T1-26
To illustrate this approach in a particular case, let's consider a connecting-rod motion of a slider-crank mechanism,
Figure 39, as in internal-combustion engines. This motion can be shown to have the following Fourier expansion:
r = crank length, in.
= connecting rod length, in.
q = crank angle, rad or deg.
x = piston placement (piston motion
in-line with crank pivot), in.
3 = crank speed, assumed constant,
rad/sec
a = piston acceleration, in/sec2
= A0 + cos q + A2 cos 2q – A4 cos 4q + A6 cos 6q ... (46)
– = cos q + A2 cos 2q – A4 cos 4q + A6 cos 6q ... (47)
where A2, A4, A6 are given as follows in Table 3 [4].
10.0 DESIGN PROBLEM EXAMPLES
The following are a number of problems intended to familiarize the reader with the basic applications of vibration isola-
tors. More advanced techniques which would result in stiffer isolators while achieving adequate isolation can be found in [1].
NOTE: In the following problems, unless otherwise stated, it is assumed that the loads are evenly distributed among the
mounting points.
x
__
r
1
__
4
1
__
16
1
__
36
a
___
r32
2
___
p
0
0
0
/r
A2
A4
A6
0.3431
0.2918
0.2540
0.2250
0.2020
0.0101
0.0062
0.0041
0.0028
0.0021
0.0003
0.0001
0.0001
—
—
3.0
3.5
4.0
4.5
5.0
TABLE 2 FOURIER EXPANSIONS FOR VIBRATORY PROCESSES IN FIGURE 38 (angles in radians)
Frequency of
Harmonics
Square wave
Saw tooth
Repeated steps
Wave Shape Function
Harmonic Amplitude as Fractions of 2h (3 = fundamental frequency)
3
23
33
43
53
63
2
___
3p
2
___
5p
1
___
p
1
___
3p
1
___
5p
1
___
2p
1
___
6p
2sin pl
______
p
2sin 3pl
_______
3p
2sin 2pl
_______
2p
2sin 4pl
_______
4p
2sin 5pl
_______
5p
2sin 6pl
_______
6p
x
r
Crank
Connecting rod
Piston or slider
q
Figure 39 Schematic of a Slider-Crank Mechanism
TABLE 3 COEFFICIENTS FOR FOURIER EXPANSION OF CONNECTING ROD MOTION
1
___
4p
