A uniform beam with simply supported ends is shown in Fig. A uniform beam w-ilh simply supported ends. Deflection curve for a simply supported beam. Consider the deflection equation 4. A simply supported beam with the Y axis at the center. For the strain energy, substituting Eq. In fact, the Rayleigh method always results in a frequency value that will be greater than the true value, unless the exact deflection curve is used.
Considerthe beam shown in Fig. A umrorm beam with simply supported ends. If the beam is made of aluminum: A closer examina- tion of the natural frequency equation shows that it depends upon the ratio of the modulus of elasticity and the density. These two factors can be examined for the beam shown in Fig. The only two items that will change in the natural-frequency equation are the modulus of elasticity and the density. An examination of the ratios shown above indicates that the natural frequency of the beam shown in Fig.
A beryllium beam, however, would have a much higher natural frequency. The Rayleigh method is very convenient for determining the resonant frequency of a uniform beam with a concentrated load Fig. A uniform beam with a con- centrated load at the center. Consider the trigonometric function shown by Eq.
This same expression can be used to approximate the deflection curve for the uni- form beam with a concentrated mass in the center.
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This method can be used uith a trigonometric or a pohnomial expression as long as the geometric boundarv" conditions of slope and deflection are satisfled. Displaceinent cun'es and natural frequency equations. Free-Free Beam wjth End Masses The resonant frequency of a free-free beam with end masses can be de- termined with the use of the Rayleigh method and a trigonometric func- tion. A free-free beam is usually considered to be a floating beam, similar perhaps to a piece of wood floating in water.
This type of structure may also be found in the vibration laboratory in the form of a vibration fixture. In this case the nodal points of the beam would be bolted to the head of the vibration machine. The beam becomes a vibration fixture that might support an optical system which requires the accurate alignment of two electronic boxes at each end of the beam Fig.
The system shown in Fig. A vibration fixture system with two electronic boxes. A free-free beam with end masses. Then At resonance, the kinetic energy must equal the strain energy so Eq. One of the characteristics of a free-free beam is that it does happen to have the same natural frequency as a fixed-fixed beam. Sample Problem Consider the situation shown in Fig.
A large amount of relative motion between the two electronic boxes in the vertical direction is undesirable. The vibration test consists of sinusoidal sweeps from 5 to Hz and back to 5 Hz several times. Care must be used in designing a fixture of this type because balance is very important, so provisions should be made for balance weights. Dynamic shifts in the center of gravity may occur if one or both electronic boxes develops a severe resonance below Hz.
A dynamic shift in the center of gravity CG might result in a rocking mode in the shaker head. See Chapter 9 for more information on vibration testing. A preliminary vibration-fixture design considered the use of a 2-in. Is this satisfactory or is a stiffer fixture plate required?
A sketch of the proposed system is shown in Fig. A vibration fixture with etectronic boxes on opposite ends. Therefore, a thicker plate might be used, or ribs could be added to the 2-m. Nonuniform Cross-Sections Electronic equipment must make use of every cubic inch of volume there is because there usually is not very much volume available. When there is a little extra space, it is usually located in an area that is drfiicult to reach.
Often structural modifications must be made to accommodate equipment that just seems to grow larger and larger with every redesign. This usually requires notching, reinforcing, or moving major load-carrying members. The net result is a chassis or a structure that supports all of the electron- ics at the expense of a cross-section with notches and cutouts along its length. If there are many different cross-sections that must be considered, most methods of analysis can become very long and time consuming. However there is a trade-off that can be made which reduces the amount of work at the expense of some accuracy.
This trade-off involves the calcula- tion of an equivalent moment of inertia for a uniform structure that will have approximately the same stiffness as the nonuniform structure. Consider a cantilever beam with two different cross-sections that form two steps Fig. The general deflection equation is see Chapter 5, Section 5. Since the natural frequency Is a function of the square root of the moment of inertia, the error in the natural frequency will be only about 8.
