________________________________________________________________________________ User's Guide to the Mathematica Package BeamFrequencies for Calculating Natural Frequencies of Uniform Single-Span Beams in Lateral Vibration ________________________________________________________________________________ Levent Kitis Department of Mechanical Engineering University of Virginia Charlottesville, VA 22901 lk3a@kelvin.seas.virginia.edu 804-924-6230 ________________________________________________________________________________ USING FrequencyEQ FrequencyEQ derives the characteristic equation for the natural frequencies of a single-span uniform beam in lateral vibration. The symbols used for the properties of beams in this guide are defined in Table 1. ________________________________________________________________________________ Table 1 Definitions of Symbols Used for Beam Properties ________________________________________________________________________________ SYMBOL | DEFINITION ________________|_______________________________________________________________ L | length of the beam ________________|_______________________________________________________________ A | cross sectional area of the beam ________________|_______________________________________________________________ EI | flexural rigidity ________________|_______________________________________________________________ ro | mass per unit volume of the beam ________________|_______________________________________________________________ u | lateral displacement of the beam ________________|_______________________________________________________________ The nondimensionalized frequency b is defined such that the natural frequency w in radians per second is given by w = (b^2/L^2) Sqrt[ EI/(ro A) ] The module FrequencyEQ expresses the left hand side of the frequency equation in terms of b, and the numerical root search module Frequency also works in terms of b. The classical boundary conditions for a beam are given in Table 2. The symbol associated with each boundary condition is a string which designates it in the programs. The definitions apply when the free, simple, fixed and sliding supports are not modified by the presence of linear or torsional springs, concentrated masses, or rotary inertia elements. ________________________________________________________________________________ Table 2 Classical Boundary Conditions for Beams in Free Vibration ________________________________________________________________________________ SYMBOL | DEFINITION ________________|_______________________________________________________________ "free" | u'' = 0 and u''' = 0 ________________|_______________________________________________________________ "simple" | u = 0 and u'' = 0 ________________|_______________________________________________________________ "fixed" | u = 0 and u' = 0 ________________|_______________________________________________________________ "sliding" | u' = 0 and u''' = 0 ________________|_______________________________________________________________ ________________________________________________________________________________ Table 3 Boundary Conditions for Beams with Attached Elements ________________________________________________________________________________ ELEMENT | DEFINITION ________________|_______________________________________________________________ "spring", K | linear spring with one end fixed to ground | K = k L^3/EI k = spring constant (force/length) ________________|_______________________________________________________________ "torsional", T | torsional spring with one grounded end | T = t L/EI t = torsional stiffness (moment/radian) ________________|_______________________________________________________________ "mass", M | concentrated mass | M = m/(ro A L) m = mass ________________|_______________________________________________________________ "rotary", J | disk free to rotate in the plane of the beam | J = I/(ro A L^3) I = moment of inertia (mass length^2) ________________|_______________________________________________________________ The arguments of FrequencyEQ are bndL, the boundary condition at the left end of the beam, bndR, the boundary condition at the right end, and b, the symbol to be given to the nondimensionalized frequency variable b defined above. Both bndL and bndR are lists containing strings and symbols only; b has head Symbol. The syntax of a bndL or bndR is illustrated by the following examples {"simple", "torsional", T} {"mass", M, "spring", K, "free"} {"torsional", T, "simple", "rotary", J} {"fixed"} Each