Dynamic Analysis of a Radius-varying Rotating Pseudo-rigid Body

本論文探討可變半徑旋轉輪的問題。在大部份的車輛動力學分析中，輪子大都被假設為半徑固定不變的旋轉剛體；然而經觀察，輪子在運動過程中旋轉半徑並非一定數，而是速度的函數。為了處理這非線性的非完整約束，Chetaev條件被用來推導該運動方程。但若與Lagrange方程推得的運動方程式相比較，我們注意到此物體可能不宜被假設為剛體，而必須要採用連續體的模型。這 塈畯怳ㄠ艦弮u性體或黏彈性體，而是選擇可允許整體性變形的擬剛體模型。擬剛體簡單而言，係指一物體在運動過程中具有空間一致的變形行為，使得變形梯度張量僅僅為時間的函數而已。此結果將使統御方程式為常微分方程式的型式而易於處理。 為了研究由不同類型物體相互組成的系統，我們採用虛功率原理及其相應之離散及連續系統變分方程式，藉其推導由剛體與擬剛體組合而成系統之變分方程式。其中，剛體的變分方程式將被用來研究受有非線性運動約束之旋轉輪的運動；同樣的問題也以Lagrange方程的方法討論。擬剛體之變分方程則用以建立具滾動不滑動約束之擬剛體的運動方程。此外，藉由將原始的非線性系統線性化，我們探討一簡化後的平面運動之穩定性，得到其穩態運動的穩定解。 基於前述單輪擬剛體的結果，我們推導了由一個平台與兩個輪子組合而成的車輛運動方程。在該系統中，兩個輪子分別假設成剛體與擬剛體兩種不同的情況。我們計算得在兩種情況下，要維持穩態運動所需施加的扭力，用來觀察擬剛體模型的效能。從數值計算結果可看出，對一假設無損耗的系統而言，擬剛體將比剛體須施加更大的扭力來維持同樣的穩態運動，而這額外所須的耗能，對未來車輛設計的考量上可能有所助益。In this dissertation, the problem of varying radius of a rolling wheel is investigated. In most analysis of vehicle dynamics, the wheels are assumed to be rigid with fixed radius of rotation; however, it has been observed that the rolling radius of wheels is not a constant during the motion but a function of velocity. To deal with the arising nonlinear nonholonomic constraints, the Chetaev condition is applied to derive the equation of motion. Nevertheless, by comparing the set of equations with that derived from Lagrange's equation, we notice that the body may not be assumed to be rigid. A continuum model needs to be adopted. Instead of the elastic or viscoelastic body, the pseudo-rigid model is chosen here which allows global deformation. A pseudo-rigid motion is simply a body with a space-wise constant deformation during motion such that the deformation gradient tensor is only a function of time. The resulting governing equations form a system of ordinary differential equations, which are easier to manage. To study the system composed of various types of bodies interacted with each other, the Principle of Virtual Power is adopted to derive the variational equations for both discrete and continuous systems. In addition, the variational equation for a system consisting of rigid bodies and pseudo-rigid bodies is deduced. The variational equation for rigid body shall be used to study the motion of a rolling wheel subject to nonlinear kinematic constraints, and the same problem is discussed by the method of Lagrange's equation. We further apply the principle to establish the equations of motion for a pseudo-rigid body with the constraint of rolling without slipping along a line. The stability of a simplified planar motion is discussed by linearizing the original nonlinear dynamic system. The stable solutions of steady motion are obtained and applied to analyze the effect of a three-connected bodies system. Based on the previous results of a pseudo-rigid body for a wheel, we derive the equations of motion for a vehicle consisting of one platform and two wheels. In the system, two wheels are assumed to be either rigid or pseudo-rigid, and the equations of motion are derived for different cases. The torque required for a steady motion is computed for both cases numerically, which demonstrates the effect of pseudo-rigid body. From the numerical computation, it is seen that to maintain the same steady motion with no dissipation, a pseudo-rigid body needs larger applied torque to rotate than that for a rigid body. The additional consumption may be useful in the design of the vehicle in the future.Contents 1 Introduction 1 2 Variational Equation for a Systemof Rigid Bodies and Pseudo- rigid Bodies 4 2.1 Principle of Virtual Power for Discrete and Continuous Systems 5 2.2 Principle of Virtual Power for a Rigid Body . . . . . . . . . . . 10 2.3 Principle of Virtual Power for a Pseudo-rigid Body . . . . . . . 12 3 Rolling of a Rigid Body with Varying E¤ective Radius 20 3.1 Equations of Motion from the Variational Equation . . . . . . . 24 3.2 Equations of Motion from Lagrange’s Equation . . . . . . . . . 28 4 Rolling of a Pseudo-rigid Body 33 4.1 General Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Plannar Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 i 4.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.4 Numerical Result . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5 Vehicle Dynamics with Pseudo-rigid Body Model 61 5.1 Equation of Motion for Rigid Wheels . . . . . . . . . . . . . . . 61 5.2 Equation of Motion for Pseudo-rigid Wheels . . . . . . . . . . . 68 5.3 E¤ect of Pseudo-rigid Body on the Motion of Vehicle . . . . . . 75 6 Conclusion 7