8 - 1

8. CALCULATION METHODS FOR DESIGNING

8. CALCULATION METHODS FOR DESIGNING

8.1 Specification symbol list

The following symbols are required for selecting the proper servo:

T

a

: Acceleration torque [N m]

T

d

: Deceleration torque [N m]

T

Md

: Servo motor torque necessary for

[N m]

acceleration

T

Mb

: Servo motor torque necessary for

[N m]

deceleration

T

LH

: Torque applied during servo motor stop [N m]

T

L

: Load torque converted into equivalent [N m]

value on servo motor shaft

T

LM

: Load torque converted into

[N m]

equivalent value on servo
motor shaft during stop

T

U

: Unbalance torque

[N m]

T

F

: Load friction torque

[N m]

T

LO

: Load torque on load shaft

[N m]

T

rms

: Continuous effective load torque

[N m]

converted into equivalent value
on servo motor shaft

J

L

: Load inertia moment converted

[kg cm

2

]

into equivalent value on servo
motor shaft

J

LO

: Load inertia moment on load shaft

[kg cm

2

]

J

M

: Servo motor's rotor inertia moment

[kg cm

2

]

N

: Servo motor speed

[r/min]

N

o

: Servo motor speed during fast feed

[r/min]

N

LO

: Load shaft speed during fast feed

[r/min]

V

: Moving part speed

[mm/min]

V

o

: Moving part speed during fast feed [mm/min]

P

B

: Ball screw lead

[mm]

Z

1

: Number of gear teeth on servo motor shaft

Z

2

: Number of gear teeth on load gear

n

: Gear ratio

n

Z

1

Z

2

Speed reduced when n 1,
Speed increased when n 1

: Drive system efficiency

g

: Gravitational acceleration (9.8[m/s

2

])

: Friction coefficient
: Circle ratio (3.14)

P

t

: Number of feedback pulses in

[pulse/rev]

position control mode

f

: Input pulse frequency in

[pps]

position control mode

f

0

: Input pulse frequency during fast

[pps]

feed in position control mode

T

psa

: Acceleration time constant of

[s]

frequency command in
position control mode

T

psd

: Deceleration time constant of

[s]

pulse frequency command in
position control mode

K

p

: Position control gain 1

[rad/s]

T

p

: Position control time constant (Tp 1/Kp) [s]

K

v

: Speed control gain

[rad/s]

T

v

: Speed control time constant (Tv 1/Kv)

[s]

: Feed per feedback pulse

in position control mode

[mm/pulse]

o

: Feed per command pulse

in position control mode

[mm/pulse]

: Feed

[mm]

P

: Number of internal command pulses [pulse]

t

s

: Internal settling time

[s]

t

o

: Positioning time

[s]

t

c

: Time at constant speed of servo

[s]

motor in 1 cycle

t

: Stopping time in 1 cycle

[s]

: Positioning accuracy

[mm]

: Number of droop pulses

[pulse]

: Load shaft rotation angle per pulse in position

control mode

[degree/pulse]

e

: Euler constant 2.718278

S

: Feed per servo motor revolution

[mm/rev]

8 - 2

8. CALCULATION METHODS FOR DESIGNING

8.2 Position resolution and electronic gear setting

Position resolution (travel per pulse ) is determined by travel per servo motor revolution S and the
number of encoder feedback pulses Pt, and is represented by Equation 8.1. As the number of feedback
pulses depends on the servo motor series, refer to Section 5.1.

Pt

S .............................................................................................................................................................(8.1)

: Travel per pulse

[mm]

S : Travel per servo motor revolution

[mm/rev]

Pt : Number of feedback pulses

[pulse/rev]

Since

has the relationship represented by Equation 8.1, its value is fixed in the control system

after the drive system and encoder have been determined. However, travel per command pulse can
be set as desired using the parameters.

CMX

CDV

SM

P

t

8192pulse/rev

Command
pulse train f

0

Electronic gear

CMX

CDV

Deviation

counter

Encoder

As shown above, command pulses are multiplied by CMX/CDV set in the parameters to be position
control pulses. Travel per command pulse

is expressed by Equation 8.2:

o PtS

CMX

CDV

CMX

CDV .......................................................................................................................(8.2)

CMX: Electronic gear (Command pulse multiplication numerator)
CDV: Electronic gear (Command pulse multiplication denominator)

Using the above relationship, travel per command pulse can be set to a value without fraction.