boundary condition contains one designation from Table 2. This designation is a single string whose position in the input list may be selected arbitrarily. In addition, elements from Table 3 may be connected at the boundary. The designation of these elements requires a string followed by a symbol. Not all combinations of conditions from Table 2 and Table 3 are meaningful. For instance {"fixed", "mass", M} is not a physically meaningful boundary condition. The program takes it to mean {"fixed"}. The input lists giving the boundary conditions are filtered in the program, and FrequencyEQ returns the boundary conditions it actually uses as part of its output. ________________________________________________________________________________ EXAMPLE 1 Beam free at one end, and with lateral spring support and torsion spring support at the other In[3]:= bnd1 = {"free"}; bnd2 = {"torsional", T, "spring", K, "free"}; In[4]:= FrequencyEQ[ bnd1, bnd2, b ] Out[4]= {{free}, {free, {torsional, T}, {spring, K}}, 4 3 (b + K T) Sech[b] + (-(b K) - b T) Sin[b] + 4 3 Cos[b] (-b + K T + (b K - b T) Tanh[b])} ________________________________________________________________________________ EXAMPLE 2 Beam with lateral spring support at each end In[5]:= bnd1 = {"spring", K1, "free"}; bnd2 = {"free", "spring", K2}; In[6]:= FrequencyEQ[ bnd1, bnd2, b ] Out[6]= {{free, {spring, K1}}, {free, {spring, K2}}, 6 3 b Sech[b] + Sin[b] (b (-K1 - K2) + 2 K1 K2 Tanh[b]) + 6 3 Cos[b] (-b + b (K1 + K2) Tanh[b])} ________________________________________________________________________________ EXAMPLE 3 Beam fixed at one end, and with lateral spring support and concentrated mass at the other In[7]:= bnd1 = {"fixed"}; bnd2 = {"mass", M, "free", "spring", K}; In[8]:= FrequencyEQ[ bnd1, bnd2, b ] Out[8]= {{fixed}, {free, {mass, M}, {spring, K}}, 3 4 3 4 b Sech[b] + (K - b M) Sin[b] + Cos[b] (b + (-K + b M) Tanh[b])} ________________________________________________________________________________ EXAMPLE 4 Beam fixed at one end, and with sliding support, concentrated mass, and and lateral spring support at the other In[25]:= bnd1={"sliding", "mass", M, "spring", K}; bnd2 = {"fixed"}; In[26]:= Timing[FrequencyEQ[ bnd1, bnd2, b ]] Out[26]= {7.21 Second, {{sliding, {mass, M}, {spring, K}}, {fixed}, 4 3 4 3 (-K + b M) Sech[b] - b Sin[b] + Cos[b] (K - b M - b Tanh[b])}} ________________________________________________________________________________ USING Frequency The module Frequency may be used to find numerical values for the natural frequencies of vibration. Its arguments are as shown below Frequency[ { bndL, bndR, lhs}, nroot, approx ] Here the first list is in the form given as output by FrequencyEQ, nroot is the number of frequencies to be found, approx is a list of starting values that will be used in the root search. Note especially that the input syntax for bndL and bndR is different from that of FrequencyEQ. If approx is omitted, the program generates a list of starting values using a finite element model, and will attempt to calculate Min[nroot, 5] of the lowest natural frequencies. If approx is provided, its length is taken as the desired number of roots, so that the value given to nroot is not used. In the first list, parameters K, M, J, T defined in Table 3 must have numerical values. The following examples show how Frequency can be used in conjunction with FrequencyEQ for studying frequency variations as a function of K, M, J, or T. ________________________________________________________________________________ EXAMPLE 5 Numerical search for the natural frequencies of a beam with torsion spring and concentrated mass In[19]:= bnd1 = {"torsional", T, "simple"}; bnd2 = {"free", "mass", M}; In[20]:= FrequencyEQ[ bnd1, bnd2, b ] Out[20]= {{simple, {torsional, T}}, {free, {mass, M}}, 2 T Sech[b] + Sin[b] (b (-1 - M T) - 2 b M Tanh[b]) + Cos[b] (T + b (1 + M T) Tanh[b])} In[23]:= f[M_, T_] = %20; In[24]:= Frequency[f[0.1, 0.1], 1, {1.0}] Out[24]= {{simple, {torsional, 0.1}}, {free, {mass, 0.1}}, {0.688747753}, {1.}} In[25]:= Frequency[f[0.1,10.0], 3] Out[25]= {{simple, {torsional, 10.}}, {free, {mass, 0.1}}, {1.591194491, 4.128864654, 7.069191843}, {1.8048, 4.52956, 7.60396, 10.7021, 13.8373, 17.8104, 21.412, 25.7032, 30.6501, 36.0499, 39.0148}} In[26]:= Frequency[f[0.1,10.0], 1, {21.4}] Out[26]= {{simple, {torsional, 10.}}, {free, {mass, 0.1}}, {22.35275019}, {21.4}} In[27]:= Frequency[f[0.1,10.0], 1, {36.0}] Out[27]= {{simple, {torsional, 10.}}, {free, {mass, 0.1}}, {34.80741993}, {36.