Offset Loads and Three-Axis Moment Analysis for Linear Actuators

Published:2026-10-10 16:00:00

Linear actuators · Loads and structural design

Describe offset loads with center of gravity and motion

A linear actuator carries both forces and moments. For the same fixture and workpiece mass, changing overhang, center-of-gravity height or acceleration changes the guide loading. Selection inputs should include payload mass, center-of-gravity coordinates and the motion profile.

Geometry Center of gravity and force application pointsDuty conditions Gravity, acceleration and process forcesChecks Guide loads, three-axis moments and stiffness

HZMotion TKA ball screw driven linear actuator product reference — TKA product reference. Capacity depends on the model, guide configuration, mounting orientation and operating conditions.

01 / Axes and moments

Define the reference point and coordinate axes

For a horizontally mounted actuator, take x along travel, y transverse to travel and z upward, normal to the mounting plane. Together they form a right-handed coordinate system. Place reference point O at the specified reference center of the guide support arrangement. For a real model, use the point defined in its drawing or load-calculation instructions, rather than arbitrarily using the center of the table surface.

Let r = (x, y, z) extend from O to a force application point, with F = (F_x, F_y, F_z) acting there. The moment about O is M = r × F. For several forces, calculate each moment at its own application point and sum them. Add any directly applied couples separately.

Component	Expression	Axis in this article	HZMotion catalog term
M_x	yF_z − zF_y	Roll about the travel axis.	M_R: static roll moment.
M_y	zF_x − xF_z	Pitch about the transverse axis.	M_P: static pitch moment.
M_z	xF_y − yF_x	Yaw about the mounting-plane normal.	M_Y: static yaw moment.

This mapping follows the mounting orientation used here and the HZMotion catalog diagrams. Moment signs follow the right-hand rule: magnitude describes the demand, while sign identifies the loading direction. When mounting orientation changes, transform gravity and other forces into the same coordinate system first.

HZMotion TKA actuator visualization preserving the reference product structure with metallic lighting — TKA product visualization based on the HZMotion reference image. Dimensions and supplied details follow confirmed drawings.

02 / Operating loads

Combine gravity, inertia and process forces at the same instant

For a fixture and workpiece undergoing rigid-body translation only, their combined mass can be represented at the combined center of gravity. In an equivalent static assessment, the load they apply to the support is F = m g − m a, with g and a expressed as vectors. This article reports the load applied to the actuator; the actuator's reaction on the payload has the opposite direction.

Source	Model	Additional conditions
Gravity	Include the workpiece, fixture, adapter plate and other moving attachments. Determine their combined center of gravity and calculate mg.	Mass is in kg and force in N. Components change with horizontal, side or vertical mounting.
Translational inertia	Apply equivalent inertial force −ma at the combined center of gravity.	Record the acceleration sign separately for acceleration, deceleration and reversal.
Process loads	Apply cutting, pressing or pulling forces at their actual points and calculate their moments separately.	These points often differ from the center of gravity. Moving all forces there without the corresponding couples loses moment information.
Cables and attachments	Include cable-chain, hose, spring and similar forces at their actual locations.	Cover position-dependent forces and directions over the full working stroke.
Rotation or flexible motion	Add the required rotational inertia and dynamic analysis.	A lumped translating mass alone is insufficient when angular acceleration, significant vibration or flexible deformation is present.

The load above describes the external payload acting on the actuator. To solve individual guide-block loads, also include the actual drive-force line, carriage weight and internal support geometry in the equilibrium model. Keep a simultaneous set of F_x, F_y, F_z, M_x, M_y and M_z for each load case. Individual component envelopes help preliminary comparisons, but maxima from different instants must not be presented as if they form a calculated physical load state.

03 / Worked example

Moments for a 20 kg payload through three motion stages

Assumptions: horizontal mounting, total external moving payload mass m = 20 kg, gravitational acceleration 9.81 m/s² and center of gravity r = (0.15, 0.10, 0.20) m relative to O. Only gravity and translation along x are included. Process forces, cable-chain forces and rotational inertia are excluded. The example demonstrates the method, not the capacity of a particular actuator.

Thus F_z = −196.2 N, F_y = 0 and F_x = −20a_x. For three stages while travelling in the +x direction:

Stage	a_x / m/s²	F_x / N	M_x / N·m	M_y / N·m	M_z / N·m
Constant speed	0	0	−19.62	29.43	0
Acceleration	+3	−60	−19.62	17.43	+6.00
Deceleration	−3	+60	−19.62	41.43	−6.00

Signed roll, pitch and yaw for steady travel, acceleration and deceleration. Deceleration pitch is 41.43 N m; yaw reverses from plus 6 to minus 6 N m. — Figure 1. Signed moments for the 20 kg example. All panels share the same vertical scale. Signs follow the defined coordinates; open hatched bars indicate negative values.

During deceleration, the pitch moment is M_y = 0.20 × 60 − 0.15 × (−196.2) = 41.43 N·m. With unchanged geometry and mass, inertia adds to the gravity term in one direction and partly offsets it in the other. Assess acceleration and deceleration separately.

Comparison after changing geometry

Keep the mass at 20 kg and move the center of gravity to r = (0.05, 0.03, 0.10) m. Repeat the same three stages. The new pitch moments are 9.81, 3.81 and 15.81 N·m. Roll is −5.886 N·m, while yaw is 0, +1.80 and −1.80 N·m. The largest absolute values of each component are:

Three-stage envelope	Original / N·m	Revised / N·m	Cause in this example
Maximum \|M_x\|, roll	19.62	5.89	Transverse offset decreases from 0.10 m to 0.03 m.
Maximum \|M_y\|, pitch	41.43	15.81	Both longitudinal overhang and center-of-gravity height decrease.
Maximum \|M_z\|, yaw	6.00	1.80	A smaller transverse offset shortens the lever arm of x-direction inertia.