Some other proportions can be examined. Compute the approximate average value with the method shown by Eq. This can be done with Eqs. There is a second method of averaging the moment of inertia as a func- tion of the length of each section for the stepped beam shown in Fig. The resonant frequency would be The resonant frequency would be about 1 1. The same type of analysis can be made for a cantilever beam with three different cross-sections that form three steps Fig. A cantilever beam with three diiTercnt cross sections.
The resonant frequency would be about The resonant frequency would be about 1 8. Using the second method for approximating the average moment of inertia as shown by Eq. There are, of course, many other methods that can be used to deter- mine an average moment of inertia for a stepped beam. For any of these methods it is generally desirable to choose one that is conservative, so that it will produce a moment of inertia that is slightly lower than the true value.
This will then result in a calculated natural frequency that is slight- ly lower than the true value. This is preferred because most electronic structures have many bolted and riveted joints which tend to reduce the stiffness and, therefore, the natural frequency. However, when the beam segment with the smallest moment of inertia is considerably longer than the other beam segments, the average moment of inertia becomes greater than the true value.
This can be demonstrated with the three-step beam shown in Fig. Sample Problem Consider, for example, an electronic box with mounting flanges at each end and a cross-section that has four diflerent moments of inertia along its length Fig. It is required to determine the approximate resonant frequency during vibration in the vertical direction along the T axis. An electronic chassis with four different cross sections.
A con- servative approximation of the natural frequency could then be made by using an approximate average moment of inertia that is known to be rela- tively low. Under these circumstances it might be desirable to use an expression similar to Eqs. Composite Beams Composite laminations are often used in electronic boxes because of electrical, thermal, and vibration requirements. Sometimes a printed- circuit board is mounted so close to a metal bulkhead that it is possible for bare metal lead wires on the circuit board to contact the metal bulk- head during vibration.
Short circuits can result, and they may damage the electronic equipment. In order to prevent possible short circuits, a thin strip of epoxy fiberglass, 0. Epoxy fiberglass is hard, tough, and can resist the pounding of many sharp points during a resonant condition. The bulkhead, of course, then be- comes a composite lamination. Printed-circuit boards often use metal-strip laminations of aluminum or copper to conduct away the heat. In some cases, thin copper strips are bonded to a circuit board and electronic component parts such as resis- tors, diodes, flatpacks, and transistors are cemented to these copper strips and their electrical lead wires are soldered to the printed-circuit board.
Aluminum plates have been laminated to printed-circuit boards in order to conduct away the heat. In some cases, the components are mounted directly on the aluminum which has clearance holes for the elec- trical lead wires to permit soldering to the circuit board on the back side of the aluminum plate.
In other cases, the components may be mounted on a circuit board only 0. The heat will then flow right through the epoxy fiberglass circuit board, with a moderate temperature rise, and be conducted away by the aluminum plate. A closely controlled heating system may be used to maintain a constant gyro temperature over a wide external tem- perature range. The natural frequency of a composite beam of these materials can be determined by considering a combination of the physical properties of both materials. Consider the case of a simply supported laminated beam that has a uni- form load distribution along its length Fig.
If the lamination of aluminum and epoxy fiberglass is side by side, as shown in the cross- section view of Fig. The width of the epoxy fiberglass lamination can be reduced to make it equivalent in its stiffiiess to an aluminum beam, using the El stiffness factor. The subscripts a and e refer to aluminum and epoxy respectively. A simply supported laminated beam. Cross section of a laminated beam. The same cross-sectional area for the aluminum and epoxy fiberglass is used to keep the same weights. The result will then be a T-section with only one material involved.
The equivalent width of an aluminum section is 1. This results in the aluminum T-section shown in Fig. If the composite lamination of aluminum and epoxy fiberglass is stacked in three layers like a sandsvich, as shown in the cross-section view of Fig. Cross section of a beam with three laminadoRs. The same cross-section area for the aluminum and epoxy fiberglass is used to keep the same weights.
