Coventry University - School of Mechanical, Aeronautical and Automotive Engineering

Module Title - Noise, Vibration and Harshness
Module Code - 334MAE

Learning Outcomes Assessed:
Application of one and two degree of freedom models to solution of dynamic problems
Application of modal analysis theory

Question 1

On the Asessments page for the module Moodle site you will find data from a slow sine sweep test conducted on a car on a "four-post" road simulator for the frequency range 0-20 Hz in the EXCEL spreadsheet "RideComfortData". The response has been measured vertically on the seat rail of the driver's seat while the front two actuators moved up and down in phase and the rear actuators remained at rest. The amplitude of the actuator motion was such that the velocity amplitude was constant throughout the frequency sweep at 100 mm/s. The response amplitude at the seat rail is measured in g.

a) Obtain the transmissibility between the input motion of the actuators as a single input and the motion of the seat rail. Provide a graph from EXCEL showing the transmissibility.

b) The transmissibility in (a) does not tend to 1 at low frequency as might be expected for an ideal one degree of freedom system. Explain this.

c) Estimate the natural frequency and damping factor for the sprung mass mode involving motion of the body on the front suspension.

d) If the diameter of the front tyres of the car is 0.6 m estimate the car speed in kph which would cause an "engine shake" problem in the car.

e) In road tests on a relatively flat test track surface at 60 kph with one front wheel deliberately unbalanced, the idle shake at the seat rail position, measured using an accelerometer, was 2.8 g rms. The other three wheels were in almost perfect balance. What displacement amplitude and frequency of the front two actuators of the road simulator would be necessary to achieve the same seat rail motion in the laboratory?

Question 2
The front of a car is lifted 100 mm and released. It moves down to a maximum position 35.4 mm below the equilibrium position in 270 milli-seconds. After one or two further oscillations up and down it comes to rest again.

a) Assuming that the front of the car behaves as a 1 dof system, estimate:
(a) The damping factor ç.
(b) The undamped natural frequency.
(c) The static deflection of the front spring due to the weight of the car.
(d) The transmissibility of the front suspension at the natural frequency.

b) If the car has its natural frequency reduced by 20% what difference would this make to the transmissibility at 10 Hz, assuming a one degree of freedom system is still accurate at 10 Hz?

You may use EXCEL in your solution of this problem.

Question 3

For low frequency response, a car can be modelled as a two degree of freedom system as shown in Figure Q2. The data for the total front and rear suspension stiffnesses, distances, car mass and rotational inertial Jo are as follows:

k₁ = 18 kN/m, k₂ = 22 kN/m,

l₁ = 1.0 m, l₂ = 1.2 m,

m = 1000 kg, J_o = 1010 kg m²

a) Compare the ratios of the distances and the stiffness. Also compare the radius of gyration with the geometric mean of the two distances. What conclusion do you come to?

b) Draw a free body diagram for the instant when the centre of gravity (C.G.) is displaced upwards z and rotated clockwise θ from the equilibrium position. Hence find the equations of motion in terms of z and θ.

c) Using the values provided calculate the two natural frequencies in Hz.

d) Also find the positions of the centre of rotation for the two modes.

Figure 3

Question 4

At 800 rpm, the idling speed, the steering wheel and column of a car vibrates violently. It can be assumed that this is because the steering wheel and column have their first natural frequency at exactly 1600 cycles per minute and this is being excited by the engine second order. To reduce the vibration, a vibration absorber comprising a mass on a rubber bush is to be attached near the top of the column. (The bush can be modelled as a spring stiffness.)

a) The mass of the absorber is chosen to be 0.8 kg. What bush stiffness is required to minimise the vibration level at an engine speed of 800 rpm.

b) Estimate the value of the bush stiffness such that the system will have its natural frequencies at 1500 cycles per minute and 2000 cycles per minute. The absorber mass is still 0.8 kg. (In this case the vibration at 800 rpm will not be the minimum.)

c) Estimate the two new natural frequencies with the vibration absorber fitted and give them in Hz.

Question 5
A steering wheel and column are modelled approximately as a three degree of freedom lumped parameter system. x1 is the steering column position. Three natural frequencies are 36.2 Hz, 48.4 Hz and 65.3 Hz. The stiffness matrix [k] is:

The three mode shapes are calculated to three decimal places as:

a) Use orthogonality to show that the mode shapes are accurate, bearing in mind that the data has been rounded to three significant figures.

b) Find the three principal stiffnesses each to three significant figures.

c) Predict the point frequency response function for the steering column position x1 based on this data and show it on a graph over an appropriate range. You only need to give the amplitude ratio for displacement divided by force in mm/N units. Use a loss factor of 0.1 for each of the modes.

You are advised to use MatLab and EXCEL for this question.