Give step-by-step solution with explanation and final answer:1. For a cascaded tanks (cross sectional areas Ay and As) shown in Figure I,
(a) Develop a complete bond graph model.
(b) Provide constit

Question

Give step-by-step solution with explanation and final answer:1. For a cascaded tanks (cross sectional areas Ay and As) shown in Figure I,
(a) Develop a complete bond graph model.
(b) Provide constitutive laws for all components.
(¢) Wite a set of state equation in the canonical format [& = A + Bu] where = [Vi Va]. And Vi and V;
are volumes of fluid contained in cach tank.
(d) Determine a transfer function Hi (s)/Qin(s). Assume the fluid is incompressible and connecting pipe of the
two tanks is negligibly short compared to tank height.
(e) Select one system parameter that is the most influential to hy’s steady state value.
Qin
Tank Vi racy ©
—
J A, Ry Af Ry
= Q:
Figure 1: A Set of Cascaded Tanks

VIPSolutions · Accepted Answer

Let's solve the problem **step-by-step**:

---

### **(a) Develop a complete bond graph model**

**Components:**
- **Capacitors (C):** Represent the tanks (fluid accumulation): $ C_1, C_2 $
- **Resistors (R):** Represent the outflow resistances: $ R_1, R_2 $
- **Source of Flow (Sf):** Represents the inflow $ Q_{in} $

#### **Bond Graph**
```
       Sf (Qin) 
         |
         1
         |
        C1
         |
        R1
         |
        C2
         |
        R2
         |
       Ground
```
- **1-junctions** (for series connection, same flow)
- **0-junctions** (for parallel connection, same effort) are not needed here as all are in series

---

### **(b) Constitutive laws for all components**

- **Capacitors (tanks):** 
  $$
  e = \frac{q}{C} \implies h = \frac{V}{A}
  $$
  where $ h $ is fluid height, $ V $ is volume, $ A $ is cross-sectional area.

- **Resistors (valves/pipe):**
  $$
  Q = \frac{h_{up} - h_{down}}{R}
  $$
  - For $ R_1 $: $ Q_1 = \frac{h_1 - h_2}{R_1} $
  - For $ R_2 $: $ Q_2 = \frac{h_2}{R_2} $

- **Source of flow:** $ Q_{in} $ (input flow)

---

### **(c) Write state equations in canonical format**

Let:
- $ V_1 = $ volume in tank 1
- $ V_2 = $ volume in tank 2
- $ h_1 = V_1 / A_1 $
- $ h_2 = V_2 / A_2 $

#### **Mass balances:**

$$
\frac{dV_1}{dt} = Q_{in} - Q_1
$$
$$
\frac{dV_2}{dt} = Q_1 - Q_2
$$

#### **Substitute flows:**

$$
Q_1 = \frac{h_1 - h_2}{R_1} = \frac{V_1/A_1 - V_2/A_2}{R_1}
$$
$$
Q_2 = \frac{h_2}{R_2} = \frac{V_2}{A_2 R_2}
$$

#### **Canonical State-Space Form:**

Let $ x = \begin{bmatrix} V_1 \ V_2 \end{bmatrix} $, $ u = Q_{in} $:

$$
\frac{d}{dt}
\begin{bmatrix}
V_1 \
V_2
\end{bmatrix}
=
\begin{bmatrix}
- \frac{1}{A_1 R_1} & \frac{1}{A_2 R_1} \
\frac{1}{A_1 R_1} & -\left(\frac{1}{A_2 R_1} + \frac{1}{A_2 R_2}\right)
\end{bmatrix}
\begin{bmatrix}
V_1 \
V_2
\end{bmatrix}
+
\begin{bmatrix}
1 \
0
\end{bmatrix}
Q_{in}
$$

Or, more simply:

$$
\boxed{
\begin{aligned}
\dot{V}_1 &= Q_{in} - \frac{V_1/A_1 - V_2/A_2}{R_1} \
\dot{V}_2 &= \frac{V_1/A_1 - V_2/A_2}{R_1} - \frac{V_2}{A_2 R_2}
\end{aligned}
}
$$

---

### **(d) Transfer function $ H_1(s)/Q_{in}(s) $ (height of tank 1 to inflow)**

Let’s use Laplace Transforms and solve for $ H_1(s) $:

Let $ H_1(s) = \frac{V_1(s)}{A_1} $, $ H_2(s) = \frac{V_2(s)}{A_2} $.

