1、AN-04-2-1 Vibration Isolation: Harmonic and Seismic Forcing Using the Wilson Theta Method James A. Carlson, P.E. Associate Member ASHRAE Joseph Turner, Ph.D. ABSTRACT It is well known that equipment can causeunwantedvibra- tions in buildings. Springs have been used to isolate these vibrations and re
2、duce transmitted forces to building struc- tures. For simple vertical harmonic motion and free vibration, theproblem is easily evaluated using simple charts andjgures. Evaluation of spring-isolated equipment responding to earth- quakes is not simple. A numerical method is included to eval- uate resp
3、onse of equipment to earthquakes and harmonic forcing functions. Conjguration of equipment on springs is simplijied for numerical analysis. A simplijied approach and associated equations of motion can be developed to evaluate the response of the equipment with vertical and horizontal forcing functio
4、ns. Response of spring isolation in the vertical direction is based on the vertical forcing function acting on the center of mass and results in the translation of the total mass (up and down). Horizontal forcing functions result in a change in the angle around the center of mass. Ifthe mass of the
5、equip- ment is ofset from the center of mass, then the vertical and angular responses are coupled. Resulting tension and compression forces at the spring can be directly related to the center of mass vertical and rotational displacements and velocities. Results of this numerical method may be used t
6、o vertfi simplijied methods. INTRODUCTION Dynamic response of equipment to earthquake ground motion is becoming an active topic in the HVAC industry. Code requirements and equipment failures during recent earthquakes drive this growing attention. Code requirements focus on the seismic evaluation of
7、the anchorage and opera- tional effects in a very simple format. New building codes have been enhanced and are adopting more stringent requirements for both when to apply a seismic restraint design and increas- ing the earthquake forces applied to equipment. In some cases, the code requirements may
8、also require a shake table to prove operability. But for most code design considerations, the seis- mic restraint andlor anchorage is the only concern. For anchorage, the International Building Code (ICC 2000) endorses a static analysis, but will allow computational methods. A simplified static anal
9、ysis for calculating earth- quake forces acting on anchor bolts has been developed by ASHRAE for anchor bolt selection. But these equations are limited to very simple installations and only look at the peak response and estimated anchor bolt forces (tension, compres- sion, and shear) based on a stat
10、ic analysis and spring acceler- ation multipliers. Finite-element analysis programs have been used to look at more complex installations. These approaches also have drawbacks for the typical HVAC design engineer selecting spring isolators and anchorage systems. Finite-element programs are very compl
11、ex and are time consuming. Simpli- fied static analysis of complex systems may be too simple with unreasonable assumptions and results. One other analysis method includes calculating or measuring the impulse response and solving by convolution. This paper looks at a rigorous implicit evaluation of s
12、pring-isolated equipment that can be employed to evaluate simple and complex installation response to any input forcing functions. Several different input excitation functions are evaluated. Harmonic forcing functions (sine waves) are applied with frequencies from 1 to 30 Hz in the vertical, horizon
13、tal, and a combination of both. The seismic forcing function used for demonstration of the methodology is in terms of acceleration James A. Carlson is with the Omaha Public Power District, Fort Calhoun, Nebraska. Joseph Turner is at the University ofNebraska, Lincoln. 02004 ASHRAE. 321 tvg L 63 X t
14、Figure 1 Simple spring-isolated equipment mass Figure 2 Free body diagram of spring-isolated equipment. representation. using the El Centro earthquake data (Donea 1980). Future evaluations can be made with input excitations developed from earthquake peak response spectrum as defined in the building
15、codes. Design engineers can gain insight into the resultant forces at the spring and anchorage by looking at the system response. SPRI NG-ISOLATED EQUIPMENT Spring-isolated equipment is simplified for the analysis. A simple configuration for equipment is two masses resting on springs and dashpots (s
16、nubbers) and attached to the ground. The forcing functions (horizontal and vertical accelerations) are applied at the ground and represent forces generated by an earthquake. Equations of motion can be developed for this system. Structural engineers, looking at buildings, have already derived similar
17、 equations of motion. But buildings are uniform with the center of mass very near the center of the building. For equipment, the equations of motion are compli- cated by offset centers of mass. Figure 1 shows a typical instal- lation with springs mounted at the comers. The total mass is separated in
