Further Mathematical Methods and Mechanics engineering (comes with solutions, need workings only)

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a lot of maths inside. So I need someone who is strong in maths and mechanics engineering

I have answers already inside. I just need the workings on how to get the correct answer in the attached to study for my exam.

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MTH215e Examination – July Semester 2017 Further Mathematical Methods and Mechanics Monday, 13 November 2017 4:00 pm – 6:00 pm ____________________________________________________________________________________ Time allowed: 2 hours ____________________________________________________________________________________ INSTRUCTIONS TO STUDENTS: 1. This examination contains SIX (6) questions and comprises FIVE (5) printed pages (including cover page). 2. You must answer ALL questions. 3. This is an Open Book examination. 4. All answers must be written in the answer book. Marks will only be awarded if FULL working is shown. At the end of the examination Please ensure that you have written your examination number on each answer book used. Failure to do so will mean that your work cannot be identified. If you have used more than one answer book, please tie them together with the string provided. THE UNIVERSITY RESERVES THE RIGHT NOT TO MARK YOUR SCRIPT IF YOU FAIL TO FOLLOW THESE INSTRUCTIONS. MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS) Examination – July Semester 2017 Page 1 of 6 Answer all questions. (Total 100 marks) Question 1 A set of simultaneous equations, in three unknowns x, y and z, is given by x – 2y + 2z = 1 –4x – y + 3z = –8 14x – y – 5z = 26. (a) Identify these equations in augmented matrix form, labelling the rows. (1 mark) (b) By using Gauss elimination method, reduce the augmented matrix to upper triangular form. (6 marks) (c) Identify the solutions for x, y and z and state whether the set of equations has • • • a unique solution, no solution, or an infinite number of solutions. If it has an infinite number of solutions, find its general solution. (5 marks) Question 2 (a) Solve the eigenvalues and its corresponding eigenvectors of a 2×2 matrix given by  23 − 14  3 − 73  . − 53  (9 marks) (b) Hence, explain the general solution in matrix form, of the system of differential equations 3x1 = 2x1 – 7x2 3x2 = –14x1 – 5x2. (3 marks) MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS) Examination – July Semester 2017 Page 2 of 6 Question 3 A particle of mass 5 kg is constrained to move horizontally on a smooth surface, and has two model dampers and a model spring attached to it as shown in Figure Q3. The spring has a stiffness of 5 Nm-1 and a natural length of 2 m. The left damper has a damping constant of 3 Nsm-1 and the right damper 6 Nsm-1. The distance between the two fixed supports is 4 m with the position vectors x(t) and y(t) are indicated as shown along the unit vector i. Figure Q3 (a) Draw a force diagram for the particle, indicating all the forces that cause the oscillations. (2 marks) (b) Express each of these forces in vector form. (3 marks) (c) Apply Newton’s second law, express the equation of motion of the particle. (4 marks) (d) What is the equilibrium position of the particle, measured from O if point A is now stationary? (2 marks) (e) If the particle is initially projected from its equilibrium position toward the right, described briefly the subsequent motion of the particle. (2 marks) Question 4 A ball of mass m is moving with velocity 2i – 2j with respect to a Cartesian coordinate system. It collides with a second ball of identical size and mass, which is initially at rest. After the collision, the two balls have velocities u1i + v1j and u2i + v2j. (a) Apply the conservation of momentum, write down the momentum equation of the system of two balls. Simplify this equation. (4 marks) (b) Find the energy of the system before and after the collision. (3 marks) MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS) Examination – July Semester 2017 Page 3 of 6 (c) Show that if, (u1i + v1j) • (u2i + v2j) = 0 then the collision is elastic. (6 marks) Question 5 A system of non-linear differential equations is given by  x = y 2 + x − 1 .  