Mathematical Methods and Mechanics

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Question I

Consider the following simultaneous linear algebraic equations,

x1 *6x2*fo1;3:5 h*3xz*bt=4 2rr+3x2+lxt=l

where ft is an unspecified constant.

(a) By using the method of Gauss elimination with partial pivoting only, rcduce the augmented matrix to upper triangular form.

(8 marks)

O) Hence, identift the value(s) of ,t, for which the equations have

- aunique solution, - no solution.

(4 marks) You are not required tofind the solution of the equationsfor ony case.

Question 2

A particle l, of mass 2 kg is connected to the left-hand wall at O by amodel spring of stiffrress 20 Nm-' and natural length 1.5 m, and on the right by another model damper of damping constant 4 Ns rn-'. The position of the particle at time / is x, as measured from o.

/o = 1.5m m:Zkg

Fieure Q2

The vibrating system is oscillating horizontally as shown in Figure Q2. The forcing functionfir) is on the displacement at B.

(a) Draw a diagram clearly expressing the forces acting on the particle, the positive xdirection, and define these forces.

(5 marks)

(b) Derive theequation of motion of the particle. (5 marks)

(3 marks)

Question 3

(a) Calculate the eigenvalues and its corresponding eigenvectors of a 2x2 matrix given by

[-+ -,ol [r 7 [

O) H€nce, find and explain the general equations

(9 marks)

solution of the system of differential

ft = 4x1 -10x2 *z= 3r1+ 7xz.

(3 marks)

Question 4

Two particles, A and B of mass I kg and 2 kg respectively, are attached to a model spnng of natural length /o and stiffness ,t as shown in Figure Q4. They rest on a rough horizontal table; the coefficient of sliding friction between the particles and the table is p. At time r = 0 the system is at rest with the spring at its natural length and particle I is at the point O. A constant horizontal force of 0.6 N is applied to particle I and at subsequent time r the positions of the particles (measured from the fixed point O) are x Ndy,as shown. -

A k'l'e, ffi0.'ru --+

(a)

Fieure 04

Write down the position of the centre of mass of the system, at time / > 0, in terms of the positions of the particles.

(2 marks)

Apply a force diagram for each particle. Express all forces in vector form, giving magnitude and diredion.

' ',{d} : , &p$,ryeur',res{il6'r&,fid,.*heqlWimnfimotiot,o#tki,egixtre ofreass of the two particles.

(2 ma*s)

.1..

, r;';!.1' ;i 5

(6 marks)

(3 mslcs)

'.i i''-

Question 5

(a) Forthefollowing3 x 3matrix

lz o rltll-r 4 -rl l-, z oJ

(i) calculate the three eigenvalues, and (9 marks)

(ii) calculate the three corresponding eigenvectors. (10 marks)

(b) The eigenvalues of a rertain 2 x 2 matrix B are 1 and -2. Let

A,=2B'3+88-1-7I

Calculate the values of the trace and the determinant of A. Show how you obtained youranswers

(6 marks)

Question 6

The question concerns the system of differential equations

I *= y-3 ty=r'-ry+2

(a) Write the system in the form

L*,i7r : V(x".y)

where V(x, y) is a vector field. (2 marks)

O) Calculate all the equilibrium points for the system of differential equations. (6 marks)

(c) For each of the equilibrium points (Xo, Io) define the excesses n and v by x= uaXo and y=v + Io andusetheseto finda linearapproximation ri = M u forthe system of equations nearthe equilibrium point.

(8 marks)

(d) In each case calculate the eigenvalues for the matrix M. Hence classiS each equilibrium point and sketch the vector fields in the neighbourhood ofthat point.

(9 marks)

Question 7

Throe model springs AB, BC and CD, each of natural length /0, have stiffrress 2k,3k and I respectively. A particle of mass 2m is attached to the springs at B and another particle of mass rn, 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 3lo apart and the system is free to oscillate along a horizontallne AD.

Fisurp O7

You may assume that the only forces acting on the particles in a horizontal direction are those due to the springs.

