The exact solution for multistepped Timoshenko beam is derived using a set of fundamental solutions. This set of solutions is derived to normalize the solution at the origin of the coordinates. The start, end, and intermediate boundary conditions involve concentrated masses and linear and rotational elastic supports. The beam start, end, and intermediate equations are assembled using the present normalized transfer matrix (NTM). The advantage of this method is that it is quicker than the standard method because the size of the complete system coefficient matrix is 4 × 4. In addition, during the assembly of this matrix, there are no inverse matrix steps required. The validity of this method is tested by comparing the results of the current method with the literature. Then the validity of the exact stepped analysis is checked using experimental and FE(3D) methods. The experimental results for stepped beams with single step and two steps, for sixteen different test samples, are in excellent agreement with those of the threedimensional finite element FE(3D). The comparison between the NTM method and the finite element method results shows that the modal percentage deviation is increased when a beam step location coincides with a peak point in the mode shape. Meanwhile, the deviation decreases when a beam step location coincides with a straight portion in the mode shape.
Stepped beamlike structure plays an important role in the construction of mechanical and civil engineering systems. Flexural vibrations were first investigated by EulerBernoulli in the eighteenth century. The rotary inertia effect was considered by Rayleigh [
The free vibrations of beams with discontinuities can be solved using either exact or approximate solution. The exact methods include the derivation of the transcendental eigenvalue equations in order to evaluate the beam natural frequencies. In the case of relatively simple problems closed form solutions for the eigenvalue problem were obtained [
For complicated problems, the beam eigenvalues are obtained by the decomposition of the domain. Numerical assembly technique (NAT) is one of the common methods used for the evaluation of the eigenvalue problem for beams with multiple discontinuity [
On the other hand, there are numerous approximate methods to approximate the eigenvalue problem for the transverse vibrations of beams [
The exact free vibration of twospan Timoshenko stepped beams has been investigated by Gutierrez et al. [
The exact free vibration of a mechanical system composed of two elastic Timoshenko segments carried on an intermediate eccentric rigid body or on elastic supports was introduced by Farghaly and ElSayed [
During the last decades, many literatures were focused on the problem of free and forced vibration analysis of Timoshenko beam and the accuracy of the natural frequency predictions. To the authors’ knowledge, there is not enough research that has tackled the experimental modal frequencies of stepped thick beams, computationally and experimentally. Therefore, the main aim of this work is to investigate the results of the modal frequencies for such beams using analytical, experimental, and the threedimensional finite element FE(3D). An analytical analysis is proposed which is based on the derivation of a set of fundamental solutions that suits the analysis of Timoshenko beams. This set of solutions is used to modify the TMM to include no inverse matrix procedure which may be called normalized transfer matrix method (NTM). The comparison between the experimental NTM and FE(3D) is done for selected singlestep and twostep application models. The percentage deviations between NTM and FE(3D) are investigated. The results show that the finite element results are very close to the experimental results. The study includes the effect of increasing the step ratio
The mathematical model for beam with multiplestepped sections is shown in Figure
Stepped multispan model.
The objective of this section is to derive the system frequency equation which represents the model shown in Figure
Substituting (
In order to introduce the current analysis, the linearly independent fundamental solutions
The beam start boundary conditions at the point of attachment 1 can be presented in nondimensional form as
Substituting the solutions presented in (
At station
Substituting the solutions in (
Substituting the solutions in (
Equation (
Among the numerical tools, finite element method is considered one of most efficient methods to perform the vibration analysis of mechanical and structural components. In this section, finite element is used to obtain the natural frequencies and mode shapes of uniform and stepped beams. ANSYS finite element commercial package is used to perform the finite element analysis. The analysis is done using threedimensional (3D) solid element models and SOLID95 elements are used for meshing. Since all the experimentally investigated samples in the current work are round and stepped. The beam crosssection is free meshed using 87 SOLID95 elements for smaller crosssection and 171 elements for the larger crosssection. This mesh is then extruded using 40 elements along the length of the beam. The total number of the element is ranging from 3480 (40 × 87) to 6840 (40 × 171) elements based on the location of the step; see Figure
Finite element 3D mesh.
In this example, the first five nondimensional natural frequencies
The first five natural frequencies












