Ordinary Differential Equations assignment help for the Kinematics of an Oscillating Pendulum by Ph.D. Experts

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What this ordinary differential equations assignment help sample will include

In this ordinary differential equation assignment help service, we have described different ways to derive a model of a dynamic system and compare model errors of both ways. It helps in a better understanding of mathematical methods in modeling and how the same physical system can be modeled with different errors. This paper provides a point of what model really is and how much model depends on mathematic we are using to describe the system.

 Help with Ordinary Differential Equations Homework for Kinematics

xB=c+c+lsin(Θ)cosCos(Ψ);yB=yc+lsin(Θ)sinsin(Ψ) {x_B=c+c+lsin(\Theta)\cos Cos (\Psi); y_B=y_c+lsin(\Theta)\sin sin(\Psi)}  

xL=xcd2sinsin(Ψ);yL=yc+d2coscos(Ψ) {x_L=x_c- \frac{d}{2}\sin sin(\Psi);y_L=y_c+ \frac{d}{2}\cos cos(\Psi)}

xR=xc+d2sinsin(Ψ);yR=ycd2coscos(Ψ) {x_R=x_c+\frac{d}{2}\sin sin(\Psi);y_R=y_c-\frac{d}{2}\cos cos(\Psi)}

zB=lcos(Θ) {z_B=lcos(\Theta)}  

   This ordinary differential equation, with the help of coordinates, describes the relationship of positions of the left, right wheel (x and y with L and R in the index for right and left wheel, respectively) and the position of point B (which is bounded for the top of the inverted pendulum) with the position of the center of a relative coordinate system bounded with pendulum. These equations specify the position of the center of mass for each robot element with respect to the fixed reference frame.

vL=(x˙c(d2)Ψ˙coscos(Ψ))n1+(y˙c(d2Ψ˙sinsin(Ψ))n2 {v^L=(\dot{x}_c-(\frac{d}{2})\dot{\Psi} \cos cos(\Psi))n_1+(\dot{y}_c-(\frac{d}{2}\dot{\Psi}\sin sin(\Psi))n_2}

vR=(x˙c(d2)Ψ˙coscos(Ψ))n1+(y˙c(d2Ψ˙sinsin(Ψ))n2 {v^R=(\dot{x}_c-(\frac{d}{2})\dot{\Psi} \cos cos(\Psi))n_1+(\dot{y}_c-(\frac{d}{2}\dot{\Psi}\sin sin(\Psi))n_2}

vB=(x˙c+lΘ˙coscos(Θ)coscos(Ψ)=lΨ˙sinsin(Θ)sin(Ψ))n1+(y˙c+lΘ˙coscos(Θcoscos(Ψ)+lΨ˙sinsin(Θ)sin(Ψ))n2(lΘsin(Θ))n3 {v^B=(\dot{x}_c+l \dot{\Theta}\cos cos (\Theta)\cos cos(\Psi)=l \dot{\Psi}\sin sin(\Theta) sin(\Psi))n_1+(\dot{y}_c+l \dot{\Theta}\cos cos (\Theta\cos cos(\Psi)+l \dot{\Psi} \sin sin(\Theta)sin(\Psi))n_2-(l \Theta sin(\Theta))n_3}

  Equations above are velocity components of point L, R, and B with respect to the fixed reference frame. Those equations represent the change of position in time (a first derivate of the position)

  The Next equation corresponds to a relative velocity and relative angular velocity of the chart with respect to point C, which represents the assembly point of the pendulum and chart.

ωL=ωc+γ˙Lc2;vL=vC+vLC {\omega ^L= \omega ^c+ \dot {\gamma}_L c_2; v ^L= v^C+v ^C_L}

ωR=ωC+γ˙Rc2;vR=vC+vRC {\omega ^R= \omega ^C+ \dot {\gamma}_R c_2 ; v ^R= v^C+v ^C_R}

ωB=ωC+Θb2;vB=vC+vBC {\omega ^B= \omega ^C+ \Theta b_2 ; v ^B= v^C+v ^C_B}

  The ordinary differential equations assignment for robot point involves the motion of the robot points. The point of kinematic equations is to give orientation about what is in correlation and make an easier development of the dynamic model. For deriving the equation of motion, they used Kane's method.

  " Instead of determining energy functions in Lagrangian approaches or examining interactive forces in Newtonian mechanics, it is based on finding generalized active forces and generalized inertia forces as the functions of generalized coordinates. "

 After implementing of Kane's method, the online differential equations expert was able to provide motion equations

x¨=sin(Θ)V1(C11g+C12Θ˙2+C13Ψ˙2)+C14(TL+TR)Θ¨=sin(Θ)V1(C21gC22Θ˙2C23Ψ˙2)C24(TL+TR)Ψ¨=sin(Θ)V2(C31Θ˙Ψ˙mBlx˙Ψ˙)C32(TL+TR) {\ddot{x}= \frac {sin(\Theta)}{V_1}(-C_{11g}+C_{12}\dot{\Theta}^2+C_{13}\dot{\Psi}^2)+C_{14}(T_L+T_R) \ddot{\Theta}=\frac{sin(\Theta)}{V1}(C_{21g}-C_{22}\dot{\Theta}^2-C_{23}\dot{\Psi}^2)-C_{24}(T_L+T_R)\ddot{\Psi}=\frac{sin(\Theta)}{V_2}(C_{31}\dot{\Theta}\dot{\Psi}-m_B l \dot{x}\dot{\Psi})-C_{32}(T_L+T_R)}

 Equations of motion represent the dynamics of the whole system with calculated origin force for every motion and mass distribution (C coefficients which depend on the moment of inertia). With these three equations, we can calculate system response and simulate any situation or change in our system. Using the above equation developed by the online differential equations expert, we can develop proper control for a specific system.