Free vibration analysis of classical single-story multi-bay planar frames

Abstract

This paper concerns free vibration analysis of single-story multi-bay planar frame structures. An exact analytical solution is obtained using a wave vibration approach, in which vibrations are described as waves propagating along uniform structural elements and are reflected and transmitted at structural discontinuities. The coupling effects between bending and longitudinal vibrations in the multi-bay frames are taken into account. Both natural frequencies and modeshapes are obtained. Numerical examples are presented along with comparisons to results available in the literature.

Keywords

Modeshape multi-bay frame natural frequency vibration wave

1. Introduction

Resonant vibration analysis of plane frameworks is of great importance. This is because resonant vibrations are the main causes of structural failure. However, owing to the inherent algebraic complexity, exact analytical solutions to vibration problems in multi-bay planar frames have rarely been sought. Approximate solutions have been obtained by considering the number of bays of the framework to be infinite, or by assuming the structure consisting of massless, elastic members with the structural mass concentrated at the column crossbar junctions. The only analytical study on multi-bay planar frames available in the literature is by Rieger and McCallion (1965), who analyzed single-story multi-bay planar frames using the conventional modal analysis approach. In their study, all the structural members are assumed to have continuously distributed mass and elasticity. However, the authors only considered bending motions in the multi-bay planar frames.

In this paper, a wave-based vibration approach is introduced, in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities (Graff, 1975; Cremer et al., 1987; Doyle, 1989). The author has successfully applied this approach in analyzing vibrations in simple built-up frames, such as H- and T-shaped (Mei, 2010), and L-shaped and portal planar frames (Mei, 2012).

In this study, the wave vibration approach is adopted for analyzing coupled bending and axial vibration analysis in complex multi-bay structures. The analysis is based on classical vibration theories with the coupled bending and axial motions, which was neglected in the previous analytical study by Rieger and McCallion, taken into account. The coupling is introduced by the joints, where incident bending waves in one beam element generate axial waves in an element normal to it and vice versa. As a result, considering the coupling effect is critical for accurately analyzing vibrations in a multi-bay structure. This accurate analytical study provides a benchmark for future numerical studies.

This paper is organized as follows. In the next section, the wave analysis approach and the equations of motion are presented. The reflection and transmission matrices at “L” and “T” joints, which were derived by the author earlier (Mei, 2010, 2012), are listed in the Appendix for the convenience of readers. In Section 3, the propagation, reflection, and transmission matrices are combined to provide a concise and systematic approach for in-plane vibration analysis of coupled flexural and longitudinal vibrations in single-story multi-bay planar frame structures. The approach is illustrated using two numerical examples, which are compared with available results in the literature. Concluding remarks are given in Section 4.

2. Wave analysis and equations of motion in a single-story multi-bay planar frame

Coupled bending and axial vibrations in single-story multi-bay planar frames are analyzed using the wave vibration approach. In this study, both bending and axial vibrations are modeled using classical vibration theories. The classical axial vibration theory assumes that the axial deformations along the neutral axis of the rod are the same in all points of the cross section, and assumes no transverse deflections. The two key assumptions in the classical Euler-Bernoulli bending theory are that the material is linear elastic according to Hooke’s law and that plane sections remain plane and perpendicular to the neutral axis during bending. As a result, the results are accurate at a lower frequency range, typically, when the transverse dimensions are much smaller than the wavelength and the lateral displacements due to axial vibrations are negligible. At higher frequencies, Timoshenko bending theory and Mindlin–Herrmann axial theory shall be adopted to take into account the effects of rotary inertia and shear distortion in bending vibration and the lateral displacements in axial vibration.

Figure 1 shows a single-story multi-bay planar frame. The n bay frame consists of n uniform cross beams and (n + 1) uniform columns. The discontinuities include two “L” joints, (n−1) “T” joints, and (n + 1) boundaries. Regardless of the complexity of a structure, from the wave standpoint, it consists of only structural elements and structural joints. Vibrations propagate along a uniform structural element, and are reflected and/or transmitted at joints and boundaries. Assembling these propagation, reflection, and transmission matrices offers a concise and systematic approach for analyzing coupled bending and longitudinal vibrations in a single-story multi-bay frame structure.

Figure 1.

A single-story multi-bay plane frame.

Figure 2.

Modeshapes of a single-story multi-bay plane frame.

The approach introduced in this section applies to general situations where all the beams and columns are of different material properties and dimensions. For convenience, however, the cross beams are assumed uniform and of the same material properties and dimensions as all columns.

The n pairs of propagation relations along the cross beam elements are

a_{i}^{+} = f (L) A_{i - 1}^{+}, A_{i - 1}^{-} = f (L) a_{i}^{-}, where i = 1, 2, \dots, n .

