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No6, 1981.

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Foundations,” Ph.D. Dissertation, University of California, Berkeley, 1998 , pp. 88-101

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表1 均質土壤性質及半無限空間

ONE-PILE IMPEDANCE-SOLID ELEMENT

Layer Hight Depth Unit S-Wave P-Wave S-Wave P-Wave

Weight Velocity Velocity Damping Ratio Damping Ratio

Halfspace = 10 112.700 300.00 734.85 0.050 0.050

1 2.00 2.00 112.700 300.00 734.85 0.050 0.050

2 2.00 4.00 112.700 300.00 734.85 0.050 0.050

3 2.00 6.00 112.700 300.00 734.85 0.050 0.050

4 2.00 8.00 112.700 300.00 734.85 0.050 0.050

5 2.00 10.00 112.700 300.00 734.85 0.050 0.050

6 2.00 12.00 112.700 300.00 734.85 0.050 0.050

7 2.00 14.00 112.700 300.00 734.85 0.050 0.050

8 2.00 16.00 112.700 300.00 734.85 0.050 0.050

9 2.00 18.00 112.700 300.00 734.85 0.050 0.050

10 2.00 20.00 112.700 300.00 734.85 0.050 0.050

11 2.00 22.00 112.700 300.00 734.85 0.050 0.050

12 2.00 24.00 112.700 300.00 734.85 0.050 0.050

13 2.00 26.00 112.700 300.00 734.85 0.050 0.050

14 2.00 28.00 112.700 300.00 734.85 0.050 0.050

15 2.00 30.00 112.700 300.00 734.85 0.050 0.050

表2 土壤與半無限空間之性質

單位重

γ

_s =112.7pcf

揚氏係數

E

= 8.82 10 ×

⁵

psf

剪力波波速 V_s =300 / secft

包松比

ν

_s =0.4

材料阻尼

β

=0.05

土層厚度 H =2ft

表3 樁的性質

揚氏係數

E

= 8.82 10 ×

⁸

psf

單位重

γ

_s =161pcf

包松比

ν

_p =0.25

材料阻尼

β

= 0

樁長 L=30ft

樁直徑 D=2.0ft

表4 單一樁運算時間

Solid Pile Method Sectional Pile Method Simplified Pile Method 互制點數運算時間(秒) 互制點數運算時間(秒) 互制點數運算時間(秒) 單一樁

129 51.6 129 51.3 1 27.6

表5 2X2 群樁運算時間

SolidPileMethod Sectional Pile Method 縮減互制點 SimplifiedPile Method 2X2 群樁

互制點數運算時間(秒) 互制點數運算時間(秒) 互制點數運算時間(秒) 互制點數運算時間(秒)

S/D=2 516 1087 516 1044 265 46.1 4 29.5

S/D=5 516 1010 516 1020 4 32.2

S/D=10 516 1016 516 998 4 55.8

表6 3X3 群樁運算時間

Solid Pile Method Sectional Pile Method 縮減互制點 Simplified Pile Method 3X3 群樁

互制點數運算時間(秒) 互制點數運算時間(秒) 互制點數運算時間(秒) 互制點數運算時間(秒)

S/D=2 1161 4389 1161 4249 409 660 9 30.3

S/D=5 1161 9808 1161 9780 9 48.7

S/D=10 1161 9606 1161 9398 9 48.5

表7 2X2 四分之一模型群樁運算時間

Solid Pile Method Sectional Pile Method Simplified Pile Method 2X2 群樁

互制點數運算時間(秒) 互制點數運算時間(秒) 互制點數運算時間(秒)