18、to two masses. The first mass is for the equipment elevated above the base by some height (hcg). The second mass represents a structural base. For equipment installed without a base, the base mass is simply zero for this analysis. The earthquake accelerations acting on the ground are represented by
19、v, and h, as seen in Figure 2. Intuitively, the horizontal oscillating forcing function will induce a rocking motion or a change in the angle defection of spring. The vertical acceleration will cause a translation of the total mass. The system selected to represent the typical equipment isolation sy
20、stem contains some information about the equip- ment and inherent features. These features include the stiff- ness (K,) and the damping coefficient (Ce) of the equipment in response to the horizontal forcing functions as shown in Figure 1. Damping coefficients for the equipment are typically assumed
21、 to be 5%. For this analysis the equipment effect and x motion are expected to be small and have an insignificant effect on the total mass translation and rocking motion. Future work will look at the stiffness and damping effects of the equipment. This work is looking at the motion at the springs an
22、d resulting spring compression and tension forces. Devel- opment of the free body diagram is completed without the equipment stiffness and damping. From Figure 2, the free body diagram, equations of motion (EOM) may be developed. SYSTEM RELATIONSHIPS There are several relationships that are used in
23、the devel- opment of the EOM. Inputs to the equations of motion use relationships of the displacement and mass as identified by the coordinate system selected. If there is an offset of the equip- ment mass as previously discussed, the center of mass of the system will not coincide with the equipment
24、 mass. This equip- ment mass offset is defined as a vector rAc as shown in Figure 2. Vector rAc is a simple summation problem determined by the relationship as shown in Equation 1. the base due to the equipment mass (M) and not the total mass (M+m). The horizontal force acts on the mass of the base
25、(m) (1) IMb + mL/4 M+ m b- ErAiMi - rAc = - but will be assumed to result only in a shear force acting on the CMi ASHRAE Transactions: Symposia 322 Equations of motion are defined in terms of the angular displacement (a) and the translation of the equipment in the vertical direction (x) at the cente
26、r of mass. The relative motion at the springs can be determined from the motion at the center of mass as defined by x and a as shown in Equations 2 and 3. xl = x-aa-rACa = x-(a+rAC)a, (2) x2 = x+ba-rAca = x+(b+rAc)a. (3) The relative velocity at the spring is simply the derivative of Equations 2 and
27、 3. i, = i-aa-rACa = i-(a+rAC)a, (4) i, = x+ba-rACa = X+(b+rAC)a. (5) The displacement and velocity of the springs that are determined from Equations 2,3,4, and 5 are used to calculate the transmitted forces (transmissibility) at the springs in tension and compression. The response at the spring is
28、one of the important results of this analysis needed to complete the anchorage design, A second important result for this analysis is the effect of damping on the system. Displacement of the spring will increase for weaker spring constants but decrease with increased damping. Displacement of the spr
29、ing may reach a maximum height where the spring is stretched beyond the design of the isolator. The displacement may also reach a minimum height and the solid length of the spring, causing a potential failure of the isolator. Typical spring constants are selected based on undamped vertical motion to
30、 reduce the transmitted forces to the building structure. This evaluation can look at the total displacement as well as the transmitted forces. Damping will reduce the total displacement of the spring without detrimentally increasing the transmissibility. The damping used in this analysis is a visco
31、us type damp- ing. Actual damping will be frictional based on the total force acting normal to the spring. The friction force can be calcu- lated by multiplying the total mass times the horizontal earth- quake acceleration times a friction factor (p) of steel to rubber. This fnctional force is a bet
32、ter assumption compared to viscous damping. However, the actual system will have a combination of both frictional and viscous type damping. A less rigorous and simplified approach is to assume the damp- ing is viscous with some damping coefficient (4). The damp- ing coefficient for equipment is usua
33、lly assumed to be 0.05 (5%). The results are acceptable when using viscous damping to see magnitudes and system response. Friction damping evaluations will be considered in future work. The mass moment of inertia of the system is used in the solution for the system rotation when summing the moments.