(1a)

The (n + 1) pairs of propagation relations along the column elements are

A_{Vi}^{-} = f (L_{V}) B_{i}^{-}, B_{i}^{+} = f (L_{V}) A_{Vi}^{+}, where i = 0, 1, 2, \dots, n .

(1b)

The reflection and transmission relations of the waves at “T” joints are

a_{i}^{-} = r_{11} a_{i}^{+} + t_{31} A_{i}^{-} + t_{21} A_{Vi}^{-}, A_{i}^{+} = r_{33} A_{i}^{-} + t_{13} a_{i}^{+} + t_{23} A_{Vi}^{-}, A_{Vi}^{+} = r_{22} A_{Vi}^{-} + t_{12} a_{i}^{+} + t_{32} A_{i}^{-}, where i = 0, 1, 2, \dots, n - 1

(1c)

The reflection and transmission relations of the waves at the two “L” joints are

A_{0}^{+} = R_{22} A_{0}^{-} + T_{12} A_{V 0}^{-}, A_{V 0}^{+} = R_{11} A_{V 0}^{-} + T_{21} A_{0}^{-}, A_{Vn}^{+} = R_{22} A_{Vn}^{-} + T_{12} a_{n}^{+}, a_{n}^{-} = R_{11} a_{n}^{+} + T_{21} A_{Vn}^{-} .

(1d)

The reflections at the boundaries are

B_{i}^{-} = r_{i} B_{i}^{+}, where i = 1, 2, \dots, n + 1,

(1e)

where

a_{i}^{\pm}

A_{i}^{\pm}

A_{Vi}^{\pm}

, and

B_{i}^{\pm}

are wave vectors as shown in Figure 1,

f (x)

is the wave propagation matrix for a distance x, R and T denote the reflection and transmission matrices at the L joints, and r and t denote the reflection and transmission matrices at the T joints. Subscripts 1, 2, and 3 denote parameters related to the involved beam elements at a joint.

Writing the above equations in matrix form gives

Az = 0,

(2)

where A is a (24n + 12) by (24n + 12) coefficient matrix, and z is a (24n + 12) wave component vector. Setting the determinant of the coefficient matrix A to zero gives the natural frequencies of the multi-bay frame.

Both bending and longitudinal vibrations exist in a multi-bay planar structure due to wave mode conversion at the “L” and “T” joints. Based on classical vibration theories, the equations of motion are as follows (Inman, 1994):

EI \frac{\partial^{4} y (x, t)}{\partial x^{4}} + ρ A \frac{\partial^{2} y (x, t)}{\partial t^{2}} = q (x, t),

(3)

ρ A \frac{\partial^{2} u (x, t)}{\partial t^{2}} - EA \frac{\partial^{2} u (x, t)}{\partial x^{2}} = p (x, t),

(4)

where x is the position along the beam axis, t is time,

y (x, t)

and

u (x, t)

are the transverse and longitudinal deflections of the centerline of the beam;

q (x, t)

and

p (x, t)

are the externally applied transverse and longitudinal forces; E and ρ are the Young’s modulus and mass density; respectively. I is the area moment of inertia of the cross section and A is the cross-sectional area.

For a free bending vibration problem, the differential equation of motion becomes

EI \frac{\partial^{4} y (x, t)}{\partial x^{4}} + ρ A \frac{\partial^{2} y (x, t)}{\partial t^{2}} = 0 .

(5)

Assuming time harmonic motion and using separation of variables, the solution to equation (5) can be written in the form

y (x, t) = y_{0} e^{- ikx} e^{i ω t}

, where ω is frequency and k is the wavenumber. Substituting this into equation (5) gives a set of wavenumbers that are functions of frequency ω as well as the properties of the structure

k = \pm \sqrt[4]{ρ A ω^{2} / EI} .

(6)

The ± sign outside the brackets indicates positive- and negative- going waves.

With the time dependence $e^{i ω t}$ suppressed, the solutions to equation (5) can be written as

y (x) = a_{1}^{+} e^{- {ik}_{1} x} + a_{2}^{+} e^{- k_{2} x} + a_{1}^{-} e^{{ik}_{1} x} + a_{2}^{-} e^{k_{2} x},

(7)

where the bending wavenumbers

k_{1} = k_{2} = \sqrt[4]{ρ A ω^{2} / EI} .

(8)

For the free longitudinal vibration problem, the differential equation is

ρ A \frac{\partial^{2} u (x, t)}{\partial t^{2}} - EA \frac{\partial^{2} u (x, t)}{\partial x^{2}} = 0

(9)

Again assuming time harmonic motion and using separation of variables, the solution to equation (9) can be written in the form $u (x, t) = u_{0} e^{- ikx} e^{i ω t}$ , where ω is frequency and k the wavenumber. Substituting this into equation (9) gives the longitudinal wavenumber, which is a function of frequency ω as well

k = \pm \sqrt{\frac{ρ}{E}} ω .