S/D=2 129 98.9 129 95.7 1 28.5 S/D=5 129 82.4 129 80.3 1 28.7 S/D=10 129 75.1 129 74.1 1 27.6

表8 鋼管樁的性質

類型鋼管樁

外徑

26.70mm

內徑

20.93mm

慣性矩 15527mm ⁴

面積 216mm ⁴

長度

1060mm

楊氏係數 200000 /N mm ²

波松比

0.3

阻尼比

0.005

單位重 7.68 10× ⁻⁵N mm/ ³

表9 試驗機性質

質量慣性矩質量底部至中心

高度( )m

中心至頂端

高度( )m X Y Z

20 片鋼版 1532.0 0.247 0.343 94.14 116.06 156.2 垂直反應 12 片鋼版 1096.4 0.200 0.272 65.14 87.06 0 水平反應

表10 102 根群樁之均質土壤性質及半無限空間

ONE-PILE IMPEDANCE-PILE ELEMENT

Layer Hight Depth Unit S-Wave P-Wave S-Wave P-Wave

Weight Velocity Velocity Damping RatioDamping Rati

Halfspace = 10 18200.000 188.00 351.72 0.010 0.010

1 0.10 0.10 17100.000 40.00 74.83 0.010 0.010

2 0.05 0.15 17100.000 55.00 102.90 0.010 0.010

3 0.15 0.30 17100.000 70.00 130.96 0.010 0.010

4 0.20 0.50 17100.000 80.00 139.67 0.010 0.010

5 0.25 0.75 17100.000 90.00 168.38 0.010 0.010

6 0.25 1.00 17100.000 100.00 187.08 0.010 0.010

7 0.25 1.25 18200.000 125.00 233.85 0.010 0.010

8 0.25 1.50 18200.000 125.00 233.85 0.010 0.010

9 0.25 1.75 18200.000 125.00 233.85 0.010 0.010

10 0.25 2.00 18200.000 125.00 233.85 0.010 0.010

11 0.25 2.25 18200.000 170.00 318.04 0.010 0.010

12 0.25 2.50 18200.000 170.00 318.04 0.010 0.010

13 0.25 2.75 18200.000 170.00 318.04 0.010 0.010

14 0.25 3.00 18200.000 170.00 318.04 0.010 0.010

表11 分析方法之特性

分析方法互制點數分析時間分析效率準確性

Simplified Pile Method 最少最快最高群樁間距越小準確性差 Sectional Pile Method 多普通低準確並可直接求得內力 Sectional Pile Method

縮減互制點少快高適用於低頻

Solid Pile Method 多普通低最準確

HS1

(a) (b)

圖1 層狀土讓結構互置系統與簡化模式示意圖

z,w

x or y, δ

Soil (G s ,ρ s ,V s )

p z (r,θ)

q _z (r,θ) Q h

P _h Foundation (E _p , m _p , υ

, h

)

r

y

R θ x

圖2 柔性基礎-層狀土壤系統承受動態外力之示意圖

圖3 土壤位移與接觸應力於極座標系統之示意圖

（a）上視圖

（b）三維視圖

（c）單一土壤-樁元素

圖4 梁元素與八個土壤-樁元素組成整體土壤-樁[52]

（a）土壤-樁元素

（b）樁體積部份

圖5 三維土壤-樁元素[52]

（a）整體座標系統

（b）局部座標系統

圖6 土壤-元素之整體及局部座標系統[52]

（a）土壤元素的局部自由度

（b）土壤-樁元素整體自由度

圖7 土壤-樁元素在局度與整體自由度[52]

圖8 梁元素在局部座標系統[52]

(a) 整體系統 (b) 基礎 (c) 上部結構圖9 Simplifed Pile Method 模擬方式[57]

圖10 Sectional Pile Method 2X2 群樁模擬方式

圖11 Sectional Pile Method 縮減互制點法

圖12 單一樁在水平層狀系統[57]

(a) 元素在整體座標系統

(b) 元素在局部座標系統圖13 元素座標系統[57]

(a) 零簡諧振態情況之局部自由度

(b) 第一簡諧振態情況之局部自由度圖14 局部自由度[57]

(a) Love 波自由度

(b) Rayleigh 波自由度圖15 自由度[57]

圖16 傳導邊界自由度圖[57]

