# Intelligent+method+of+determining+dimension+of+mortise+and+tenon+joint+based+on+parameterization

```北

2021 年 3 月

Vol. 43，No. 3
JOURNAL OF BEIJING FORESTRY UNIVERSITY
Mar. ，2021
DOI:10.12171/j.1000−1522.20200104

(1. 南京工业职业技术大学艺术设计学院，江苏 南京 210046；2. 南京林业大学家居与工业设计学院，江苏 南京 210037)

Guan Huiyuan. Intelligent method of determining dimension of mortise and tenon joint based on parameterization[J]. Journal of
Beijing Forestry University, 2021, 43(3): 145−154.
Intelligent method of determining dimension of mortise and
tenon joint based on parameterization
Tang Lin1
Guan Huiyuan2
(1. College of Art and Design, Nanjing Vocational University of Industry Technology, Nanjing 210046, Jiangsu, China;
2. College of Furnishings and Industrial Design, Nanjing Forestry University, Nanjing 210037, Jiangsu, China)
Abstract: [Objective] The determination of dimensions of mortise and tenon joint is the premise of editing
NC(numerical control) codes of mortise and tenon joint. It is also the foundation of realizing CNC (computer
numerical control) machining of mortise and tenon joint. But the existing method of determining the
dimensions of mortise and tenon joint by modifying the geometric models repeatedly is low efficiency and
high difficulty, which has seriously affected the development of digital machining of mortise and tenon joint.
Therefore, it is necessary to design an intelligent method of determining dimension of mortise and tenon
joint. [Method] This paper takes a modified lattice shoulder tenon as example. In the first, the mathematical
models of lattices shoulder tenon were established and the dimension parameters of latticed shoulder tenon

146

were extracted by parametric design ideas. Secondly, the correlation functions between dimension
parameters of lattice tenon were established based on assembly constraints, which included two perspectives:
position relation and match relation. Then, the correlation functions between dimension parameters and
process constraints were established, and value ranges and preset values of part dimensions were settled
based on process constraints, which included three aspects: NC machining technology, market research and
process knowledge. Finally, taking the end dimensions of parts as the input parameters, integrating with
associated functions, value ranges, and preset values, the relevant parameter function table was established.
[Result] By establishing the mathematical model of mortise and tenon joint, the dimension parameters of
mortise and tenon joint were extracted. According to the assembly and process constraints of mortise and
tenon joint, the correlation functions between dimension parameters were established successfully, and the
value ranges and the default values were obtained. By founding the correlation function table of dimension
parameters, the system can automatically output other dimension parameters by just inputting the crosssection dimensions of the mortise and tenon parts. [Conclusion] The intelligent method of determining
dimension of mortise and tenon joint was established on the combination of assembly and process
constraints, that was also the scientific arrangement and reuse of process knowledge. This method can not
only help to realize the intelligent determination of dimension, but also provide basic condition for
intelligent manufacturing of mortise and tenon joints, even solid wood furniture processing.
Key words: mortise and tenon joint; dimension design; parameterization; intelligent determination;
assembly constraint; process constraint

。但榫卯加工的
[1−2]

[3]

（1）全面综合的原则

（2）手动输入参数尽可能少的原则

[4]

[5]

，实现了仅需要输入榫卯零件的断面尺寸，
[6−7]

1 榫卯尺寸智能确定方法设计原则

（3）定量与定性方法相结合的原则

147
（4）模块化的原则

2 榫卯尺寸智能确定方法设计流程
3.1

，建立榫卯数学模型，提取榫卯

[10−11]

3 榫卯尺寸智能确定方法实施
Z 轴正方向，材料长度方向在 X 轴正方向，材料厚度

Establishing a parametric mathematical
model of mortise and tenon

Extracting dimension parameters of mortise and tenon

Based on assembly constraints

Establishing relation functions based on assembly constraints

Designing association of dimension
parameters based on position relation

Designing association of dimension
parameters based on match relation

Based on machining technology

Establishing and optimizing relation functions based on process constraints

Optimizing ranges of parameters
based on market research results

Constraining dimension parameters
based on NC machining

Optimizing dimension parameters
based on process knowledge

Relevant function table of dimension parameters

Determining the dimension

Fig. 1 Intelligent dimension determination flow of mortise and tenon joints

148

W3

W1
G
A W2
W3
L
B
W4

Lm
Wm
Z
u3
X
Dm
u1
Y
a
u2
Wm 为榫眼材料断面宽度；Dm 为榫眼材料断面厚度；Lm 为榫眼中心

W3 为椭圆榫眼左右边距；W4 为榫眼下偏置（椭圆榫眼下边线于材

thickness of mortise material, Lm is the distance between the center of
mortise and the end face in X direction, G is the length of breach, A is
the wide of mortise, B is the length of mortise, L is the depth of
mortise, W1 is the thickness of triangular mortise, W2 is the offset of
mortise (distance between the border-bottom of triangular mortise and
border-top of mortise), W3 is the margins of mortise, and W4 is the
under offset of mortise (distance between the border-bottom of mortise
and the underside of material).