}} In[28]:= Timing[ Frequency[f[0.1,10.0], 1, {39.0}]] Out[28]= {0.81 Second, {{simple, {torsional, 10.}}, {free, {mass, 0.1}}, {37.93099906}, {39.}}} In[29]:= Timing[ Frequency[f[0.1, 0.01], 1, {1.0}] ] Out[29]= {0.73 Second, {{simple, {torsional, 0.01}}, {free, {mass, 0.1}}, {0.3895095383}, {1.}}} In this calculation, In[25] generates finite element approximations to the first 11 frequencies. A model with 5 elements, which has 11 degrees of freedom with these boundary conditions, is used in finding these approximate frequencies. The last list in each output list from Frequency is either the finite element approximation or the list of starting values supplied by the user. ________________________________________________________________________________ EXAMPLE 6 Numerical search for the natural frequencies of a beam with a torsional spring and pin support at one end, and free at the other In[30]:= bnd1 = {"free"}; bnd2 = {"torsional", T, "simple"}; In[31]:= F[T_] = FrequencyEQ[bnd1, bnd2, b] Out[31]= {{free}, {simple, {torsional, T}}, -(T Sech[b]) + b Sin[b] + Cos[b] (-T - b Tanh[b])} In[32]:= Frequency[ F[0.01], 5 ] Out[32]= {{free}, {simple, {torsional, 0.01}}, {0.4159342407, 3.927804795, 7.069291233, 10.21066552, 13.35214311}, {0.620512, 3.93307, 7.08062, 10.2597, 13.4898, 17.3266, 20.8841, 25.0479, 29.777, 34.4044, 38.7081}} In[33]:= Frequency[ F[0.01], 1, {17.3} ] Out[33]= {{free}, {simple, {torsional, 0.01}}, {16.49366449}, {17.3}} In[34]:= Frequency[ F[1.0], 5 ] Out[34]= {{free}, {simple, {torsional, 1.}}, {1.24791741, 4.031139437, 7.134132241, 10.25662107, 13.38775633}, {1.61641, 4.2678, 7.32842, 10.4524, 13.6401, 17.5216, 21.0796, 25.2608, 30.0292, 34.7702, 38.7132}} ________________________________________________________________________________ SHORTCUTS The frequency equations in symbolic form for 38 cases are tabulated below. ________________________________________________________________________________ Table 4 Frequency Equations (lhs==0) for 38 Cases { bndL, bndR, lhs } ________________________________________________________________________________ 1 {{"free"}, {"free"}, -Cos[b] + Sech[b]} ________________________________________________________________________________ 2 {{"fixed"}, {"free"}, Cos[b] + Sech[b]} ________________________________________________________________________________ 3 {{"fixed"}, {"fixed"}, -Cos[b] + Sech[b]} ________________________________________________________________________________ 4 {{"simple"}, {"simple"}, -(Sin[b] Tanh[b])} ________________________________________________________________________________ 5 {{"fixed"}, {"simple"}, -Sin[b] + Cos[b] Tanh[b]} ________________________________________________________________________________ 6 {{"simple"}, {"free"}, -Sin[b] + Cos[b] Tanh[b]} ________________________________________________________________________________ 7 {{"free"}, {"sliding"}, -Sin[b] - Cos[b] Tanh[b]} ________________________________________________________________________________ 8 {{"fixed"}, {"sliding"}, Sin[b] + Cos[b] Tanh[b]} ________________________________________________________________________________ 9 {{"sliding"}, {"simple"}, Cos[b]} ________________________________________________________________________________ 10 {{"sliding"}, {"sliding"}, Sin[b] Tanh[b]} ________________________________________________________________________________ 11 {{"free"}, {"simple", {"torsional", T}}, -(T Sech[b]) + b Sin[b] + Cos[b] (-T - b Tanh[b])} ________________________________________________________________________________ 12 {{"simple"}, {"simple", {"torsional", T}}, T Cos[b] Tanh[b] + Sin[b] (-T - 2 b Tanh[b])} ________________________________________________________________________________ 13 {{"fixed"}, {"simple", {"torsional", T}}, -(T Sech[b]) - b Sin[b] + Cos[b] (T + b Tanh[b])} ________________________________________________________________________________ 14 {{"simple"}, {"free", {"spring", K}}, b^3 Cos[b] Tanh[b] + Sin[b] (-b^3 + 2 K Tanh[b])} ________________________________________________________________________________ 15 {{"fixed"}, {"free", {"spring", K}}, b^3 Sech[b] + K Sin[b] + Cos[b] (b^3 - K Tanh[b])} ________________________________________________________________________________ 16 {{"simple"}, {"free", {"mass", M}}, Cos[b] Tanh[b] + Sin[b] (-1 - 2 b M Tanh[b])} ________________________________________________________________________________ 17 {{"fixed"}, {"free", {"mass", M}}, Sech[b] - b M Sin[b] + Cos[b] (1 + b M Tanh[b])} ________________________________________________________________________________ 18 {{"free"}, {"simple", {"rotary", J}}, b^3 