Analysis of suspension of a Formula Student race car by a multibody-dynamics simulation software

UNIVERSIDAD POLITÉCNICA DE CARTAGENA

Escuela Técnica Superior de Ingeniería Industrial

Analysis of suspension of a Formula Student race car by a multibody-dynamics simulation

software

TRABAJO FIN DE GRADO GRADO EN INGENIERÍA MECÁNICA

Cartagena, Enero 2018 Autor: Javier Bermejo Gozálvez Director: José Andres Moreno Nicolás Codirector: Patricio Franco.

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𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝐿𝑜𝑎𝑑 𝑜𝑛 𝑡𝑖𝑟𝑒 =𝐹_𝑦

𝐹_𝑧 = 𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝜇 =𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑏𝑜𝑑𝑖𝑒𝑠

𝑁𝑜𝑟𝑚𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑏𝑜𝑑𝑖𝑒𝑠

𝛼

𝑆_𝐵 =𝜔_𝐵− 𝜔₀ 𝜔₀

ω ω

𝑆_𝐷 =𝜔_𝐷− 𝜔₀ 𝜔₀

ω ω

𝐹_𝑅 = √𝐹_𝑥²+ 𝐹_𝑦²

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 𝑉_𝑦 𝜓

𝑑𝜓

𝑑𝑡 > 0 𝑑𝑉_𝑦 𝑑𝑡 > 0



𝑑𝜓

𝑑𝑡 = 0 𝑑𝑉_𝑦 𝑑𝑡 = 0

 𝜓 𝑉_𝑦

𝑑𝜓

𝑑𝑡 < 0 𝑑𝑉_𝑦 𝑑𝑡 < 0

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𝐶𝐹 = 𝑌_𝑅 + 𝑌_𝐹 = 𝐶_𝐹𝛼_𝐹+ 𝐶_𝑅𝛼_𝑅

𝐶𝐹 𝑌_𝐹 𝑌_𝑅

𝐶_𝐹 𝐶_𝑅

𝛼_𝐹 𝛼_𝑅

𝐶_𝐹𝛼_𝐹𝑎 = 𝐶_𝑅𝛼_𝑅𝑏

𝑎 𝑏

𝛿

𝛿 ≈𝐿

𝑅+ 𝛼_𝐹 − 𝛼_𝑅

𝑌_𝐹 = 𝑊 · 𝑉²· 𝑎

𝑔 · 𝑅 · (𝑎 + 𝑏) 𝑌_𝑅 = 𝑊 · 𝑉²· 𝑏 𝑔 · 𝑅 · (𝑎 + 𝑏)

𝑅 𝐿 𝑖𝑠

𝛼

=

2·𝐶_𝐹

=

𝑔·𝑅

·

𝐶_𝐹

𝛼

=

2·𝐶_𝐹

=

𝑔·𝑅

·

𝐶_𝑅

𝛿 = 𝐿

𝑅+ (𝑊_𝐹 𝐶_𝐹 −𝑊_𝑅

𝐶_𝑅) · 𝑉² 𝑔 · 𝑅 𝐾_𝑐

𝐾_𝑐 = (^𝑊𝐹

𝐶𝐹 −^𝑊𝑅

𝐶𝑅) 𝛼_𝐹− 𝛼_𝑅 𝐾_𝑐

𝛿 = 𝐿

𝑅+ 𝐾_𝑐· 𝑉² 𝑔 · 𝑅

𝛿



𝐾_𝑐 = 0

𝛿 = 𝐿 𝑅



𝐾_𝑐 > 0

𝛿 = 𝐿

𝑅+ 𝐾_𝑐 𝑉² 𝑔𝑅



𝐾_𝑐 < 0

𝛿 = 𝐿

𝑅− |𝐾_𝑐|𝑉² 𝑔𝑅

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αF α

∆𝐶_𝐵= 𝑎𝑟𝑐 𝑡𝑎𝑛 ( 1 𝐹𝑟. 𝑉𝑆𝐴𝐿)

𝑖𝑓 𝐹𝑟. 𝑉𝑆𝐴𝐿 = ∞ ∆𝐶_𝐵 = 0

𝛾_𝑔 ∅

𝛾_𝑔 = 𝛾_𝑏+ ∅

𝛾_𝑏

∆𝐶_𝑅 = ∅ · (1 − ( 𝑇𝑟𝑎𝑐𝑘 2 𝐹𝑟. 𝑉𝑆𝐴𝐿))

𝑖𝑓 𝐹𝑟. 𝑉𝑆𝐴𝐿 =𝑇𝑟𝑎𝑐𝑘

2 ∆𝐶_𝑅 = 0

𝑇𝑟𝑎𝑐𝑘 2⁄

∆𝐶𝑎𝑠𝑡_𝑐 = 𝑎𝑟𝑐 𝑡𝑎𝑛 ( 1 𝐿𝑎𝑡. 𝑉𝑆𝐴𝐿)