[ 
0.25  1.0  0.8  3.010  9.696  34.010  74.430  132.341 
Present NTM  3.0098  9.6956  34.0101  74.4300  132.3410  
[ 
0.8  0.8  3.249  9.664  34.315  74.645  131.803  
Present NTM  3.2490  9.6638  34.3151  74.6445  131.7507  
[ 
0.8  0.6  3.385  8.117  28.630  60.997  103.439  
Present NTM  3.3851  8.1171  28.6302  60.9947  103.4075  
[ 
0.50  1.0  0.8  2.958  10.046  34.993  80.145  139.666 
Present NTM  2.9579  10.0460  34.9931  80.1455  139.6667  
[ 
0.8  0.8  3.124  10.165  34.688  80.570  139.175  
Present NTM  3.1240  10.1655  34.6882  80.5707  139.1759  
[ 
0.8  0.6  3.284  9.499  28.501  70.123  118.482  
Present NTM  3.2841  9.4989  28.5006  70.1232  118.4830  
0.0036  
[ 
0.25  1.0  0.8  3.007  9.668  33.571  72.393  126.111 
Present NTM  3.0066  9.6678  33.5711  72.3926  126.1107  
[ 
0.8  0.8  3.245  9.635  33.862  72.607  125.617  
Present NTM  3.2454  9.6354  33.8621  72.6068  125.6169  
[ 
0.8  0.6  3.381  8.103  28.364  59.839  100.245  
Present NTM  3.3812  8.1031  28.3643  59.8385  100.2447  
[ 
0.50  1.0  0.8  2.946  9.919  33.205  71.469  116.725 
Present NTM  2.9462  9.9192  33.2046  71.4693  116.7245  
[ 
0.8  0.8  3.112  10.030  32.926  71.741  116.362  
Present NTM  3.1120  10.0304  32.9259  71.7415  116.3618  
[ 
0.8  0.6  3.272  9.391  27.449  64.020  102.697  
Present NTM  3.2721  9.3908  27.4490  64.0200  102.6966  
0.01  
[ 
0.25  1.0  0.8  3.001  9.619  32.841  69.229  117.238 
Present NTM  3.0009  9.6192  32.8407  69.2292  117.2380  
[ 
0.8  0.8  3.239  9.586  33.109  69.444  116.798  
Present NTM  3.2390  9.5858  33.1092  69.4435  116.7981  
[ 
0.8  0.6  3.374  8.078  27.914  57.970  95.370  
Present NTM  3.3744  8.0784  27.9139  57.9705  95.3698  
[ 
0.50  1.0  0.8  2.926  9.710  30.711  61.768  95.623 
Present NTM  2.9255  9.7104  30.7111  61.7680  95.6230  
[ 
0.8  0.8  3.019  9.809  30.464  61.912  95.326  
Present NTM  3.0908  9.8086  30.4643  61.9117  95.3257  
[ 
0.8  0.6  3.251  9.211  25.893  56.578  86.647  
Present NTM  3.2511  9.2111  25.8933  56.5776  86.6473 
Verification Example 1 [
In this example, Timoshenko beam with threestep round crosssection presented in [
Comparison of the first five natural frequencies, in rad/sec, using the present NTM results with those obtained in [

