Vibration- Quarter car analysis

Question 1

How do we model suspension?

Answer 1

As a linear spring and damper, no friction

Question 2

How do we model tyres?

Answer 2

As a linear spring, no damping

Question 3

What is the mean square spectral density MSSD?

Answer 3

The MSSD provides a quantitative measure of the frequency content of a random signal.

Often referred to as a power spectrum:
The power spectrum Sx(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range.

Question 4

What is the area under the MSSD equal to?

Answer 4

The mean square value (equal to the RMS value squared)

Question 5

How to find the MSSD

Answer 5

For a random signal, the MSSD can be constructed as follows: pass the sinal through a narrow band pass filter having passband ∆w centered at some frequency w. The result is approximately the contribution of frequency w in the total signal. Now calculate the mean-square of such contribution. One can show that the mean square of qi(t) is the sum of the mean squares of the individual contributing sinusoids:

Question 6

What is the method for finding RMS?

Answer 6

First square all the values, then find the average (mean) of these square values over a complete cycle, and finally find the square root of this average.

Question 7

What are 3 traits of the MSSD curve?

Answer 7

It is a real, even and non-negative function

Question 8

How do we find the response of a system in terms of spectral densities?

Answer 8

Multiply the spectral density of the input signal by the transfer function (frequency response function) of the system SQUARED, which gives you the MSSD of the output signal. (datasheet)

Question 9

How do you get the mean square response from the MSSD?

Answer 9

Recall that the area under the MSSD graph is equal to the mean square value. Therefore we take the MSSD and integrate the MSSD between + and - infinity. However this integral doesn’t need to be performed due to the white noise analytical solution that can be found.

Question 10

In general, what type of components have larger amplitudes in a vertical displacement profile?

Answer 10

Long wavelength components have larger amplitude and vice versa.

Question 11

What is wavenumber n?

Answer 11

A measure of spatial frequency (cycles/m) and is what the Displacement MSSD is plotted against

Question 12

Describe the process of obtaining the MSSD for frequency

Answer 12

First you measure the vertical displacement of the vehicle in terms with distance travelled.
The MSSD of this curve is than calculated, plotting the displacement MSSD (log axes) against wavenumber (cycles/m). This makes a straight line from which an analytical solution can be derived. For vehicle response calculations this needs to be defined in terms of frequency, to do this we use the fact that the mean square value and thus area under each MSSD does not change, and then analyse the mean square integral, substituting dn for dω (by using an expression for ω in terms of wavenumber n (cycles per meter) and the speed of the vehicle V and 2pi rad/cycle (n2πV =ω)

Question 13

What are the assumptions are inherent in the linear quarter car model (and state how this may be differ from reality)?

Answer 13

• Motion is only vertical (the can also be translation and rotation in 3 axes)
• There is only 2 masses (other masses In the car like the engine will be flexibly mounted and have their own vibration modes)
• Tyre modelled as a spring (there may also be damping in reality)
• Suspension modelled as a linear spring and damper (spring stiffness may be non constant and different damping rates in compression and extension)
• Tyre contact modelled as point follower (may actually loose contact)

Question 14

What is the analytical solution for the displacement MSSD?

Answer 14

Sz(n) = kn^-w

k=roughness
w=downwards slope of the MSSD
n=wavenumber

Question 15

What are the 3 performance criteria when assessing the performance of a vehicle?

Answer 15

• Body acceleration - measure of ride discomfort
• Working space - the space available for relative displacement of the sprung and unsprung masses is usually limited
• Tyre force - the ability of the tyres to generate braking or cornering force is reduced if the tyre force oscillates

Question 16

How to calculate the transfer functions for the sprung and unsprung masses using their equations of motion:

Answer 16

Take Laplace transforms of the equations of motion, taking 0 as the initial conditions, and then replace s with jω. These equations can then be manipulated into equations for body acceleration, working space and tyre force outputs by rearranging for Zs/Zr and Zu/Zr and then plugging into the form of the transfer functions (remember to convert to frequency domain as well)

Question 17

What is the transfer function equation for Body acceleration?

Answer 17

H(BA) = Zs (double dot/acceleration) / Zr (dot/input velocity)
Zs=sprung mass displacement
Zr=displacement of bottom of tyres

Question 18

What is the transfer function equation for working space?

Answer 18

H(WS) = (Zs - Zu) / Zr (dot/input velocity)
Zs=sprung mass displacement
Zu=unsprung mass displacement

Question 19

What is the transfer function equation for tyre force?

Answer 19

H(TF) = (Zu - Zr)k / Zr (dot/input velocity)
Zu=unsprung mass displacement
Zr=displacement of bottom of tyres
k = tyre stiffness (modelled as linear spring)

Question 20

Why is it beneficial to have the input spectral density as white noise?

Answer 20

As the spectral density for white noise is constant with frequency, which is achieved by considering the vertical velocity rather than displacement as the input

Question 21

What is angular temporal frequency?

Answer 21

Just another name for frequency related to angles and time, therefore is rad/s.

Question 22

When is the input velocity equal to white noise in terms of w, the downwards slope of the displacement MSSD?

Answer 22

When w = 2, so the displacement MSSD has gradient -2

Question 23

What are the 2 modes of vibration that show up in the transfer functions as peaks in their plots?

Answer 23

Body bounce mode and Wheel hop mode