[Setting example]
Find a parameter value for

o 0.01 [mm] in a drive system where ball screw lead PB 10 [mm] and

reduction ratio 1/n 1.
The encoder feedback pulses Pt of the HC MF 8192 [pulses/rev].
Since s 10 [mm/rev], the following is obtained according to Equation 8.2:

CMX

CDV o

Pt

S 0.01

8192

10

1024

125

<Relationship between position resolution and overall accuracy>
Positioning accuracy of machine is the sum of electrical errors and mechanical errors. Normally,
provisions should be made so that positioning accuracy are not affected by electrical system errors. As a
guideline, Equation 8.3 should be satisfied:

15

1

10

D .........................................................................................................................................(8.3)

where,

:

Travel per feedback pulse [mm/pulse]

:

Positioning accuracy [mm]

8 - 3

8. CALCULATION METHODS FOR DESIGNING

8.3 Speed and command pulse frequency

The servo motor is run at a speed where the command pulses and feedback pulses are equivalent.
Therefore, the command pulse frequency and feedback pulse frequency are equivalent. The relation
including the parameter settings (CMX, CDV) is as indicated below (refer to the following diagram):

f0 CMX

CDV Pt

No

60

.................................................................................................................................... (8.4)

CMX

CDV

f

0

Electronic

gear

Feedback pulse
frequency

Servo motor

f0

:Command pulse frequency [pps]

CMX

:Electronic gear
(Commanded pulse multiplication numerator)

CDV

:Electronic gear
(Commanded pulse multiplication denominator)

No

:Servo motor speed [r/min]

Pt

:Number of feedback pulses [pulses/rev]
(Pt 8192 for HC-MF)

According to Equation 8.4, the following equations may
be used to obtain the electronic gear and command
pulse frequency to rotate the servo motor at No.

Electronic gear

CMX

CDV Pt

No

60

1

f0 ...........................................................................................................................................(8.5)

Command pulse frequency

f0 Pt No

60

CDV

CMX

...........................................................................................................................................(8.6)

[Setting example]
Obtain the command pulse frequency required to run the HC-MF at 3000r/min.
When the electronic gear ratio 1 (initial parameter value) is used, the following result is found according
to Equation 8.6:

f0 8192 No

60

CDV

CMX

(Command pulse frequency)

8192

3000

60

1

409600[pps]

However, as the maximum input command pulse frequency in the open collector system is 200kpps, for
general-purpose servo 409600pps cannot be entered.
To run the servo motor at the speed of 3000r/min at not more than 200kpps, the electronic gear setting
must be changed. This electronic gear is found by Equation 8.5:

CMX

CDV 8192

3000

60

1

200 10

3

(Electronic gear)

256

125

Therefore, the parameters are set to CMX 256 and CDV 125.

8 - 4

8. CALCULATION METHODS FOR DESIGNING

8.4 Stopping characteristics

(1) Droop pulses ( )

When a pulse train command is used to run the servo motor, there is a relationship between the
command pulse frequency and servo motor speed as shown in the figure. The difference between the
command pulses and feedback pulses during acceleration are called droop pulses, which are
accumulated in the servo amplifier's deviation counter. Equation 8.7 defines a relationship between
the command pulse frequency (f) and position control gain 1(Kp).

fo

Kp[pulse] ...............................................................................................................................................(8.7)

Supposing that the value of position control gain 1 is 70 [rad/s], the droop pulses during operation will
be as follows at the command pulse frequency of 200 [kpps] according to Equation 15.1:

1

200 10

3

2858[pulse]

Com

m

and p

u

ls

e

fr

eq

u

e

n

cy f

S

e

rv

o m

o

to

r speed

[pps]

[r/min]

0

Time

Command Droop pulses

Servo motor
speed

T

psa

T

psd

t

s

t

s

3

1

70

0.04

T

p

(2) Settling time (ts) during linear acceleration/deceleration

Since droop pulses still exist when there are no command pulses, settling time (ts) is required until the
servo motor stops. Set the operation pattern in consideration for the settling time.
The ts value is obtained according to Equation 8.8:

ts 3 Tp

3

1

Kp[s] ....................................................................................................................................................(8.8)

*When Kp 70 [rad/s], ts 0.04 [s]. (Refer to the above diagram.)

The settling time (ts) indicates the time required for the servo motor to stop in the necessary
positioning accuracy range. This does not always mean that the servo motor has stopped completely.
Thus, especially when the servo motor is used in high-duty operation and positioning accuracy has no
margin for travel per pulse (

), the value obtained by Equation 8.8 must be increased.

ts will vary with the moving part conditions. Especially when the load friction torque is large,
movement may be unstable near the stopping position.

8 - 5

8. CALCULATION METHODS FOR DESIGNING

8.5 Capacity selection

As a first step, confirm the load conditions and temporarily select the servo motor capacity. Then,
determine the operation pattern, calculate required torques according to the following equations, and
check that the servo motor of the initially selected capacity may be used for operation .