0 10 20 30

t(sec) -0.12

-0.08 -0.04 0 0.04 0.08 0.12

Acceleration (g)

圖17 El Centro 地震加速度歷時(最大加速度 0.1g)

0 1 2 3 4 5

a₀ 0.5

0.6 0.7 0.8 0.9 1

Absolute Value of Sz

H_s1/R=1.0 H_E/R=0.5 V_s1/V_sh=0.8 V_s1/V_sh=0.3

圖18 散射函數之絕對值

0 1 2 3 4 5 a₀

-0.1 0 0.1 0.2 0.3 0.4 0.5

Phase Angle of Sz

Hs1/R=1.0 HE/R=0.5 V_s1/V_sh=0.8 V_s1/V_sh=0.3

圖19 散射函數之相位角

0 2 4 6 8 10

t (sec) -0.12

-0.08 -0.04 0 0.04 0.08 0.12

Acceleration (g)

bz=5.0 Hs1/R=1.0 HE/R=0.5 Vs1/Vsh=0.8 SASSI

Simplified Modal

圖20 基礎加速度歷時比較圖(Vs1/Vsh=0.8)

0 2 4 6 8 10

t (sec) -0.15

-0.1 -0.05 0 0.05 0.1 0.15

Acceleration (g)

bz=5.0 Hs1/R=1.0 H_E/R=0.5 V_s1/V_sh=0.3 SASSI

Simplified Modal

圖21 基礎加速度歷時比較圖(Vs1/Vsh=0.3)

圖22 圓形基礎板於SASSI 之有限元素模型

Normalized Radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Displacement, Re[w] Gs*R

-0.1 0.0 0.1 0.2

0.3

ν

=0.33, α=0.003, b

=0, Vertical Loading

Normalized Radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

(Welded Contact Boundary)

ν

=0.33, α=0.003, b

=0, Vertical Loading

Normalized Radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

Normal ize d Dis p la ce me nt, Im[w] G s *R

-0.2 -0.1 0.0 0.1

a₀=0 (Eff. Ring Method, 15 rings) a₀=1 (Eff. Ring Method, 15 rings) a₀=3 (Eff. Ring Method, 15 rings) a₀=5 (Eff. Ring Method, 15 rings)

Normalized Radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

a₀=0 (SASSI) a₀=1 (SASSI) a₀=3 (SASSI) a₀=5 (SASSI)

(Welded Contact Boundary)

圖23 圓形柔性基礎於土壤半空間承受均勻垂直載重之垂直位移剖面圖(α=0.003)

Normalized Radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Displacement, Re[w] Gs*R

-0.1 0.0 0.1

0.2

ν

=0.33, α=0.003, b

=0, Rocking Loading

Normalized Radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

(Welded Contact Boundary)

ν

=0.33, α=0.003, b

=0, Rocking Loading

Normalized Radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

Normal ize d Dis p la ce me nt, Im[w] G s *R

-0.2 -0.1 0.0 0.1

a₀=0 (Eff. Ring Method, 15 rings) a₀=1 (Eff. Ring Method, 15 rings) a₀=3 (Eff. Ring Method, 15 rings) a₀=5 (Eff. Ring Method, 15 rings)

Normalized Radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

a₀=0 (SASSI) a₀=1 (SASSI) a₀=3 (SASSI) a₀=5 (SASSI)

(Welded Contact Boundary)

圖24 圓形柔性基礎於土壤半空間承受線性翻轉垂直載重之垂直位移剖面圖(α=0.003)

a₀=0

a₀=2.69 (Efficient ring method, 10 rings) a₀=4.04 (Efficient ring method, 10 rings) a₀=5.39 (Efficient ring method, 10 rings)

(Relaxed Contact Boundary)

ν s=0.35, bz=0, α=0.035, Vertical Loading P

^0,0

=1

Normalized radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

Normal ized displ a ceme nt Re[w] / w(a

=0, r= 0 )

