'******************************************************************************* '* Resonance Frequencies of a bending beam * '* - Modal Mass and Stiffness * '* - Deformation modes and Maximum Strain * '* --------------------------------------------------------------------------- * '* SAMPLE RUN: * '* * '* Fixed-Free Beam, M=1 * '* Fixed-Supported, M=2 * '* Fixed-Fixed, M=3 * '* Free-Free, M=4 * '* Free-Supported, M=5 * '* Supported-Supported, M=6 * '* * '* Limit Conditions, M= 3 * '* * '* Young Modulus: 2D11 * '* * '* Volumic Mass.: 7800 * '* * '* Beam Width....: 0.4 * '* Beam Thickness: 0.5 * '* Beam Length...: 10 * '* * '* Frequency (Hz) = 26.025373 * '* Frequency (Hz) = 71.739944 * '* Frequency (Hz) = 140.638973 * '* Frequency (Hz) = 232.483361 * '* Frequency (Hz) = 347.290041 * '* * '* * '* Do you want the modes, modal masses & Stiffnesses (y/n): y * '* * '* How many modes (Maximum 5): 5 * '* * '* How many points for deviation, slope & max. strain: 11 * '* * '* Do you want automatic divisions (y/n): y * '* * '* * '* MODE #1 * '* ------- * '* * '* OMEGA= 163.5222 FREQUENCY= 26.0254 * '* * '* Modal Mass= 5400027.436371487 Modal Stiffness= 144394161046.433 * '* * '* X Deviation Slope Strain (x 1e6) * '* ------------------------------------------------ * '* 0.0000 0.0000 0.0000 22771.74 * '* 0.5000 0.1925 0.3498 12232.17 * '* 1.0000 0.6304 0.4927 2225.41 * '* 1.5000 1.1155 0.4494 -6194.01 * '* 2.0000 1.4814 0.2635 -11846.94 * '* 2.5000 1.6164 0.0000 -13841.17 * '* 3.0000 1.4814 -0.2635 -11846.94 * '* 3.5000 1.1155 -0.4494 -6194.01 * '* 4.0000 0.6304 -0.4927 2225.41 * '* 4.5000 0.1925 -0.3498 12232.17 * '* 5.0000 -0.0000 -0.0000 22771.74 * '* * '* MODE # 2 * '* ---------- * '* * '* OMEGA= 450.7554 FREQUENCY= 71.7399 * '* * '* Modal Mass= 1643844717.577793 Modal Stiffness= 333997017195898.7 * '* * '* X Deviation Slope Strain (x 1e6) * '* ------------------------------------------------ * '* 0.0000 0.0000 0.0000 61624.92 * '* 1.0000 0.4554 0.7516 14015.32 * '* 2.0000 1.2058 0.6202 -24480.45 * '* 3.0000 1.5043 -0.0778 -40796.43 * '* 4.0000 1.0338 -0.8261 -29766.62 * '* 5.0000 0.0000 -1.1411 -0.00 * '* 6.0000 -1.0338 -0.8261 29766.62 * '* 7.0000 -1.5043 -0.0778 40796.43 * '* 8.0000 -1.2058 0.6202 24480.46 * '* 9.0000 -0.4554 0.7516 -14015.31 * '* 10.0000 -0.0000 0.0000 -61624.92 * '* * '* MODE # 3 * '* ---------- * '* * '* OMEGA= 883.6607 FREQUENCY= 140.6390 * '* * '* Modal Mass= 630221918268.5579 Modal Stiffness= 4.921127477872044D+17 * '* * '* X Deviation Slope Strain (x 1e6) * '* ------------------------------------------------ * '* 0.0000 0.0000 0.0000 120907.45 * '* 1.0000 0.7701 1.1128 -6282.87 * '* 2.0000 1.5079 0.1215 -77726.96 * '* 3.0000 0.8687 -1.2982 -47991.93 * '* 4.0000 -0.6284 -1.3976 39638.73 * '* 5.0000 -1.4060 -0.0000 85988.24 * '* 6.0000 -0.6284 1.3976 39638.73 * '* 7.0000 0.8687 1.2982 -47991.92 * '* 8.0000 1.5079 -0.1215 -77726.96 * '* 9.0000 0.7701 -1.1128 -6282.87 * '* 10.0000 -0.0000 -0.0000 120907.44 * '* * '* MODE # 4 * '* ---------- * '* * '* OMEGA=1460.7360 FREQUENCY= 232.4834 * '* * '* Modal Mass= 262456494665287 Modal Stiffness= 5.600164860063141D+20 * '* * '* X Deviation Slope Strain (x 1e6) * '* ------------------------------------------------ * '* 0.0000 0.0000 0.0000 199859.15 * '* 1.0000 1.0745 1.2736 -58760.62 * '* 2.0000 1.3192 -0.9913 -120007.61 * '* 3.0000 -0.4227 -1.9219 45103.94 * '* 4.0000 -1.3935 0.3075 139911.06 * '* 5.0000 -0.0000 1.9969 0.02 * '* 6.0000 1.3935 0.3075 -139911.05 * '* 7.0000 0.4227 -1.9219 -45103.98 * '* 8.0000 -1.3192 -0.9913 120007.59 * '* 9.0000 -1.0745 1.2736 58760.66 * '* 10.0000 -0.0000 0.0000 -199859.09 * '* * '* MODE # 5 * '* ---------- * '* * '* OMEGA=2182.0877 FREQUENCY= 347.2900 * '* * '* Modal Mass= 1.149904814147907D+17 Modal Stiffness= 5.475279419775085D+23 * '* * '* X Deviation Slope Strain (x 1e6) * '* ------------------------------------------------ * '* 0.0000 0.0000 0.0000 298555.55 * '* 1.0000 1.3218 1.1293 -144271.18 * '* 2.0000 0.6736 -2.2318 -91130.35 * '* 3.0000 -1.3394 -0.7648 201616.08 * '* 4.0000 -0.2202 2.4118 33178.41 * '* 5.0000 1.4146 0.0000 -211057.80 * '* 6.0000 -0.2202 -2.4118 33178.38 * '* 7.0000 -1.3394 0.7648 201616.09 * '* 8.0000 0.6736 2.2318 -91130.32 * '* 9.0000 1.3218 -1.1293 -144271.20 * '* 10.0000 -0.0000 -0.0000 298555.50 * '* * '* --------------------------------------------------------------------------- * '* REFERENCE: "Mécanique des vibrations linéaires By M. Lalanne, * '* P. Berthier, J. Der Hagopian, Masson, Paris 1980" [16]. * '* * '* Quick Basic Release By J-P Moreau, Paris. * '* (www.jpmoreau.fr) * '******************************************************************************* 'NOTE: For a bending beam of constant section S, the deviation v(x) is given 'by: ' EI d4v/dx4 + rho*S d2v/dt2 - Tex = 0 (1) 'where: ' E = Young's modulus of beam material (steel: 2E11 Pa) ' I = Moment of inertia of S (rectangle: B*H^3/12) ' rho = Volumic Mass (steel: 7800 kg/m3) ' Tex = External force per length unit ' x = beam abscissa from x=0 to x=L ' ' d4v/dx4: 4th partial derivative with respect to x, ' d2v/dt2: 2nd partial derivative with respect to time, t. ' 'We seek solutions of v(x,t) having the form: v(x) * f(t). This leads to solving ' ' d2f(t)/dt2 + w2 f(t) = 0 (2) 'and ' d4v(x)/dx4 - rho.S/(EI) w2 v(x) = 0 (3) 'with ' w = pulsation of force excitation ' 'The solution of (2) has the form: f(t) = A sin(wt) + B cos(wt) (4) 'The solution of (3) is obtained by considering v(x) = V e^rx (5) ' 'the characreristic equation of which is: r^4 - rho.S w2 /(EI) = 0 (6) ' 'The complex roots of (6) are: beta, -beta, j.beta and -j.beta, with ' ' beta = 4th Root of (rho.s w2 / (EI)) (7) ' 'From (5) and the the 4 complex roots of (6), we can write: ' 'v(x) = C sin(beta.x) + D cos(beta.x) + E sh(beta.x) + F ch(beta.x) (8) ' 'The resonance pulsations are given by: wn = Xn^2 /L^2 * sqrt(EI/(rho.S)) (9) ' 'The Xn values are presented below for various limit conditions: ' ' Limit Conditions X1^2 X2^2 X3^2 X4^2 X5^2 '----------------------------------------------------------------------- 'Fixed-free: 1+ch(X)*cos(X)=0 3.516 22.03 61.69 120.9 199.8 'Sup.-Sup.: sin(X)=0 9.869 39.47 88.82 157.9 246.7 'Fixed-Fixed or Free-Free: '1 - ch(X) cos(X) = 0 22.37 61.67 120.9 199.8 298.5 'Fixed-Sup. or Free-Sup.: 'tan(X) = tanh(X) 15.41 49.96 104.2 178.2 272.0 '----------------------------------------------------------------------- ' 'The max. strain is given by: sigma = E * H/2 * d2vi(x) / dx2 ' '------------------------------------------------------------------------------- 'PROGRAM BEAM1 DEFDBL A-H, O-Z DEFINT I-N CLS PRINT PRINT " Fixed-Free Beam, M=1" PRINT " Fixed-Supported, M=2" PRINT " Fixed-Fixed, M=3" PRINT " Free-Free, M=4" PRINT " Free-Supported, M=5" PRINT " Supported-Supported, M=6" PRINT INPUT " Limit Conditions, M= ", M DIM A1(5) DIM I1 AS