(1) Initial selection of servo motor capacity

After calculating the load torque (T

L

) and load inertia moment (J

L

), select a servo motor which will

satisfy the following two relationships:
Servo motor's rated torque T

L

Servo motor J

M

J

L

/m

m 3

: High duty (more than 100 times/min.)

Settling time 40ms or less

m 5

: Middle duty (60 to 100 times/min.)

Settling time 100ms or less

m permissible load inertia moment

: Low duty (less than 60 times/min.)

Settling time more than 100ms

Find the acceleration and deceleration torques and continuous effective load torque as described in (2)
to make a final selection. For high-duty positioning, the JL value should be as small as possible. If
positioning is infrequent as in line control, the JL value may be slightly larger than in the above
conditions.

(2) Acceleration and deceleration torques

The following equations are used to calculate the acceleration and deceleration torques in the
following operation pattern:

Com

m

and pul

se

S

e

rv

o m

o

to

r s

peed [

r/

m

in

]

0

0

Nofo

Time

Time

Command

Servo motor
speed

T

psa

T

psd

T

a

T

d

Deceleration
torque

Acceleration
torque

f

requency f

[

p

p

s

]

Acceleration torque T

a

9.55 10

4

(J

L

J

M

) No

T

psa

1

........................................................................................(8.9)

Deceleration torque T

b

9.55 10

4

(J

L

J

M

) No

T

psd

1

.....................................................................................(8.10)

8 - 6

8. CALCULATION METHODS FOR DESIGNING

(3) Torques required for operation

Torques required for the servo motor are the highest during acceleration. If any of the torques
obtained with Equations 8.11 to 8.13 exceeds the maximum servo motor torque, the servo motor speed
cannot be increased as commanded. Confirm that the calculated value is lower than the servo motor's
maximum torque. Since a friction load is normally applied during deceleration, only the acceleration
torque needs to be considered. In the regenerative mode, the value found by Equation 8.13 is negative.

S

e

rvo

m

o

to

r t

o

rque

0

Time

T

Ma

T

Md

T

1

T

2

T

L

T

a

T

d

f

re

q

u

e

ncy

f [

pps]

S

e

rvo

m

o

to

r s

p

ee

d

[r

/m

in

]

0

Nofo

Time

Command

Servo motor
speed

T

psa

T

psd

C

o

mma

n

d

p

u

lse

T

1

T

Ma

Ta T

L

.................................................................................................................................... (8.11)

T

2

T

L

........................................................................................................................................................ (8.12)

T

3

T

Md

T

d

T

L

.................................................................................................................................. (8.13)

8 - 7

8. CALCULATION METHODS FOR DESIGNING

(4) Continuous effective load torque

If the torque required for the servo motor changes with time, the continuous effective load torque
should be lower than the rated torque of the servo motor. There may be a servo motor torque delay at
the start of acceleration or deceleration due to a delay in the control system. To simplify the
calculation, however, it is assumed that constant acceleration and deceleration torques are applied
during Tpsa and Tpsd. The following equation is used to calculate the continuous effective load torque
in the following operation pattern. T

LH

indicates the torque applied during a servo motor stop. A large

torque may be applied especially during a stop in vertical motion applications, and this must be fully
taken into consideration. During vertical drive, the unbalanced torque TU will become T

LH

.

S

e

rvo

m

o

to

r

speedTi

m

e

S

e

rv

o

m

o

to

r to

rq

u

e

N

[r/min]

0

Time

0

Time

T

psa

t

c

T

psd

t

f

(1 cycle)

T

Ma

T

Md

T

d

T

a

T

L

T

LH

t

Trms

tf

T

Ma

T

psa

T

L

tc T

Md

T

psd

T

LH

t

2

2

2

2

.................................................................................(8.14)

8 - 8

8. CALCULATION METHODS FOR DESIGNING

8.6 Load torque equations

Typical load torque equations are indicated below:

Type

Mechanism

Equation

Linear

movement

Servo motor

W

Z

1

F

C

F

G

Z

2

T

L

2 10

3

F

N

V

2 10

3

F

S

..................................(8.15)

F : Force in the axial direction of the machine in linear motion [N]
F in Equation 15.9 is obtained with Equation 8.16 when the table
is moved, for example, as shown in the left diagram.

