Math / Algebra

Question- Consider a rack-and-pinion system. The rotational motion of the pinion is transformed into transitional motion of the rack
For simplicity, the spring effects are ignored $T_{\text {in }}-T_{\text {out }}=J \frac{d \omega}{d t}+c_{1} \omega$
Dr. Terek A. Tutanf
Example continued
The rotational equation is $\begin{array}{l} T_{\text {in }}-T_{\text {out }}=J \frac{d \omega}{d t}+c_{1} \omega \\ F-c_{2} v=m \frac{d v}{d t} \end{array}$
The transitional equation is
Using the equations And manipulating the rotational and transitional equations with the input torque, Tin, as inputs and velocity, v , as output, we get $\begin{array}{l} T_{\text {out }}=r F \\ \omega=v / r \\ T_{\text {in }}=\left(c_{1} / r+c_{2} r\right) v+(J / r+m r) \frac{d v}{d t} \end{array}$ Dr. Tarek A. Tutunji
Example continued
Let us take a look at the state space equations In general, where x is the states vector, y is the output vector, and $u$ is the input vector
In our example, we will use the states: $\omega$ and $v$ , the inputs: $\mathrm{T}_{\mathrm{in}}$ and F the output: v $\begin{array}{l} {\left[\begin{array}{l} d \omega / d t \\ d v / d t \end{array}\right]=\left[\begin{array}{cc} -c_{1} / J & 0 \\ 0 & -c_{2} / m \end{array}\right]\left[\begin{array}{l} \omega \\ v \end{array}\right]+\left[\begin{array}{cc} 1 / J & -r / J \\ 0 & 1 / m \end{array}\right]\left[\begin{array}{c} T_{i n} \\ F \end{array}\right]} \\ v=\left[\begin{array}{ll} 0 & 1 \end{array}\right]\left[\begin{array}{l} \omega \\ v \end{array}\right] \end{array}$

Studdy Solution

STEP 1

What is this asking? We're looking at a rack-and-pinion system and figuring out how the input torque affects the velocity of the rack, ignoring spring effects.
We'll also represent the system using state-space equations. Watch out! Don't mix up rotational and translational variables!
Also, keep track of your $r$ (radius) – it connects the rotational and translational worlds.

STEP 2

1. Relate Torque and Force
2. Relate Angular and Linear Velocity
3. Derive the Input-Output Relationship
4. Define State Variables
5. Construct the State-Space Representation

STEP 3

We know that torque ( $T_{out}$ ) applied to the pinion creates a force ( $F$ ) on the rack.
They're related by the radius ( $r$ ) of the pinion: $T_{out} = r \cdot F$

STEP 4

The angular velocity ( $\omega$ ) of the pinion and the linear velocity ( $v$ ) of the rack are also related by the radius ( $r$ ): $\omega = \frac{v}{r}$

STEP 5

We're given these two equations: $T_{in} - T_{out} = J \cdot \frac{d\omega}{dt} + c_1 \cdot \omega$ $F - c_2 \cdot v = m \cdot \frac{dv}{dt}$ where $J$ is the moment of inertia, $c_1$ and $c_2$ are damping coefficients, and $m$ is the mass of the rack.

STEP 6

Substitute the expressions for $T_{out}$ and $\omega$ from the previous relationships into the first equation: $T_{in} - r \cdot F = J \cdot \frac{d}{dt} \left( \frac{v}{r} \right) + c_1 \cdot \frac{v}{r}$ $T_{in} - r \cdot F = \frac{J}{r} \cdot \frac{dv}{dt} + \frac{c_1}{r} \cdot v$

STEP 7

From the second given equation, we can express $F$ as: $F = m \cdot \frac{dv}{dt} + c_2 \cdot v$

STEP 8

Substitute this expression for $F$ back into the equation we derived in 2.3.2: $T_{in} - r \cdot \left( m \cdot \frac{dv}{dt} + c_2 \cdot v \right) = \frac{J}{r} \cdot \frac{dv}{dt} + \frac{c_1}{r} \cdot v$ $T_{in} - mr \cdot \frac{dv}{dt} - rc_2 \cdot v = \frac{J}{r} \cdot \frac{dv}{dt} + \frac{c_1}{r} \cdot v$

STEP 9

Now, rearrange the equation to isolate $T_{in}$ : $T_{in} = \left( \frac{J}{r} + mr \right) \cdot \frac{dv}{dt} + \left( \frac{c_1}{r} + rc_2 \right) \cdot v$ This equation shows how the input torque ( $T_{in}$ ) relates to the rack's velocity ( $v$ ).

STEP 10

We'll choose $\omega$ and $v$ as our state variables, represented as a vector: $x = \begin{bmatrix} \omega \\ v \end{bmatrix}$

STEP 11

We're given the state-space equations: $\frac{dx}{dt} = Ax + Bu$ $y = Cx$ where $x$ is the state vector, $u$ is the input vector, and $y$ is the output vector.

STEP 12

In our case, $u = \begin{bmatrix} T_{in} \\ F \end{bmatrix}$ and $y = v$ .

