University of Mumbai 2007-3rd Sem M.E Mechanical Engineering System Modelling & Analysis -e - Question Paper
Ofi-Ex-Nk-O/. 17
Con. 3348-07. . / 1 ' ' l'~( ' 'J ' 1 ' ' ' BB-1657
<4 Hours) [ Total Marks : 100
N.B. (1) Attempt any five questions.
(2) Assume any unspecified data it required. _ a ., . #, * iv/i /I vH
1. (a) According to Newton's taw of cooling, the rate at which the temperature of a body changes is 12 proportional to the difference between the instantaneous temperature of the body and the temperature of the surrounding medium. If a body whose temperature is initially 100 BC is allowed to cool in air which 'emains at ihe constant temperature 20 *C and if it Is observed that Is 10 min. Ihe body has cooled to 60 C, find the temperature of the body as a function of lime.
(b) Evaluate each complex function at the indicated value of S and determine its magnitude and phase. 8
(s +2)
(t) G<s) = fsz when s 12
2. (a) Governing differential equation is given as :
y + 4y = f(t) with y (0) = y (0) = 0 where f(t) * 0 , t < 2 = t, t 2: 2 Solve using L.T. method.
State the theorems clearly.
(b) The governing equation for a first order dynamic system is given as k + 2x 8 (t - 1). Assuming $ that the system is subjected to zero initial conditions, determine the response x{t). Roughly sketch
the graph of the response curve.
(c) Repeat the problem Q. 2(b) for x + x u(t - 1) subjected to zero initial conditions. 6
3. (a) OeMne system dynamics. 2
(b) Describe he significance of system modelling in engineering design. Explain the steps involved g in the modelling of a physical system.
(c) Solve the differential equation 12
1 _2f
y + 4y + 13y * - e sin3t
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for which y(0) 1 and y (0) a - 2.
4. (a) Determine the inverse Laplace transform of 6
FW = (s + 3)( sa +2s +5)
Using (i) Partial Fraction Method
(ii) Convolution Method.
(b) A mechanical system experiencing translational and rotational motion is given by 5
JO + B0 + K0 + Rk (R0 - x) - Ru(t)
mx + bx - k {RQ - x) 0 Where J, K, B, R, m, k, and b denote parameters and are regarded as constants, x and 8 represent displacement and angular displacement, respectively and u(t) is an applied force. Express the equations in second order matrix form.
(c) Governing differentia! equation is given as 7
2x + 3x + x * 1{t)
Obtain a state space form.
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(b) Relate various parameters of a mechanical system to analogous electrical system using voltage force analogy. Name the law governing each system.
Con. 3345-BB-1557-07. , , 2 , I I tT]
5. $1) Findthe state-space relation for (he electric network shown below. Output is V0{1)
(c) Define :
(i) State of dynamic system
(ii) State variable
(iii) State space
(Hi) State equation
(iv) Output equation.
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6. (a) A machine having mass m * 1500 kg and a mass moment of inertia of J0 400 kg m2 is supported on elastic sipports as shown In figure. If the stiffness of supports are K, * 2800 N mm" 1 and kf 2100 N mm' \ and the supports are located at /, - -4 m and is * -5 m, determine the natural frequencies and mode shapes of the machine tool.
(b) Explain the significance ol :
(I) Elasticially conjugate points
(ii) Dynamically conjugate points
(iii) Doubly conjugate points.
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7. (a) Consider a dynamic system with input f(t) and output x. whose state variable equations are
xa = f - 3x, - 2x2 + f(t) )
Directly team these equations, determine input-output equation,
(b) Determine the transfer function for the single degree of freedom mechanical system shown in 10 figure below using its state-space form.
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| Earning: Approval pending. |