𝐹𝑟𝑜𝑛𝑡 𝑎𝑛𝑡𝑖 − 𝑑𝑖𝑣𝑒 (%) = 𝑡𝑎𝑛 𝜃_𝐹 ℎ

𝑙

· 100

𝑅𝑒𝑎𝑟 𝑎𝑛𝑡𝑖 − 𝑙𝑖𝑓𝑡(%) = 𝑡𝑎𝑛 𝜃_𝑅 ℎ

𝑙

· 100

𝐹𝑟𝑜𝑛𝑡 𝑎𝑛𝑡𝑖 − 𝑑𝑖𝑣𝑒 𝑓𝑜𝑟𝑐𝑒 = 𝑡𝑎𝑛 𝜃_𝐹∗ 𝐹_𝑓𝑥 ℎ

𝑙

· 100

𝑅𝑒𝑎𝑟 𝑎𝑛𝑡𝑖 − 𝑙𝑖𝑓𝑡 𝑓𝑜𝑟𝑐𝑒 = 𝑡𝑎𝑛 𝜃_𝑅 ∗ 𝐹_𝑟𝑥 ℎ

𝑙

· 100

𝑅𝑒𝑎𝑟 𝑎𝑛𝑡𝑖 − 𝑠𝑞𝑢𝑎𝑡 (%) = 𝑡𝑎𝑛 𝜃_𝑅 ℎ

𝑙

· 100

𝑅𝑒𝑎𝑟 𝑎𝑛𝑡𝑖 − 𝑠𝑞𝑢𝑎𝑡 𝑓𝑜𝑟𝑐𝑒 = 𝑡𝑎𝑛 𝜃_𝑅 · 𝐹_𝑥 ℎ

𝑙

· 100

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∆𝑊 =𝑀 · 𝐴_𝑦· 𝐻 𝑡

∆𝑊_{𝑢𝑛𝑠𝑝𝑟𝑢𝑛𝑔𝐹} =𝑀_𝑢𝐹· 𝐴_𝑦 · 𝐻_𝑢𝐹 𝑡_𝐹

∆𝑊_{𝑢𝑛𝑠𝑝𝑟𝑢𝑛𝑔𝑅} =𝑀_𝑢𝑅· 𝐴_𝑦 · 𝐻_𝑢𝑅 𝑡_𝑅

∆𝑊_{𝑢𝑛𝑠𝑝𝑟𝑢𝑛𝑔} 𝑀_𝑢

𝐻_𝑢

∆𝑊_{𝑠𝑝𝑟𝑢𝑛𝑔𝐹} =𝑀_𝑠 · 𝑏_𝑠

𝑙 ·𝐴_𝑦 · 𝐻_𝐹𝑟 𝑡_𝐹

∆𝑊_{𝑠𝑝𝑟𝑢𝑛𝑔𝑅} = 𝑀_𝑠· 𝑎_𝑠

𝑙 ·𝐴_𝑦· 𝐻_𝑅𝑟 𝑡_𝑅

∆𝑊_{𝑠𝑝𝑟𝑢𝑛𝑔} 𝑏_𝑠

𝑎_𝑠 𝐻_𝐹𝑟

𝐻_𝑅𝑟

𝑀_∅ = 𝑀_𝑠 · ℎ_𝑠𝑟 · 𝐴_𝑦

ℎ_𝑠𝑟

𝐾_∅ = 𝐾_∅𝐹+ 𝐾_∅𝑅

𝑀_∅ = 𝐾_∅· ∅ = (𝐾_∅𝐹+ 𝐾_∅𝑅) · ∅

𝐾_∅ 𝐾_∅𝐹 𝐾_∅𝑅

𝑀_𝑠 · ℎ_𝑠𝑟· 𝐴_𝑦 = (𝐾_∅𝐹+ 𝐾_∅𝑅) · ∅

∅

𝐴_𝑦 = 𝑀_𝑠· ℎ_𝑠𝑟 𝐾_∅𝐹+ 𝐾_∅𝑅

𝑀_∅𝐹 = 𝐾_∅𝐹· 𝑀_𝑠· ℎ_𝑠𝑟 · 𝐴_𝑦 𝐾_∅𝐹 + 𝐾_∅𝑅

𝑀_∅𝑅 = 𝐾_∅𝑅· 𝑀_𝑠· ℎ_𝑠𝑟· 𝐴_𝑦 𝐾_∅𝐹 + 𝐾_∅𝑅 𝑀_{∅𝑇𝑜𝑡𝑎𝑙}= 𝑀_∅𝐹+ 𝑀_∅𝑅

∆𝑊_{𝑟𝑜𝑙𝑙𝐹} =

𝑀_𝑠· 𝐴_𝑦· (𝐻_𝑠 − 𝐻_𝐹𝑟) · 𝑀_∅𝐹 𝑀_{∅𝑇𝑜𝑡𝑎𝑙} 𝑡_𝐹

∆𝑊_{𝑟𝑜𝑙𝑙𝑅}=

𝑀_𝑠· 𝐴_𝑦· (𝐻_𝑠 − 𝐻_𝑅𝑟) · 𝑀_∅𝑅 𝑀_{∅𝑇𝑜𝑡𝑎𝑙} 𝑡_𝑅

∆𝑊_𝐹 = ∆𝑊_{𝑠𝑝𝑟𝑢𝑛𝑔𝐹}+ ∆𝑊_{𝑢𝑛𝑠𝑝𝑟𝑢𝑛𝑔𝐹}+ ∆𝑊_{𝑟𝑜𝑙𝑙𝐹}

∆𝑊_𝐹 = (𝑀_𝑢𝐹· 𝐴_𝑦· 𝐻_𝑢𝐹

𝑡_𝐹 ) + (𝑀_𝑠· 𝑏_𝑠

𝑙 ·𝐴_𝑦· 𝐻_𝐹𝑟 𝑡_𝐹 ) + (

