Oil of viscosity μ and density ρ drains steadily down the side of a tall, wide vertical plate, as shown in Fig, C1.4 In the region shown, fully developed conditions exist', that is, the velocity profile shape and the film thickness δ are independent of distance z along the plate. The vertical velocity w becomes a function only of x, and the shear resistance from the atmosphere is negligible. (a) Sketch the approximate shape of the velocity profile w(x), considering the boundary conditions at the wall and at the film surface. (b) Suppose film thickness δ , and the slope of the velocity profile at the wall, (dw/dx) w a l l , are measured by a laser Doppler anemometer (to be discussed in Chap. 6). Find an expression for the viscosity of the oil as a function of ρ , δ (dw/dx) w a l l , and the gravitational acceleration g. Note that, for the coordinate system given, both w and (dw/dx) w a l l are negative.
Oil of viscosity μ and density ρ drains steadily down the side of a tall, wide vertical plate, as shown in Fig, C1.4 In the region shown, fully developed conditions exist', that is, the velocity profile shape and the film thickness δ are independent of distance z along the plate. The vertical velocity w becomes a function only of x, and the shear resistance from the atmosphere is negligible. (a) Sketch the approximate shape of the velocity profile w(x), considering the boundary conditions at the wall and at the film surface. (b) Suppose film thickness δ , and the slope of the velocity profile at the wall, (dw/dx) w a l l , are measured by a laser Doppler anemometer (to be discussed in Chap. 6). Find an expression for the viscosity of the oil as a function of ρ , δ (dw/dx) w a l l , and the gravitational acceleration g. Note that, for the coordinate system given, both w and (dw/dx) w a l l are negative.
Oil of viscosity
μ
and density
ρ
drains steadily down the side of a tall, wide vertical plate, as shown in Fig, C1.4 In the region shown, fully developed conditions exist', that is, the velocity profile shape and the film thickness
δ
are independent of distance z along the plate. The vertical velocity w becomes a function only of x, and the shear resistance from the atmosphere is negligible.
(a) Sketch the approximate shape of the velocity profile w(x), considering the boundary conditions at the wall and at the film surface.
(b) Suppose film thickness
δ
, and the slope of the velocity profile at the wall, (dw/dx)wall, are measured by a laser Doppler anemometer (to be discussed in Chap. 6). Find an expression for the viscosity of the oil as a function of
ρ
,
δ
(dw/dx)wall, and the gravitational acceleration g. Note that, for the coordinate system given, both w and (dw/dx)wallare negative.
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A union feedback control system has the following open loop transfer function
where k>0 is a variable proportional gain
i. for K = 1 , derive the exact magnitude and phase expressions of G(jw).
ii) for K = 1 , identify the gaincross-over frequency (Wgc) [where IG(jo))| 1] and phase cross-overfrequency [where <G(jw) = - 180]. You can use MATLAB command "margin" to obtain there quantities.
iii) Calculate gain margin (in dB) and phase margin (in degrees) ·State whether the closed-loop is stable for K = 1 and briefly justify your answer based on the margin . (Gain marginPhase margin)
iv. what happens to the gain margin and Phase margin when you increase the value of K?you
You can use for loop in MATLAB to check that.Helpful matlab commands : if, bode, margin, rlocus
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The 120 kg wheel has a radius of gyration of 0.7 m. A force P with a magnitude of 50 N is applied at the edge of the wheel as seen in the diagram. The coefficient of static friction is 0.3, and the coefficient of kinetic friction is 0.25. Find the acceleration and angular acceleration of the wheel.
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Using MATLAB , find the magnitude and phase plot of the compensators
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