2  y 1 y 2 x x = + − +  (a) Calculate all the equilibrium points for the system of differential equations. (8 marks) (b) For each of the equilibrium points (X0, Y0), define the excesses u and v by x = u + X0 and y = v + Y0 and use these to explain in matrix form, a linear approximation u = Mu for the system of equations near the equilibrium point where u = (u,v)T. (8 marks) (c) Hence in each case, find the eigenvalues and explain these equilibrium points. Sketch the vector fields in the neighbourhood of each point. (9 marks) Question 6 Consider the longitudinal vibrations of two particles B and C, of mass 4 kg and 1 kg respectively. Particle B is attached by a model spring of stiffness 6 Nm-1 to a fixed point A. The two particles are connected in between by a model spring of stiffness 2 Nm-1. The displacements of the two particles from their equilibrium positions O1 and O2, are denoted by x and y respectively, as shown in Figure Q6. You may ignore air resistance and any other frictional forces. Figure Q6 MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS) Examination – July Semester 2017 Page 4 of 6 (a) Draw force diagrams showing the forces acting on each particle. (2 marks) (b) Describe in vector form, the changes in the spring forces acting on the particles, when the particles are displaced from their equilibrium positions by distances x and y respectively in the direction as shown, for the case y > x > 0. (3 marks) (c) Describe the equation of motion of the two particles and show that it can be written as  x  y = A    x  y ,   and hence write down the dynamic matrix A of the system in numerical form. (5 marks) (d) Relate the eigenvalues of matrix A and hence write down the normal mode angular frequencies of the system. (4 marks) (e) Relate the eigenvectors of matrix A and hence write down the normal mode displacement ratios of the system. State whether each normal mode is in-phase or phase-opposed. (6 marks) (f) Write down the displacement of each particle from its equilibrium position as a function of time in matrix form. (Note: There should be four arbitrary constants). (2 marks) (g) If the particles are released from rest with the initial displacements  x  − 0.2  y  =  0.2      relate qualitatively the subsequent motion. (3 marks) ----- END OF PAPER ----- MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS) Examination – July Semester 2017 Page 5 of 6 Answers 1 −2 2 1  Q1(b) 0 −9 11 −4  0 0  0 0 1 Q1(c) x = ( 4 k + 17 ) 9 1 y = ( 4 k + 11) z =k 9 1 Q2(a) Eigenvalues are λ = 3 with Eigenvector   and  −1 1  λ = -4 with Eigenvector   2  x1   1  3t 1  −4 t Q2(b)= + C e C x   −1 2 e      2  y ) i R2 = −3 ( x−6 x i H = 5(2 − x) i Q3(b) R1 = Q3(c) 4  x + 9 x + 5 x = 3 y + 10 Q3(d) xeq = 2 m Q3(e) Damping Ratio = 0.9 This is a weak damping, so there is some initial oscillation Q4(a) −2j 2i = ( u1+u2 ) i + ( v1+ v2 ) j Q4(b) Before collision E1 = 4 m , E= 2 1 1 m ( u12 + v12 ) + m ( u22 + v22 ) 2 2 Q4(c) E1 = E2 Collision is elastic Q5(a) (1,0) and (0, -1) are equilibrium points u  1 0  u  u   1 −2  u  Q5(b) at (1,0)   =  . at (0, -1)    v  =  −2 1   v   v  0 1   v       Q5(c) (1,0) is unstable source. (0,-1) is an unstable saddle point −6 x i , ∆H 2 = 2 ( y- x ) i , ∆H 3 = −∆H 2 = −2 ( y- x ) i Q6(b) ∆H1 = 1  y, = y 2 x− 2 y 2 Q6(d) Normal mode angular frequencies are w1 = 1 , w2 = 3 Q6(e) When λ = −1 Displacement Ratio =2, normal mode is in-phase When λ = −3 Displacement Ratio =-2, normal mode is phase-opposed.  x 1  1 Q6(f)=  y  C  2  cos ( t + φ1 ) + D  −2  cos 3 t + φ2       Q6(c)  x= −2 x + ( )  x   −0.2  Q6(g) The displacement   =   does not corresponds to any two of the normal  y   0.2  modes, so the oscillation is non-sinusoidal MTH215e Copyright © 2017 Singapore University of Social Sciences (SUSS) Examination – July Semester 2017 Page 6 of 6 MTH215 – 2016 Exam Final Answers x1 = 10 , x2 =  5 and x3 =  5 There is virtually no change in the solutions between essential row interchanges and partial pivoting which is usually stable. So, there is no evidence of induced instability. However, a change of 0.001 in one of the coefficient results in a maximum change of approximately 0.002905 in the value of x1, which gives a scale factor of 2.905, so there is no evidence that the equations are ill-conditioned. 