FEEG2002W1
SEMESTER 2 EXAMINATIONS 2023-24
TITLE: MECHANICS, MACHINES AND VIBRATION
Q1
(i) In the 5-bar linkage mechanism with a suspension link that is shown in Figure 1.1, the crank AD that rotates about fixed axis A, has a pin D which slides in the straight slot of link CE. Link BC turns about joint B.
Figure 1.1 Schematic of a hydraulic excavator mechanism
Determine the mobility of this mechanism and state any assumptions you make in calculating the mobility. [5 marks]
(ii) In the crank and slotted-lever, quick-return mechanism shown in Figure 1.2, the link OA with 40 mm length rotates at a constant angular speed of 100 rad/s. A sliding link which is pin joined to OA at A, slides along the link BC and hence makes BC rotate about B, as shown in the figure:
Figure 1.2. Crank and slotted-lever, quick-return mechanism
The ground link length is 70 mm. For the position shown, i.e., when the angle of OA is 60°, calculate the angular velocity of the link BC. [16 marks]
(iii) For the same four-bar linkage mechanism in Figure 1.2:
a) Locate all the possible instantaneous centres of velocities. [7 marks]
b) Calculate the angular velocity of the slider, using the “angular velocity ratio” theorem. [6 marks]
TOTAL [34 MARKS]
Q2
A technician is given two identical elastic springs of stiffness k equal to 2500 Nm-1 and a single viscous damper c. The technician is asked to support a motor of mass m equal to 5 kg and cannot decide which of the two configurations shown in Figure 2.1 to use. For any calculations take g the acceleration due to gravity to be 10 ms-2.
Figure 2.1. Mounting configurations using two identical springs, a single viscous damper and a supported mass.
(i) Mounting options and comparisons.
(a) What is the frequency above which any motor vibration to the ground will begin to be isolated for the two configurations? [6 marks]
(b) What will be the static deflection of the mount which gives the widest frequency range of isolation? [2 marks]
(c) What is the value for the viscous damping constant in the two cases such that the combined loaded mount (mass
supported by the springs plus damper) has critical damping? [4 marks]
(d) The technician uses both mount options and measures equal vibration on the motor mass when the motor is operating at 3000 rpm. Why is the vibration level the same irrespective of the mount option chosen? Estimate the motor out of balance (mee) when the motor mass acceleration is 10g at this speed. [4 marks]
(ii) A novel mount installation
An academic proposes to use a massless bar that is pivoted at its midpoint with one bar end supporting the mass via one spring and the other bar end restrained by an identical elastic spring k as shown in Figure 2.2. One can assume that the bar is free to rotate by an angle θ and displacements are small from the equilibrium position.
Figure 2.2. Isolation comprising a massless bar and the two identical springs and the attached mass.
(a) What are the displacements of points A and B at the bar ends in terms of the bar rotation? Hence draw the free body diagram for the supported mass m and the massless bar from the static equilibrium position. [6 marks]
(b) Considering the equation of motion for the bar, show that the ‘novel’ system is identical in its dynamic behaviour to one of the systems presented in part (i) when the damper is not present. [7 marks]
(c) A single viscous damper is to be added either in parallel with the upper or lower spring. Explain why there is no preferable position to add it. [4 marks]
TOTAL [33 MARKS]
Q3
A uniform horizontal sign of length 8L is represented by a rigid bar supported by two vertical elastic springs each of stiffness Kv as shown in Figure 3.1 below. The bar has a mass M equal to 8PAL , where PA is the mass per unit length. The bar has a moment of inertia J about its centre equal to 6ML2 .
Figure 3.1. A horizontal sign in its equilibrium position.
(i)
(a) Derive expressions for the potential and kinetic energies of the system in terms of the small vertical displacement y of its centre and small rotationθ in radians about its centre. [4 marks]
(b) Using Lagrange’s equations, or otherwise, obtain the equations of motion for the system in matrix form. [8 marks]
(c) Show that the system has no rigid body modes in this plane. [3 marks]
(ii) The bar is observed to be bending when the wind is gusting and it is necessary to estimate the fundamental natural frequency treating the bar as having a bending stiffness EI.
(a) Show that the function φ(x) = W (x2 + Lx) is suitable to use in Rayleigh’s method. [2 marks]
(b) Using Rayleigh’s method, show that the estimated fundamental natural frequency w0 , in rad/s, is given by the expression:
[11 marks]
(c) The elastic springs have stiffness Kv equal to 40 kNm-1. The sign has a total length 8L= 8 m, mass per unit length PA equal to 10 kgm-1 and a bending stiffness EI about its neutral axis of 2 MNm-2. Evaluate the estimated fundamental natural frequency in Hz to two decimal places. [3 marks]
(d) To avoid a resonance due to the wind loading both springs are repositioned to be at the left-hand end of the beam.
Why is this a wrong decision? [2 marks]
TOTAL [33 MARKS]