It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by
where m is the mass of the helicopter, g is the gravitational constant, I is the rotational inertia of the helicopter, h is the height of the rotor plane above the center of gravity, q1and q2are stabilizer flapping parameters, k, ki, and kd, are controller parameters: all constants > 0. The UAV is supposed to pick up a payload: when this occurs, the mass, height. and inertia change to m’, h’, and I’, respectively, all still > 0. Show that the helicopter will remain stable as long as
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- Equation of motion of a suspension system is given as: Mä(t) + Cx(t) + ax² (t) + bx(t) = F(t), where the spring force is given with a non-linear function as K(x) = ax²(t) + bx(t). %3D a. Find the linearized equation of motion of the system for the motion that it makes around steady state equilibrium point x, under the effect of constant F, force. b. Find the natural frequency and damping ratio of the linearized system. - c. Find the step response of the system ( Numerical values: a=2, b=5, M=1kg, C=3Ns/m, Fo=1N, xo=0.05marrow_forward6. Consider the mechanical system shown in Fig. 8. Let V(t) be the input and the acceleration of the mass be the output. Derive the state equations and the output equation using linear graphs and normal trees. B m V₁(t) Figure 8: A mechanical system with an across-variable sourcearrow_forwardA mass of 2 kilograms is on a spring with spring constant k newtons per meter with no damping. Suppose the system is at rest and at time t = 0 the mass is kicked and starts traveling at 2 meters per second. How large does k have to be to so that the mass does not go further than 3 meters from the rest position? use 2nd order differential equations to solve (mechanical vibrations)arrow_forward
- 6. The electro-mechanical system shown below consists of an electric motor with input voltage V which drives inertia I in the mechanical system (see torque T). Find the governing differential equations of motion for this electro-mechanical system in terms of the input voltage to the motor and output displacement y. Electrical System puthiy C V V₁ R bac (0) T bac T Motor - Motor Input Voltage - Motor Back EMF = Kbac ( - Motor Angular Velocity - Motor Output Torque = K₂ i Kbacs K₁ - Motor Constants Mechanical System M T Frictionless Supportarrow_forwardFor the system shown in the figure below: 1. Derive the system differential equations of motion. 2. Use Laplace transform to solve for the displacement x:(t) and x2(t), when K,=k2 =k3=1, m,= m;=1, and x1(0) =0, x1(0) = -1, x2(0) =0, and x{0) =1 3. Sketch x:(t) and X2(t) m, X1 X2arrow_forwardmechanical vibrations 3m i+4cx+ 2kx = 4cj+3ky For the system given above, obtain the state-s pace representation,arrow_forward
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- Given the following translational mechanical system, find the transfer function G(s): -x, (t) +-x₂(t) f(t) 3 kg 1 N/m -0000 1 N-s/m ㄸ 1 N-s/m 2 kg X₂ (8) F(s)arrow_forwardRouth-Hurwitz criterion determines: No. of roots in the bottom half of s-plane No. of roots in the top half of s-plane No. of roots in the left half of s-plane No. of roots in the right half of s-planearrow_forward• The unity feedback control structure has the following block diagram: w C(s) P(s)arrow_forward
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