Fig. 2
l
1
Ds
Ls
X
Y
Ws 为榫头材料断面宽度；Ds 为榫头材料断面厚度； Ls 为材料端面到

material, Ls is the distance between the end face and the bottom of
tenon, a is the thickness of tenon, b is the width of tenon, l is the length
of tenon, u1 is the thickness of triangular tenon, u2 is the offset of
tenon, and u3 is the margins of tenon.

Fig. 3 Geometrical model of modified lattice shoulder tenon
Geometrical model of modified lattice shoulder mortise
m
Ws
Z



L − 0.5G≤x≤Lm + 0.5G


 m
−0.5G≤y≤0
f (x, y, z) = 


W − W ≤z≤D
b

(1)
m



−u1 ≤x≤0



−Ls ≤y≤ − Ls + 0.5Ws
f (x, y, z) = 
(3)


0≤z≤W
s

f (x, y, z) =


[x − (Lm − 0.5 (B − A))]2 + [z − (Wm − W1 − W2 − 0.5A)]2






≥(0.5A)2





2
2



[x − (Lm + 0.5 (B − A))] + [z − (Wm − W1 − W2 − 0.5A)]
2


≥(0.5A)





L

m − 0.5G + W3 ≤x≤Lm + 0.5G − W3





−L≤y≤0



Wm − W1 − W2 − A≤z≤Wm − W1 − W2
(2)

f (x, y, z) =
−u − u − a &lt; x≤ − u − u ,

1
2
1
2




−L
≤y≤
−
L
+
l

s
s




[x − (−u1 − u2 − 0.5a)]2 + [z − (u3 + 0.5a)]2 ≤(0.5a)2




[x − (−u1 − u2 − 0.5a)]2 + [z − (Ws − u3 − 0.5a)]2 ≤(0.5a)2
(4)

u2 为榫头偏置；u3 为椭圆榫头上下边距。
3.2

L 为椭圆榫眼的深度；W2 为榫眼偏置（三角肩缺下边

149

Tab. 1

Component
name

Cross-section
dimension parameter

Mortise
part

Material
width
(W m )

Material
thickness
( Dm )

Tenon
part

Material
width
(Ws)

Material
thickness
( Ds )
Parameter lists of lattice shoulder tenon

Location dimension parameter

Location of
mortise
( Lm )

Form size parameter

Thickness of
triangular
mortise (W1)

Offset of
mortise
(W2 )

Margin
between
bottom (W3 )

Under
offset
(W4 )

Thickness of
triangular
tenon (u1)

Offset of
tenon
(u2)

Margin
between
bottom (u3)

Back
offset
(u4)

Positional relationship

W3 = 0.5 (G − B)
(5)

W4 = Wm − W1 − W2 − A

Width of
mortise
(A)

Thickness
of tenon
(a)

Length of
mortise
( B)

Width of
tenon
(b )

Depth of
mortise
(L )

Length
of tenon
(l )

Assembly relationship

Length of
breach
(G )

Match relationship

Parallel

Interference fit

Vertical

Transition fit

Skew

Clearance fit

Coplane
(6)

Coaxial

u3 = 0.5(Ws − b)
(7)
……

u 4 = Ds − u 1 − u 2 − a
Fig. 4 Assembly relations in mortise and tenon joints
(8)

Coplane
3.3 基于装配约束的榫卯尺寸关联设计
3.3.1 榫卯构件的装配关系分析

Coaxial

Coplane

3.3.2 基于榫卯位置关系的尺寸参数关联设计

。在
[19]

Fig. 5 Position matching relationship of mortise-tenon
(taking lattice shoulder tenon for example)

G = Ws
(9)
W1 = u1
(10)

l≤Dm
(11)

150

u3

ΔB

L = Dm − 2
W3
W2
(12)
u2

B

b
W1 = u1

ΔA

3.3.3 基于榫卯配合关系的尺寸参数关联设计

a

A

A 为榫眼宽度，B 为榫眼长度，L 为榫眼深度，a 为榫头厚度，b 为榫

mortise，B is the length of mortise，L is the depth of mortise，a is the
thickness of tenon, b is the width of tenon， l is the length of mortise,
ΔA is the fitting parameter of the thickness of mortise，ΔB is the fitting
parameter of the width of mortise, and ΔL is the fitting parameter of the
length of mortise.