(a)

(b)

(c)

WritE down and describe in vector form, the changes in the spring forces acting on the particles, when the particles at .B and C arc displaced from their equilibrium positions by distances xr and.r2 respectively in the direction away from the fixed poirrt l, for the case.r2 > xr > 0.

(3 marks)

Write down the equations of motion of the two particles. (6 marks)

Find and relate the normal mode angular frequencies of the system to the corresponding normal mode displacement ratios.

(12 marks)

The initial positions of the particles are such that .B is f lo to the left of its equilibrium position while C is { /o to the right of its equilibrium position. The particles are released from rest fmm these positions. Describe the resultant motion is one of the normal mode, and write down the angular frequency of the motion'

(4 marks)

(d)

Question I

Consider the following simultsreous linear algebraic equations,

xt * 4xz * 3r: = 3 rt-9xz-2h=-7 3xt- xz*44=k

wherp &is an unspecified cofftaflt

(a) By usrng the mettrod of Gauss elimination with essential row interehanges only, reduce ttre augmented maix to upper friangular form.

(9 ma*s)

(b) Hence, find the value(s) of,t, for which the equations are consistent, explaining the reasons for your choice.

(3 ma*s)

Qucstion 2

(a) Find the eigenvalues and its corresponding eigenvectors of a 2x?, marix given by

l-+ 6l L-, r rl'

(b) Hence, find th.e gene,ral solution of ihe system of differential equations

(9mad$)

(3 ma*s)

*r=4x1 + 6xz *2 = 4x1 +llxz.

Question 3

A particle of mass m is connected to the left-hand wall by a model spring of stiffiress ft and natural length /0, and a model damper with damping constant r, and on the right by another model damper of damping constant r. The position of the particle at time / is x (measured fromthe left-hand wall).

fieure O3

The vibrating system is oscillating horizontally as shown above, with symbols in their usual meaning. The forcing functionflr) is on the displacement at B.

(a) Draw a diagram clearly indicating the forpes acting on the particle and the positive x- direction.

(3 marks)

(b) Write down the vector equations of all the forces acting on the particle. (4 marks)

mi + 2r* + kx = klo+r!. (6 marks)

Question 4

Two identical particles, each having a mass of 2kg are moving with velocities -4i + 3j and 2i + 5j respectively (measured in ms-l). The two particles collide and coalesce to form a composite particle.

(a) Using the conseryation of linear momentum, write down the vector equation which will determine the velocity of the composite particle after the collision.

(3 marks)

(b) Find the final velocity of the composite particle. (3 marks)

Question 5

The question concems the system of differelrtial equdions

{ *=3y-xy lj'=z*-*Y'

(a) Write the system inthe form

l*,rjr = v(r,Y)

where V(A y) is a vector field. (2 marks)

(b) Find all the equilibrium points forfte system of differential equations. (6 marks)

(c) For each of the equilibrium points (Xu Yo\ define the excesses u and v byr = u* Xo and y=v + Ioand usettreseto find alinear approximation t = M u for the system of equations near the equilibrium point.

(7 marks)

(d) ln each case find the eigenvalues for the marix M. Hence classifu each equilibrium point and sketch the vector fields in the neighbourhood of that point.

(10 marks)

Question 6

Two model springs OA and AB have stiffiress 4k and 3& respectively, and equal natural lengths /6. A particle ofmass 2z is attached to the springs atA and another, of mass 3m at B.T\e end O of the first spring is fixed and the syste& is free to oscillate along a horizontal line as shown in the figure.

OAB Fisure 06

You may assume that the only forces acting on the particles in a horizontal direction are those due to the springs.

(a)

{c)

Let rr and xz be the displacements of the particles ,{ and ,B from their respective equilibrium positions, away from the fixed point O. Show that the equations of motion of the particles are

Zmir-4b1 +35s, friz= fut- laz.

(8 marks)

Find the normal mode angular frequencies of the system and the corresponding normal mode displacement ratios.

(12 marks)

The idtial positions of the particles are such that the springs OA and AB each have length * /0. The particles are released from rest tom these positions. Show that the resultant motion is one of the normal modes, and write down expressions for the displacements of the two particles at time r for this motion.

