the relative angular acceleration between two connected
splines, `spline_a` and `spline_b`. The relative angle, \$\\phi_{rel}\$,
is first determined as the difference between the absolute angles of the two
splines:

math \phi_{rel} = \phi_b - \phi_a

The relative angular velocity, \$w_{rel}\$, is then calculated as the time
derivative of this relative angle:

math w_{rel} = \frac{d\phi_{rel}}

Finally, the output, relative angular acceleration \$a_{rel}\$, is obtained
as the time derivative of the relative angular velocity:

math a_{rel} = \frac{dw_{rel}}{dt} = \frac{d^2\phi_{rel}}

The sensor is ideal, which implies it does not exert any torque back on the connected splines.
This is explicitly modeled by the equation \$\tau_a = 0\$ for `spline_a`.
"""
component RelativeAccelerationSensor
  "Left spline connector for the sensor."
  spline_a = Spline() {"Dyad": {"placement": {"icon": {"x1": -50, "y1": 450, "x2": 50, "y2": 550}}}}
  "Right spline connector for the sensor."
  spline_b = Spline() {"Dyad": {"placement": {"icon": {"x1": 950, "y1": 450, "x2": 1050, "y2": 550}}}}
  "Relative angular acceleration between two splines as output signal"
  a_rel = RealOutput() {
    "Dyad": {
      "placement": {"icon": {"x1": 450, "y1": 950, "x2": 550, "y2": 1050, "rot": 90}}
    }
  }
  "Relative angle between two splines"
  variable phi_rel::Angle
  "Relative angular velocity between two splines"
  variable w_rel::AngularVelocity
relations
  0 = spline_a.tau + spline_b.tau
  phi_rel = spline_b.phi - spline_a.phi
  w_rel = der(phi_rel)
  a_rel = der(w_rel)
  0 = spline_a.tau
metadata {
  "Dyad": {
    "icons": {"default": "dyad://RotationalComponents/RelSensor-Angle-Vel-Acc.svg"}
  }
}
end

Test Cases

No test cases defined.

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