F Fc

(W g F

G

) ........................................................(8.16)

Fc : Force applied in the axial direction of the moving part [N]
F

G

: Tightening force of the table guide surface [N]

W : Full weight of the moving part [kg]

Rotary

movement

Servo motor

Z

1

Z

2

T

LO

T

L

n

1 1

T

LO

T

F

...........................................................(8.17)

T

F

: Load friction torque converted into equivalent value on servo

motor shaft [N m]

Vertical

movement

Counter

weight

Servo motor

W

2

W

1

1/n

Load

Guide

During rise
T

L

T

U

T

F

.........................................................................(8.18)

During fall
T

L

T

U

2

T

F

................................................................(8.19)

T

F

: Friction torque of the moving part [N m]

T

U

=

2 10

3

(W

1

W

2

)

N

V

2 10

3

g

S

(W

1

W

2

) g

...........................(8.20)

T

F

=

10

3

2

g

(W

1

W

2

)

S .........................................................(8.21)

W

1

: Weight of load [kg]

W

2

: Weight of counterweight [kg]

8 - 9

8. CALCULATION METHODS FOR DESIGNING

8.7 Load inertia moment equations

Typical load inertia moment equations are indicated below:

Type

Mechanism

Equation

Axis of rotation is on the cylinder
center

Axis of rotation

L

D

1

D

2

J

LO

=

32

L (D D )

4
1

4
2

8

W (D D )

2
1

2
2

..........................(8.22)

ρ

: Cylinder material density [kg/cm

3

]

L : Cylinder length [cm]
D

1

: Cylinder outside diameter [cm]

D

2

: Cylinder inside diameter [cm]

W : Cylinder weight [kg]

Reference data: material density
Iron

: 7.8 10

3

[kg/cm

3

]

Aluminum : 2.7 10

3

[kg/cm

3

]

Copper

: 8.96 10

3

[kg/cm

3

]

Cylinder

Axis of rotation is off the cylinder

Axis of rotation

D

R

center

J

LO

8

W

(D 8R )

2

2

..............................................................(8.23)

Square block

Axis of rotation

R

a

a

b

b

J

LO

W

3

a

2

b

2

R

2

.........................................................(8.24)

W

: Square block weight [kg]

a, b, R : Left diagram [cm]

Object which

moves linearly

Servo motor

V

W

N

J

L

W

600

V

2

1

W

N 10

V

2

W

20

2

S

..............(8.25)

V

: Speed of object moving linearly [mm/min]

S

: Moving distance of object moving linearly per servo

motor revolution [mm/rev]

W

: Object weight [kg]

Object that is

hung with pulley

Servo motor

W

D

J

L

W

2

D

2

Jp ..................................................................(8.26)

J

P

: Pulley inertia moment [kg cm

2

]

D

: Pulley diameter [cm]

W

: Object weight [kg]

Converted load

Load A

J

A

J

31

J

B

N

3

J

21

J

11

J

22

N

2

N

1

J

L

J

11

(J

21

J

22

J

A

)

N

1

N

2

2

(

J

31

J

B

)

N

1

N

3

2

................... (8.27)

J

A

, J

B

: Inertia moments of loads A, B [kg cm

2

]

J

11

to J

31

: Inertia moments [kg cm

2

]

N

1

to N

3

: Speed of each shaft [r/min]

8 - 10

8. CALCULATION METHODS FOR DESIGNING

8.8 Precautions for zeroing

When a general positioning unit is used, the sequence of events is as shown in the following figure.

ON

Actuator signal

When determining the ON duration of the
actuator, consider the deceleration time so
that the speed reaches the creep speed.

Considering the variations of the
actuator signal, adjust the actuator
so that it switches off near the center
of the High of the zero pulse signal.

Zero pulse signal

Zeroing speed V

1

Deceleration started by actuator signal

Creep speed V

2

OFF

(1) When determining the ON duration of the actuator, consider the delay time of the control section and

the deceleration time so that the creep speed is attained. If the actuator signal switches off during
deceleration, precise home position return cannot be performed.

ON duration of the actuator L

D

Deceleration time t

d

Zeroing speed V

1

Travel distance gained after
detecting the zeroing dog until
the creep speed is reached L

1

Control delay
time t

1

Creep speed V

2

Travel distance L1 in the chart can be obtained by Equation 8.28

L

1

1

60

V

1

t

1

1

120

V

1

t

d

1

V

1

V

2

2

1

60

V

1

T

p

........................................................................ (8.28)

ON duration of the actuator LD [mm] must be longer than L1 obtained by Equation 8.28, as indicated
in Equation 8.29
L

D

L

1

...................................................................................................................................................... (8.29)

where,
V

1,

V

2

: As shown in the chart [mm/min]

t

1,

t

d

: As shown in the chart [s]

L

1

: As shown in the chart [mm]

L

D

: As shown in the chart [mm]

8 - 11

8. CALCULATION METHODS FOR DESIGNING

(2) Set the end (OFF position) of the actuator signal at the middle of two ON positions (Lows) of the zero

pulse signal. If it is set near either ON position of the zero pulse signal, the positioning unit is liable to
misdetect the zero pulse signal. In this case, a fault will occur, e.g. the home position will shift by one
revolution of the servo motor.
The zero pulse output position can be confirmed by OP (encoder Z-phase pulse) on the external I/O
signal display.

