T1-22
Hz. When the machine is installed on five linear isolators with rubber flexible elements selected in accordance with the
manufacturer's recommendations, different for different mounting points (line 2, f_{n} = 15 Hz), the maximum amplitude of the
relative vibrations (resulting in waviness of the ground surface) was 0.35 3m. However, when the grinder was installed on
five indentical CNF isolators with rubber flexible elements (line 1, f_{n} = 20 Hz, or about two times stiffer than the linear
isolators), the maximum relative vibration amplitudes was 0.25 3m, about 30% lower.
7.0 CONNECTIONS OF SPRING ELEMENTS
7.1 Springs in Parallel
These combine like electrical resistance in series. This is the case when several springs
support a single load, as shown in Figure 34. The springs are equivalent to a single spring,
the spring constant of which is equal to the sum of the spring constants of the constituent
springs. The spring constant k of the single equivalent spring is given by:
k = k_{1} + k_{1} + k_{1}. (27)
7.2 Springs in Series
The series connected springs in Figure 35 combine like electrical resistances in parallel.
The equivalent single spring is softer than any of the component springs. The spring con-
stant k of the equivalent single spring is given by:
= + . (28)
If n springs are in series, this formula is readily extended to:
= + + + ..... + . (29)
7.3 Spring Connected Partly in Parallel and Partly in Series
Obtain equivalent spring constants for each set of parallel or series springs separately
and then combine. For example, in Figure 36, the springs k_{1} and k_{2} are equivalent to a single
spring, the spring constant of which, k_{e1}, is given by:
= + = or k_{e1} = (30a)
The three springs, k_{3}, k_{4}, k_{5} in parallel, are equivalent to a single spring, the spring constant
of which, k_{e2}, is given by:
k_{e2} = k_{3} + k_{4} + k_{5} (30b)
Now equivalent springs k_{e1} and k_{e2} are in series. Hence, the spring constant k of the equiva-
lent spring for the entire system is:
= + or k = (30c)
0.5
0.2
0.25
0.35
0.1
0.05
10
15
Frequency (Hz)
20
25
30
1
2
35
Figure 33 Amplitude of Relative Motion in Work Zone with: 1 - Regular (Linear) Isolators; 2 - CNF Isolators
1
__
k
1
__
k_{1}
1
__
k_{2}
1
__
k
1
__
k_{1}
1
__
k_{2}
1
__
k_{3}
1
__
k_{n}
1
___
k_{e1}
1
__
k_{1}
1
__
k_{2}
k_{1} + k_{2}
_______
k_{1}k_{2}
k_{1}k_{2}
______
k_{1 }+
k_{2}
1
__
k
1
___
k_{e1}
1
___
k_{e2}
(k_{1}k_{2})(k_{3} + k_{4} + k_{5})
______________________
k_{1}k_{2 }+ (k_{1 }+ k_{2})(k_{3 }+ k_{4}+ k_{5})
k1
k2
k1
k2
k3
Figure 34 Parallel
Connection
of Springs
Figure 35 Series
Connection
of Springs
k1
k2
k3
k4
k5
Figure 36 Mixed
Connection
of Springs