An investigation of the dynamic response of architectural kerf structures

Abstract

Kerfing is a subtractive manufacturing method to create flexible surfaces from stiff planar materials. While kerf structures are ubiquitous in indoor and outdoor architectures due to their pleasing aesthetics, they have potential applications for tuning the indoor acoustics and altering the dynamic response of a building from wind, traffic, etc., by varying their geometrical parameters (kerf pattern, cut density, cut thickness, etc.) and locally deforming the kerf cells. However, the dynamic responses of kerf structures have never been explored before. This research presents an investigation of the dynamic response, in terms of mode shapes, natural frequencies, and stress wave propagation, of the flexible kerf cells. The influence of the material behavior, i.e., elastic and viscoelastic, on the dynamic response of the kerf cells is also investigated. Mathematical models are used to understand the interplay between material behavior, geometrical kerf pattern, and dynamic responses. Experimental tests using scanning laser vibrometry are performed to study the mode shapes and frequencies on two kerf cells with stainless steel and medium density fiber materials. Responses from the mathematical models are compared to experimental results in order to validate the modeling approach. Understanding the dynamic responses of kerf cells in association with their geometrical and material characteristics can lead to a systematic design of kerf structures exposed to various dynamic loadings.

Appendices

Appendix 1

This section discusses the influence of viscoelastic materials on the resonant frequency of a system. The partial differential equations for the beam in Eqs. (4) or (6) with the displacement vector in Eq. (9) can be written in general as:

$$ {\mathbf{M}}({\ddot{\mathbf{q}}}) + {\mathbf{L}}({\mathbf{q}}) = {\mathbf{F}}(x,t). $$

(A.1)

To present an analytical solution, we ignore the transverse shear and bending coupling, so we can reduce Eq. (A.1) to:

$$ M(\ddot{q}_{i} ) + L(q_{i} ) = F_{i} (x,t), $$

(A.2)

where M() and L() are linear differential operator,¹$q_{i} = \phi_{i} (x)y_{i} (t)$ and $F_{i} = \phi_{i} (x)f_{i} (t)$, and thus Eq. (A.2) with a viscoelastic material is rewritten as:

$$ \ddot{y}_{i} + C_{i} * dy_{i} = f_{i} (t), $$

(A.3)

where $C_{i} = \frac{{L(\phi_{i} )}}{{M(\phi_{i} )}}$. Since Eq. (A.3) is written for each scalar component of the displacement, to reduce complexity we further eliminate the subscript i in the rest of the formulation. Consider an input $f(t) = f_{o} \sin \omega t$, at the steady state the displacement takes the following form:

$$ y(t) = y_{1} \sin \omega t + y_{2} \sin \omega t = \sqrt {y_{1}^{2} + y_{2}^{2} } \sin \left( {\omega t + \delta (\omega )} \right). $$

(A.4)

Substituting Eq. (A.4) into Eq. (A.3) and with the complex property $C*(\omega ) = C^{\prime } (\omega ) + iC^{\prime \prime } \left( \omega \right)$, we have:

$$ y_{1} = f_{o} \frac{{C^{\prime } - \omega^{2} }}{{\left( {C^{\prime } - \omega^{2} } \right)^{2} + C^{\prime \prime 2} }};\quad y_{2} = f_{o} \frac{{C^{\prime \prime } }}{{\left( {C^{\prime } - \omega^{2} } \right)^{2} + C^{\prime \prime 2} }}. $$

(A.5)

The displacement amplitude is

$$ y = \frac{{f_{o} }}{{\sqrt {\left( {C^{\prime } - \omega^{2} } \right)^{2} + C^{\prime \prime 2} } }} = \frac{{f_{0} /C(0)}}{{\sqrt {\left( {\frac{{C^{\prime } }}{C(0)} - \frac{{\omega^{2} }}{{\omega_{n}^{2} }}} \right)^{2} + \left( {\frac{{C^{\prime \prime } }}{C(0)}} \right)^{2} } }}.$$

(A.6)