(5 marks)

Question I

Consider the simultaneous equations

h-?*tz* 5xr=6 h*3xz- 44=7

L\ + 1xz -lb3 = 12

(a) By using the method of Gauss elimination tfihpartial pivoting only,rcducethe augmented matrix to upper triangular form.

(8 marks)

(b) Hence, using back zubstitutien, find the solution of the equations. (4 marks)

Question 2

The variable x in a population model satisfies the differential equation

*-:0.+*(r-'l-zs dt [ 1000/

(a) Find the two equilibrium values of the population. (7 marks)

O) For each of the trvo equilibrium values, determine whether it is stable or unstable'

(5 marks)

Question 3

A particle of mass 5 kg is suspended from the ceiling by a model spring of stiffiress 100 Nm-I and natural length 0.5 metres. A model damper with damping constant 50 Nsrn-l is attached to the bottom of the particle. The other end of the damper is attached to the floor at a point directly below the point of suspension of the spring. The height of the ceiling above the floor is 2.5 metres. The particle moves in a vertical line. The acceleration due to gravity can be taken as 10 ms''. Take the upward pointing x-axis with origin at the floor.

(a) Draw a diagram clearly indicating the forces acting on the particle and the positive x- direction.

(3 marks)

(b) Write down the vector equations of all the forces acting on the particle. (3 marks)

i + 10i +20x = 30. (4 marks)

(d) Find the equilibrium position of the particle. (3 marks)

Question 4

Two particles having masses 3 kg arld 5 kg are moving with velocity vectors i and -2i +; respectively (measured in ms-'). The two particles collide and coalesce to form a composite particle.

(a) Using the conservation of linear momentum, write down the vector equation which will determine the velocity of the composite particle after the collision.

(4 marks)

(b) Find the final velocity of the composite particle. (3 marks)

Question 5

(a) Find the eigenvalues and the corresponding eigenvectors for the matrix

lz -r -21 lo r -41. [-, -r ,.]

O) The eigenvalues of a certain 2 x 2mafrix A are -2 and 3. Let

(17 marks)

B=A2 - 2A - 3I.

What are the values of the trace and the determinant of B? Show how you obtained your answers.

(8 marks)

Question 6

Three model springs AB, BC and CD each have stiffiress ft and natural length /o are arrange horizontally as shown in Figure Q6. Two particles of equal mass nl are attached to the springs at B and C and the ends A ard D are fixed to tvro points a horizontal distance 3lnapart. The system is free to move in the horizontal lne AD.

4ieure O6

You may assume that the only forces acting on the panicies in a horizontal direction are those due to the springs.

(a) Let xr md xz be the displacements of the particles .B and C from their equilibrium positions. Show that the equations of motion of the particles are

frir= -2fu1 + fuz fr*z: lur-2b2.

(8 marks)

(b) Find the normal mode angular frequencies of the system and the corresponding normal mode displacement ratios.

(l I marks)

(c) The initial positions of the particles are such that B is { Io from I and C is } Io from D. The particlec are released from rsst from these positions. Show that the reultant motion is one of tlre normal mode, and urrite down the angular ftequency ofthe motion.

(6 madcs)

!=i t'tiiri';.; : fr. ,-.i;1i1.:.i{}{}1/i i:;1.r' 'i-i. ::r}:i"i;{'l . j ,{i. ..1', , '+, ..

].,. ;r1..;r:rilt.r r^).: l.; r.,'i::r:iil;11 .,..,,,:.,

!.r. r'.! .j'i:Ii:.:I .:. ,'-i,ii l'r,ifl-i- ::rg;.;i:Ji ,:1:;',i.l1lrii

: ' ii ",i,' i !i..;i:'11:r' :.i.:'i:.I i."'r1..''? i;1!:;i' :i:rir:.:I,. ::.