153 14 21 1. (a) (b) 2. (a) (b) Eigenvalues are 4 and 10. Eigenvectors are [1 –1]T and [1 1]T respectively. (c) trace B = 3. (a) (b) (c) H = – 10(x – 1)i and R = 3( y – x )i (use vector notations) x  3x  10 x  3 y  10 xeq = 1m 4. (a) W1i = –2mg (sin 6 i – cos 6 j), (b) 1 2 5. , det B = 9700 6 W2i = 3mg i (|T| – mg) For particle 1, ma = For particle 2, 1 ma = mg  3 |T| 2 5 a= (a) Eigenvalues are 1 and 2 with eigenvectors are [1 2]T and [1 1]T respectively.  x  3  1  x  0 t  y   2 0   y   6e        (c) (d) (a) (b) (c) g, |T| = 9 5 (c) (b) 6. 697 6 mg  xC  1 t 1 2t  y   A2 e  B 1e     C  x  1 t 1 2t 1 t  y   32e  21e  4e         x   x( A  x  y )   y    y ( B  x  y ) = V(x,y)     (0,0), (0,B) and (A, 0) A 0 Near (0,0), M =   0 B A  B 0  Near (0,B), M =    B  B (d)  A  A  M =   0 B  A (0,0) with eigenvalues 2 and 1, is an unstable source, (0,1) with eigenvalues 1 and –1, is an unstable saddle point, (2,0) with eigenvalues –2 and –1, is a stable sink. (a) (a) Near (A,0), 7. (b) (c) (d) ΔH1 = –kx1 i, ΔH2 = 4k(x2 – x1)i, ΔH4 = –kx2 i  x1  k  5 4   x1   x   m  4  5  x    2   2 ΔH3 = –ΔH2 = –4k(x2 – x1)i k k and 2  3 m m with displacement ratios 1 and –1 respectively.    x1  1  k  1  k      A cos t    B cos 3 t   x  1  m 1  m         2   Normal modes angular frequencies are 1  MTH215e Examination – July Semester 2016 Further Mathematical Methods and Mechanics Monday, 14 November 2016 4:00 pm – 6:00 pm ____________________________________________________________________________________ Time allowed: 2 hours ____________________________________________________________________________________ INSTRUCTIONS TO STUDENTS: 1. This examination contains SEVEN (7) questions and comprises SIX (6) printed pages (including cover page). 2. You must answer ALL questions in section A and any TWO (2) questions in section B. 3. This is an Open Book examination. 4. All answers must be written in the answer book. Marks will only be awarded if FULL working is shown. At the end of the examination Please ensure that you have written your examination number on each answer book used. Failure to do so will mean that your work cannot be identified. If you have used more than one answer book, please tie them together with the string provided. THE UNIVERSITY RESERVES THE RIGHT NOT TO MARK YOUR SCRIPT IF YOU FAIL TO FOLLOW THESE INSTRUCTIONS. MTH215e Copyright © 2016 SIM University Examination – July Semester 2016 Page 1 of 6 SECTION A (Total 50 marks) Answer all questions. Question 1 (a) Identify a set of simultaneous equations, in three unknowns x1, x2 and x3, by Gauss elimination with partial pivoting, the stage below was reached. 2 3 6 − 3   0 2 − 3 7  0 3 − 2 0  Complete the process of solution by the same method, and obtain values for x1, x2 and x3. (7 marks) (b) A certain set of three simultaneous linear algebraic equations in three unknowns x1, x2 and x3, by Gauss elimination in various ways. In the first place only essential row interchanges are made. Secondly, the equations are solved by Gauss elimination with partial pivoting. Thirdly, one coefficient is changed, from 15 to 15.001, and the equations solved again using essential row interchanges only. In each case the calculations are carried out to eight figure accuracy, and the answers rounded. The results are summarised in Table Q1(b). Method Essential row interchanges Partial pivoting Changed coefficient x1 6.789232 6.789233 6.786328 Solutions x2 3.676731 3.676731 3.674532 x3 –0.234767 –0.234768 –0.237341 Table Q1(b) Decide, on the basis of these results, whether or not the equations are likely to be ill-conditioned or whether there might have been induced instability in any of the methods. Give reasons for your answer. (5 marks) Question 2 A 2 × 2 matrix is given by (a) 7 3  A=  . 3 7  Solve the eigenvalues of the matrix. (4 marks) (b) Solve and explain the corresponding eigenvectors of the matrix. (4 marks) MTH215e Copyright © 2016 SIM University Examination – July Semester 2016 Page 2 of 6 (c) Let B be a matrix defined by B = (10I – A)–1 + A2. Solve the trace and the determinant of B? Explain how you obtained your answers. (4 marks) Question 3 A particle A, of mass 1 kg, moves along a frictionless horizontal track. The particle is attached to a fixed point O by a model spring, and on the right by a model damper as shown in Figure Q3. The damping constant is 3 Ns m-1. The spring has stiffness 10 Nm-1 and natural length 1 metre. The displacement of the particle, x, is measured from O and the forcing function y(t) is on the displacement at B. Figure Q3 (a) Draw a force diagram for