Method  BernoulliEuler 
Timoshenko 


Pinnedpinned 

NAT [ 
319.4341  316.5288 
NTM  319.4340  316.4855  

NAT [ 
1853.3864  1789.4207  
NTM  1853.3864  1784.8723  

NAT [ 
4110.1341  3825.8438  
NTM  4110.1341  3836.5347  

NAT [ 
7709.5714  6642.9094  
NTM  7709.5712  6639.2170  

NAT [ 
11621.7699  9886.6243  
NTM  11621.7698  9918.8328  


Freeclamped 

NAT [ 
371.5354  367.9440 
NTM  371.5354  367.1487  

NAT [ 
1243.9063  1216.6220  
NTM  1243.9065  1211.5531  

NAT [ 
3082.0846  2827.0352  
NTM  3082.0845  2849.1465  

NAT [ 
5541.1410  4957.9240  
NTM  5541.1409  5003.7433  

NAT [ 
9599.3810  7673.7578  
NTM  9599.3810  7751.3336  


Clampedfree 

NTM  55.1257  55.0130 

NTM  616.6018  598.5917  

NTM  2702.3238  2532.1483  

NTM  5483.2379  4969.1659  

NTM  9570.9586  7956.5299  


Clampedpinned 

NTM  480.8915  469.8444 

NTM  2098.0556  1991.9611  

NTM  5007.3315  4589.5676  

NTM  8366.8073  7049.5393  

NTM  13678.7112  11215.0349  


Clampedclamped 

NTM  836.3527  807.0997 

NTM  3019.5886  2772.2732  

NTM  5583.6469  4980.5538  

NTM  9625.8943  7734.8319  

NTM  14403.3163  11538.1581 
Verification Example 2 [
The third verification example is shown in Figure
Comparison of the present NTM results with those obtained in [
Mode number  DQEM [ 
Exp. [ 
NTM 
FE(3D)  % error 

I  II  III  IV  (II & IV)  
1  292.440  291  290.316  290.120  0.302 
2  1181.300  1165  1162.158  1167.100  −0.180 
3  1804.100  1771  1767.321  1772.200  −0.068 
Verification Example 3 [
The fourth example is shown in Figure
Validation of the present NTM using 20span uniform beam with 19 equally spaced concentrated masses
Method 





Computational time (sec) 

NAT [ 


3.7177  6.1716  8.6392  48.17 
Present NTM 


3.7177  6.1716  8.6392  13.53 
NAT [ 


2.3871  4.7742  7.1612  34.67 
Present NTM 


2.3871  4.7742  7.1612  12.14 
NAT [ 


3.5941  5.9671  8.35471  40.40 
Present NTM 


3.5941  5.9671  8.35471  13.47 
NAT [ 


1.5989  2.5329  4.5969  22.13 
Present NTM 


1.5989  2.5329  4.5969  10.23 
Verification Example 4, twenty equal span uniform beams carrying equally spaced nineteen concentrated masses with general end flexibilities conditions.
In order to measure the natural frequencies of the system under study, the freefree test samples were put in free oscillations by using an instrumented hammer model B and K 8202. An accelerometer model B and K 4366 is fixed to the shaft in order to capture the vibration signal. The output of the charge amplifier B and K 2635 is connected to NI 6216 data acquisition card. This card is connected to the PC and managed by Lab VIEW software. Figure
Schematic drawings for the experimental samples Group S2.
Schematic drawings for the experimental samples Group S3.
Real image for typical experimental test samples.
Test setup.
Twospan twelve test samples are shown in group S2, namely, S240/40, S240/100, and S240/140, shown in Figure
Typical samples dimension of group S2 with
Group S2  (Singlestep beam samples of steel rod 









 
S240/40  200  40  0.2  160  40  30  1.282  0.002 
S240/100  200  100  0.5  100  40  30  1.375  0.002 
S240/140  200  140  0.7  60  40  30  1.541  0.002 
Experiment first four natural frequencies for test sample S240/40.
Group S3 consists of four different samples, namely, S380/250, S380/200, S380/150, and S380/100. More details about the geometrical dimensions and material properties for these samples are shown in Table
The sample dimensions of group S3 as shown in Figure
Group S3  (Four cylindrical samples have two steps of steel rod 









 
S380/250  500  200  250  20  80  10.480  0.5  0.000100 
S380/200  500  225  200  20  80  7.398  0.4  0.000100 
S380/150  500  250  150  20  80  6.781  0.3  0.000100 
S380/100  500  275  100  20  80  4.932  0.2  0.000100 
Percentage error between the computational and experimental results for singlestep test samples (Group S2). Lowest four nonzero freefree modes.