34、 The mass moment of inertia can be calculated by the relation- ships of the mass and their offsets added to the mass moment of the base. The total mass moment of inertia for this scheme is given by Equation 6. 2 2 I, = Z, + Mh, + mrAC In Equation 6, Io is the mass moment of the base. The example use
35、d in this paper assumes the base is a solid bar. For simplicity, the center of the bar mass is assumed to be at the mass center of the system. The mass of the equipment (M) and the mass of the base (m) are simply added for the total mass acting in the vertical direction. my = M+m (7) EQUATIONS OF MO
36、TION The previous relative relationships are inputs into the equations of motion for this two degrees of freedom system (2DOF). To solve for the motion, two equations are required. The first equation is obtained by summing forces in the verti- cal direction. The result is CF, = m$, ,(t. m,-kln - (a
37、+ rAc). a - c,x- (a + rAc). b, = 3/r;b2 = 2b1;b, = 2;b, = 2;b5 = t/2. (22) The incremental displacement is substituted into Equa- tion 17 to obtain the acceleration at r = t + O.At and directly plugged into Equations 12,13, and 14 for z = t + At. The final result for each incremental time step is gi
38、ven by where 2 b, = bO/;b7 = -b2/0;b8 = 1 - 3/0;b9 = At/2;bl0 = At /6. (26) Equations 2 1 23,24, and 25 only are required to be solved for any system. As in the seismic restraint design for spring- isolated equipment, the solution can be easily programmed on a spreadsheet. SOLUTION OF THE SPRING-ISO
39、LATED EQUIPMENT An example is used to illustrate this numerical solution. Consider a fan (1000 lb) with dimensions as given in Table 1 and selected so that the center of mass corresponds to the equipment mass center. The result is two uncoupled equations of motion where M, C, and K are identified in
40、 Equation 27. The vertical forcing function acts on the translation of the equip- ment and the horizontal forcing function acts on the rotation of the equipment base. For this example the equilibrium displacement is 0.75 inch as the 1000 pounds (lb) displace two springs with a 667 iblin. spring cons
41、tant. Input data for the calculation spreadsheet for analysis is shown in Table 1. The matrix form of the example given in Equation 27: M= The complete solution for this example of the spring- isolated fan is a very large spreadsheet. The actual table has Table 1. System Parameters Physical Dimensio
42、ns M= 2.33 Slugs m = 0.26 Slugs mt = 2.59 Slugs xes = 0.75 inches Icg = 34.1 slugs-in.2 Hcg = 35 inches L = 40 inches b = 20 inches 6 = 0.1 (damping coefficient) dt = 0.02 seconds e= 1.0 Wilson Theta Factors bo = 15000 b, = 150 b, = 2 b, = 2 b, = 300 b5 = 0.01 b6 = 15000 b7 = -300 b, = -2 b, = 0.01
43、bin = 6.667E-05 ASHRAE Transactions: Symposia 325 ._ Spring Deflection O 15 - 0.10 kb- Figure 4 Deflection ofthe springs. over 2500 time steps. Figure 4 is the total response of the equipment over the time history lasting 53 seconds. The maxi- mum displacement is O. 15 in. Since the equilibrium disp
44、lace- ment is 0.75 in., the response of this damped equipment is well within the capability of the spring. The equivalent acceleration of the El Centro earthquake is approximately 0.3 19 gs. The equivalent g-level of the earthquake is used in the static anal- ysis. This earthquake is minor compared
45、to the code require- ments. Higher accelerations and resultant forces would be expected for an acceptable evaluation to meet code require- ments. The resultant spring forces can be calculated from the transmissibility as defined by The transmissibility is positive where the spring is in tension or n
46、egative where the spring is in compression. The maximum forces with a viscous damping coefficient of 0.1% at spring 1 are 159 lb in tension and 106 lb in Compression. The maximum forces at spring 2 are 258 lb in tension and 386 lb in compression. The maximum forces with a viscous damping coefficient
47、 of 0.3% at spring 1 are 155 lb in tension and 1 14 lb in compression. The maximum forces at spring 2 are 27 1 lb in tension and 385 lb in compression. A comparison is made with the static analysis. Based on the International Building Code, the maximum horizontal force developed by an earthquake wou
48、ld be defined by Equa- tion 29. I Fp = (0.4.a .SD,.W)l(1+2$ RP P = (0.4.2.5 .0.319. IOOo)( 1 + 2 .3 = 127.6 . (29) Earhquake Acceleration E .- Z. 50 f -50 50 Figure 5 Acceleration time history of the El Centro earthquake. I Earthquake Velocity Profile Figure 6 Velocity of the ground during the El Ce
49、ntro earthquake. The compression and tension of the spring-isolated equipment are calculated using the ASHRAE guideline as defined in Equation 30, where Fpv = FJ3. I Fh) F eq- 2LI - 4 T,C = F iuR+Lv T, C = Feqk(L 127.6.35 + -) 127.6/3 = , 40 2 The static analysis, as defined by the code, results in a tension and compression about equilibrium of h133 lb. The analysis completed using the numerical method resulted in higher forces than would be allowed by a static analysis. The tension for both analyses is within 10%. But the compression allowed by code is approximately one-th
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