𝑀_𝑠· 𝐴_𝑦· (𝐻_𝑠− 𝐻_𝐹𝑟) · 𝑀_∅𝐹 𝑀_{∅𝑇𝑜𝑡𝑎𝑙}

𝑡_𝐹 )

∆𝑊_𝑅 = ∆𝑊_{𝑠𝑝𝑟𝑢𝑛𝑔𝑅}+ ∆𝑊_{𝑢𝑛𝑠𝑝𝑟𝑢𝑛𝑔𝑅}+ ∆𝑊_{𝑟𝑜𝑙𝑙𝑅}

∆𝑊_𝑅 = (𝑀_𝑢𝑅· 𝐴_𝑦· 𝐻_𝑢𝑅

𝑡_𝑅 ) + (𝑀_𝑠· 𝑎_𝑠

𝑙 ·𝐴_𝑦· 𝐻_𝑅𝑟 𝑡_𝑅 ) + (

𝑀_𝑠· 𝐴_𝑦 · (𝐻_𝑠− 𝐻_𝐹𝑟) · 𝑀_∅𝑅 𝑀_{∅𝑇𝑜𝑡𝑎𝑙}

𝑡_𝑅 )

𝐾_∅ = 𝐾_{∅ 𝑠𝑝𝑟𝑖𝑛𝑔𝑠}+ 𝐾_{∅ 𝐴𝑅𝐵}

𝐾_{∅𝑠𝑝𝑟𝑖𝑛𝑔𝑠} 𝐾_{∅ 𝐴𝑅𝐵}

𝛿₀ = 𝑡𝑎𝑛⁻¹ 𝐿 (𝑅 +𝑡

2)

≅ 𝐿

(𝑅 +𝑡 2)

𝛿_𝑖 = 𝑡𝑎𝑛⁻¹ 𝐿 (𝑅 − 𝑡

2)

≅ 𝐿

(𝑅 −𝑡 2)

𝛿₀ 𝛿_𝑖

 ℝ

𝑓(𝑞) = [

𝑓₁(𝑞) 𝑓₂(𝑞)

⋮ 𝑓_𝑛−1(𝑞)

𝑓_𝑛(𝑞) ]

= 0

ℝ ℝ

𝑓(𝑞) ≈ 𝑓(𝑞⁽⁰⁾) + 𝐹(𝑞⁽⁰⁾)(𝑞 − 𝑞⁽⁰⁾)

𝐹(𝑞⁽⁰⁾) = |𝜕𝑓

𝜕𝑞|

𝑞=𝑞⁽⁰⁾

= [𝜕𝑓_𝑖

𝜕𝑞_𝑗]

𝑞=𝑞⁽⁰⁾

𝑓(𝑞⁽⁰⁾) + 𝐹(𝑞⁽⁰⁾)(𝑞⁽¹⁾− 𝑞⁽⁰⁾) = 0

𝑞⁽¹⁾=𝑞⁽⁰⁾−[𝐹(𝑞⁽⁰⁾)]⁻¹𝑓(𝑞⁽⁰⁾)

𝑝 = [ 𝑥 𝑦 𝑧 ]

  

𝜀 = [





]

𝑞_𝑖 = [𝑝_𝑖 𝜀_𝑖]

𝑢 = 𝑝̇

𝜔̅ = 𝐵𝜀̇ ≡ 𝐵𝜁

𝐵 = [

𝑠𝑖𝑛∅𝑠𝑖𝑛𝜃 0 𝑐𝑜𝑠∅

𝑐𝑜𝑠∅𝑠𝑖𝑛𝜃 0 −𝑠𝑖𝑛∅

𝑐𝑜𝑠𝜃 1 0

]

𝝎̅

𝝎̅

𝐴̇ = 𝐴𝜔̅̃

ψ θ φ

𝐴 = [

𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜙 − 𝑠𝑖𝑛𝜓𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 −𝑐𝑜𝑠𝜓𝑠𝑖𝑛𝜙 − 𝑠𝑖𝑛𝜓𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙 𝑠𝑖𝑛𝜓𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜓𝑐𝑜𝑠𝜙 + 𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 −𝑠𝑖𝑛𝜓𝑠𝑖𝑛𝜙 + 𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙 −𝑐𝑜𝑠𝜓𝑠𝑖𝑛𝜃

𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜙 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜙 𝑐𝑜𝑠𝜃

]

𝑎̃ = [

0 −𝑎₃ −𝑎₂ 𝑎₃ 0 −𝑎₁

−𝑎₂ 𝑎₁ 0 ]

𝑞 = [𝑞₁ ^𝑇 𝑞₂^𝑇… 𝑞_𝑛𝑏^𝑇 ]^𝑇 = [𝑞₁ 𝑞₂… 𝑞_𝑛]^𝑇

𝛷(𝑞) = 0



𝛷(𝑞) = [𝛷₁^𝑇(𝑞) 𝛷_2(𝑞)^𝑇 … 𝛷_𝑛𝑗^𝑇 (𝑞) ]^𝑇 = [𝛷₁(𝑞) 𝛷₂(𝑞) … 𝛷_𝑚(𝑞)]^𝑇

𝛷_𝑞𝑞̇ = 0

𝛷_𝑞𝑞̈ = −(𝛷_𝑞𝑞̇)_𝑞𝑞̇ ≡ 𝜏

𝛷(𝑞, 𝑡) = 0

𝛷(𝑞, 𝑡) = 0

𝛷_𝑞(𝑞, 𝑡). 𝑞̇ = −𝛷_𝑡(𝑞, 𝑡)

𝛷_𝑞(𝑞, 𝑡). 𝑞̈ = −(𝛷_𝑞𝑞̇)_𝑞𝑞̇ − 2𝛷_𝑞𝑡𝑞̇ − 𝛷_𝑡𝑡(𝑞, 𝑡)

𝒒̇

𝛷(𝑞, 𝑡₀) = 0

𝛷_𝑞(𝑞, 𝑡). 𝑞̇ = −𝛷_𝑡(𝑞, 𝑡)

𝑓(𝑞₁, … , 𝑞_𝑛) = 1

2𝑊₁(𝑞₁− 𝑞₁⁰)²+ ⋯ + 1

2𝑊_𝑛(𝑞_𝑛− 𝑞_𝑛⁰)²





𝑓(𝑞) =1

2(𝑞 − 𝑞⁰)^𝑇𝑊(𝑞 − 𝑞⁰)

𝛷(𝑞, 𝑡₀) = 0

𝑊 = 𝑑𝑖𝑎𝑔(𝑤_1, 𝑤₂, … , 𝑤_𝑛)

𝛷(𝑞, 𝑡₀) = 𝛷(𝑞⁰, 𝑡₀) + 𝛷_𝑞(𝑞⁰, 𝑡₀)(𝑞 − 𝑞⁰)



𝑓(𝑑) =1

2𝑑^𝑇𝑊𝑑

𝛷(𝑞⁰, 𝑡₀) + 𝛷_𝑞(𝑞⁰, 𝑡₀)𝑑 = 0

𝐹(𝑑, 𝜆) = 𝑓(𝑑) + 𝜆^𝑇(𝛷(𝑞⁰) + 𝛷_𝑞(𝑞⁰)𝑑)

(𝜕𝐹

𝜕𝑑)

𝑇

= 0 (𝜕𝐹

𝜕𝜆)

𝑇

= 0

[ 𝑊 𝛷_𝑞^𝑇(𝑞⁰) 𝛷_𝑞(𝑞⁰) 0 ] [𝑑

𝜆] = [ 0

−𝛷(𝑞⁰)]

𝑞 = 𝑞⁰+ 𝑑

𝑞̇_𝑖

𝑓(𝑞̇₁, … , 𝑞̇_𝑛) = 1

2(𝑞̇ − 𝑞̇₀)^𝑇𝑊(𝑞̇ − 𝑞̇₀)

𝛷_𝑞(𝑞, 𝑡₀). 𝑞̇ + 𝛷_𝑡(𝑞, 𝑡₀) = 0

[ 𝑀 𝛷_𝑞^𝑇(𝑞⁰) 𝛷_𝑞(𝑞⁰) 0 ] [𝑞̈

𝜆] = [𝐹 𝜏]

𝒒̈



𝐹^𝐶 = (𝜕𝛷^(𝑗) 𝑣_𝑖 )

𝑇

𝜆^(𝑗)

𝑇^𝐶 = − (𝜕𝛷^(𝑗) 𝜔_𝑖 )

𝑇

𝜆^(𝑗)

 

 

𝛷_𝑞(𝑞₀, 𝑡₁)𝛥^(𝑗)= −𝛷(𝑞₁^(𝑗), 𝑡₁)



 



𝒒̈

𝒒̈

𝛷_𝑞^𝑇𝜆 = 𝐹 − 𝑀𝑞̈

𝐾 =1

2𝑢^𝑇𝑀𝑢 +1 2𝜔̅^𝑇𝐽̅𝜔̅ 𝑱̅

𝔽(𝑞 , 𝑞̇, 𝑡) = [𝑓 𝑛̅] ∈ 𝑅⁶

𝒏̅

𝑄(𝑞 , 𝑞̇, 𝑡) ∈ 𝑅⁶

𝑄 = [(𝛱^𝑃)^𝑇 𝑓 (𝛱^𝑅)^𝑇 𝑛̅]