Method  Modal frequencies in Hz  

1  2  3  4  
20/40  40  0.2  (A) EXP  1848.940  5028.140  9854  16303 
(B) NTM  1873.981  5194.229  10211.79  16776.400  
(C) FE(3D)  1859.490  5056.749  9893.345  16380.170  
(D) % error (A, B)  1.354  3.303  3.630  2.903  
(E) % error (A, C)  0.570  0.568  0.399  0.473  
100  0.5  (A) EXP  2153.900  7713.530  13701.800  19845.900  
(B) NTM  2244.601  7923.127  13661.990  20230.700  
(C) FE(3D)  2142.012  7700.169  13663.580  19793.360  
(D) % error (A, B)  4.211  2.717  −0.290  1.938  
(E) % error (A, C)  −0.551  −0.173  −0.278  −0.264  
140  0.7  (A) EXP  3596.820  8103.960  15773.100  20733.800  
(B) NTM  3813.426  8160.003  15739.160  21174.340  
(C) FE(3D)  3581.521  8090.723  15705.070  20634.360  
(D) % error (A, B)  6.022  0.691  −0.215  2.124  
(E) % error (A, C)  −0.425  −0.163  −0.431  −0.479  


25/40  40  0.2  (A) EXP (Figure 




(B) NTM  2444.111  6638.069  12492.520  19588.580  
(C) FE(3D)  2430.512  6518.094  12223.080  19314.540  
(D) % error (A, B)  0.570  1.957  2.509  1.184  
(E) % error (A, C)  0.011  0.114  0.298  −0.231  
100  0.5  (A) EXP  2796.050  8693.500  14542.200  22134  
(B) NTM  2912.321  8781.311  14633.780  22182.430  
(C) FE(3D)  2806.782  8698.963  14551.070  22095.920  
(D) % error (A, B)  4.158  1.010  0.629  0.218  
(E) % error (A, C)  0.383  0.062  0.060  −0.172  
140  0.7  (A) EXP  4015.620  8665.980  16238.9  22684  
(B) NTM  4167.385  8797.248  16074.330  22814.460  
(C) FE(3D)  4021.237  8655.476  16159.230  22610.690  
(D) % error (A, B)  3.779  1.514  −1.013  0.575  
(E) % error (A, C)  0.139  −0.121  −0.490  −0.323  


30/40  40  0.2  (A) EXP  3023.690  7906  14381.400  21856.600 
(B) NTM  3023.321  7946.457  14429.850  21806.140  
(C) FE(3D)  3014.442  7878.443  14307.960  21781.750  
(D) % error (A, B)  −0.012  0.511  0.336  −0.230  
(E) % error (A, C)  −0.305  −0.348  −0.510  −0.342  
100  0.5  (A) EXP  3401.940  9329.910  15681.400  23607.700  
(B) NTM  3478.703  9293.555  15691.770  23326.460  
(C) FE(3D)  3405.324  9313.349  15645.690  23513.650  
(D) % error (A, B)  2.256  −0.389  0.066  −1.191  
(E) % error (A, C)  0.099  −0.177  −0.227  −0.398  
140  0.7  (A) EXP  4201.680  9322.630  16596.400  23967.700  
(B) NTM  4268.272  9431.643  16437.830  23729.240  
(C) FE(3D)  4211.100  9322.286  16530.200  23881.400  
(D) % error (A, B)  1.584  1.169  −0.955  −0.994  
(E) % error (A, C)  0.224  −0.003  −0.398  −0.360  