If bolted joints are used, relative motion will usually occur between the aluminum and the epoxy, which will reduce the stiff- ness and the natural frequency. It takes a large number of large bolts to prevent relative motion between two members at high frequencies and high G forces. Electronic Components Mounted on Circuit Boards Electronic boxes are being required to occupy less space while providing more functions, so the emphasis has been put on reducing the size of electronic component parts. The development of solid-state electronic parts, such as integrated circuits, sharply reduced the physical size of the parts and permitted more functions to be included in a smaller volume.
Even small electronic component parts, however, must be mounted to provide the proper heat removal, accessibility, and structural integrity, depending upon the environment. Because space is very limited in most aircraft, spacecraft, submarines, and even automobiles, electronic equipment must be supported by many different types of structures that can be adapted to the geometry of the system. Also, the physical size and shape of many electronic component parts themselves may permit them to be analyzed as structural members.
This is a common practice in the electronics industry because it permits low-cost production and easy maintenance.
These component parts are generally soldered to the printed-circuit board by dip-soldering, wave-soldering, and even hand- soldering. The printed-circuit boards are usually of the plug-in type which are guided along the edges to permit easy connector engagement. Under these conditions, the edges of the printed-circuit board can usually be considered as simply supported. A typical installation might be like that shown in Fig. During vibra- tion in an axis perpendicular to the plane of the printed-circuit board, the acceleration forces will produce deflections in the circuit board.
As the circuit board bends back and forth, bending stresses are developed in the electrical lead wires that fasten the electronic component parts to the circuit board Fig. Electronic component parts can be mounted on printed-circuit boards in many different ways. Bending in the component lead wires on a vibrating circuit board.
AND RINGS severe vibration environment will depend upon many difTerent factors such as component size, resonant frequency of the circuit board, accel- eration C forces, method of mounting components, type of strain relief in the electrical lead wires, location of the component, and duration of the vibration environment. Most component failures in a scvere-vibration environment will be due to cracked solder joints, cracked seals, or broken electrical lead wires.
These failures are usually due to dynamic stresses that develop because of relative motion between the electronic component body, the electrical lead wires, and the printed-circuit board. This relative motion is generally most severe during resonant conditions that can develop in the electronic component part or in the printed-circuit board. Resonances may develop in the component part when the body of the component acts as the mass and the electrical lead wires act as the springs.
These resonances are usually not too severe if the body of the compon- ent is in contact with the printed-circuit board, since this contact will sharply reduce the relative motion of the component. If resonances of this type do develop, it is an easy task to tic or cement the component part to the circuit board. If resonances develop in the circuit board, large displacements can force the electrical lead wires to bend back and forth as the circuit board vibrates up and down Fig. If the stress levels are high enough and if the number of fatigue cycles is great enough, then fatigue failures can be expected in the solder joints and the electrical lead wires.
The most severe stress condition will be for a component part mounted at the center of a printed-circuit board. For a rectangular board, the most severe condition will occur when the body of the component part is paral- lel to the short side of the circuit board. This is due to the more rapid change of curvature for the short side of the board compared to the long side of the board, for the same displacement. Tying the electronic com- ponent part to the circuit board with lacing cord, at the center of the com- ponent body, will generally have very little affect on the relative motion between the component body and the circuit board when the circuit board is in resonance.
Cementing the component body to the circuit board or tying both ends of the component part to the circuit board will greatly improve the fatigue life in the solder joints and in the electrical lead wires. Conformal coatings are quite often used on printed-circuit boards to protect the circuitry from moisture. Although most conformal coatings are quite thin, from about 0. The stiffness of the compo- nent body is therefore much greater than the stiffness of the lead wire.
Because of this, a good approximation of the mounting geometry can be obtained by assuming that all of the deflection between the component body and the circuit board is due to bending in the electrical lead wires. Under these conditions, the loads, deflections and stresses in the electri- Combined loading a Vertical load acting alone b Sm nents mounted on circuit boards. The most severe stress conditions will usually occur at the fundamental resonant mode of the circuit board, because displacements are usually the greatest then.
