Structural Safe-Load Limits for Cargo Cranes (Mechanical Engineering) Solution

Problem Overview

We analyze a motorized shipyard gantry crane hoisting a cargo container along a vertical track. The winch drum is offset horizontally by a fixed distance $x = 24.0\text{ m}$. At a cable length $z = 30.0\text{ m}$, the winch reels cable inward at $\frac{dz}{dt} = -2.0\text{ m/s}$. We calculate the vertical lift velocity $\frac{dy}{dt}$ and verify adherence to dynamic hoist load safety limits ($v_{\text{max}} \le 3.0\text{ m/s}$).

Step-by-Step Mathematical Solution

Step 1: Geometric Setup & Differentiation

The cable geometry forms a right triangle where:

• Horizontal drum offset $x = 24.0\text{ m}$ is a static constant ($\frac{dx}{dt} = 0$).
• Vertical distance from drum level down to the cargo spreader is $y(t)$ (dynamic variable).
• Active cable length is the hypotenuse $z(t)$ (dynamic variable).

The governing formula is the Pythagorean Theorem:

Substitute the static horizontal offset $x = 24.0$ before differentiating:

Differentiating both sides implicitly with respect to time $t$ using the Chain Rule:

Isolating the vertical velocity $\frac{dy}{dt}$:

Step 2: Snapshot Calculation of $y$ and $\frac{dy}{dt}$

1. Find the vertical position $y$ when $z = 30.0\text{ m}$:
   $$576.0 + y^2 = (30.0)^2 = 900.0$$
   $$y^2 = 900.0 - 576.0 = 324.0 \implies y = \sqrt{324.0} = 18.0\text{ m}$$
2. Compute the instantaneous vertical velocity $\frac{dy}{dt}$:
   $$\frac{dy}{dt} = \frac{30.0}{18.0} (-2.0) = \frac{5}{3} (-2.0) = -\frac{10}{3}\text{ m/s} \approx -3.33\text{ m/s}$$
   The negative sign indicates the vertical distance $y$ from the gantry drum down to the container is decreasing, meaning the container is moving upward at a speed of $3.33\text{ m/s}$.

Step 3: Structural Safety Compliance Evaluation

• Regulatory Threshold: Maximum vertical hoist velocity $v_{\text{safe}} \le 3.0\text{ m/s}$.
• Calculated Vertical Velocity: $|v_{\text{vertical}}| = 3.33\text{ m/s}$.
• Safety Verdict: NON-COMPLIANT / UNSAFE EXCURSION. The container's vertical ascent speed ($3.33\text{ m/s}$) exceeds the $3.0\text{ m/s}$ structural safety limit by $11\%$, despite the winch reel speed being only $2.0\text{ m/s}$.

Engineering Conclusion

In mechanical hoisting and crane automation, the winch spool tachometer measures only the line reel-in rate ($\frac{dz}{dt}$), not the vertical cargo velocity ($\frac{dy}{dt}$).

Because $\frac{dy}{dt} = \frac{z}{y}\frac{dz}{dt}$, the vertical velocity is amplified by the geometric factor $\frac{z}{y} = \frac{1}{\cos\phi}$ (where $\phi$ is the angle of the cable relative to the vertical track). As the container is hoisted higher, $y$ decreases while the fixed horizontal offset $x = 24\text{ m}$ remains constant. Consequently, the cable angle $\phi$ widens, and the ratio $\frac{z}{y}$ increases rapidly:

• When $y = 32\text{ m}$ ($z = 40\text{ m}$): $\frac{z}{y} = 1.25 \implies |v_y| = 2.5\text{ m/s}$ (safe).
• When $y = 18\text{ m}$ ($z = 30\text{ m}$): $\frac{z}{y} = 1.67 \implies |v_y| = 3.33\text{ m/s}$ (exceeds $3.0\text{ m/s}$ limit).
• When $y = 10\text{ m}$ ($z = 26\text{ m}$): $\frac{z}{y} = 2.60 \implies |v_y| = 5.20\text{ m/s}$ (extreme hazard).

If crane control microcontrollers do not execute this related rates calculation in real time, a winch operator maintaining a constant motor spool speed of $2.0\text{ m/s}$ will unknowingly subject the crane structure and cargo rigging to severe shock accelerations and dynamic pendulum swinging. Microcontrollers must automatically derate and throttle down winch motor RPM as $y$ decreases to enforce a strict cap on $\frac{dy}{dt}$.