35/40  40  0.2  (A) EXP  3608.280  9126.360  16128.600  23817.3 
(B) NTM  3607.895  9104.530  16000.810  23500.960  
(C) FE(3D)  3607.465  9106.544  16054.460  23720.110  
(D) % error (A, B)  −0.010  −0.239  −0.792  −1.328  
(E) % error (A, C)  −0.022  −0.217  −0.459  −0.408  
100  0.5  (A) EXP  3885.030  9778.800  16719.900  24538.800  
(B) NTM  3906.518  9705.047  16577.640  24125.780  
(C) FE(3D)  3882.419  9759.382  16658.180  24417.41  
(D) % error (A, B)  0.553  −0.754  −0.850  −1.683  
(E) % error (A, C)  −0.067  −0.198  −0.369  −0.495  
140  0.7  (A) EXP  4248.240  9888.540  17028  24738.400  
(B) NTM  4253.660  9878.822  16849.850  24311.230  
(C) FE(3D)  4247.271  9870.286  16963.900  24616.760  
(D) % error (A, B)  0.127  −0.098  −1.046  −1.726  
(E) % error (A, C)  −0.022  −0.184  −0.376  −0.491 
Percentage error between the computational and experimental results for twostep samples (Group S3). Lowest four nonzero freefree modes.
Method  Modal frequencies in Hz  

1  2  3  4  
S380/250  
(A) EXP  403.148  2126.460  4139.830  5153.500  
(B) NTM  420.212  2203.984  4334.192  5605.556  
(C) FE(3D)  404.180  2126.900  4153.700  5169.800  
(D) % error (A, B)  −3.966  3.517  4.484  −8.428  
(E) % error (A, C)  −0.255  −0.020  −0.333  −0.315  
S380/200  
(A) EXP (Figure 





(B) NTM  367.381  1788.472  2551.555  4660.983  
(C) FE(3D)  355.230  1728.300  2318.400  4532.200  
(D) % error (A, B)  −3.549  3.386  10.192  −2.860  
(E) % error (A, C)  0.124  0.023  1.160  0.018  
S380/150  
(A) EXP  335.859  1360.050  1545.130  3778.200  
(B) NTM  346.243  1434.113  1638.174  3889.033  
(C) FE(3D)  336.440  1364.200  1548.500  3781.900  
(D) % error (A, B)  −3.091  −5.445  −6.021  −2.933  
(E) % error (A, C)  0.172  0.305  0.218  0.097  
S380/100  
(A) EXP  337.400  1018.240  1371.070  3221.200  
(B) NTM  345.942  1074.574  1424.992  3303.837  
(C) FE(3D)  337.480  1024  1372.900  3215  
(D) % error (A, B)  −2.531  −5.532  −3.932  −2.565  
(E) % error (A, C)  0.023  0.565  0.133  −0.192 
Experimental first four natural frequencies for test sample S380/200.
The results of the previous section show the accuracy of the FE(3D) model in evaluating the natural frequencies of stepped beam. Therefore, in this section, a freefree twospan model is deeply investigated using (
Typical relative deviation between present NTM and finite element (3D) results. Lowest three nonzero modes for two categories of singlestep FF beam; see Figure

Method 

Modal frequencies in Hz  


 
1  2  3  1  2  3  
0.5  NTM  0.1  217.034  635.604  1267.590  1878.302  5163.860  9645.650 
0.4  232.036  797.543  1834.110  1995.497  6510.740  13831.40  
0.7  447.629  1037.270  2253.940  3813.426  8160  15739.200  
FE(3D)  0.1  216.987  634.850  1263.259  1876.267  5123.790  9448.138  
0.4  229.554  785.722  1819.305  1930.166  6271.645  13642.314  
0.7  437.591  1033.915  2245.125  3583.014  8091.330  15706.439  
% dev.  0.1  −0.022  −0.119  −0.343  −0.108  −0.782  −2.090  
0.4  −1.082  −1.504  −0.814  −3.385  −3.812  −1.386  
0.7  −2.294  −0.324  −0.392  −6.431  −0.849  −0.208  