DOUBLE DIM I2 AS DOUBLE DIM L AS DOUBLE DIM L0 AS DOUBLE DIM J AS DOUBLE DIM K1 AS DOUBLE DIM M1 AS DOUBLE PI = 4# * ATN(1#) IF M = 1 THEN A1(1) = 3.5160152# A1(2) = 22.034491# A1(3) = 61.697214# A1(4) = 120.90191# A1(5) = 199.85953# ELSEIF M = 2 OR M = 5 THEN A1(1) = 15.418205# A1(2) = 49.964862# A1(3) = 104.24769# A1(4) = 178.26972# A1(5) = 272.03097# ELSEIF M = 3 OR M = 4 THEN A1(1) = 22.373285# A1(2) = 61.672822# A1(3) = 120.90339# A1(4) = 199.85944# A1(5) = 298.55553# ELSEIF M = 6 THEN A1(1) = 9.8696044# A1(2) = 39.478417# A1(3) = 88.82643899999999# A1(4) = 157.91367# A1(5) = 246.74011# END IF PRINT INPUT " Young's Modulus: ", E PRINT INPUT " Volumic Mass...: ", R PRINT INPUT " Beam Width....: ", B INPUT " Beam Thickness: ", H INPUT " Beam Lenght...: ", L PRINT J = B * H ^ 3 / 12# S = B * H F$ = "####.######" FOR I = 1 TO 5 B1 = SQR(A1(I)) F = A1(I) / (2# * PI * L ^ 2) * SQR(E * J / R / S) PRINT " Frequency (Hz) = "; PRINT USING F$; F NEXT I PRINT INPUT " Do you want the modes, modal masses & Stiffnesses (y/n): ", ANS$ IF ANS$ = "n" THEN END PRINT INPUT " How many modes (Maximum 5): ", N1 PRINT INPUT " How many points for deviation, slope & max. strain: ", N2 PRINT DIM X(N2) INPUT " Do you want automatic divisions (y/n): ", ANS$ IF ANS$ = "n" THEN FOR I = 1 TO N2 PRINT " X("; I; ") = "; : INPUT "", X(I) NEXT I ELSE L0 = L / (N2 - 1) X(1) = 0# FOR I = 2 TO N2 X(I) = X(I - 1) + L0 NEXT I END IF FOR I = 1 TO N1 B1 = SQR(A1(I)) B2 = B1 / L O1 = COS(B1): I1 = SIN(B1) O2 = COS(2# * B1): I2 = SIN(2# * B1) C1 = (EXP(B1) + EXP(-B1)) / 2# S1 = (EXP(B1) - EXP(-B1)) / 2# C2 = (EXP(2# * B1) + EXP(-2# * B1)) / 2# S2 = (EXP(2# * B1) - EXP(-2# * B1)) / 2# A = 1# IF M = 1 THEN B = -(I1 + S1) / (O1 + C1) C = -1# D = -B ELSEIF M = 2 OR M = 3 THEN B = (-I1 + S1) / (O1 - C1) C = -1# D = -B ELSEIF M = 4 THEN B = (I1 - S1) / (-O1 + C1) C = 1# D = B ELSEIF M = 5 THEN B = -(I1 + S1) / (O1 + C1) C = 1# D = B ELSEIF M = 6 THEN B = 0# C = 0# D = 0# END IF T1 = A ^ 2 / 2 * (L - 1 / 2# / B2 * I2) T2 = B ^ 2 / 2 * (L + I2 / 2# / B2) T3 = C ^ 2 / 2 * (S2 / 2# / B2 - L) T4 = D ^ 2 / 2 * (S2 / 2# / B2 + L) T5 = A * B / 2# / 2# / B2 * (1# - O2) T6 = C * D / 2# / 2# / B2 * (C2 - 1#) T7 = EXP(B1) / 2# / 2# / B2 * (A * C + A * D) * (I1 - O1) T7 = T7 + EXP(-B1) / 2# / 2# / B2 * (A * D - A * C) * (I1 + O1) T7 = T7 + A * C / 2# / B2 T8 = EXP(B1) / 2# / 2# / B2 * (B * C + B * D) * (I1 + O1) T8 = T8 + EXP(-B1) / 2# / 2# / B2 * (B * D - B * C) * (I1 - O1) T8 = T8 - B * C / 2# / B2 T = T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 M1 = R * S * T K1 = M1 * B2 ^ 4 * E * H ^ 2 / 12# / R W = SQR(K1 / M1) F = W / 2# / PI F$ = "####.####" PRINT PRINT " MODE #"; I PRINT " ----------" PRINT PRINT " OMEGA="; : PRINT USING F$; W; PRINT " FREQUENCY="; : PRINT USING F$; F PRINT PRINT " Modal Mass="; M1; PRINT " Modal Stiffness="; K1 PRINT PRINT " X Deviation Slope Strain (x 1e6) " PRINT " ------------------------------------------------" FOR K = 1 TO N2 Z = X(K) I1 = SIN(B2 * Z): O1 = COS(B2 * Z) S1 = (EXP(B2 * Z) - EXP(-B2 * Z)) / 2# C1 = (EXP(B2 * Z) + EXP(-B2 * Z)) / 2# Y = A * I1 + B * O1 + C * S1 + D * C1 P = B2 * (A * O1 - B * I1 + C * C1 + D * S1) M1 = B2 ^ 2 * (-A * I1 - B * O1 + C * S1 + D * C1) M1 = E * H * M1 / 2# PRINT " "; PRINT USING F$; Z; PRINT " "; PRINT USING F$; Y; PRINT " "; PRINT USING F$; P; PRINT USING " ######.##"; M1 / 1000000# NEXT K INPUT "", ANS$ NEXT I END 'end of file beam1.bas