-0.60

ν s=0.35, bz=0, α=0.035, Vertical Loading P

^0,0

=1

0.0 0.2 0.4 0.6 0.8 1.0

Normalized radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

Normalized displacement Im[w] / w(a

=0, r=0)

-0.75

(Relaxed Contact Boundary)

圖25 圓形柔性基礎於土壤半空間承受均勻垂直載重之垂直位移剖面(α=0.035)

Normalized radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

Normalized displacement, Re[ w ] / | w

Rg,max

|

-0.2 0.0 0.2 0.4 0.6 0.8

1.0

ν s=0.33, β

=2%, bz=0, α=0.028, Rocking Loading P

^1,0

=1

0.0 0.2 0.4 0.6 0.8 1.0

(Relaxed Contact Boundary)

Normalized radius, r / R

0.0 0.2 0.4 0.6 0.8 1.0

Nor m aliz ed dis p la cem ent, Im[ w ] / | w

Rg,max

|

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

a₀=0 (Gucunski ) a₀=2 (Gucunski ) a₀=4 (Gucunski ) a₀=6 (Gucunski )

0.0 0.2 0.4 0.6 0.8 1.0

a₀=0 (Eff. Ring Method, 10 rings) a₀=2 (Eff. Ring Method, 10 rings) a₀=4 (Eff. Ring Method, 10 rings) a₀=6 (Eff. Ring Method, 10 rings)

圖26 圓形柔性基礎於土壤半空間承受翻轉垂直載重之垂直位移剖面(α=0.028)

圖27 位移影響因子模擬方式

L/D=10 K=E

/E

S/D

0 2 4 6 8 10 12 14 16 18 20 22

位移影響因子

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Simple K=100 Simple K=1000 Solid K=100 Solid K=1000 Sectional K=100 Sectional K=1000 Poulos K=100 Poulos K=1000

圖28 L/D=10 之位移影響因子

Simple K=100 Simple K=1000 Solid K=100 Solid K=1000 Sectional K=100 Sectional K=1000 Poulos K=100 Poulos K=1000

圖29 L/D=25 之位移影響因子

Simple K=100 Simple K=1000 Solid K=100 Solid K=1000 Sectional K=100 Sectional K=1000 Poulos K=100 Poulos K=1000