Revised maximum absolute roll: 19.62 to 5.89, pitch: 41.43 to 15.81, yaw: 6 to 1.8 N m. — Figure 2. CG moves from (0.15, 0.10, 0.20) m to (0.05, 0.03, 0.10) m. Bars show the largest absolute moment across the three example cases, not model ratings.

These results compare external moment demand only. Suitability still depends on actual rail and block layout, combined loads, motion life, mounting and stiffness calculations. A changed adapter plate also requires reassessment of its own mass, deflection and the combined center of gravity.

04 / Catalog ratings

Separate individual block ratings from assembly moments

The HZMotion linear actuator catalog distinguishes individual guide-block load ratings, overall guide loading directions, and static pitch, yaw and roll moments. Guide structures vary across TKA, TKK, TKS, HKK, TKC and belt-driven families. Use the table for the actual series and size.

Catalog item	Confirm	Application
Individual block dynamic and static ratings	Whether the value describes one block or an assembly, and the life-rating basis.	Calculate actual block loads before applying the corresponding method. Multiplying by the number of blocks is not a load-distribution calculation.
M_P, M_Y, M_R	Directions, reference point, mounting conditions and whether the values are permissible static moments.	Evaluate combined loading for the same load case. Static moment ratings are not continuous dynamic allowances.
Guide spacing B and L	B is transverse spacing and L longitudinal spacing, as defined in the applicable structural drawing.	Use in load distribution and moment assessment. Overall housing width and carriage length are not substitutes.
Drive and motion parameters	Screw or belt transmission, thrust, speed, acceleration and mounting orientation.	Check separately from the guide system, then confirm the complete actuator meets all requirements.
Stiffness and accuracy	Displacement through the adapter plate, carriage, guides, housing, base and connections.	Assess deflection and settling at the tool or workpiece point, beyond load ratings alone.

Keep units consistent: 1 kN·m = 1,000 N·m. Use the model-specific combined-load criterion or obtain a confirmed engineering assessment. Do not invent a universal rule that the sum of three moment ratios must be below one. When only static catalog values are available, dynamic operation still requires load-distribution and life calculations.

05 / Design and selection

Adjust geometry and motion around the load path

Reduce overhang and improve mass placementWithin process-space constraints, bring the combined center of gravity closer to the guide support region and shorten transverse and longitudinal lever arms. Lowering it reduces overturning moments from horizontal inertial forces.
Check guide support geometryCompare the actual number of blocks, transverse and longitudinal spacing, and mounting conditions. Confirm with model data whether a larger actuator envelope improves capacity in the required direction.
Check process forces and connection stiffnessCalculate moments at the real tool contact point and verify the adapter plate, fasteners and base. Elastic displacement at an overhang may govern accuracy before the load rating governs selection.
Assess the motion profile togetherCheck acceleration, deceleration, reversal and process dwell separately. After changing acceleration or jerk, reassess cycle time, vibration and settling rather than comparing maximum speed alone.
Verify loads and operationComplete model-specific static, dynamic-life and drive checks. Then measure loaded deflection, repeatability, settling and temperature at representative stroke positions.

For drive selection, see the screw-versus-belt actuator guide. Stacked axes also require assessment of inter-axis geometry and overall stiffness. Guide-system moments and motor drive torque refer to different calculation objects.

06 / Input data

Include position, direction and time in the selection brief

Input	Provide	Documentation
Layout and coordinates	Mounting orientation, fastening, load reference O, x/y/z axes and stroke range.	Use a coordinate-marked assembly drawing and one consistent reference.
Mass and center of gravity	Masses of the workpiece, fixture, adapter plate and moving attachments; their combined center of gravity.	State kg and mm or m. Cover workpiece changes and center-of-gravity movement through the stroke.
Motion cycle	Speed, acceleration, time, repetition rate and dwell for every stage.	Preserve direction signs and distinguish normal operation from emergency stops.
External actions	Process forces, cable-chain forces, couples and their application points.	Identify simultaneous conditions and directions, rather than providing only a total force.
Performance targets	Life, allowable working-point displacement, repeatability, cycle time and settling time.	State measurement points, loads, stroke positions and acceptance conditions.

07 / FAQ

Applying offset-load calculations

Is a 20 kg payload specification enough to select an actuator?

No. Include mounting orientation, center-of-gravity position, speed, acceleration, process forces and accuracy requirements. The same mass creates different guide demands with different lever arms and motion.

Are moment checks needed when the mass is directly above the carriage?

Yes. Even with zero x and y offset, a center-of-gravity height z lets x- or y-direction inertia generate pitch or roll. Forces acting elsewhere require separate moment calculations.

Can permissible static moments be used directly for continuous reciprocation?

Not as a complete assessment. Motion also requires actual guide-load distribution, dynamic life, operating-condition and stiffness checks.

Does a larger motor increase offset-load capacity?

A motor affects drive capability and motion performance. It does not automatically change guide spacing, block ratings or structural stiffness. Offset-load capacity follows the guide and structural configuration.

Is selection complete if each maximum moment is below its catalog value?

No. Check simultaneous forces and moments, static conditions, dynamic life, drive capacity and working-point displacement. Separate component comparisons are only a preliminary check.