(10)

Similarly, the ± sign indicates that longitudinal waves in the beam travel in both the positive and negative directions.

With the time dependence $e^{i ω t}$ suppressed, the solutions to equation (9) can be written as

u (x) = c^{+} e^{- {ik}_{3} x} + c^{-} e^{{ik}_{3} x},

(11)

where the longitudinal wavenumber

k_{3} = \sqrt{\frac{ρ}{E}} ω

(12)

Consider two points A and B on a uniform beam a distance x apart. Waves propagate from one point to the other, with the propagation being determined by the appropriate wavenumber. Denoting the positive and negative going wave vectors at points A and B as $a^{+}$ and $a^{-}$ and $b^{+}$ and $b^{-}$ , respectively, they are related by

a^{-} = f (x) b^{-}; b^{+} = f (x) a^{+} .

(13)

where

f (x) = [e^{- {ik}_{1} x} 000 e^{- k_{2} x} 000 e^{- {ik}_{3} x}]

(14)

is the propagation matrix for a distance x, and

a^{+} = [a_{1}^{+} a_{2}^{+} c^{+}], a^{-} = [a_{1}^{-} a_{2}^{-} c^{-}], b^{+} = [b_{1}^{+} b_{2}^{+} d^{+}], b^{-} = [b_{1}^{-} b_{2}^{-} d^{-}] .

(15)

where

a_{i}^{\pm}

and

b_{i}^{\pm}

(

i = 1, 2

), and

c^{\pm}

and

d^{\pm}

denote the involved bending and longitudinal wave components, respectively.

Waves propagate along uniform waveguide and are reflected and transmitted at structural discontinuities. Details on reflection at classical boundaries, as well as reflection and transmission at “L” and “T” joints can be found in the Appendix. Putting the propagation, reflection, and transmission equations in matrix form of equation (2) and setting the determinant of the coefficient matrix A to zero gives the natural frequencies of the single-story multi-bay frame.

It has been observed that for periodical beam structures, there exist the so-called “passing” and “stopping” bands. That is, the natural frequencies are clustered. As will be seen in the numerical studies that follow, the “passing” and “stopping” bands are observed in multi-bay frames as well. The number of natural frequencies in a cluster equals the number of bays. This observation can be applied to optimal structural design from a vibration suppression standpoint. For example, a multi-bay structure can be so designed as to allow the disturbance frequencies to fall into the gaps between two clustered natural frequencies. Such a design can be achieved by carefully choosing the materials, dimensions, and boundary conditions of a multi-bay frame structure, which together determine the natural frequencies of the frame.

3. Numerical examples

For comparison purposes, the same single-story multi-bay planar frame structure studied using the conventional modal approach in (Rieger and McCallion, 1965) is analyzed using the proposed wave approach. As a result, the physical properties are chosen to be the same as those in Rieger and McCallion (1965) and they are as follows:

Lengths of columns are $6.0 in$ , cross section (height by width) of column and cross beam elements are $3 / 16 in$ by $5 / 16 in$ (4.8 mm by 7.9 mm), and Young’s modulus $E = 28.3 \times 10^{6} lb / {in}^{2}$ ( $195.12 GN / m^{2}$ ). Unfortunately, the mass density was not given in Rieger and McCallion (1965), which made the comparison difficult. The only related information given was that the material used was mild steel. Depending on the grade of steel and the alloying elements added, the density of steel can vary between $0.2782 lbm / {in}^{3}$ and $0.2908 lbm / {in}^{3}$ ( $7700 kg / m^{3}$ and $8050 kg / m^{3}$ ). In this study, mass density is assumed $ρ = 0.2782 lbm / {in}^{3}$ ( $7700 kg / m^{3}$ ), since it better matches the experimental results in Rieger and McCallion (1965).

The length of the cross beams is assumed to be the same as that of the columns in the first example, and it is then assumed to be half the length of the columns in the second example. In both examples, the boundary conditions are clamped, and the cross-sectional dimensions and the material properties of the cross beams and columns are assumed the same. The results are tabulated in Tables 1 and 2, respectively. It can be seen that the numerical results obtained in the present study agree quite well with the experimental and the numerical results obtained by Rieger and McCallion (1965). The natural frequencies obtained from the present study are in general smaller than the computed natural frequencies and closer to the experimental results in Rieger and McCallion (1965), which represents a higher accuracy. This is because the coupled bending and axial vibrations, which were neglected in Rieger and McCallion’s study are considered in the current study.

Table 1.