圖30 L/D=50 之位移影響因子

單一樁垂直勁度

無因次化頻率a₀

0.0 0.1 0.2 0.3 0.4 0.5

0 1 2 3 4

Sectional-Pile Solid-Pile Simple-Pile Kaynia1982

正規化垂直勁度

圖31 單一樁垂直勁度

單一樁垂直阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 2 4 6 8 10

Sectional-Pile Solid-Pile Simple-Pile Kaynia1982

無因次化頻率a₀

正規化垂直阻尼

圖32 單一樁垂直阻尼

單一樁水平勁度

0.0 0.1 0.2 0.3 0.4 0.5

0 1 2 3 4

Sectional-Pile Solid-Pile Simple-Pile Kaynia1982

無因次化頻率a₀

正規化水平勁度

圖33 單一樁水平勁度

單一樁水平阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 2 4 6 8 10

Sectional-Pile Solid-Pile Simple-Pile Kaynia1982

無因次化頻率a

₀

正規化水平阻尼

圖34 單一樁水平阻尼

2X2 S/D=2垂直勁度

0.0 0.1 0.2 0.3 0.4 0.5

-2 -1 0 1 2 3 4

Sectional-Pile-1 Solid-Pile Simple-Pile Kaynia1982 Sectional-Pile-2

無因次化頻率a₀

正規化垂直勁度

圖35 2X2-S/D=2 群樁垂直勁度

2X2 S/D=2垂直阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 5 10 15 20

0.0 0.1 0.2 0.3 0.4 0.5

0 5 10 15 20 25

Sectional-Pile Solid-Pile Simple-Pile Kaynia1982

₀

正規化垂直勁度

圖51 2X2-S/D=20 群樁垂直勁度

2X2 S/D=20垂直阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 5 10 15 20

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a₀

正規化水平勁度

圖65 3X3-S/D=5 群樁水平勁度

3X3 S/D=5水平阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 2 4 6 8 10

Sectional-Pile Solid-Pile Simple-Pile Kaynia1982

無因次化頻率a

正規化水平阻尼

圖70 3X3-S/D=10 群樁水平阻尼

3X3 S/D=16垂直勁度

0.0 0.1 0.2 0.3 0.4 0.5

0 2 4 6 8 10

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a

₀

正規化垂直勁度

圖71 3X3-S/D=16 群樁垂直勁度

3X3 S/D=16垂直阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 10 20 30 40 50 60

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a

₀

正規化垂直阻尼

圖72 3X3-S/D=16 群樁垂直阻尼

102

3X3 S/D=16水平勁度

0.0 0.1 0.2 0.3 0.4 0.5

0 1 2 3 4 5 6

3X3 S/D=20水平勁度

0.0 0.1 0.2 0.3 0.4 0.5

0 1 2 3 4 5 6

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a

₀

正規化水平勁度

圖77 3X3-S/D=20 群樁水平勁度

3X3 S/D=20水平阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 5 10 15 20 25 30

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a₀

正規化水平阻尼

圖78 3X3-S/D=20 群樁水平阻尼

3X3 S/D=26垂直勁度

0.0 0.1 0.2 0.3 0.4 0.5

0 2 4 6 8 10

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a

₀

正規化垂直勁度

圖79 3X3-S/D=26 群樁垂直勁度

3X3 S/D=26垂直阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 5 10 15 20 25 30

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a

₀

正規化垂直阻尼

圖80 3X3-S/D=26 群樁垂直阻尼

106

3X3 S/D=26水平勁度

0.0 0.1 0.2 0.3 0.4 0.5

0 1 2 3 4 5 6

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a

₀

正規化水平勁度

圖81 3X3-S/D=26 群樁水平勁度

3X3 S/D=26水平阻尼

0.0 0.1 0.2 0.3 0.4 0.5

0 5 10 15 20 25 30

Sectional-Pile Solid-Pile Simple-Pile

無因次化頻率a₀

正規化水平阻尼

圖82 3X3-S/D=26 群樁水平阻尼

無因次化頻率a

₀

正規化垂直勁度

2X2群樁各間距之垂直勁度變化

0.0 0.1 0.2 0.3 0.4 0.5

0 2 4 6

單一樁 2x2 S/D= 2 2x2 S/D= 5 2x2 S/D=10 2x2 S/D=16 2x2 S/D=20 2x2 S/D=26

圖83 2X2 群樁各間距之垂直勁度變化

無因次化頻率a

₀

正規化垂直阻尼

2X2群樁各間距之垂直阻尼變化

0.0 0.1 0.2 0.3 0.4 0.5

0 5 10 15 20

單一樁 2x2 S/D= 2 2x2 S/D= 5 2x2 S/D=10 2x2 S/D=16 2x2 S/D=20 2x2 S/D=26

圖84 2X2 群樁各間距之垂直阻尼變化

108 2x2 S/D=10 2x2 S/D=16 2x2 S/D=10 2x2 S/D=16 2x2 S/D=20 2x2 S/D=26

圖86 2X2 群樁各間距之水平阻尼變化

無因次化頻率a

₀ 3x3 S/D=10 3x3 S/D=16 3x3 S/D=20 3x3 S/D=26

圖87 3X3 群樁各間距之垂直勁度變化 3x3 S/D=10 3x3 S/D=16 3x3 S/D=20 3x3 S/D=26

圖88 3X3 群樁各間距之垂直阻尼變化

110 3x3 S/D=10 3x3 S/D=16 3x3 S/D=20 3x3 S/D=26

圖89 3X3 群樁各間距之水平勁度變化 3x3 S/D=10 3x3 S/D=16 3x3 S/D=20 3x3 S/D=26

圖90 3X3 群樁各間距之水平阻尼變化

在文檔中建築結構動態簡化模式及參數分析 (頁 57-146)

