Position and velocity vectors are calculated by integrating the perturbed equation of motion (Cowell's equation). To update the state vectors, equation (9) is integrated numerically by using 4th order Runge-Kutta's as [11]: h Xo + (k₁ + 2 k₂ + 2k + k₁₂ ) step 6 (17) where vxo is the initial velocity at epoch, vx₁ is the predicated velocity, k₁ = axo, k₂ hstep = axo -k₁, k₁ = ax。+ k₂, ka xo+ hstep h3,, axo is the acceleration at epoch, 2 h. step 2 t step hstep is the step for the method, tstep m = Τ m is the sub steps number during one nnn revolution the used values 10,20,30,40, and 50, Tp is the satellite's orbital period, the used value 5317.5 second, and nnn is the step number during 400 revolutions, the used value is 800,1000,1200,1400, 1600, and 1800. X₁ = X + Вер 6 - hep (kk, +2 kk₂+ 2 kk, + kk₁) (18) where X is the initial position at epoch, X₁ is the predicated position, kk₁ = vx, kk₂ = vx+ hap kk,, kk, = vxo+ h 2 step 2 -kk₂, kk₁ = vxo+hstep kk3. In addition, Equations (17) and (18) can be used to calculate the other velocities and positions components by the same way.

Here we have a very simple one dimensional model of a stroke rehabili- tation patient supported by a robot. Often, when you have a controlled electromechanical system like a robot and a human working together, you want to separate them by a compliant mechanism (spring with stiffness k) so that the robot has some "give". Otherwise, if there are errors in the control of the robot it can be very uncomfortable for a patient. The spring has zero force when x, = xp = 0. The patient provides viscous damping with damping Xp coefficient c. Fp + C Patient mp k Xp Robot Arm Xr Figure 2: Schematic diagram of a very simplified model of a patient pushing against a robot. The robot supports the patient via a spring for safety. In this case we want to eventually determine the force that the patient is exerting on the robot as a function of time, so we'll do some steps towards that goal. Your tasks: A Draw the free body diagram for the patient mass (mass mp). B Write the equations of motion for the…

A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): 1. What is the order of this system?

A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): A. Use Laplace transform of the differential equation to determine the transfer function of the system.

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The governing equation of motion for a base motion system is given by (assume the units are Newton) mä(t)+ci(t)+kx(t) =cY@, cos(@,t) +kY sin(@t) %3D Given that m = 180 kg, c = 30 kg/s, Y = 0.02 m, and о, — 3.5 rad/s %3| 1. Use Excel or Matlab to find the largest value of the stiffness, k that makes the transmissibility ratio less than 0.85 2. Using the value of the stiffness obtained in part (1), determine the transmitted force to the base motion system using Matlab or Excel. 3. Display and discuss your results.

Convert the following 4th order ODE into a system of four first order ODES of the form dy/dt f(t,y). Remember to account for initial conditions in the system you create. 4y"'(t) – 8y"' (t) + 2y"(t) + 16y'(t) – 4y(t) = 20e2t y(0) = 3, y'(0) = 4, y"(0) = 7 and y"'(0) = = 13 2:36 PM 73°F Mostly cloudy 3/4/2022

The relative displacement u(t) of a single-storey shear building subjected to an earthquake ground motion is represented by the following second-order linear ordinary differential equation: d?u dt2 du + c + ku = a, (t) m dt where m, c, and k are the mass (kg), damping constant (Ns/m), and stiffness of the structure (N/m), respectively. Meanwhile, a,(t) is the function of earthquake ground acceleration. Suppose the building with a mass of 2000 kg and supported by columns of combined stiffness of 32 x 103 N/m is subjected to earthquake with ground acceleration given by the following function: ag(t) = 36000 cos 2t Find the equation of the displacement, u(t), given the damping is not installed to the building. Then, find how much the building is displaced from t =30 seconds to t =60 seconds of earthquake.

Problem 5: Impulse-Momentum no picture necessary Use the Buckingham method to derive the impulse-momentum equation for steady flow of an incompressible fluid, following the 5-step process on pages 221-222 of Song (2018). Assume the sum of forces EF depends on the discharge Q, the change in velocity AV, and the density p. Use mass-length-time (MLT) dimensions.