The points where the electrical lead wires are soldered to the circuit board will tend to remain perpendicular to the circuit board as it bends back and forth. The points where the electrical lead wires are joined to the body of the component will also tend to remain fixed, be- cause the component body is so much stilTer than the electrical lead wires. This results in a complex bending mode in the electrical lead wires that can be approximated as a combination of two separate loading patterns on rectangular bents, as shown in Fig.
Bent with a Vertical Load and Fixed Ends CASE 1 Although the electrical lead wires will usually have the same cross- section, consider a more general case where the horizontal member has a different cross-section than the two vertical members, as shown in Fig. Superposition can be used to determine the various loads, moments, and deflections. The method of analysis is to assume that deflections are L. Bent with a vertical load and fixed ends. Free body force diagram of a bent with fixed ends and a concen- trated load.
A free-body force diagram of the rectangular bent is shown in Fig. Superposition can be used to determine the angular rotation. The effect of the vertical force V is small Fig. The angles at the bottom of the vertical leg can be determined from a standard handbook. Free body force diagram of a vertical leg. Superposition can be used to determine the angular rotation on the horizontal leg. Angular changes due to Hr will also be small Fig.
Angles can be determined from a standard handbook. Free body force diagram of a horizontal leg. Deflection diagram of the horizontal leg. Bending moment diagram of the horizontal leg. Bent with hinged ends and end moments. A free-body diagram of the rectangular bent appears as shown in Fig. Consider the vertical leg CD. Superposition can be used to determine the angular rotation Fig.
Free body force diagram of bent with hinged ends and end moments. Free body diagram of a vertical leg. Free body deflection of the horizontal leg. The deflections as shown in Fig. The combined deflection at the top leg must therefore be the difference between these two deflections. The total moment at point 2 in Fig. AMPLE Problem Consider a printed-circuit board with fiatpack integrated circuits mounted on one side of the circuit board, shown in Fig.
This printed-circuit board is mounted in an electronic box that will be vibrated in a direction perpendicular to the plane of the circuit board. The natural frequency for the circuit board in the problem, as shown by test data, is as foUow's: Vibration tests show' that the plug-in connector acts like a simply supported side so that the circuit board can be approx- imated as a fiat plate with four simply supported edges.
However, since there are a wide varietj' tllCTKO'. If there is no lest data as'ailable on a new group of circuit boards, for example, Eq. The single-amplitude displacement can then be determined from Chapter 2. The box is very rigid so that it will not amplify the input to the circuit board at a frequency of Hz. The displacement at the center of the circuit board is 9. Deflection of a printed circuit board with flat packs during vibration. The wires will then tend to deform, as shown in Fig. The relative displacement of the electrical lead wires, for the combined loading condition shown in Fig.
The bending stress in the wire at this point can be determined from the standard bending stress equation neglecting stress concentration factors 5. Bent with a Lateral Load— Fixed Ends When the vibration direction is in the same plane as the pnnted circuit board and parallel to the axis of the resistor, the inertia force on the re- sistor can result in another type of bending in the component lead wires, as shown in Fig.
Since the body of the resistor is much stiffer than the electrical lead wires, almost all of the bending deflection will occur in the lead wires. If it is assumed that deflections are small and stresses do not exceed the elastic limit, then a good approximation of the system can be obtained by considering only the deflection of the electrical lead wires. The inertia force acting on the resistor body can then be treated as a concentrated load.
There must be enough clearance between the body of the resistor and the printed-circuit board, so that the resistor does not contact the board during vibration, in order to make the following analysis valid. However, since some manufacturers still mount components this way, it is desirable to know the characteristics Fig. Superposition can be used to determine the loads, moments, and deflec- tions. Assume that deflections are small and that stresses do not exceed the elastic limit. A free-body force diagram of the rectangular bent will appear as shown in Fig.