From above:

$$
sV_1 = Q_{in}(s) - \frac{H_1(s) - H_2(s)}{R_1}
$$
$$
sV_2 = \frac{H_1(s) - H_2(s)}{R_1} - \frac{H_2(s)}{R_2}
$$

But $ V_1 = A_1 H_1 $, $ V_2 = A_2 H_2 $:

$$
A_1 s H_1 = Q_{in}(s) - \frac{H_1 - H_2}{R_1}
$$
$$
A_2 s H_2 = \frac{H_1 - H_2}{R_1} - \frac{H_2}{R_2}
$$

Solve the two equations:

Equation 1:
$$
A_1 s H_1 + \frac{H_1}{R_1} = Q_{in}(s) + \frac{H_2}{R_1}
$$
$$
\left(A_1 s + \frac{1}{R_1}\right) H_1 - \frac{1}{R_1} H_2 = Q_{in}(s)
$$

Equation 2:
$$
A_2 s H_2 + \frac{H_2}{R_1} + \frac{H_2}{R_2} = \frac{H_1}{R_1}
$$
$$
- \frac{1}{R_1} H_1 + \left(A_2 s + \frac{1}{R_1} + \frac{1}{R_2}\right) H_2 = 0
$$

Write in matrix form:

$$
\begin{bmatrix}
A_1 s + \frac{1}{R_1} & -\frac{1}{R_1} \
-\frac{1}{R_1} & A_2 s + \frac{1}{R_1} + \frac{1}{R_2}
\end{bmatrix}
\begin{bmatrix}
H_1 \ H_2
\end{bmatrix}
=
\begin{bmatrix}
Q_{in}(s) \ 0
\end{bmatrix}
$$

Solve for $ H_1 $:

$$
H_1(s) = \frac{
\left(A_2 s + \frac{1}{R_1} + \frac{1}{R_2}\right)
}{
\left(A_1 s + \frac{1}{R_1}\right)\left(A_2 s + \frac{1}{R_1} + \frac{1}{R_2}\right) - \left(\frac{1}{R_1}\right)^2
} Q_{in}(s)
$$

So,

$$
\boxed{
\frac{H_1(s)}{Q_{in}(s)} = \frac{A_2 s + \frac{1}{R_1} + \frac{1}{R_2}}{\left(A_1 s + \frac{1}{R_1}\right)\left(A_2 s + \frac{1}{R_1} + \frac{1}{R_2}\right) - \left(\frac{1}{R_1}\right)^2}
}
$$

---

### **(e) Most influential system parameter for $ h_1 $'s steady-state value**

At steady state ($ s \to 0 $), set derivatives to zero:

$$
0 = Q_{in} - \frac{h_1 - h_2}{R_1}
$$
$$
0 = \frac{h_1 - h_2}{R_1} - \frac{h_2}{R_2}
$$

Solve the second for $ h_2 $:

$$
\frac{h_1 - h_2}{R_1} = \frac{h_2}{R_2} \implies (h_1 - h_2)R_2 = h_2 R_1 \implies h_1 R_2 = h_2 (R_1 + R_2) \implies h_2 = \frac{h_1 R_2}{R_1 + R_2}
$$

Substitute into first:

$$
Q_{in} = \frac{h_1 - h_2}{R_1} = \frac{h_1 - \frac{h_1 R_2}{R_1 + R_2}}{R_1} = \frac{h_1 (R_1 + R_2) - h_1 R_2}{R_1 (R_1 + R_2)} = \frac{h_1 R_1}{R_1 (R_1 + R_2)} = \frac{h_1}{R_1 + R_2}
$$
$$
h_1 = Q_{in} (R_1 + R_2)
$$

**Thus, the sum $ R_1 + R_2 $ is the most influential parameter for $ h_1 $'s steady-state value!**

---

## **Summary of Answers**

### (a) **Bond Graph Model:** (as drawn above)

### (b) **Constitutive Laws:**
- $ h = V/A $
- $ Q_1 = (h_1 - h_2)/R_1 $
- $ Q_2 = h_2/R_2 $

### (c) **State Equations:** (see boxed equations above)

### (d) **Transfer Function:**

$$
\boxed{
\frac{H_1(s)}{Q_{in}(s)} = \frac{A_2 s + \frac{1}{R_1} + \frac{1}{R_2}}{\left(A_1 s + \frac{1}{R_1}\right)\left(A_2 s + \frac{1}{R_1} + \frac{1}{R_2}\right) - \left(\frac{1}{R_1}\right)^2}
}
$$

### (e) **Most Influential Parameter:**

$$
\boxed{R_1 + R_2}
$$
is the most influenti

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