To solve this problem of determining the angular velocity of the rocker (BY) for the given four-bar mechanism, let's proceed step by step. Here's the process:

1. Given Data

• Link lengths:

  • $Z\!A = 54 \, \text{cm}$
  • $A\!B = 104 \, \text{cm}$
  • $B\!Y = 124 \, \text{cm}$
  • $Z\!Y = 100 \, \text{cm}$

• Angular configuration:

  • $\angle Y\!Z\!A = 120^\circ$
  • $\angle Z\!A\!B = 100.53^\circ$
  • $\angle Z\!Y\!B = 67.25^\circ$

• Angular velocity of the crank $Z\!A$:

  • $\omega_{Z\!A} = 3 \, \text{rpm} = \frac{3 \cdot 2\pi}{60} = 0.314 \, \text{rad/s}$

2. Solution Method

The four-bar mechanism requires applying loop closure equations in the velocity analysis. We'll use the following steps:

1. Relative Angular Velocities:

   • The velocity relationship for the links $Z\!A$, $A\!B$, $Z\!Y$, and $B\!Y$ depends on their geometric relationships.

2. Vector Loops and Velocity Analysis:

   • Based on the angles and dimensions of the four-bar, we calculate the relative velocity components of each link.
   • Using the principle of relative motion, relate the angular velocities.

3. Trigonometric Calculations:

   • Use the sine and cosine laws in triangles to determine any unknown angles or magnitudes required.

Let me now process these calculations. math

Given data

ZA = 54 # cm AB = 104 # cm BY = 124 # cm ZY = 100 # cm

Angles in degrees

YZA = 120 # degrees ZAB = 100.53 # degrees ZYB = 67.25 # degrees

Convert angles to radians for calculations

YZA_rad = math.radians(YZA) ZAB_rad = math.radians(ZAB) ZYB_rad = math.radians(ZYB)

Angular velocity of crank Z-A

omega_ZA_rpm = 3 # rpm omega_ZA = omega_ZA_rpm * 2 * math.pi / 60 # rad/s

Step 1: Calculate the relative angular velocities using velocity loop equations.

Applying sine law and geometry to determine the angular velocity of the rocker

Velocity of point A relative to Z: V_A = omega_ZA * ZA

V_A = omega_ZA * ZA # cm/s

Use vector closure equations and calculate angular velocity of rocker (BY)

BY's angular velocity will depend on ZY and its relative velocity contributions.

V_A_magnitude = V_A # as magnitude of crank's angular velocity is a primary driver

V_BY = V_A_magnitude * (math.sin(ZAB_rad) / math.sin(ZYB_rad)) # Velocity relation using geometry

Angular velocity of rocker (omega_BY)

omega_BY = V_BY / BY

Final result rounded to 2 decimal places

omega_BY_rounded = round(omega_BY, 2) omega_BY_roundedThe angular velocity of the rocker $B\!Y$ is $0.15 \, \text{rad/s}$.

Would you like a detailed explanation of the steps or the derivations used? Let me know if you have additional questions.

Here are five related questions to explore further:

1. How do velocity loop equations apply to four-bar mechanisms in general?
2. What is the significance of angles in determining relative velocities?
3. How do you convert between angular velocity and linear velocity in mechanisms?
4. Can the direction of rotation be derived for all links in this configuration?
5. How would the solution change if the input angular velocity was doubled?

Tip: Always ensure all angles are in consistent units (degrees or radians) before calculations involving trigonometric functions.