Natural frequencies of the first example frame

No. of bays	Natural frequencies (Hz)
No. of bays	Present				Experimental (Rieger and McCallion, 1965)				Computed (Rieger and McCallion, 1965)
1	152.0	598.8	978.1	1057.3	152.7	602.8	980	1069	N/A
2	140.8	579.5	984.8	1053.6	142.3	583	986	1067	142.3	583	990	1072
2	140.8	728.5	1.0592	736	734	1072
3	137.1	571.6	987.9	1050.6	138.6	574	986	1064	138.3	575	994	1072
		657.4	1057.3	663	663	1072
		803.9	1059.7	808.5	812	1072
4	135.0	567.6	989.7	1048.4	137.3	571.5	989	1064	136.0	571	995	1072
		622.4	1055.4	614.5	626	1072
		729.5	1058.6	739.5	736	1072
		842.1	1060.0	849	850	1072
5	133.7	565.1	990.8	1046.8	136.6	571	991	1066	134.5	567	995	1072
		603.8	1053.6	611	607	1072
		680.7	1057.3	680	684	1072
		779.1	1059.2	787	785	1072
		862.7	1060.1	870	874	1072
6	132.8	563.4	991.5	1045.5	135.8	571	990	1065	133.6	566	997	1072
		592.2	1051.9	601	595	1072
		650.2	1056.0	660	654	1072
		728.5	1058.3	740	736	1072
		812.7	1059.5	820	820	1072
		873.9	1060.2	879	888	1072
7	132.2	562.2	992.1	1044.6	134.9	569	990	1068	132.9	564	997	1072
		584.7	1050.6	591	587	1072
		629.5	1054.7	638	633	1072
		693.0	1057.3	702	697	1072
		765.1	1058.8	773	772	1072
		835.8	1059.7	844	844	1072
		879.5	1060.2	881	897	1072
8	131.7	561.4	992.5	1043.8	132	565	998	1069	132.5	565	998	1072
		579.4	1049.4	580	584	1072
		615.2	1053.5	619	620	1072
		666.8	1056.3	672	673	1072
		728.8	1058.1	742	736	1072
		792.8	1059.2	792	801	1072
		852.2	1059.9	827	862	1072
		881.1	1060.2	871	904	1072

Table 2.

Natural frequencies of the second example frame

No. of bays	Natural frequencies (Hz)
No. of bays	Present				Experimental (Rieger and McCallion, 1965)				Computed (Rieger and McCallion, 1965)
1	192.4	841.5	1125.3	2219.8	191.3	829	1113	2142	N/A
2	181.6	870.5	1118.0	2193.7	181.7	861	1119	2147	181.7	870	1160	2195
2	181.6	905.3	2467.1	892	2379	905	2486
3	177.8	866.8	1116.0	2179.7	178.8	853	1116	2144	177.8	866	1145	2180
		899.8	2359.0	890	2290	897	2385
		952.0	2583.2	949	2515	950	2600
4	175.6	875.7	1115.0	2173.6	176.5	863	1118	2145	175.2	875	1140	2172
		881.3	2290.1	872	2235	882	2305
		940.5	2497.1	932	2412	939	2515
		966.2	2625.1	966	2590	965	2650
8	172.0	874.4	1113.6	2162.5	174.0	809	1108	2180	171.6	876	1138	2160
		877.3	2202.1	878	2220	876	2220
		907.8	2284.9	885	2303	906	2300
		927.6	2391.9	907	2480	926	2416
		949.2	2508.6	909	2511	947	2530
		961.9	2547.2	970	2570	961	2615
		975.4	2617.5	989	2650	975	2680
		975.7	2653.5	N/A	2670	980	2705

The modeshapes are obtained by deleting one raw of the coefficient matrix A in equation (2) and solving the remaining components in terms of an arbitrarily chosen element, which are presented in Figure 2.

4. Discussions and conclusions

In this paper, in-plane vibrations in single-story multi-bay planar frames are analyzed using the wave approach. The vibrations are modeled using classical vibration theories. The coupled bending and longitudinal vibrations are taken into account. With the availability of the propagation, reflection, and transmission matrices, vibration analysis of single-story multi-bay planar frames becomes systematic and concise: it involves a simple assembly of the involved matrices. The procedures are illustrated using two numerical examples, and both show good agreements with the results available in the literature.

Footnotes

Acknowledgements

The author gratefully acknowledges the support on this project from the Civil, Mechanical and Manufacturing Innovation Division of the National Science Foundation.

Funding

This work was supported by the Civil, Mechanical and Manufacturing Innovation Division of the National Science Foundation (grant number #0825761).

References

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McCallion

(1965) The natural frequencies of portal frames. International Journal of Mechanical Science 7: 263–276.