Fig. 6 Matching relationship and related dimension

parameters of lattice shoulder tenon

ΔL 可设置系统预设值为−2 mm。

A = a + ∆A
(13)
B = b + ∆B
(14)

L = l + ∆L
(15)

3.4

u2 = W2 − 0.5∆A
(16)
u3 = W3 − 0.5∆B
(17)

u2 和榫头上下边距 u3 以及配合参数 ΔA 和 ΔB，就可

151
(A≥d) ∩ (u2 ≥d)

3.4.2

(18)

3.4.1 数控加工方式对榫卯零件尺寸约束的分析

46 件，桌案类家具 21 件，箱柜类家具 28 件，床榻类

。例如，改良型格肩榫，就必须满足榫孔宽度

[23]

Tab. 2

Common size range of lattice shoulder tenon

Wm ，Ws
13 ~ 74
24、30、32、42
Dm ， Ds
9 ~ 65
28、30、35、42
u1，W1
4 ~ 12，以5 ~ 8内居多
4−12，far more prevalent in 5−8
6
a
4 ~ 13，以6 ~ 8内居多
4−13，far more prevalent 6−8
6、7、8
b
13 ~ 74
28、32、33
u2
6~8
8

13≤W ≤74

s




9≤D
≤65

s



4≤a≤13






13≤b≤74
4≤W1 ≤12

3.4.3

(19)

W1 = u1 = 6
(20)

u2 = d = 8
(21)

u3 = 0.5
(22)
b = Ws − 2u3 = Ws − 1
(23)

3.5

152

Tab. 3

Technological knowledge and equal dependent function of mortise and tenon joint

Process knowledge

Equivalent correlation function

Remark

A short tenon takes less strain
l ≥ Dm/2
l为榫头长度，Dm为榫眼厚度
l is the mortise length, and Dm is the
thickness of mortise material

Strong tenon and weak mortise
Dm/4 ≤ A ≤ Dm/3.5
A为榫眼宽度
A is the tenon width

Small lattice shoulder height is less
than half width of material
h ≤ 0.5Dm，推荐值h = Dm/3
When h ≤ 0.5Dm, recommended value h = Dm/3
h为小格肩肩高
h is the small lattice shoulder height

Recommended value of margin
thickness is 5−6 mm
W1 = 5 mm∪W1 = 6 mm
W1为榫边距
W1 is the margin thickness

Double tenon structure has stronger
tensile force
a ≥ 25 mm，推荐双榫结构；b ≥ 40 mm，推荐双榫结构
If a ≥ 25 mm, double tenon structure is recommended; if b ≥
40 mm, double tenon structure is recommended
a为榫头厚度，b为榫头宽度
a is tenon thickness, and b is tenon
width

Tab. 4

Relevant function table of dimension parameters (taking lattice shoulder tenon for an example)

Parameter type

Input parameter from user

Preset value (system allows users to
modify the default value)

Dependent parameter

Parameter name

Value range and preset value/mm
Ws, Wm, Ds, Dm
13 ≤ Ws ≤ 74，13 ≤ Wm ≤ 74，9 ≤ Ds ≤ 65，9 ≤ Dm ≤ 65
Lm

Design value of the distance between middle of tenon and end
W1
W1 = 6，4 ≤ W1 ≤ 12
u2
u2 = 8
u3
u3 = 0.5
A
A=8
ΔA

Library of fit parameter in the mortise thickness direction
ΔB

Library of fit parameter in the mortise width direction
ΔL
ΔL = −2
G
G = Ws
B
B = Ws – 1 – 0.5ΔB
L
L= Dm – 2
u1
u1= W1
W2
W2 = u2 + 0.5ΔA
a
a = A – ΔA，4 ≤ a ≤ 13
b
B = B – ΔB，13 ≤ b ≤ 74
l
l = L – ΔL
W3
W3 = u3 + 0.5ΔB
W4
W4 = Wm – W1 – W2 – A
u4
u4 = Ds – u1 – u2 – a

153
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