The design of a vibration mount, or shock mount, involves a calculation of the force- or motion transmission from equipment to foundation or vice versa. In order to facilitate this
computation, it has been recognized that many mechanical systems can be represented by reasonably simple mathematical models. Once these models have been analyzed, the results can be tabulated and used with minimum effort. Below, we have selected 6 mathematical models of vibrating machinery and systems, which we believe will be applicable to a large number of vibration-isolation problems encountered by the engineer. The basic results are summarized for each case. Solved problems are given in the subsequent section and illustrate the application of these cases to vibration-mount design.
CASE A: UNBALANCED SINUSOIDAL FORCE ACTING ON EQUIPMENT
p = frequency of applied force or forcing frequency, rad/sec ξ = damping ratio = c/cc, where cc = critical damping coefficient ω = (undamped natural frequency of system, rad/sec) g = gravitational constant, 386 in/sec²
p/ω = frequency ratio (ratio of forced to natural)
x = equipment displacement, measured from equilibrium, in.
The performance of the vibration moment can be measured by factors of transmissibility. The force transmissibility, TF, is defined as follows:
TF = Max. force transmitted to base Max. unbalanced force acting on equipment
The quantity (1 - TF) is sometimes called the isolation or insulation and is expressed as a percentage.
The motion amplification of the equipment can be measured by comparing the max. equipment displacement to the static displacement under the unbalanced force Fmax. This leads to the displacement transmissibility of the equipment, TDE:
TDE = Max. displacement of equipment from static Static displacement of equipment under force Fmax
(A-1)
The condition p/ω = 1 is known as resonance when ξ = 0.
When the damping ratio is less than 0.1 (10%) the max value of TF still occurs very nearly when p/ω =1 and the corresponding value of TF is 1/2ξ very nearly. TF is less than 1 only when p/ω, is greater than 1.41.
The following basic vibration chart, Table 9, gives static deflection vs. frequency and % vibration isolation (1 - TF). It is basic to all vibration-mount circulations (see problem section).
TABLE 9
The following equations are useful in calculating the natural frequency (ω) and the damping ratio (ξ) of the system:
Let xst = static deflection of spring, in.
Ω = natural frequency of system with damping, rad/sec.
ΩN = same in cycles/min.
ωN = natural frequency of system without damping, cycles/min.
ξ = damping ratio of system = c/cc.
an = nth max. amplitude of displacement. x, of equipment (on same side of mean position).
Then: xst = W (in.) (A-4) K
The sketch below shows a displacement-time curve of a free, damped vibration with successive amplitudes (a1, a2.. .). Equation (A-6) states that the natural undamped frequency depends only on the static deflection of the system, and this is often readily measured. The damping ratio, ξ, can be measured by allowing the damped system to vibrate and measuring the rate of decay of maximum amplitudes. Equation (A-9) then shows how the damping ratio can be determined.
CASE B: FOUNDATION MOVES WITH SINUSOIDAL DISPLACEMENT
W = equipment weight. lbs
k = vibration mount spring constant, lbs/in c = system damping coefficient
w = undamped natural frequency of system, rad/sec.
ξ = damping ratio, c/cu
Y = Ymaxsinpt = forced motion of base Ymax = max. displacement of base, in
p = forced frequency of base section (rad/sec)
x = equipment displacement from equilibrium position, in
Here we are interested in reducing the displacement transmitted from the base to the equipment. The transmission factors are:
TD = Max. displacement of equipment Max. displacement of base TDl = Max. vibration - mount deflection Max. displacement of base
TD is numerically equal to TF [eg. (A-1), base A].
When the damping ratio is less than about 0.1, the maximum force transmitted to the equipment is given very nearly by kTDlYmax.
CASE C: SHOCK MOTION OF BASE (BASE SUDDENLY BROUGHT TO REST OR BASE ACQUIRES SUDDEN VELOCITY)
W = equipment weight, lbs.
k = vibration mount spring constant, lbs/in.
c = system damping coefficient.
x = equipment displacement from equilibrium, in.
y = displacement of base, in.
t = time, secs.
The results in this section are due to R.D. Mindlin, "Dynamics of Package Cushioning", Bell System Technical Journal, Vol.24, pp., 353-361, July-Oct. 1945.
This case is applicable to equipment which drops from a height, or to equipment which acquires a sudden velocity.
Let V = sudden velocity change of base, in/sec x = damping ratio = c/cc
w = = undamped natural frequency of system, rad/sec g = gravitational constant, 386 in/sec²
dmax = max. isolator deflection, measured from equilibrium position, in dst = static isolator def lection = W/k, in.
amax = max. equiment acceleration, in/sec² When 0 ≤ξ≤ 0.2, the following is nearly exact:
(C-1) Figure C-1 illustrates eq. (C-1).
When the damping is small, max force transmitted to equipment is very nearly kdmax.
CASE D: SUDDEN IMPACT ON EQUIPMENT
The results in this section are due to R.E.
Newton: Shock and Vibration Handbook, McGraw Hill Book Co., Inc., N.Y.("Theory of Shock Isolation Vol. II, p. 3I-28) and R.D.
Mindlin: "Dynamics of Package Cushioning".
