posted 10 months ago

posted 10 months ago

\dot{\mathrm{x}}=\left[\begin{array}{cc} -0.2 & -0.6 \\ 2 & -4 \end{array}\right] \mathrm{x}+\left[\begin{array}{c} 0 \\ 1.5 \end{array}\right] u \quad y=\left[\begin{array}{ll} 1 & 0 \end{array}\right] \mathbf{x}

Compute the eigenvalues "by hand."a.

b. Use MATLAB to verify your answer in part (a).

c. Describe the free response of the output y(t) given an arbitrary initial state x(0).

Use MATLAB or Simulink to verify your answer in part (c). The initial state vector is

\mathbf{x}(0)=\left[\begin{array}{ll} x_{1}(0) & x_{2}(0) \end{array}\right]^{T}=\left[\begin{array}{ll} -2 & -1 \end{array}\right]^{T}

posted 1 years ago

\ddot{x}+15 \pi x+20 x^{3}=\sin (150 \pi t) ; \quad x_{0}=1 ; \dot{x}_{0}=1 ; \quad t \in[010]

4 \ddot{x}+5 x \cdot \dot{x}+6 x=\sin \left(t^{2}\right) \quad x_{0}=2 ; \dot{x}_{0}=2 ; \quad t \in[0300]

(b) Describe the method of using the trajectory graph (velocity vs.displacement) to identify the stability of the system defined by a differential equation.

(c) In the motion equation of a MEMS (Micro-electromechanical systems)resonator, identify the factors affecting the natural frequency, and factors that contribute to the system damping. Write down the analytical equation of the fundamental natural frequency.

(d) Describe the procedure of fabricating a radio frequency (RF) MEMS switch with the top-down process, sketch out process flow.(6 mark)

posted 1 years ago

(b) A voltage of 100 V is generated from a PZT5A (a piezoelectric material)cube of 40 mm length (piezoelectric voltage constant g33 = 22 x 103 Vm/N). Calculate the exerted force.

(c) For a two-mass, three-spring system shown in Figure Q3(c), derive the equation for the resonant frequencies.

(d) Write general forms for the following two second order differential equations that describe the motions of micro resonators and identify the linearity of these systems.

34 \ddot{x}+17 \dot{x}+3 x=5

2 \ddot{x}-\dot{x} x^{3}+9 x^{2}=4

posted 1 years ago

(b) For the three differential equations below describing MEMS dynamic systems, write their transfer functions based on the Laplace Transform.Work out fundamental frequency for each case (neglect the damping effect).

3 \ddot{x}(t)+11 \dot{x}(t)+300 x=3 f(t)

a \dot{x}(t)+25 a x(t)=c f(t) ; \quad \text { a and } c \text { are non-zero constants }

\frac{1}{3} \ddot{x}(t)+27 x(t)=5 f(t)

(c) There is a parallel plate MEMS actuator shown in Figure Q 3(c). The initial gap go is 12 µm, and the area of the plate A is 200 µm x 100 µm. (Note:free space permittivity is 8.85 x 10-12 F/m)

(1) Derive the formula for the electrostatic force F;

(2) Assume a 5 volts potential is applied to the actuator. Calculate F when the top plate is in its initial position.

(3) Derive the equation for Vvs. g and draw a graph showing V vs. g (g varies from 0 to 12 um with 2 µm step size; given spring constant kis 30 N/m). Give a brief explanation about the pull-in voltage.

(d) Describe the working principle of a strain sensor that is made of a piezoresistive material and a piezoelectric material respectively.

posted 1 years ago

(b) To design a cantilever type piezoelectric energy harvester, calculate one set of dimensions to match with the mechanical vibration of 3500 Hz. (UsePZT5A as the piezoelectric material, its density p is 7750 kg/m³) The stiffness of a cantilever is given as k = wt3E/(4L³), where w=30 µm, t, Lare width, thickness, and length of the cantilever, E is the Young's modulus of the material (1.96×1011 Nm-2).[4 marks]

(c) Derive the state-space representation (or system matrix) for the two microresonators' dynamic motion equations shown below. Write MATLAB functions for these two equations and solve them with boundary conditions provided. (MATLAB codes need to be included in the answer sheet). Plot xvs. t, i vs. t, and i vs. x, respectively.

13 \ddot{x}+9 \dot{x}+x=\sin (0.28 t) ; \quad x_{0}=0 ; \dot{x}_{0}=0 ; \quad t \in[0200]

\check{5 x}+3 x+70 x=4 \quad x_{0}=1 ; \dot{x}_{0}=0 ; \quad t \in[020]

Sketch out a process flow for fabricating a MEMS resonator using the bulk micromachining process.

posted 1 years ago

The code has numerous errors in it. It is supposed to compute the values of y = ax? + bx + c for various input parameters a, b, c, for x in the range of [s,e] (start to end) and save to a file called results.txt. Spot and correct the various errors. Your goal is to revise codes to run without error. Submit m. file with revised codes and use % to comment how you revised right above the revised coding.

posted 1 years ago

posted 1 years ago

The height can be modeled by the logistic function:

H=\frac{100.8}{1+23 e^{-0.093 t}}

Where H is the height (in.) and t is the time (days). Make a plot of the height versus time. The figure should show the data from the table above as points only. Use red circles to plot each data point.

On the same figure, plot the height modeled by the equation as a solid blue line.

Add a legend in the southeast corner of your figure, labeling the data from the table as "Data Set" and the line from the equation as “Model". Include x and y labels for time and height. Make sure to include units in your labels. Include a title to your figure.

posted 1 years ago

G(s)=\frac{50}{(s+1)(s+3)(s+10)}

design a controller to yield a 10% overshoot and a settling time of 0.5 second. Place the third pole10 times as far from the imaginary axis as the dominant pole pair. Use the phase variables for state-variable feedback.