Consider the vertical leg AB. Using superposition, the concentrated load and the moment at the top of the vertical leg can be considered separ- ately, as showm in Fig. Bending mode of a resistor mounted on a circuit board. Deflection in the lead wires on a vibrating resistor. Free body force diagramofabcntinlhe lateral direction. Free body force diagram of the vertical leg. On Hit- Mnh lEl. Superposition can be used to deter- mine the angular rotation at point B Fig. Free body force diagram of the horizontal leg. Using standard handbook equations 8. Deflection diagram of the vertical leg.
The natural frequency can be determined from the static deflection equation, using Eq. If the transmissibility of the resistor at resonance is approximated by A - then the dynamic load acting on the resistor can be determined from the input G force. Assuming this dynamic load turns out to be one pound, bending moments, forces, and stresses in the electrical lead wire can be determined. Bent with Transverse Load-Fixed Ends The third axis of vibration that must be considered for electronic compo- nents mounted on printed-circuit boards is in the plane of the circuit board, but perpendicular to the axis of the electronic component part Fig.
There must be enough clearance between the body of the component and the printed-circuit board to ensure that they do not make contact during Wbration in order to make the following analysis valid. It is much better to have the component cemented, tied, or fastened more securely to the circuit board. However, since some manufacturers still mount compo- nents in this fashion, it is desirable to know' the characteristics of the structure.
Since the body of most resistors, capacitors, diodes, and flatpack integrated circuits is much stiifer then the electrical lead w'ires, the body of the component can be ignored when the geometry of the structure is being analyzed. The weight of the component can then be used to deter- mine the magnitude of the dynamic load acting on the system Fig. Using superposition, loads, moments, and deflections can be deter- mined by considering a free-body force diagram, as shown in Fig.
When the top horizontal leg BC bends, a torsional moment is imposed on the vertical leg. Torsional deflection in the vertical leg. The torsional displacement of the vertical leg will be approximately the same as the angular displacement of the horizontal leg, at their junction, if the displacements are small. A standard handbook can be used to deter- mine this angle, as showm in Fig. Free body force diasTHin of tfae horizontal leg.
Superposition can be used by considering the action of the force on the horizontal leg and vertical leg separately. Consider the hori- zontal leg first as shown in Fig. Using equations from a standard handbook 5. Next consider the vertical leg, as shown in Fig. Bending deflection of the vertical leg. The bending moment at point E on the horizontal leg BC can be deter- mined from Eq. End Conditions During High-Frequency Resonances Many different types of bents are used as structural members in elec- tronic packages. These bents must usually be analyzed to determine bending stresses and deflections.
Since the bending stresses and deflec- tions, to a great extent, depend upon the end conditions, it is important to estimate accurately the end conditions of the bent. If this bent had its ends bolted to the plate instead of welded, then there is a good possibility the ends of the bent might be much closer to a simply supported condition than a fixed condi- tion. The coefficient of fixity for bolted ends will depend upon the number and size of the bolts used to fasten the ends of the bent. In general, it might be stated that the stiffer the bent the more difficult it is to achieve fixed ends with bolts.
For example, vibration tests were nm on a rectangular bent 10 in. Since the bent was to be used as part of a vibration fixture, it was desirable to keep the resonant frequency above Hz, so the design goal was 1 Hz. The vibration tests showed a much lower resonant frequency than expected. The analysis showed the results were much closer to a hinged end condition than fixed ends for vibration in the vertical direction, as shown in Fig.
A strobe light and a microscope were used to e. A microscope was required to see clearly the small deflections at the base. No attempt wtis made to measure these deflec- tions. Even when the bolts were torqued extra tight the deflections could not be eliminated. Experiences such as this have shown that it is verj- difficult to prevent relative motion at bolted Interfaces when high-frequency resonances FIGURE 5. Rigid vibration fixture in the shape of a bent. Therefore, when an analysis of a rigid structure with bolted inter- faces is being made, it is better to be conservative by estimating the end fixity to be closer to a simply supported condition than to a fixed condi- tion, unless test data show otherwise.
Bent with a Vertical Load— Hinged Ends The bending moments, loads, and deflections of a rectangular bent with hinged ends can be determined with the use of superposition, considering a concentrated load in the vertical direction as shown in Fig. A free- body force diagram of the rectangular bent will appear as shown in Fig.
