Linear Motor Theory

Theory & Formulas

Core formulas

Cruise (holding) force:

F_L = μ · (M_L + M_p) · g + F_n

Achievable acceleration time from rated peak force:

T_a* = (M_L + M_p) · V_m · K_p / (F_pk − F_L)

Acceleration force:

F_a = (M_L + M_p) · V_m / T_a + F_L

Deceleration force:

F_d = (M_L + M_p) · V_m / T_c − F_L

RMS continuous force over one cycle:

F_rms = √[ (F_a²·T_a + F_L²·T_b + F_d²·T_c + F_w²·T_w) / (T_a + T_b + T_c + T_w) ]

Apply safety factors:

F_req,cont = K · F_rms

Symbols

M_L	Load mass
M_p	Mover mass
S	Required stroke
V_m	Max velocity
T_a, T_b, T_c	Accel / cruise / decel time
T_w	Cycle / dwell time
μ	Friction coefficient
g	Gravity
F_n	Cable-chain drag force
K	Safety factor

Worked example

Load 10 kg …

Verification criteria

Stroke:

Time:

Peak:

Continuous:

Servo Drive / Amplifier Sizing

From the motor force constant, resistance and inductance plus the required forces, estimate the amplifier currents, back-EMF at speed, required DC bus voltage and power.

Amplifier / servo sizing equations

I = F / K_f → I_pk = F_pk / K_f , I_rms = F_rms / K_f

E = K_f · V_m (back-EMF at max speed; K_f = K_e in SI)

V_bus,req = E + I_pk·R + 2π·f_c·L·I_pk

P_wind = I_rms² · R (copper loss, per phase) ; P_amp ≈ V_bus·I_rms

Linear Guide / Bearing

Select a linear guideway and compute its L10 rating life from the dynamic load rating, type, load factor and the equivalent load in your application.

L10 rating life equations

F_eq = f_s · (F_process + M · a_peak)

L₁₀ = base · (C / (f_w·F_eq))^p (ball: p=3, base=50 km; roller: p=10/3, base=100 km)

N_cycles = L₁₀·1000 / (2·S) ; t_life = N_cycles / (cycles per hour)

Method consolidated from supplier notes …

Linear Motor Selector · educational engineering tool