STEP 13

From the given equations and relationships, we can derive the matrices $A$ , $B$ , and $C$ : $A = \begin{bmatrix} -\frac{c_1}{J} & 0 \\ 0 & -\frac{c_2}{m} \end{bmatrix}, \quad B = \begin{bmatrix} \frac{1}{J} & -\frac{r}{J} \\ 0 & \frac{1}{m} \end{bmatrix}, \quad C = \begin{bmatrix} 0 & 1 \end{bmatrix}$

STEP 14

The relationship between input torque and velocity is: $T_{in} = \left( \frac{J}{r} + mr \right) \cdot \frac{dv}{dt} + \left( \frac{c_1}{r} + rc_2 \right) \cdot v$ The state-space representation is: $\frac{dx}{dt} = \begin{bmatrix} -\frac{c_1}{J} & 0 \\ 0 & -\frac{c_2}{m} \end{bmatrix}x + \begin{bmatrix} \frac{1}{J} & -\frac{r}{J} \\ 0 & \frac{1}{m} \end{bmatrix}u$ $y = \begin{bmatrix} 0 & 1 \end{bmatrix}x$

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Studdy Solution

STEP 1

STEP 2

1. Relate Torque and Force
2. Relate Angular and Linear Velocity
3. Derive the Input-Output Relationship
4. Define State Variables
5. Construct the State-Space Representation

STEP 3

We know that torque ( $T_{out}$ ) applied to the pinion creates a force ( $F$ ) on the rack.
They're related by the radius ( $r$ ) of the pinion: $T_{out} = r \cdot F$

STEP 4

The angular velocity ( $\omega$ ) of the pinion and the linear velocity ( $v$ ) of the rack are also related by the radius ( $r$ ): $\omega = \frac{v}{r}$

STEP 5

We're given these two equations: $T_{in} - T_{out} = J \cdot \frac{d\omega}{dt} + c_1 \cdot \omega$ $F - c_2 \cdot v = m \cdot \frac{dv}{dt}$ where $J$ is the moment of inertia, $c_1$ and $c_2$ are damping coefficients, and $m$ is the mass of the rack.

STEP 6

Substitute the expressions for $T_{out}$ and $\omega$ from the previous relationships into the first equation: $T_{in} - r \cdot F = J \cdot \frac{d}{dt} \left( \frac{v}{r} \right) + c_1 \cdot \frac{v}{r}$ $T_{in} - r \cdot F = \frac{J}{r} \cdot \frac{dv}{dt} + \frac{c_1}{r} \cdot v$

STEP 7

From the second given equation, we can express $F$ as: $F = m \cdot \frac{dv}{dt} + c_2 \cdot v$

STEP 8

STEP 9

Now, rearrange the equation to isolate $T_{in}$ : $T_{in} = \left( \frac{J}{r} + mr \right) \cdot \frac{dv}{dt} + \left( \frac{c_1}{r} + rc_2 \right) \cdot v$ This equation shows how the input torque ( $T_{in}$ ) relates to the rack's velocity ( $v$ ).

STEP 10

We'll choose $\omega$ and $v$ as our state variables, represented as a vector: $x = \begin{bmatrix} \omega \\ v \end{bmatrix}$

STEP 11

We're given the state-space equations: $\frac{dx}{dt} = Ax + Bu$ $y = Cx$ where $x$ is the state vector, $u$ is the input vector, and $y$ is the output vector.

STEP 12

In our case, $u = \begin{bmatrix} T_{in} \\ F \end{bmatrix}$ and $y = v$ .

STEP 13

STEP 14

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Studdy Solution

STEP 1

STEP 2

1. Relate Torque and Force2. Relate Angular and Linear Velocity3. Derive the Input-Output Relationship4. Define State Variables5. Construct the State-Space Representation

STEP 3

We know that torque (ToutT_{out}Tout​) applied to the pinion creates a force (FFF) on the rack.They're related by the radius (rrr) of the pinion: Tout=r⋅FT_{out} = r \cdot FTout​=r⋅F

STEP 4

The angular velocity (ω\omegaω) of the pinion and the linear velocity (vvv) of the rack are also related by the radius (rrr): ω=vr\omega = \frac{v}{r}ω=rv​

STEP 5

STEP 6

STEP 7

From the second given equation, we can express FFF as: F=m⋅dvdt+c2⋅vF = m \cdot \frac{dv}{dt} + c_2 \cdot vF=m⋅dtdv​+c2​⋅v

STEP 8

STEP 9

STEP 10

We'll choose ω\omegaω and vvv as our state variables, represented as a vector: x=[ωv] x = \begin{bmatrix} \omega \\ v \end{bmatrix} x=[ωv​]

STEP 11

We're given the state-space equations: dxdt=Ax+Bu\frac{dx}{dt} = Ax + Budtdx​=Ax+Bu y=Cxy = Cxy=Cxwhere xxx is the state vector, uuu is the input vector, and yyy is the output vector.

STEP 12

In our case, u=[TinF]u = \begin{bmatrix} T_{in} \\ F \end{bmatrix}u=[Tin​F​] and y=vy = vy=v.

STEP 13

STEP 14

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1. Relate Torque and Force
2. Relate Angular and Linear Velocity
3. Derive the Input-Output Relationship
4. Define State Variables
5. Construct the State-Space Representation

We know that torque ( $T_{out}$ ) applied to the pinion creates a force ( $F$ ) on the rack.
They're related by the radius ( $r$ ) of the pinion: $T_{out} = r \cdot F$

The angular velocity ( $\omega$ ) of the pinion and the linear velocity ( $v$ ) of the rack are also related by the radius ( $r$ ): $\omega = \frac{v}{r}$

From the second given equation, we can express $F$ as: $F = m \cdot \frac{dv}{dt} + c_2 \cdot v$

We'll choose $\omega$ and $v$ as our state variables, represented as a vector: $x = \begin{bmatrix} \omega \\ v \end{bmatrix}$

We're given the state-space equations: $\frac{dx}{dt} = Ax + Bu$ $y = Cx$ where $x$ is the state vector, $u$ is the input vector, and $y$ is the output vector.

In our case, $u = \begin{bmatrix} T_{in} \\ F \end{bmatrix}$ and $y = v$ .