The angle at the top of the leg, consider- ing small displacements is: Free body force diagram ofa bent with fi. Bent with a Lateral Load— Hinged Ends Consider a bent with hinged ends and a concentrated load acting in the lateral direction. The deflection can be broken up into two parts. The second part considers bending of the horizontal leg BC only, with no bending in the vertical legs. When the horizontal leg bends, the angular displacement causes an additional dis- placement of the vertical leg through pure rotation Fig.
Bent with a lateral load and hinged ends. Free body force diagram of the y vertical leg. Free body force diagram of ihehorizonl. On MnL 6EI, 5. Enclosed Frame— Hinged Ends A completely enclosed frame can be evaluated using the same superposi- tion methods. The simply supported frame with a concentrated load in the lateral direction is shown in Fig. The vertical leg AB is shown in Fig. Enclosed frame with hinged supports and a lateral load. Free body force diagram oflhe lop horizontal leg. Free body force diagram oflhe bottom horizontal leg.
This module is usually made up of component parts such as resistors, capacitors, and diodes standing up between two small printed- circuit boards that are sometimes called jig wafers see Fig. The cordwood module subassembly is usually dip-soldered to a master circuit board which can be plugged into an electronic box for easy service and maintenance see also Fig. A cordwood module mounted on a circuit board. If he number of fatigue cycles is great enough, fatigue failures may occur in the solder joints. Consider the case of a small cordwood module with the electronic component parts in a symmetrical arrangement.
The angular displacement of the master circuit board will induce the same angular displacement in the electrical lead wires, where the lead wires are soldered to the master circuit board Fig. The bending moments in the electrical lead wires can be approximated by ignoring the body of the electronic component. Since the body is so much stiffer than the leads, most of the bending will be in the electrical leads and printed-circuit boards, as shown in Fig. Deflection mode of the electrical lead wires in a covdwood module. A free-body force diagram of the bent appears as shown in Fig.
The vertical leg AS is shown in Fig.
Using superposition, the concentrated load and the moment at the top of the vertical leg can be considered separ- ately, as showm in Fig. Open Preview See a Problem? Alexa Actionable Analytics for the Web. The same cross-sectional area for the aluminum and epoxy fiberglass is used to keep the same weights. To see what your friends thought of this book, please sign up.
Mn-i lE-sh Me-i 5. Free body force diagram of the lower horizontal leg. Free body force diagram of the upper vertical lec. Free body force diagram of the upper horizontal leg. Once the moment Mb-i is known, all of the other bending moments can be determined. A cordwood module at the center of the circuit board will be forced to deflect as shown in Fig. The master board deflection curve can be approximated by a trigono- metric function as follows: VGS The slope at any point along the circuit board is 0. P mass H' 0. The deflection can be determined from Eq.
Further assume the electronic box will have a transmissibility of 2 at a frequency of 94 Hz. The iransmissibillty of the circuit board can be approximated by considering the energy dissi- pation in the circuit board guides, the connector, and the components FIGURE 5. Cordwood modules mounted on a printed-circuit board that has stiffening ribs fastened with screws courtesy Norden division of United Aircraft.
Printed circuit board supported on four sides. If the component body is ignored and only the lead wires are considered, the geometry of the cordwood module will appear as shown in Fig. The horizontal force due to the kinetic energy of the component body is ignored here. Geometry of a cordwood nuxluleon acircuit board. The bending stress in the electrical lead wire can be determined from the standard bending stress equation. An examination of the fatigue curves for solder see Chapter 10, Fig.
High stresses in the electrical lead wires are developed because large displacement amplitudes occur during circuit-board resonances. Increas- ing the resonant frequency will reduce the displacements and stresses very rapidly as shown by Eqs. These engineers know how difScult it is to ensure the reli- ability of eveiy solder joint. They also know that if a solder joint is not properly made, or if a solder joint should fail, it may result in an intermit- tent electrical connection that can be very difficult to locate. Even if a solder joint is properly made, high bending or shear stresses coupled with many stress-reversal cycles can lead to fatigue failures.