𝛱^𝑃 =𝜕𝑣^𝑃

𝜕𝑢

𝛱^𝑅 = 𝜕𝜔̅

𝜕𝜁

𝑑 𝑑𝑡[(𝜕𝐾

𝜕𝑞̇)

𝑇

] − (𝜕𝐾

𝜕𝑞)

𝑇

+ 𝛷_𝑞^𝑇𝜆 = 𝑄

𝑑 𝑑𝑡[(^𝜕𝐾

𝜕𝑢)^𝑇 (^𝜕𝐾

𝜕𝜁)^𝑇 ] [(^𝜕𝐾

𝜕𝑝)^𝑇 (^𝜕𝐾

𝜕𝜀)^𝑇

] + [𝛷_𝑝^𝑇𝜆

𝛷_𝜀^𝑇𝜆] = [(𝛱^𝑃)^𝑇 𝑓 (𝛱^𝑅)^𝑇 𝑛̅]

𝑑 𝑑𝑡(𝜕𝐾

𝜕𝑢)

𝑇

= 𝑀𝑢̇

(𝜕𝐾

𝜕𝑝)

𝑇

= 0

𝛤 ≡𝜕𝐾

𝜕𝜁 = 𝐵^𝑇𝐽̅𝐵𝜁

𝑀𝑢̇ + 𝛷_𝑃^𝑇𝜆 = (𝛱^𝑃)^𝑇𝑓

𝛤̇ −^𝜕𝐾

𝜕𝜀 𝛷_𝜀^𝑇𝜆 = (𝛱^𝑅)^𝑇𝑛̅

𝑀𝑢̇ + 𝛷_𝑃^𝑇𝜆 − (𝛱^𝑃)^𝑇𝑓 = 0

𝛤 − 𝐵^𝑇𝐽̅𝐵𝜍

𝛤̇ −^𝜕𝐾

𝜕𝜀 𝛷_𝜀^𝑇𝜆 − (𝛱^𝑅)^𝑇𝑛̅ = 0

𝑝̇ − 𝑢 = 0

𝜀 − 𝜁 = 0

𝒚̇₁

𝑦₁̇ =1

ℎ𝑦₁−1 ℎ𝑦₀

𝒚̇

1

ℎ𝑦₁−1

ℎ𝑦₀− 𝑔(𝑡₁, 𝑦₁) = 0

1

ℎ𝑀𝑢 −1

ℎ𝑀𝑢₀ + 𝛷_𝑝^𝑇𝜆 − (𝛱^𝑝)^𝑇𝑓 = 0 𝛤 − 𝐵^𝑇𝐽̅𝐵𝜍 = 0

1 ℎ𝛤 −1

ℎ𝛤₀− (𝜕𝐾

𝜕𝜀)

𝑇

+ 𝛷_𝜀^𝑇𝜆 − (𝛱^𝑅)^𝑇𝑛̅ = 0 1

ℎ𝑝 −1

ℎ𝑝₀− 𝑢 = 0 1

ℎ𝜀 −1

ℎ𝜀₀− 𝜁 = 0 𝛷(𝑝, 𝜀, 𝑡₁) = 0 𝑓 − 𝐹(𝑢, 𝜁, 𝑝. 𝜀, 𝑓, 𝑛̅, 𝑡₁) = 0 𝑛̅ − 𝑇(𝑢, 𝜁, 𝑝. 𝜀, 𝑓, 𝑛̅, 𝑡₁) = 0

    𝒏̅

𝜓(𝑦) = 0

𝜓_𝑦(𝑦₀)𝛥^(𝑗) = −𝜓(𝑦^(𝑗)) 𝑦^(𝑗+1)= 𝑦^(𝑗)+ 𝛥^(𝑗)





 

𝛷_𝑝^𝑇𝜆 − (𝐴^𝑝)^𝑇𝑓 = 0 𝛤 − 𝐵^𝑇𝐽̅𝐵𝜍 = 0

− (𝜕𝐾

𝜕𝜀)

𝑇

+ 𝛷_𝜀^𝑇𝜆 − (𝐴^𝑅)^𝑇𝑛̅ = 0 𝑢 = 0

𝜁 = 0 𝛷(𝑝, 𝜀, 𝑡₀) = 0

𝑓 − 𝐹(𝑢, 𝜁, 𝑝. 𝜀, 𝑓, 𝑛̅, 𝑥, 𝑡₀) = 0 𝑛̅ − 𝑇(𝑢, 𝜁, 𝑝. 𝜀, 𝑓, 𝑛̅, 𝑥, 𝑡₀) = 0

𝑑(𝑢, 𝜁, 𝑝. 𝜀, 𝑓, 𝑛̅, 𝑥, 𝑡₀) = 0

𝑥̇ − 𝑑(𝑢, 𝜁, 𝑝. 𝜀, 𝑓, 𝑛̅, 𝑥, 𝑡) = 0

𝒙̇

𝑦^𝑇 = [𝑢 𝛤 𝜁 𝑝 𝜀 𝜆 𝑓 𝑛̅ 𝑥]

