Assignment 3 1. In a combustor, butane (C4H10) is used as the fuel with an air fuel ratio of 22, by mass. Calculate the fuel utilization efficiency of this process if the exhaust gas has a temperature of 800K. In a constant pressure combustion, a stoichio
The Hong Kong Polytechnic University
Department of Mechanical Engineering
2008/09 Semester 1 ME4204 Mechatronic Systems
Quiz 1 Student Name: Programme Code: Student ID: Date:
Please answer ALL the questions and write your answer on separate sheets if nece
ME4204 08/09 Semester 1
The Hong Kong Polytechnic University
Department of Mechanical Engineering
2008/09 Semester 1
ME4204 Mechatronics Systems Mini-Project Objective To enhance students understanding and design experience to integrate the components of
ma2176 a4 Brief Sol Integration and its Applications 1.
x 3 x if 0 x < 1 . Find F ( x) = f (t )dt , 0 x 2 . Let f ( x) = 2 + x if 1 x 2 0
Solution: For 0 x < 1 , F ( x) = f ( t ) dt = ( 3 t ) dt = 3 x
0 0 x x
x2 . 2
x
For 1 x 2 ,
F ( x) = f ( t ) dt = f
ma2176 a4 Integration and its Applications
x 3 x if 0 x < 1 . Find F ( x) = f (t )dt , 0 x 2 . 1. Let f ( x) = 2 + x if 1 x 2 0
2. Evaluate the following integrals: x +1 (a) dx x 3. Evaluate the following integrals: (a)
(b)
x2 1 + x 2 dx
(c)
e3x + 1 e x +
ma2176 a3 sol Applications of Derivatives 1. An airplane, flying horizontally at an altitude of 1 km, passes directly over an observer. If the constant speed of the plane is 240 km per hour, how fast is its distance from the observer increasing 30 seconds
ma2176 a3 1.
Applications of Derivatives
An airplane, flying horizontally at an altitude of 1 km, passes directly over an observer. If the constant speed of the plane is 240 km per hour, how fast is its distance from the observer increasing 30 seconds lat
ma2176 a2 sol Derivatives 1. If y = x sin x , prove that x 2 y ' 2 xy '+ 2 + x 2 y = 0 .
(
)
Proof: y = x sin x y ' = sin x + x cos x y " = cos x + cos x x sin x = 2 cos x x sin x . So x 2 y ' 2 xy '+ (2 + x 2 ) y = x 2 ( 2 cos x x sin x ) 2 x ( sin x + x
ma2176 a2 1.
Derivatives
If y = x sin x , prove that x 2 y ' 2 xy '+ 2 + x 2 y = 0 .
(
)
2. 3.
If u = ax 2 + 2bx + c , prove that
d 2ax 2 + 3bx + c ( xu ) = . dx u
Differentiate the following functions with respect to x: (a)
(1 + 2 x 5x )
2 3000
,
(b) sin
ma2176 a1 sol Functions, Limits and Continuity 1. Find the largest possible domain of each of the following functions: 2x , (b) y = f ( x) = 25 x 2 , (c) (a) y = f ( x) = 2 x 4x 5
( x) =
x2 1 x 1
Solution: (a) x 2 4 x 5 = ( x 5)( x + 1) = 0 x = 1 or x =
ma2176 a1 1.
Functions, Limits and Continuity
Find the largest possible domain of each of the following functions: x2 1 2x , (b) y = f ( x) = 25 x 2 , (c) ( x) = (a) y = f ( x) = 2 x 1 x 4x 5
2.
Let f ( x ) = x 3 + 2, g ( x ) =
2 . x 1
g Find formulas for
Design and Manufacturing I (ME3204)
Test Name: ID: Question (a) Complete the profit model for a consumer product as shown in the following table. (8 marks)
Year Sales Price Number of units sold Net Sales Cumulative net sales Unit cost Cost of product sold
ma2176 a5 Brief Sol Complex Numbers 1. Find the square roots of 5 12i .
Solution: 2 If 5 12i = a + bi , then 5 12i = ( a + bi ) = a 2 b2 + 2abi .
a 2 b 2 = 5 . Thus, the solutions to Equating the real and imaginary parts, we have ab = 6 3 2i, 3 + 2i . 2.
ME 4905 2008 Advanced Numerical Methods for Engineers Homework 1 (Due date: 16 Sept in the class) 1. If the determinant of the coefficient matrix A is zero, what can you say about the system? 2. Consider the following linear system: 4x y + z = 8 2x + 5 y
Supplement: Finite Difference
1. Finite difference method
The complete governing equations in boundary layer:
u v + =0 x y
(1)
u
u u 2u +v = g(T - T) + x y y 2
(2)
u
2T T T +v = y x y 2
(3)
For an analytical solution, it allows for temperature determinat
ME 4905 Advanced Numerical Methods
Fourier Approximation
Lecturer: T.Y. Ng (PhD) Email: mmeng@polyu.edu.hk
Introduction
Engineers often deal with systems that oscillate or vibrate. Therefore trigonometric functions play a fundamental role in modeling such
ME 4905 Advanced Numerical Methods
Least-Squares Regression
Lecturer: T.Y. Ng (PhD) Email: mmeng@polyu.edu.hk
Curve Fitting
Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates. There are two general approac
ME 4905 Advanced Numerical Methods
Solutions of Linear Systems: Iterative Methods
Lecturer: T.Y. Ng (PhD) Email: mmeng@polyu.edu.hk
2 Equations 2 Unknowns
3 possibilities: 1.No solution 2.one solution 3.Infinite many solutions
Linear Systems
Many mathemat
The Hong Kong Polytechnic University
Department of Mechanical Engineering
2008/09 Semester 1 ME4204 Mechatronics Systems
Assignment 2 Name: Student ID:
Notes: 1. Please answer all the following questions. 2. Submission Deadline: 1 November, 2008 3. Marks
The Hong Kong Polytechnic University
Department of Mechanical Engineering
2008/09 Semester 1 ME4204 Mechatronic Systems
Quiz 2 Student Name: Programme Code: Student ID: Date:
Notes: Please answer ALL the questions. You may write your answer on separate sh
The Hong Kong Polytechnic University
Department of Mechanical Engineering
2008/09 Semester 1 ME4204 Mechatronics Systems
Assignment 1 Name: Student ID:
1. Find out the currents through R1 and R3 in the circuit of Fig. 1. Given the voltage sources of V 1 =
The Hong Kong Polytechnic University
Department of Mechanical Engineering
2008/09 Semester 1 ME4204 Mechatronics Systems
Assignment 2 Name: Student ID:
Notes: 1. Please answer all the following questions. 2. Submission Deadline: 1 November, 2008 3. Submis
The Hong Kong Polytechnic University
Department of Mechanical Engineering
2008/09 Semester 1 ME4204 Mechatronics Systems
Assignment 1 Name: Student ID:
1. Find out the currents through R1 and R3 in the circuit of Fig. 1. Given the voltage sources of V1 =
ME4204 Mechatronic Systems
2008/09 Semester 1 Review
Topics
1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction of Mechatronic Systems Electrical Components of Mechatronic Systems Sensors and Transducers Part 1 Sensors and Transducers Part 2 Actuators Part 1 Actuator