Since there are many different ways in which a solder joint can be formed, it is rather difficult to generalize on the size and shape of these solder joints. Also, many printed-circuit boards have printed circuits on only one side of the board. Many circuit boards with circuits on both sides, or even multiple- layer printed-circuit boards, may use plated-through holes or eyelets to improve the integrity of the soldered joint. Printed-circuit boards with circuits on one side of the board only will usually have the weakest solder joints.
A bending moment acting on the single-sided circuit board will have to fracture only one solder joint. A bending moment acting on the double- sided circuit board will have to fracture a double solder joint on the electrical lead wire, thus this joint is much stronger also see Chapter 6, Section 6.
The shear-out type of failure in the single-sided circuit board appears, in many cases, to occur at a diameter in the solder fillet that is approxim- ately 1. Some typical measured dimensions are shown in Fig. The maximum shear tear-out stress in the solder joints on the cordwood module will occur at paints W and F, as shosvn in Figs. This stress can be determined from the following equationU 2] S. Fracture point in solder on a single sided circuit board.
Solder-joint stresses can be reduced in several dilferent ways; 1. Increasing the resonant frequency of the printed circuit board will reduce the displacement and solder stresses or resonance. Increasing the damping in the circuit board will decrease the trans- missibility at resonance. This might be done with the use of lamina- tions, to introduce shear damping, when the circuit board bends. Using double-sided printed circuits increases the strength of the solder joint, Fig.
Increasing the diameter of he copper pads to which the electrical lead wires are soldered will prevent lifting the pad off of the circuit board, which is another form of vibration failure. Increasing THE Cordwood Jig-Wafer Thickness When vibration fatigue failures have occurred in cordwood-module solder joints, suggestions have often been made to increase the thickness of the jig-waferprinted-circuit boards to decrease the solder-joint stresses.
This possibility can be examined by assuming the jig-wafer thickness in the previous problem is doubled, to 0.
Increasing the thickness oflhecordwood-modulejig-wafers from 0. A thicker jig wafer will increase the stiffness of the master circuit board in only a small local area. Since this stiffness is not carried across the entire board, as it would be in the case of a rib, the resonant frequency can be assumed to stay about the same as that shown by Eq. The physical properties of the cordwood module shown in Fig. Therefore the bending stresses in the cordwood- module electrical-lead-wire solder joints will also increase, if the jig- wafer thickness is increased.
The resonant frequency of the master circuit board should be increased by adding ribs, for example, to decrease the dynamic displacements and solder-joint stresses in the cordwood modules. Bent with Uniform Lateral Load and Hinged Ends Superposition can be used to determine the loads, moments, and deflec- tions of a rectangular bent with hinged ends subjected to a uniform lateral load. The first part will consist of bending of the vertical legs with no bending of the horizontal leg. The second part will consist of the bending of the horizontal leg with no bending of the vertical legs.
However, when the horizontal leg bends, each end will rotate through an angle Ob which will permit the vertical legs to rotate not bend to that same angle as shown Fig. A free- body force diagram of the bent is shown in Fig. Dent with uniform lateral load and hinged ends. Free body force diagram of bent with lateral load.
For small displacements, the vertical force V will result in small moments so this force can be ignored with very little error. Formulas for Natural Frequency and Mode Shape. Peterson's Stress Concentration Factors. Sponsored products related to this item What's this?
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Dave Steinberg's books are some of the best books in the topic. If you are in the business you most likely know this already. If you are just entering into the field of mechanical design for electronic equipment, this book is indispensable. It is worth its weight in gold! One person found this helpful. High-frequency vibration durability testing is more mythology than science when executed by many manufacturers. Steinberg lays out a straightforward process for working through the issues related to shaker testing. Very practical with lots of good. Such a good book. Very practical with lots of good, easy-to-follow guidance.
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