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

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



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

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

𝑓 (𝑥) = 0

ℝ  ℝ

𝑥⁽¹⁾= 𝑥⁽⁰⁾− 𝑓(𝑥⁽⁰⁾) 𝑓^′(𝑥⁽⁰⁾)

𝑓(𝑥) ≈ 𝑓(𝑥⁽⁰⁾) + 𝑓^′(𝑥⁽⁰⁾)(𝑥 − 𝑥⁽⁰⁾)

𝑓(𝑥⁽⁰⁾) + 𝑓^′(𝑥⁽⁰⁾)(𝑥⁽¹⁾− 𝑥⁽⁰⁾) = 0 ⇒ 𝑥⁽¹⁾= 𝑥⁽⁰⁾− 𝑓(𝑥⁽⁰⁾) 𝑓^′(𝑥⁽⁰⁾)



0.3𝑆𝑛 = 0.3 · 122.5 = 36.75 𝑚𝑚 0.7𝑆𝑛 = 0.7 · 122.5 = 85.75 𝑚𝑚

𝐹₂− 𝐹₁

𝑆₂− 𝑆₁ =2572.596 − 2100.563

66.27612 − 58.9788 = 64.68𝑁/𝑚𝑚

2000 2100 2200 2300 2400 2500 2600 2700

59 60 61 62 63 64 65 66

Force vs Displacement

∅

𝐴_𝑦 =𝑚 · 𝑔 · 𝐻_{𝐶𝑜𝐺−𝑅𝐶} 𝐾_∅𝐹+ 𝐾_∅𝑅

𝐾∅𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑇𝑜𝑡𝑎𝑙 = 𝐾_∅𝐹 + 𝐾_∅𝑅 = 703.10 𝑁𝑚/º

𝑁₁ 𝑁₂

𝑁₁ = 49.04 %

𝑁₂ = 50.96 %

𝐾∅𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝐹𝑟𝑜𝑛𝑡=

𝐾∅𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑇𝑜𝑡𝑎𝑙· 𝑁₁ 100

𝐾∅𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝐹𝑟𝑜𝑛𝑡 = 344.80 𝑁𝑚/º

𝐾∅𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑅𝑒𝑎𝑟=

𝐾∅𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑇𝑜𝑡𝑎𝑙· (100 − 𝑁₂) 100

𝐾∅𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑅𝑒𝑎𝑟 = 358.30 𝑁𝑚/º

𝜔_{𝐹𝑟𝑜𝑛𝑡} = 2.5 𝐻𝑧 𝜔_{𝑅𝑒𝑎𝑟} = 2.65 𝐻𝑧

𝐾𝑊ℎ𝑒𝑒𝑙 𝐹𝑟𝑜𝑛𝑡 = 4𝜋²· 𝜔_{𝐹𝑟𝑜𝑛𝑡}²· 𝑚_𝐹𝑤

𝐾𝑊ℎ𝑒𝑒𝑙 𝐹𝑟𝑜𝑛𝑡 = 16.842 𝑁

𝑚𝑚= 16842 𝑁 𝑚

𝐾

= 4𝜋

· 𝜔

· 𝑚

𝐾_{𝑊ℎ𝑒𝑒𝑙 𝑅𝑒𝑎𝑟} = 23.59 𝑁

𝑚𝑚= 235900 𝑁 𝑚

𝐾_{𝑆𝑝𝑟𝑖𝑛𝑔𝑠} = 64.68 𝑁

𝑚𝑚 = 646800 𝑁 𝑚

𝑀𝑅_𝐹 = √𝐾𝑊ℎ𝑒𝑒𝑙 𝐹𝑟𝑜𝑛𝑡

𝐾_{𝑆𝑝𝑟𝑖𝑛𝑔𝑠} 𝑀𝑅_𝐹 = 0.510

𝑀𝑅_𝑅 = √𝐾_{𝑊ℎ𝑒𝑒𝑙 𝑅𝑒𝑎𝑟} 𝐾_{𝑆𝑝𝑟𝑖𝑛𝑔𝑠}

𝑀𝑅_𝑅 = 0.604

𝐾∅ 𝑆𝑝𝑟𝑖𝑛𝑔𝑠 𝐹𝑟𝑜𝑛𝑡=

𝑡_𝐹² · 𝐾𝑊ℎ𝑒𝑒𝑙 𝐹𝑟𝑜𝑛𝑡

2 · 𝜋

180

𝐾∅ 𝑆𝑝𝑟𝑖𝑛𝑔𝑠 𝐹𝑟𝑜𝑛𝑡 = 211.64 𝑁𝑚/º

𝐾∅ 𝑆𝑝𝑟𝑖𝑛𝑔𝑠 𝑅𝑒𝑎𝑟=

𝑡_𝑅²· 𝐾_{𝑊ℎ𝑒𝑒𝑙 𝑅𝑒𝑎𝑟}

2 · 𝜋

180 𝐾∅ 𝑆𝑝𝑟𝑖𝑛𝑔𝑠 𝑅𝑒𝑎𝑟 = 296.49 𝑁𝑚/º

𝐾_{∅𝐴𝑅𝐵 𝑇𝑜𝑡𝑎𝑙} = 𝐾∅𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑇𝑜𝑡𝑎𝑙 − 𝐾∅ 𝑆𝑝𝑟𝑖𝑛𝑔𝑠 𝑇𝑜𝑡𝑎𝑙