0.625  NTM  0.1  283.930  812.357  1608.220  2446.347  6532.940  11979.500 
0.4  308.616  991.955  2074.890  2639.638  7837.580  15025.700  
0.7  495.792  1128.820  2333.290  4167.385  8797.250  16074.300  
FE(3D)  0.1  283.898  811.795  1604.956  2445.458  6509.821  11855.709  
0.4  305.955  981.347  2072.554  2570.408  7647.979  15037.839  
0.7  489.234  1121.452  2336.177  4021.301  8655.042  16159.248  
% dev.  0.1  −0.011  −0.069  −0.203  −0.036  −0.355  −1.044  
0.4  −0.870  −1.081  −0.113  −2.693  −2.479  0.080  
0.7  −1.341  −0.657  0.123  −3.633  −1.643  0.526  


0.750  NTM  0.1  355.027  993.279  1944.090  3030.222  7818.400  14004 
0.4  381.639  1149.720  2248.110  3236.367  8841.600  15842.900  
0.7  511.227  1230.890  2406.190  4268.272  9431.640  16437.800  
FE(3D)  0.1  355.036  993.179  1943.016  3031.657  7819.722  13978.716  
0.4  379.583  1144.159  2251.351  3181.291  8759.634  15934.664  
0.7  508.439  1224.396  2410.039  4205.930  9312.425  16529.197  
% dev.  0.1  0.002  −0.010  −0.055  0.047  0.017  −0.181  
0.4  −0.542  −0.486  0.144  −1.731  −0.936  0.576  
0.7  −0.548  −0.530  0.160  −1.482  −1.280  0.553  


0.875  NTM  0.1  429.273  1177.820  2274.860  3616.212  9013.570  15741.600 
0.4  448.134  1269.840  2419.690  3759.295  9566.420  16569.700  
0.7  511.380  1312.660  2498.460  4253.660  9878.820  16849.800  
FE(3D)  0.1  429.347  1178.446  2277.013  3621.275  9047.347  15829.045  
0.4  447.258  1269.273  2423.695  3739.834  9595.934  16695.492  
0.7  510.778  1310.738  2502.153  4247.181  9870.045  16963.843  
% dev.  0.1  0.017  0.053  0.094  0.140  0.373  0.552  
0.4  −0.196  −0.045  0.165  −0.520  0.308  0.753  
0.7  −0.118  −0.146  0.148  −0.153  −0.089  0.672 
Long and short stepped samples at three values of
Variation of the percentage modal deviation between FE(3D) and NTM results as a function of
The results of Table
Percentage modal deviation between FE(3D) and NTM as a function of
Typical first three modal shapes at the peak points shown in Figure
First mode
Second mode
Third mode
Figures
Due to the importance of tapered or conical beams in many engineering applications, the current section is devoted to show how to use the present analysis to solve the problem of taper beam. The current analysis is based on uniform beams, while the partial differential equation which represents the lateral vibration of tapered or conical beams is fourthorder Bessel equation [
Stepped tapered beam with varying depth and width.
To verify the suitability of the current model to represent conical beams, the results of the current model are compared with the exact solution for cantilevered (CF) conical beam with variable taper ratio
The first three eigenvalues for CF conical beam using NTM in comparison with those of the exact solution presented in [

[ 
NAT [ 
NTM 
NTM 
NTM 
NTM 


Time (s)  Time (s)  Time (s)  Time (s)  Time (s)  