the particle, indicating only the forces that cause the oscillations, and define these forces. (5 marks) (b) Express the equation of motion of the particle. (5 marks) (c) What is the equilibrium position of the particle, measured from O if point B is now stationary? (3 marks) Question 4 A particle of mass 2m lies on a frictionless plane which is inclined at an angle of 16 π to the horizontal as shown in Figure Q4. The particle is connected by a light inextensible string, passing over a small light frictionless pulley at the top of the inclined plane, to a second particle of mass 3m, which is hanging freely. When release from rest, the hanging particle moves downwards. Figure Q4 MTH215e Copyright © 2016 SIM University Examination – July Semester 2016 Page 3 of 6 (a) Draw two force diagrams showing all the forces acting on each of the two particles. (4 marks) (b) Apply Newton’s second law and write down the equations of motion of each of the two particles. (5 marks) (c) Calculate the common acceleration of the two particles and the tension in the string in terms of m and g, the magnitude of the acceleration due to gravity. (4 marks) MTH215e Copyright © 2016 SIM University Examination – July Semester 2016 Page 4 of 6 SECTION B (Total 50 marks) Answer 2 of the following 3 questions. Question 5 (a) Solve the eigenvalues and its corresponding eigenvectors of a 2×2 matrix given by 3 − 1 2 0  .   (8 marks) For the system of differential equations, x = 3x – y y = 2x + 6e–t. (b) Write and explain the system of differential equations in matrix form. (2 marks) (c) Find the complementary function. (3 marks) (d) Calculate the solutions of the system of differential equations which satisfy the initial conditions, x(0) = 0 and y(0) = 0. (12 marks) Question 6 The question concerns a flock of sheep and a herd of cows sharing a field and competing for the same food source, i.e. the grass. let x be the number of sheep and y be the number of cows. The evolution of the two populations can be modelled by the system of differential equations,  x = x( A − x − y ) .   y = y ( B − x − y ) where A and B are non-equal, positive constants representing the total food value of the field of grass to the sheep and cows respectively. (a) Explain the system of differential equations in matrix form [ x , y ]T = V(x, y) where V(x, y) is a vector field. (2 marks) (b) Calculate all the equilibrium points for the system of differential equations. (6 marks) MTH215e Copyright © 2016 SIM University Examination – July Semester 2016 Page 5 of 6 (c) For each of the equilibrium points (X0, Y0) define the excesses u and v by x = u + X0 and y = v + Y0 and use these to find a linear approximation u = M u for the system of equations near the equilibrium point. Specify this matrix M for each equilibrium point in terms of A and B. (8 marks) (d) In each case, A = 2 and B = 1, deduce the eigenvalues for the matrix M. Hence classify each equilibrium point and state if it is stable. (9 marks) Question 7 Three model springs AB, BC and CD, each of natural length l0, have stiffness k, 4k and k respectively. A particle of mass m is attached to the springs at B and another particle of mass m is attached at C as shown in Figure Q7. The end A and D of the springs are fixed to two points which are a horizontal distance 3l0 apart and the system is free to oscillate along a horizontal line AD. Figure Q7 You may assume that the only forces acting on the particles in a horizontal direction are those due to the springs. (a) Describe in vector form, the changes in the spring forces acting on the particles, when the particles at B and C are displaced from their equilibrium positions by distances x1 and x2 respectively in the direction away from the fixed point A, for the case x2 > x1 > 0. (3 marks) (b) Relate the equations of motion of the two particles. (6 marks) (c) Relate the normal mode angular frequencies of the system and the corresponding normal mode displacement ratios. (10 marks) (d) Write down the general solution of the motion. (2 marks) (e) Assuming the phase angle φ is zero, and with initial conditions, x1(0) = 0.1 and x2(0) = 0.2, find the required solution of the motion. (4 marks) ----- END OF PAPER ----MTH215e Copyright © 2016 SIM University Examination – July Semester 2016 Page 6 of 6
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Explanation & Answer