𝐾_{∅𝐴𝑅𝐵 𝑇𝑜𝑡𝑎𝑙} = 222.53 𝑁𝑚/º

𝐾_{∅𝐴𝑅𝐵 𝐹𝑟𝑜𝑛𝑡} = 150 𝑁𝑚/º

𝐾_{∅𝐴𝑅𝐵 𝑅𝑒𝑎𝑟} = 50 𝑁𝑚/º

𝐾_{∅ 𝐹𝑟𝑜𝑛𝑡} = 361.64 𝑁𝑚 º

𝐾_{∅ 𝑅𝑒𝑎𝑟} = 349.49 𝑁𝑚 º

∆_𝑊𝐹= 𝐴_𝑦·𝑀

𝑡_𝑓(𝐻 𝐾_∅𝐹 𝐾_{∅𝑇𝑜𝑡𝑎𝑙}+𝑏

𝑙 ℎ_{𝑟𝑜𝑙𝑙 𝐹})

𝑊_𝐹 = 454.81 𝑁/𝐺

∆_𝑊𝑅= 𝐴_𝑦·𝑀

𝑡_𝑟(𝐻 𝐾_∅𝑅 𝐾_{∅𝑇𝑜𝑡𝑎𝑙}+𝑏

𝑙 ℎ_{𝑟𝑜𝑙𝑙 𝑅})

𝑊_𝑅 = 447.89 𝑁/𝐺

𝑊_{𝐹𝐿 𝑑𝑦𝑛𝑎𝑚𝑖𝑐} = 𝑊_{𝐹𝐿 𝑠𝑡𝑎𝑡𝑖𝑐} − ∆_𝑊𝐹 𝑊_{𝐹𝑅 𝑑𝑦𝑛𝑎𝑚𝑖𝑐} = 𝑊_{𝐹𝑅 𝑠𝑡𝑎𝑡𝑖𝑐}+ ∆_𝑊𝐹 𝑊_{𝑅𝐿 𝑑𝑦𝑛𝑎𝑚𝑖𝑐} = 𝑊_{𝑅𝐿 𝑠𝑡𝑎𝑡𝑖𝑐}− ∆_𝑊𝐹 𝑊_{𝑅𝑅 𝑑𝑦𝑛𝑎𝑚𝑖𝑐} = 𝑊_{𝑅𝑅 𝑠𝑡𝑎𝑡𝑖𝑐} + ∆_𝑊𝐹