8.9 Selection example

Speed of moving part during fast feed

Vo

30000[mm/min]

Command resolution

10[ m]

Travel

400[mm]

Positioning time

to

within 1[s]

Number of feeds

40[times/min]

Operation cycle

tf

1.5[s]

Gear ratio

n

8/5

Moving part weight

W

60[kg]

Drive system efficiency

0.8

Friction coefficient

0.2

Ball screw lead

Pb

16[mm]

Ball screw diameter

20[mm]

Ball screw length

500[mm]

Gear diameter (servo motor)

25[mm]

Gear diameter (load shaft)

40[mm]

Machine specifications

Servo
motor

Servo

amplifier

Gear ratio 5:8

Gear face width

10[mm]

(1) Selection of control parameters

Setting of electronic gear (command pulse multiplication numerator, denominator)
There is the following relationship between the multiplication setting and travel per pulse

∆

.

8192 (gear ration)

(ball screw lead)

CDV

CMX

When the above machining specifications are substituted in the above equation:

CDV

CMX

10 16 1000

8192 8/5

1000

8192

Acceptable as CMX/CDV is within 1/20 to 20.

(2) Servo motor speed

No

Pb

Vo n

16

30000

5

8

3000[r/min]

8 - 12

8. CALCULATION METHODS FOR DESIGNING

(3) Acceleration/deceleration time constant

T

psa

T

psd

to

Vo/60

ts 0.05[s]

*ts: settling time.(Here, this is assumed to be 0.15s.)

(4) Operation pattern

S

e

rv

o m

o

to

r s

p

e

e

d

[r/min]

3000

Time

0.05

t

o

1.0

t

f

1.5

0.05

t

s

T

psa

T

psd

0.15

[s]

(5) Load torque (converted into equivalent value on servo motor shaft)

Travel per servo motor revolution

S P

b

1

n 16

5

8

10[mm]

T

L

2 10

3

W g

S

2 10

3

3.14

0.2 60 9.8 10

0.8

0.23[N m]

8 - 13

8. CALCULATION METHODS FOR DESIGNING

(6) Load inertia moment (converted into equivalent value on servo motor shaft)

Moving part

J

L1

W

20

2

S

1.52[kg cm

2

]

Ball screw

J

L2

32

L D

4

n

2

1

0.24[kg cm

2

]

* 7.8 10

3

[kg cm

3

]

Gear (servo motor shaft)

J

L3

32

L

D

4

0.03[kg cm

2

]

Gear (load shaft)

J

L4

32

L

D

4

n

2

1

0.8[kg cm

2

]

Full load inertia moment (converted into equivalent value on servo motor shaft)

J

L

J

L1

J

L2

J

L3

J

L4

2.59[kg cm

2

]

(7) Temporary selection of servo motor

Selection conditions
1) Load torque

servo motor's rated torque

2) Full load inertia moment 30 servo motor inertia moment

From the above, the HC-MF23 (200W) is temporarily selected.

(8) Acceleration and deceleration torques

Torque required for servo motor during acceleration

T

Ma

(J

L

J

M

) No

9.55 10

4

T

psa

T

L

1.7[N m]

Torque required for servo motor during deceleration

T

Md

(J

L

J

M

) No

9.55 10

4

T

psd

+ T

L

1.2[N m]

The torque required for the servo motor during deceleration must be lower than the servo motor's
maximum torque.

8 - 14

8. CALCULATION METHODS FOR DESIGNING

(9) Continuous effective load torque

Trms

tf

T

T

psa

T tc

T

T

psd

2

2

2

Ma

L

Md

0.42[N m]

The continuous effective load torque must be lower than the servo motor's rated torque.

(10) Torque pattern

To

rq

ue

1.7

[N m]

0.23

1.2

Time [s]

0.05

0.75

0.05

1.5

0.15

(11) Selection results

The HC-MF23 servo motor and MR-J2-20A servo amplifier are selected.
1) Electronic gear setting

CMX 8192
CDV 1000

2) During rapid feed

Servo motor speed

N

o

3000 [r/min]

3) Acceleration/deceleration time constant

T

psa

T

psd

0.05[s]