Attached.

MTH215e
Examination – July Semester 2017
Further Mathematical Methods and
Mechanics

Questions 1:
Answer to A:
{x - 2 y + 2 z = 1, -4 x - y + 3 z = -8, 14 x - y - 5 z = 26}
Result:
{x - 2 y + 2 z = 1, -4 x - y + 3 z = -8, 14 x - y - 5 z = 26}
Alternate forms:
{x + 2 z = 2 y + 1, 4 x + y = 3 z + 8, 14 x = y + 5 z + 26}
{z = -x\/2 + y + 1\/2, z = (4 x)\/3 + y\/3 - 8\/3, z = (14 x)\/5 - y\/5 - 26\/5}
y = (11 x)\/4 - 19\/4, z = (9 x)\/4 - 17\/4
y = (11 x)\/4 - 19\/4, z = (9 x)\/4 - 17\/4
Integer solution:
x = 4 n + 1,
y = 11 n - 2,
z = 9 n - 2,

Answer to B
Multiply first equation by 4 and add the result to the second equation. The result is:
𝑥
0
14𝑥

−2𝑦
−9𝑦
−𝑦

+2𝑧 = 1
+11𝑧 = −4
−5𝑧 = 26

Multiply first equation by −14 and add the result to the third equation. The result is:
𝑥
0
0

−2𝑦
−9𝑦
+27𝑦

+ 2𝑧 = 1
+ 11𝑧 = −4
−33𝑧 = 12

Multiply second equation by 3 and add the result to the third equation. The result is:

𝟏
𝟎
𝟎

−𝟐
−𝟗
𝟎

𝟐
𝟏
𝟏𝟏 − 𝟒
𝟎 𝟎

Notice the last equation is always true. So the system has infinitely many solutions.
Answer to C:
14𝑥
= ( −4
𝑥

−𝑦
−𝑦
−2𝑦

−5𝑧 26
3𝑧 − 8 )
2𝑧 1

Cancel the leading coefficient in row R2 by performing 𝑅2 ← 𝑅2 +
14𝑥

−𝑦
−7𝑥𝑦 − 2𝑦
7𝑥
−2𝑦

=( 0
𝑥
1

Cancel the coefficient 𝑅3 ← 𝑅3 −

14

0
0

Now, swap the matrix rows; R2

0
(

=( 0

0
14𝑥
=( 0

−𝑦
−27𝑦
14
27𝑦
− 14

−𝑦
−27𝑦
14

−5𝑥
33𝑧
14
33𝑧
14

−5𝑥
0

0

0

1

14𝑥

−𝑦

0

0

−27𝑦
14

0

0

0

1

=

(

−5𝑥
26
21𝑥𝑧 − 10𝑧 −56𝑥 + 52
)
7𝑥
7𝑥
2𝑧
1

−𝑦
−7𝑥𝑦 − 2𝑦
7𝑥
27𝑦

14

−5𝑥
26
21𝑥𝑧 − 10𝑧 −56𝑥 + 52
7𝑥
7𝑥
33𝑧
6

)
14
7

R3
14𝑥

14𝑥

∙ 𝑅1

∙ 𝑅1

14𝑥

(

2
7𝑥

0

26



6
7 )
17
4𝑧

26
513
56 )
17

4𝑧
19
4
513
56
17
− 4𝑧
)

−𝑦
−7𝑥𝑦 − 2𝑦
7𝑥
27𝑦

14

−5𝑥
26
21𝑥𝑧 − 10𝑧 −56𝑥 + 52
7𝑥
7𝑥
33𝑧
6

)
14
7

14𝑥
0
=(
0

0
1

0

0
19
− 4𝑦

0

0

1

1 0
1
0 9
=(



0
0

0 0

)

17
4𝑧

0
1
− 9)
17
− 4𝑘

1

𝟏

x = 𝟗 (𝟒𝒌 + 𝟏𝟕) 𝒚 =

𝟏
(𝟒𝒌 +
𝟗

𝟏𝟏) 𝒛 = 𝒌

Question 2:
Answer to A:
2
det ( 3
−14
( 3

−7
3 ) − 𝜆 (1
−5
0
3

0
) ; 𝜆2 + 𝜆 − 12
1
)

Now, solve for λ+𝜆 − 12 = 0: 𝜆 = 3, 𝜆 = −4
Eigenvectors for λ = 3:
2

(𝐴 −

−7

3
𝜆𝐼):(−14
3

−7
3
= (−14
3

1
− 3(
0

0
3
) = (−14
1

1
(
0

−7
3
−14)

3

3

−7
3
−14):

−7

3
−5)

3

𝑥
0
(𝑦) = ( ⋮ ) , 𝑟𝑒𝑑𝑢𝑐𝑒 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥

0

𝑥
1
1
): The system associated with the eigenvalue λ = 3: (𝐴 − 3𝐼) (𝑦) = (
0
0

1 𝑥
0
)( ) = ( )
0 𝑦
0

3

𝑦
𝟏
This reduces to equation: 𝑥 + 𝑦 = 0: Isolate: 𝑥 = −7, now, let ′ s plug: 𝑣 = (−𝑦) 𝑦 ≠ 0. Let y = 1, so: ( )
−𝟏...


Anonymous
This is great! Exactly what I wanted.

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