𝑚_𝑈 𝑚_𝑆 𝐾_𝑆

𝐶_𝐵 𝐾_𝑇

𝑅𝑖𝑑𝑒 𝑅𝑎𝑡𝑒 = 𝐾_𝑅 = 𝐾_𝑊· 𝐾_𝑇 𝐾_𝑊+ 𝐾_𝑇

𝐾_𝑇 = 200 𝑁 𝑚𝑚

𝐾_𝑊𝐹 = 16.84 𝑁 𝑚𝑚

𝐾_𝑊𝑅 = 23.59 𝑁 𝑚𝑚

𝐾_𝑅𝐹 = 15.53 𝑁 𝑚𝑚

𝐾_𝑅𝑅 = 21.10 𝑁 𝑚𝑚

𝜔_𝑆 = 1 2𝜋√𝐾_𝑅

𝑀_𝑆

𝜔_𝑆𝐹 =

𝜔_𝑆𝑅 =

𝜔_𝑆 = 1

2𝜋√𝐾_𝑠+ 𝐾_𝑇 𝑀_𝑈𝑆

𝜔_𝑈𝑆𝐹 = 16.83 𝐻𝑧

𝜔_𝑈𝑆𝑅 = 17.09 𝐻𝑧

𝐶_𝑐𝑠 = 2√𝐾𝑅· 𝑀_𝑠

𝐶_𝐶𝑆𝐹 = 1.7825 𝑁𝑠 𝑚𝑚

𝐶_𝐶𝑆𝑅 = 2.25 𝑁𝑠 𝑚𝑚

𝐶_𝑐𝑢 = 2√(𝐾_𝑠+ 𝐾_𝑇) ∗ 𝑀_𝑈𝑠

𝐶_𝐶𝑈𝐹 = 4.09 𝑁𝑠 𝑚𝑚

𝐶_𝐶𝑈𝑅 = 4.16 𝑁𝑠 𝑚𝑚









𝐷𝑅 = 𝐶_𝑐 ∗ 𝜀

𝜀

𝐷𝑅_{𝐿𝑆𝐶𝐹} = 𝐶_𝐶𝑆𝐹· 𝜀_{𝐿𝑆𝐶𝐹} = 7.128 𝑁𝑠 𝑚𝑚

𝐷𝑅_{𝐿𝑆𝐶𝑅} = 𝐶_𝐶𝑆𝑅· 𝜀_{𝐿𝑆𝐶𝑅} = 7.2 𝑁𝑠 𝑚𝑚

𝐷𝑅_{𝐻𝑆𝐶𝐹} = 𝐶_𝐶𝑈𝐹· 𝜀_{𝐻𝑆𝐶F} = 3.68 Ns mm

DR_HSCR= C_CUR· ε_HSCR= 3.32 Ns mm

DR_LSRF = C_CSF· R_CR = 10.69 Ns mm

DR_LSRR = C_CSR· R_CR = 10.8 Ns mm

DR_HSRF = C_CUF· R_CR = 5.52 Ns mm

DR_HSRR= C_CUR· R_CR = 4.98 Ns mm

7.128 Ns

mm 10.69 Ns mm 7.2 Ns

mm 10.8 Ns

mm

3.68 Ns

mm 5.52 Ns

mm 3.32 Ns

mm 4.98 Ns

mm

DF = DR · V

DF V

-1500 -1000 -500 0 500 1000

0 25 50 75 100 125 150 175 200

Compression Rebound Force [N]

Velocity [mm/s]

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800

0 25 50 75 100 125 150 175

Compression Rebound

Velocity[mm/s]

Force [N]

-1500 -1000 -500 0 500 1000

0 25 50 75 100 125 150 175 200

Compression Rebound Velocity [mm/s]

Force [N]

-1500 -1000 -500 0 500 1000

0 25 50 75 100 125 150 175

Compression Rebound

Velocity [mm/s]

Force [N]

loc_x loc_y loc_z

hpl_BC_axis -600.0 -220.0 500.85

hpl_BC_center -535.0 -220.0 490.0

hpl_damper_inboard -531.4 -10.34 541.7

hpl_damper_outboard -531.46 -209.37 559.0

hpl_lca_front -677.33 -196.5 24.55

hpl_lca_outer -525.0 -570.0 71.51

hpl_lca_rear -345.78 -196.5 24.55

hpl_prod_inboard -531.41 -330.0 550.0

hpl_prod_outboard -525.0 -570.03 71.51

hpl_ride_height -766.0 -219.1 21.85

hpl_tirerod_inner -430.0 -196.5 38.0

hpl_tierod_outer -430.0 -570.03 115.0

hpl_uca_front -678.7 -223.0 108.12

hpl_uca_outer -518.0 -570.03 217.17

hpl_uca_rear -345.78 -223.0 108.12

hpl_wheel_center -512.0 -600.0 155.2

hps_camber_adj_orient -512.0 0.0 27.4

loc_x loc_y loc_z

hpl_arb_bend -450.0 -330.0 450.0

hpl_drop_link -531.4 -330.0 550.0

hpl_leaf_link -531.4 -330.0 450.0

hps_arb_center -450.0 0.0 450.0

loc_x loc_y loc_z hpl_rack_house_mount -430.0 -196.5 38.0

hpl_tierod_inner -430.0 -196.5 38.0

hps_intermediate_shaft_forward -100.0 0.0 400.0 hps_intermediate_shaft_rearward 0.0 0.0 500.0

hps_pinion_pivot -430.0 0.0 38.0

hps_steering_wheel_center 100.0 0.0 550.0

loc_x loc_y loc_z

hpl_BC_axis 1070.0 -226.0 301.4

hpl_BC_center 1065.87 -226.0 301.4 hpl_damper_inboard 1038.86 -18.63 338.66 hpl_damper_outboard 1050.0 -224.65 381.0

hpl_lca_front 936.6 -188.93 9.69

hpl_lca_outer 1041.1 -585.03 33.18

hpl_lca_rear 1167.0 -188.93 9.69

hpl_prod_inboard 1049.6 -328.2 338.79 hpl_prod_outboard 1041.1 -585.03 33.18 hpl_ride_height 1000.0 -381.0 63.5 hpl_tirerod_inner 1250.0 -188.93 26.0 hpl_tierod_outer 1070.0 -585.03 33.18

hpl_uca_front 817.2 -208.09 98.38

hpl_uca_outer 1050.41 -585.03 202.32

hpl_uca_rear 1166.42 -208.09 98.39

hpl_wheel_center 1030.4 -600.0 154.0

hps_camber_adj_orient 1030.4 0.0 51.6

loc_x loc_y loc_z

hpl_arb_bend 1150.0 -328.0 250.0

hpl_drop_link 1049.6 -328.0 338.79

hpl_leaf_link 1049.0 -328.0 250.0

hps_arb_center 1150.0 0.0 250.0