See Case C.
W = equipment weight, lbs
K = vibration mount spring constant, lbs/in C = system damping coefficient
x = displacement of equipment from equilibrium, (in)
Sudden impact, or a sharp blow is characterized by a large force (Fo) acting for a short period of time (to) as shown in the sketch. For practical purposes, suddenness is taken to mean that to is small in comparison with the natural period of vibration of the system. The impulse, l is defined as the area under the force-time curve, i.e.
l = Foto lb-secs. (D-1) The impulse, l, results in a sudden downward velocity V of the equipment, given by
(D-2)
The maximum mount deflection and the maximum equipment acceleration (dmax and amax) can be calculated by substituting V into equation (C-1) of Case C.
CASE E: SUDDEN IMPACT (VELOCITY CHANGE OF BASE) WITH EQUIPMENT CONTAINING RESILIENT COMPONENT The results of this are due to RD. Mindlin (Ref. of Case C).
WZ = weight of lightweight component (lbs) z = displacement of WZ from equilibrium (in)
T228
kzxCzx Wx kc yV
= spring constant of component support system (lbs/in)
= damping constant of component support system
= main equipment weight, (lbs)
= main-equipment displacement from equilibrium (in)
= vibration mount spring constant (lbs/in)
= main equipment damping constant
= displacement of base (in)
= velocity of base (in/sec)
The sudden impact considered is that in which the base is either moving and suddenly brought to rest, or is at rest and acquires a sudden upward velocity. The former, for example, is an approximation of what happens to a package which drops to ground (say) and remains in contact with the ground. A sudden velocity change, V, of the base is shown in the sketch. Since the equipment and base are assumed always to remain in contact, we refer to this case as one of inelastic impact. It is also assumed that the component weight, Wz, is small in comparison with the main equipment weight, W.
In view of this assumption we can neglect the forces exerted on the main equipment by the component support system (but not the converse). This means that the base-mount-main-equipment system is identical to that of Case C.
In the event Case E represents a system falling from a height h to ground, the velocity V is given by:
V = 2gh in/sec (E-1) where: g = 386 in/sec²
h = height of fall, inches
The amplification factor, A0, is defined by R.D. Mindlin as follows:
The term "quasistatic" means that the component deflection is calculated for the condition in which the acceleration pulse is very slow in comparison with the natural period of vibration of the component.
The following curves (R.D. Mindlin) show plots of the amplification factor against the frequency ratio ω1/ω2 where:
ω1 = = undamped natural frequency of component support system (E-3) and
ω2 = kg/w = undamped natural frequency of main equipment and vibration mount (E-4) Various values of the damping of the component support system are considered:
Figure E-1: 1% main equipment damping.
Figure E-2: 5% main equipment damping.
Figure E-3: 10% main equipment damping.
Sometimes the impact involved may be elastic, i.e. there may be rebound of the package. It is then often a fair approximation, which is on the conservative side, to evaluate the isolator design on the basis of no rebound (inelastic impact) as above.
However, if a more exact analysis is required, see R.D. Mindlin (Ref. Case C).
Figure E-1 Amplification Factors For Linear Damped Cushioning With No Rebound. β2 = 0.01 (From R.D. Mindlin:
Dynamics of Package Cushioning, p.82)
Figure E-2 Amplification Factors For Linear Damped Cushioning With No Rebound. β2 = 0.05 (From R.D. Mindlin:
Dynamics of Package Cushioning, p.82)
Figure E-3 Amplification Factors For Linear Damped Cushioning With No Rebound. β2 = 0.1 (From R.D. Mindlin:
Dynamics of Package Cushioning, p.83)
CASE F: EQUIPMENT SUBJECT TO A DISTURBING FORCE AND/OR DISTURBING TORQUES - 4-POINT MOUNTING
The results of this case are due to E.H. Hull: "The Use of Rubber in Vibration Isolation," ASME Transactions (J. Applied Mechanics) 4,3, Sept. 1937, pp.(A-109)-(A-114).
The figure shows equipment, center of gravity C, mounted on 4 supports, which may represent vibration mounts, and acted upon by a disturbing force, Fy, in the y-direction and/or by torques, Tx, Ty, Tz acting singly or in combination about the x, y, z axes, which are principal axes through the center of gravity, C.
The four supports are symmetrically disposed relative to the center of gravity, their location defined by distances bx, by, bz from the axes, as shown. The mass moments of inertia through C about the coordinate axes are lx, lz respectively. As a result of the external force and torques, the equipment motion is a displacement of C, maximum values of which are denoted (a) by the coordinates (Cx, Cy, Cz) and (b) by the rotation of the equipment (from equilibrium) about the coordinate axes (θx, θy, θz). This displacement is generally small relative to the major dimensions of the equipment.
Let M = mass of equipment (equipment weight/g, where g = 386 in/sec²).
ky = total vertical stiffness of the four supports i.e. 4 times the stiffness of each support, lbs/in ks = total horizontal or shear stiffness of the four supports, i.e. 4 times the horizontal stiffness of each support, lbs/in ω = frequency of sinusoidally applied force and torques (rad/sec)
Damping is assumed to be negligible.