J Sech[b] + Sin[b] + Cos[b] (b^3 J - Tanh[b])} ________________________________________________________________________________ 19 {{"simple"}, {"simple", {"rotary", J}}, Sin[b] (b^3 J - 2 Tanh[b]) - b^3 J Cos[b] Tanh[b]} ________________________________________________________________________________ 20 {{"fixed"}, {"simple", {"rotary", J}}, b^3 J Sech[b] - Sin[b] + Cos[b] (-(b^3 J) + Tanh[b])} ________________________________________________________________________________ 21 {{"free"}, {"free", {"mass", M}}, Sech[b] + b M Sin[b] + Cos[b] (-1 - b M Tanh[b])} ________________________________________________________________________________ 22 {{"fixed"}, {"free", {"mass", M}, {"spring", K}}, b^3 Sech[b] + (K - b^4 M) Sin[b] + Cos[b] (b^3 + (-K + b^4 M) Tanh[b])} ________________________________________________________________________________ 23 {{"simple"}, {"free", {"mass", M}, {"spring", K}}, b^3 Cos[b] Tanh[b] + Sin[b] (-b^3 + (2 K - 2 b^4 M) Tanh[b])} ________________________________________________________________________________ 24 {{"simple", {"rotary", J}, {"torsional", T}}, {"fixed"}, (-(b^4 J) + T) Sech[b] + b Sin[b] + Cos[b] (b^4 J - T - b Tanh[b])} ________________________________________________________________________________ 25 {{"simple", {"rotary", J}, {"torsional", T}}, {"simple"}, (-(b^4 J) + T) Cos[b] Tanh[b] + Sin[b] (b^4 J - T - 2 b Tanh[b])} ________________________________________________________________________________ 26 {{"simple", {"rotary", J}, {"torsional", T}}, {"free"}, (-(b^4 J) + T) Sech[b] + b Sin[b] + Cos[b] (b^4 J - T - b Tanh[b])} ________________________________________________________________________________ 27 {{"fixed"}, {"simple", {"torsional", T}}, -(T Sech[b]) - b Sin[b] + Cos[b] (T + b Tanh[b])} ________________________________________________________________________________ 28 {{"simple"}, {"simple", {"torsional", T}}, T Cos[b] Tanh[b] + Sin[b] (-T - 2 b Tanh[b])} ________________________________________________________________________________ 29 {{"free"}, {"simple", {"torsional", T}}, -(T Sech[b]) + b Sin[b] + Cos[b] (-T - b Tanh[b])} ________________________________________________________________________________ 30 {{"free", {"spring", K}}, {"free", {"spring", K2}}, b^6 Sech[b] + Sin[b] (b^3 (-K - K2) + 2 K K2 Tanh[b]) + Cos[b] (-b^6 + b^3 (K + K2) Tanh[b])} ________________________________________________________________________________ 31 {{"simple", {"torsional", T}}, {"free", {"spring", K}}, b^3 T Sech[b] + Sin[b] (-b^4 + K T + 2 b K Tanh[b]) + Cos[b] (b^3 T + (b^4 - K T) Tanh[b])} ________________________________________________________________________________ 32 {{"simple", {"torsional", T}}, {"simple", {"torsional", T2}}, -(T T2 Sech[b]) + Sin[b] (b (-T - T2) - 2 b^2 Tanh[b]) + Cos[b] (T T2 + b (T + T2) Tanh[b])} ________________________________________________________________________________ 33 {{"simple", {"rotary", J}}, {"free", {"spring", K}}, -(b^6 J Sech[b]) + Sin[b] (b^3 (-1 - J K) + 2 K Tanh[b]) + Cos[b] (-(b^6 J) + b^3 (1 + J K) Tanh[b])} ________________________________________________________________________________ 34 {{"simple", {"rotary", J}}, {"simple", {"torsional", T}}, b^3 J T Sech[b] + Sin[b] (b^4 J - T - 2 b Tanh[b]) + Cos[b] (-(b^3 J T) + (-(b^4 J) + T) Tanh[b])} ________________________________________________________________________________ 35 {{"simple", {"torsional", T}}, {"free", {"mass", M}}, T Sech[b] + Sin[b] (b (-1 - M T) - 2 b^2 M Tanh[b]) + Cos[b] (T + b (1 + M T) Tanh[b])} ________________________________________________________________________________ 36 {{"simple", {"rotary", J}}, {"simple", {"rotary", J2}}, -(b^6 J J2 Sech[b]) + Sin[b] (b^3 (J + J2) - 2 Tanh[b]) + Cos[b] (b^6 J J2 + b^3 (-J - J2) Tanh[b])} ________________________________________________________________________________ 37 {{"simple", {"rotary", J}}, {"free", {"mass", M}}, -(b^3 J Sech[b]) + Sin[b] (-1 + b^4 J M - 2 b M Tanh[b]) + Cos[b] (-(b^3 J) + (1 - b^4 J M) Tanh[b])} ________________________________________________________________________________ 38 {{"free", {"mass", M}}, {"free", {"mass", M2}}, Sech[b] + Cos[b] (-1 + b (-M - M2) Tanh[b]) + Sin[b] (b (M + M2) + 2 b^2 M M2 Tanh[b])} ________________________________________________________________________________ REFERENCES Blevins, R., D., Formulas for Natural Frequency and Mode Shape, 1979, Van Nostrand Reinhold Company. Gorman, D., J., Free Vibration Analysis of Beams and Shafts, 1975, John Wiley and Sons. ________________________________________________________________________________