ONE-PILE IMPEDANCE-SOLID ELEMENT

1 2.00 2.00 112.700 300.00 734.85 0.050 0.050

2 2.00 4.00 112.700 300.00 734.85 0.050 0.050

3 2.00 6.00 112.700 300.00 734.85 0.050 0.050

4 2.00 8.00 112.700 300.00 734.85 0.050 0.050

5 2.00 10.00 112.700 300.00 734.85 0.050 0.050

6 2.00 12.00 112.700 300.00 734.85 0.050 0.050

7 2.00 14.00 112.700 300.00 734.85 0.050 0.050

8 2.00 16.00 112.700 300.00 734.85 0.050 0.050

9 2.00 18.00 112.700 300.00 734.85 0.050 0.050

10 2.00 20.00 112.700 300.00 734.85 0.050 0.050

11 2.00 22.00 112.700 300.00 734.85 0.050 0.050

12 2.00 24.00 112.700 300.00 734.85 0.050 0.050

13 2.00 26.00 112.700 300.00 734.85 0.050 0.050

14 2.00 28.00 112.700 300.00 734.85 0.050 0.050

15 2.00 30.00 112.700 300.00 734.85 0.050 0.050

γ

E

= 8.82 10 ×

psf

ν

β

E

= 8.82 10 ×

psf

γ

ν

β

26.70mm

20.93mm

1060mm

0.3

0.005

ONE-PILE IMPEDANCE-PILE ELEMENT

Layer Hight Depth Unit S-Wave P-Wave S-Wave P-Wave

Weight Velocity Velocity Damping RatioDamping Rati

Halfspace = 10 18200.000 188.00 351.72 0.010 0.010

1 0.10 0.10 17100.000 40.00 74.83 0.010 0.010

2 0.05 0.15 17100.000 55.00 102.90 0.010 0.010

3 0.15 0.30 17100.000 70.00 130.96 0.010 0.010

4 0.20 0.50 17100.000 80.00 139.67 0.010 0.010

5 0.25 0.75 17100.000 90.00 168.38 0.010 0.010

6 0.25 1.00 17100.000 100.00 187.08 0.010 0.010

7 0.25 1.25 18200.000 125.00 233.85 0.010 0.010

8 0.25 1.50 18200.000 125.00 233.85 0.010 0.010

9 0.25 1.75 18200.000 125.00 233.85 0.010 0.010

10 0.25 2.00 18200.000 125.00 233.85 0.010 0.010

11 0.25 2.25 18200.000 170.00 318.04 0.010 0.010

12 0.25 2.50 18200.000 170.00 318.04 0.010 0.010

13 0.25 2.75 18200.000 170.00 318.04 0.010 0.010

14 0.25 3.00 18200.000 170.00 318.04 0.010 0.010

(a) (b)

z,w

x or y, δ

Soil (G s ,ρ s ,V s )

p z (r,θ)

q z (r,θ) Q h

P h Foundation (E p , m p , υ

, h

)

r

y

R θ x

Normalized Radius, r / R

Normalized Displacement, Re[w] Gs*R

ν

=0.33, α=0.003, b

=0, Vertical Loading

Normalized Radius, r / R

(Welded Contact Boundary)

ν

=0.33, α=0.003, b

=0, Vertical Loading

Normalized Radius, r / R

Normal ize d Dis p la ce me nt, Im[w] G s *R

Normalized Radius, r / R

(Welded Contact Boundary)

Normalized Radius, r / R

Normalized Displacement, Re[w] Gs*R

q _z (r,θ) Q h

P _h Foundation (E _p , m _p , υ

正規化水平阻尼

正規化垂直阻尼

正規化水平勁度

正規化垂直勁度

正規化水平勁度

正規化水平阻尼