* i'"'r'lY:.1."1i1{..,,: ,::a j;r?,. i;;rl1:i,:: .": -'t: '.'.., ,.:.,tk:.::,:i,

1t-i!ji;i;i11.-!fill:*!. tt' ; i.t,:r;.{,i 1:: r.-' ttl.,.i .; '-a;}i-}iijl..t ;,' r'.'t

. 1.. . -:, l:

,..' 'iiiii {}, l-:'';r rli:!'i;.ll ii;*ii{.-:-i:

Questiotr 1

Use Gaussian elimination with essentful row inlerch*ages, to solvo a sot of simultaneous equations, in three unknowns x1, 12 and.r3, given in the augmonted mafiix by,

(12 marks)

Question 2

'\--l (a) Find *re eigenvalues, and coneqponding eigenvectors, of the matrix

lt -lolB=l l. L5 -8J

(9 marks)

O) Findthe general solution ofthe rystem of differential equations

.* = 7x-I$y i * 5r- 8y

(3 marks)

[o I zl -21

Lr -:;l -;J

Question 3

A particle A, of mass I kg, moves along a frictionless horizontal track. The particle is attached to a fixed point O by a model damper, and to another point B by a model spring as shown in Figure Q3. The two points O and B arc a distance 2 metres.apart on the track. The damping constant is 6 Ns m-1. The spring has stiffiress 9 Nm-r and natural length I metre. The displacement of the particle, x, is measured from O.

r=6 a k:9m=l h=l Fisure 03

(a) Draw a force diagram for the particle, indicating only the forces that eause the oscillations, and define these forces.

(3 marks)

(b) Derive the equation ofmotion of the particle. (5 marks)

(d) The particle is initially projected from its equilibrium position towards the point B. Describe briefly and qualitatively the subsequent motion of the particle.

(2 marks) Ngte : Do not solve the equation of motion.

Question 4

A particle of mass z lies on a frictionless plane which is inclined at an angle of f r 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 plme, to a second particle of mass rr, which is hanging freely. When release from rest, the hanging particle moves downwards.

(a)

(b)

(c)

Fisure O4

Drawtwo force diagrams showing all the forces acting on each of the two particles.

(4 marks)

Write down the equations of motion of each of the two particles. (5 marks)

Find the coflrmon acceleration ofthe two particles and the tension in the string in terms of m andg the magnitude of the acceleration due to gravity.

(4 marks)

Question 5

Consider the non-linear system of differential equations

f+x+t l+x+y+l

(a) Find the two equilibrium points of the system. (8 marks)

(b) For each of the equilibrium points (Xo, Yo), define the excesses z and v by x = u * Xo and y: v * Yo and use these to find a linear approximation u : M u for the system of equations near the equilibrium point where u: (zl, v)r.

(9 marks)

(8 marks)

Question 6

Three model springs AB, BC and CD have stiffness k, 4k and k respectively, and equal natural lengths /e. Particles of equal mass tn are attached to the springs at B and C, and the ends A and D are fixed to two points a horizontal distance 6ls apart as shown in Figure Q6. The question is concerned with the longitudinal vibrations of the system.

6lo

(a)

Fisure 06

Suppose that when the particles are in equilibrium, the length of the spring AB is x.o. The length of the spring CD when the particles are in equilibrium will also be x.o.

(i) Draw a force diagram for the forces acting on each particle. (2 marks)

(ii) Write down the spring forces H1 and H2 acting on the mass at B, in terms of x"o and /0, and use a unit vector to speciff its direction.

(2 marks)

(iiD By considering the spring forces acting on the mass at B, find in terms of /e alone, the lengths of the three springs when the particle is in equilibrium.

(3 marks)

x= i=

O) By applying Newton's second law to each of the particles in turn, show that the displacem€ilts xr and xc of the particles at B and C from their equilibrium positions satis$/ the differential equations

ffiin=-Sfua+Afuc tnic:4fua*Sbc

(6 marks)

(c) Write down the dynamic matrix of the system when the mass of each of the panicles is 5 kg anA *r" stiffiress of each of the springs,4B and CD is 125 Nm-l. Find the normal modes angular frequencies for this system and the corresponding normal mode displacement ratios.

(8 marks)

(d) Describe the general initial conditions that will give normal mode motion, for each of the two normal modes, in cases where:

(i) the particles are initially stationery; (2 marks)

(ii) the particles are initially at their equilibrium positions. (2 marks)