0.2 

6.1964  6.1683  6  6.1706  1.8  6.1954  5.3  6.1962  10.5  6.1963  32.6 

18.3855  18.2513  18.2513  18.3801  18.3840  18.3853  

39.8336  39.4809  39.4809  39.8194  39.8300  39.8336  


0.5 

4.6252  4.6155  6.2  4.6175  1.9  4.6249  5.4  4.6250  10.9  4.6251  34 

19.5476  19.5074  19.5074  19.5460  19.5472  19.5477  

48.5789  48.4725  48.4725  48.5746  48.5779  48.5788  


0.7 

4.0669  4.0615  6.2  4.0635  1.9  4.0668  5.6  4.0669  11.1  4.0670  35 

20.5554  20.5360  20.5361  20.5547  20.5553  20.5554  

—  53.9625  53.9625  54.0131  54.0147  54.0152 
The results of Table
To investigate the capability of the present model to evaluate the natural frequencies of taper or conical beam with variable boundary conditions, some cases of nonuniform beams are selected form [
The first four eigenvalues for taper or conical beam subjected to variable end conditions using the present NTM and [
Beam parameters  Frequency parameter 


[ 
Present NTM  

21.94  21.9518 
60.44  60.5461  
118.7  118.7589  
196.4  196.4162  



4.98  5.0292 
12.50  12.4289  
40.13  39.8154  
88.28  88.0372  



13.63  13.6709 
24.59  24.9270  
48.37  48.7915  
97.28  97.5762  



25.83  25.8327 
71.00  71.1461  
139.3  139.4223  
230.3  230.4320 
A new proposed normalized transfer matrix NTM uses new set of fundamental solution in combination with the transfer matrix method. This method has the advantage of the TMM in that the determinant of the frequency equation is 4 × 4 for
The current work introduces a comparison between the experimental and analytical NTM and threedimensional finite element analyses for stepped thick beams. Different system parameters such as the step diameters ratio
An excellent agreement between the present experimental results, using several test samples, and those of FE(3D) predictions has been recorded.
The results show that the threedimensional finite element can be trusted in the prediction of the modal frequencies of stepped thick beams in structural and mechanical system.
An interesting percentage of modal deviations between the FE(3D) results and those obtained using the present frequency equation (
Typical example shows that the computational time using normalized transfer matrix (NTM) is greatly reduced in comparison with numerical assembly technique (NAT).
In addition, to the above importance of conclusion, one can obtain accurate results for any combination of classical and elastic end conditions.
Based on the stepped beam, the current analysis can be used to evaluate the results of tapered beam. The present NTM show a very good stability at very large number of spans which enables the accurate evaluation of the tapered beam results using stepped beam analysis.
Crosssectional area of the beam
Polynomial roots
Segment diameter
Segment diameter ratio
Young’s modulus of elasticity
Frequency (Hz)
Shear modulus of rigidity
Moment of inertia of the beam crosssection about the neutral axis
Rotational moment of inertia of the station mass
Shear deformation shape coefficient
Elastic stiffness
End translational spring stiffness
Length of the beam (between points 1 and
Ratio
Concentrated mass at
Total mass of beam
Rotary inertia parameter
Shear deformation parameter
Nondimensional lateral deflection
System coordinate of the beam
Nondimensional stiffness parameters defined as
Nondimensional stiffness parameters defined as
Nondimensional term
Set of nondimensional terms defined as in (
Frequency parameter
Frequency parameter
Nondimensional beam length
Poisson’s ratio
Nondimensional beam length
Mass density of the beam material (
End rotational spring stiffness
Nondimensional rotational spring parameters defined as
Nondimensional rotational spring parameters defined as
Slope due to bending
1st derivative with respect to
2nd derivative with respect to
3rd derivative with respect to
4th derivative with respect to
Clamped (fixed) end
Free end
Pinned (hinged) end
Twospan with singlestep sample
Threespan with twostep sample
The authors declare that they have no conflicts of interest.
The authors gratefully thank the assistants, Eng. M. Zain, workshop technician, and office secretary for their technical support during all stages of this work.