Luffing mechanisms can be classified into two types based on the method of operation: trolley-type luffing and boom-type luffing. Boom-type luffing can be further subdivided into telescopic boom and luffing boom (pivoting boom) types.
Counterbalanced (rack-and-pinion) luffing-jib crane
Based on operational requirements, luffing can be classified into non-working luffing and working luffing; based on performance requirements, it can be classified into non-counterbalanced luffing and counterbalanced luffing.
Balanced vs. Unbalanced Luffing
Balanced luffing refers to a system where the load's center of gravity moves along a horizontal or near-horizontal line during the luffing process. The dead weight of the boom (or boom system) is counterbalanced by a movable counterweight, ensuring that the combined center of gravity of the load and the boom moves along a horizontal (or near-horizontal) line or remains stationary at a single point. Alternatively, the system can rely on the structural design of the boom assembly itself-without a movable counterweight-to ensure the boom's center of gravity moves horizontally or nearly horizontally during luffing. Balanced luffing is commonly employed in "operational luffing" mechanisms-such as those found on portal cranes and other port cranes-that require frequent luffing while carrying a load.
In unbalanced luffing, the centers of gravity of both the swinging boom and the load rise and fall during the luffing process. Reducing the operating radius consumes significant drive power, while increasing the radius results in the release of potential energy, which affects operational performance. Unbalanced luffing is primarily used in "non-operational luffing" mechanisms, such as those found on mobile cranes.
There are three main types of luffing resistance in cranes: the luffing resistance of non-balanced boom-type luffing mechanisms, the equivalent luffing resistance of balanced boom-type luffing mechanisms, and the steady-state luffing resistance of trolley-type luffing mechanisms. Calculations for luffing mechanisms are complex; this article focuses on the second type: the equivalent luffing resistance of balanced boom-type luffing mechanisms.
Luffing Resistance F for Balanced Jib-Type Luffing Mechanisms
For balanced jib-type luffing mechanisms, the total luffing resistance varies significantly across different operating radii throughout the luffing process. At any given radius, the maximum luffing resistance acting on the traction component (such as a rack, screw, or hydraulic cylinder piston rod) is determined by the direct summation of the various resistance components (adding their algebraic values based on their directions of action). Specifically, under the calculated operating condition where both the luffing and slewing mechanisms are functioning simultaneously and moving steadily, the total luffing resistance F that the traction component must overcome to swing the jib about its pivot axis can be summarized as follows:
Note: In the above, F represents the total luffing resistance at a specific state or the average total luffing resistance over a specific segment. (3-5-35) FQ-luffing resistance caused by the non-horizontal displacement of the load during the luffing process (N); Fb-luffing resistance caused by incomplete counterbalancing of the boom system's dead weight (N);
Fw-luffing resistance caused by wind loads acting on the boom system (N); FH-luffing resistance resulting from the deflection angle α of the load's hoisting rope (N); this accounts for wind loads and centrifugal forces acting on the load, as well as horizontal inertial forces generated during luffing, slewing, or the starting/braking of travel motions; Fc-luffing resistance caused by the centrifugal force of the boom system during crane slewing (N);
Fi-luffing resistance caused by the radial inertial force of the boom system during the luffing process (N) (the inertial force considered here is that generated during constant-speed luffing motion; inertial forces arising from the starting or braking of the luffing drive mechanism are accounted for when verifying starting and braking times); Ff-luffing resistance caused by friction in the boom hinge pins and friction losses in the compensating pulley block (N); Fp-luffing resistance caused by factors such as crane track gradients (N). The entire luffing range is divided into several segments to determine the forces acting on the luffing traction components within each segment. Luffing resistances for various load conditions across these segments should be tabulated to facilitate load combination calculations. In general calculations, luffing resistances caused by the boom system's slewing centrifugal force, luffing inertial force, and hinge pin friction are often disregarded; the primary components considered are FQ, Fb, and Fw.
The luffing resistance of unbalanced luffing-jib mechanisms can also be calculated using the formulas above.
A simplified calculation method for the luffing resistance $F_u$ of a rack-driven, rigid tie-rod combined boom assembly
The loads acting on the various components of the luffing system-including the counterweight system-consist of vertical forces $F_{Vi}$ and horizontal forces $F_{Hi}$. The points of application of these forces shift during the luffing process, undergoing vertical displacements $\Delta y_i$ and horizontal displacements $\Delta x_i$. The total work performed by these forces equals the work done by the rack force $F_u$ as it moves the rack by a distance $\Delta l$ (neglecting frictional losses in the bearings at the boom assembly's pivot points), as expressed by the following equivalence formula (3-5-36):
As the operating radius changes, the projected wind area of each component varies; therefore, the wind force acting on a component within a specific radius interval should be calculated using the average of the projected wind areas at the two endpoints of that interval. The rack length and total travel are determined based on the maximum and minimum operating radii. The total travel is divided into several segments (typically 8 to 10), with the rack travel partitioned accordingly; each segment corresponds to a specific time interval. For the boom system at each radius position, the vertical and horizontal components of external loads are plotted at their respective points of application. For each incremental shift Δl in rack travel, the vertical displacement Δyi and horizontal displacement Δxi (accounting for directional signs) of the force application points between adjacent radius positions can be measured directly from the diagram. By multiplying the vertical and horizontal forces by their respective displacement paths, the rack force Fu for each time interval can be calculated using Equation (3-5-36).
Rack-driven, rigid tie-rod combined luffing mechanism (bottom view) | Source: DOC163
This method offers a simple and clear calculation process that is well-suited for computer implementation; parameters such as luffing power, tension in luffing members, luffing speed, and acceleration can all be calculated based on this same principle.
Equivalent luffing resistance $F_{Id}$ for a balanced jib luffing mechanism
The luffing resistance at each position of a balanced jib luffing mechanism can be calculated using the methods described in Section 01 or Section 02 above-specifically, by calculating $F$ or $F_u$ for each interval. However, the equivalent luffing resistance $F_{Id}$ for the entire luffing process should be calculated as follows:
FId - Equivalent luffing resistance for the luffing mechanism of a balanced boom system, in Newtons (N);
PIi - Average luffing resistance between two adjacent calculation points within the operating range from position i to position i+1, in Newtons (N); (F or Fu for each interval)
ti - Duration of action of FIi, in seconds (s).
Equivalent Luffing Power ($P_e$) of the Motor – Balanced Jib Luffing Mechanism
The equivalent luffing power of the motor for a balanced jib luffing mechanism is calculated using the following formula:
Pe - Equivalent luffing power of the luffing mechanism motor, in kilowatts (kW);
FId - Equivalent luffing resistance of the balanced-jib luffing mechanism, in newtons (N);
Vb - Linear speed of the traction component (wire rope, rack, screw, hydraulic cylinder, piston, etc.), in meters per second (m/s);
η - Total transmission efficiency of the luffing mechanism.
Reference for the total transmission efficiency (η) of a rack-and-pinion driven, rigid tie-rod combined boom luffing system:
η = η₁ × η₂ × η₃
The efficiency values for the individual components are approximately as follows: Gear reducer efficiency (η₁): typically 0.94 to 0.98 for hardened gears. Gear-rack meshing efficiency (η₂): typically 0.95 to 0.98. Pivot point friction efficiency (η₃): typically close to 0.98 to 0.99. Consequently, the total transmission efficiency (η) generally falls within the range of 0.88 to 0.95.
Based on the calculated result Pe, preliminarily select the required motor from the motor catalog.