It is noted that $\omega_{n}^{2} = C(0)$. The variable C(t) is a function of the modulus of the material and inertial property. For example, for the first component of the displacement vector in Eq. (9), $C_{1} (t) = E(t)/\rho$, the second component neglecting the rotational coupling $C_{2} (t) = kG(t)/\rho$, etc. Consider a viscoelastic material whose relaxation modulus is described by $E(t) = E(\infty ) + E_{1} e^{{ - t/\tau_{R} }}$, where $E(\infty )$ is the long-term (relaxed) modulus and $\tau_{R}$ is the characteristics of relaxation time that indicates how quickly the stress relaxes. The instantaneous (initial) modulus is given as $E(0) = E(\infty ) + E_{1}$, which corresponds to a modulus of elastic materials. The ratio $E\infty /E(0)$ measures the extent of stress relaxation. The corresponding complex moduli are:

$$ E^{\prime } = \frac{{E(\infty ) + E(0)(\omega \tau_{R} )^{2} }}{{1 + (\omega \tau_{R} )^{2} }};\quad E^{\prime \prime } = \frac{{\omega \tau_{R} (E(0) - E(\infty ))}}{{1 + (\omega \tau_{R} )^{2} }}. $$

(A.7)

It is seen that $C^{\prime } /C(0) = E^{\prime } /E(0); \, C^{\prime \prime } /C(0) = E^{\prime \prime } /E(0)$. We define a parameter $\xi = \tau_{R} \omega_{n}$, where $\omega_{n}$ is the natural frequency of an undamped system and thus $\xi$ is interpreted as the ratio of the material relaxation time to the natural period of the system. A low value $\xi$ indicates the material relaxes faster than the natural period. Thus, Eq. (A.7) is rewritten as:

$$ \frac{{C^{\prime } }}{C(0)} = \frac{{E(\infty )/E(0) + \left( {\frac{\omega }{{\omega_{n} }}\xi } \right)^{2} }}{{1 + \left( {\frac{\omega }{{\omega_{n} }}\xi } \right)^{2} }};\quad \frac{{C^{\prime \prime } }}{C(0)} = \frac{{\frac{\omega }{{\omega_{n} }}\xi (1 - E(\infty )/E(0))}}{{1 + \left( {\frac{\omega }{{\omega_{n} }}\xi } \right)^{2} }}. $$

(A.8)

To illustrate the implication of the viscoelastic material on the resonant frequency of the system, we constructed the plots of the normalized displacement amplitude against the normalized excitation frequency by substituting Eq. (A.8) into Eq. (A.6). Figure 

Fig. 22

Resonant frequency responses of a system with a viscoelastic material

22 shows the resonant frequency responses of a system with a viscoelastic (dissipative) material for different $\xi$ and $E\infty /E(0)$. The use of viscoelastic materials can lower the resonant frequency of the systems in addition to attenuate the responses. With a proper choice of a viscoelastic characteristic of the material compared to the natural frequency of the system, it is possible to tune the resonance in the system, which would be beneficial for flexible facades under dynamic loads.

Appendix 2

With the understanding of the modal response of the straight continuous beam, we consider continuous folded beams with a constant angle, $\theta$ (see Fig. 4). The folded beams are a combination of identical straight beams where $i=\mathrm{1,2},\dots .N+1$, with $N$ folds connect the beams at an arbitrary angle. The displacements for each beam segment (i) are:

$$ u_{1}^{\left( i \right)} \left( {x_{1}^{\left( i \right)} ,t} \right), u_{3}^{\left( i \right)} \left( {x_{1}^{\left( i \right)} ,t} \right) \quad 0 \le x_{1}^{\left( i \right)} \le l^{\left( i \right)}. $$

(B.1)

To derive the equations of motion for the folded beams, continuity conditions at ${x}_{1}^{(i)}=l$ and ${x}_{1}^{(i+1)}=0$ are used. The continuity conditions imply that the resultants of internal moments and forces are equal and the displacements are continuous at ${x}_{1}^{(i)}=l$ and ${x}_{1}^{(i+1)}=0$. The continuity conditions are:

$$ \begin{gathered} \left\{ {\begin{array}{*{20}c} {u_{1}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right)} \\ {u_{3}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right)} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} { - \cos \theta } & { - \sin \theta } \\ {\sin \theta } & { - \cos \theta } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {u_{1}^{{\left( {i + 1} \right)}} \left( {0,t} \right)} \\ {u_{3}^{{\left( {i + 1} \right)}} \left( {0,t} \right)} \\ \end{array} } \right\};\hfill \\ \frac{{\partial u_{2}^{\left( i \right)} }}{{\partial x_{1}^{\left( i \right)} }}\left( {l^{\left( i \right)} ,t} \right) = \frac{{\partial u_{2}^{{\left( {i + 1} \right) }} }}{{\partial x_{1}^{{\left( {i + 1} \right)}} }}\left( {0,t} \right) \equiv \varphi_{3}^{{\left( {i + 1} \right)}} \left( {0,t} \right) + \gamma_{12}^{{\left( {i + 1} \right)}} \left( {0,t} \right); \hfill \\ \frac{{\partial u_{3}^{\left( i \right)} }}{{\partial x_{1}^{\left( i \right)} }}\left( {l^{\left( i \right)} ,t} \right) = \frac{{\partial u_{3}^{{\left( {i + 1} \right)}} }}{{\partial x_{1}^{{\left( {i + 1} \right)}} }}\left( {0,t} \right) \equiv - \varphi_{2}^{{\left( {i + 1} \right)}} \left( {0,t} \right) + \gamma_{13}^{{\left( {i + 1} \right)}} \left( {0,t} \right); \hfill \\ E^{\left( i \right)} I_{22}^{\left( i \right)} \left( {\frac{{\partial \varphi_{2}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right)}}{{\partial x_{1}^{\left( i \right)} }}} \right) = E^{{\left( {i + 1} \right)}} I_{22}^{{\left( {i + 1} \right)}} \left( {\frac{{\partial \varphi_{2}^{{\left( {i + 1} \right)}} \left( {0,t} \right)}}{{\partial x_{1}^{{\left( {i + 1} \right)}} }}} \right); \hfill \\ E^{\left( i \right)} I_{33}^{\left( i \right)} \left( {\frac{{\partial \varphi_{3}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right)}}{{\partial x_{1}^{\left( i \right)} }}} \right) = E^{{\left( {i + 1} \right)}} I_{33}^{{\left( {i + 1} \right)}} \left( {\frac{{\partial \varphi_{3}^{{\left( {i + 1} \right)}} \left( {0,t} \right)}}{{\partial x_{1}^{{\left( {i + 1} \right)}} }}} \right); \hfill \\ kG^{\left( i \right)} A^{\left( i \right)} \left( { - \varphi_{3}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right) + \frac{{\partial u_{2}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right)}}{{\partial x_{1}^{\left( i \right)} }}} \right) = kG^{{\left( {i + 1} \right)}} A^{{\left( {i + 1} \right)}} \left( { - \varphi_{3}^{{\left( {i + 1} \right)}} \left( {0,t} \right) + \frac{{\partial u_{2}^{{\left( {i + 1} \right)}} \left( {0,t} \right)}}{{\partial x_{1}^{{\left( {i + 1} \right)}} }}} \right) ;\hfill \\ kG^{\left( i \right)} A^{\left( i \right)} \left( {\varphi_{2}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right) + \frac{{\partial u_{3}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right)}}{{\partial x_{1}^{\left( i \right)} }}} \right) = E^{{\left( {i + 1} \right)}} A^{{\left( {i + 1} \right)}} \frac{{du_{1}^{{\left( {i + 1} \right)}} \left( {0,t} \right)}}{{dx_{1}^{{\left( {i + 1} \right)}} }}\sin \theta \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\, - kG^{{\left( {i + 1} \right)}} A^{{\left( {i + 1} \right)}} \left( {\varphi_{2}^{{\left( {i + 1} \right)}} \left( {0,t} \right) + \frac{{\partial u_{3}^{{\left( {i + 1} \right)}} \left( {0,t} \right)}}{{\partial x_{1}^{{\left( {i + 1} \right)}} }}} \right)\cos \theta ; \hfill \\ E^{\left( i \right)} A^{\left( i \right)} \frac{{du_{1}^{\left( i \right)} \left( {l^{\left( i \right)} ,t} \right)}}{{dx_{1}^{\left( i \right)} }} = E^{{\left( {i + 1} \right)}} A^{{\left( {i + 1} \right)}} \frac{{du_{1}^{{\left( {i + 1} \right)}} \left( {0,t} \right)}}{{dx_{1}^{{\left( {i + 1} \right)}} }}\cos \theta \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \, + kG^{{\left( {i + 1} \right)}} A^{{\left( {i + 1} \right)}} \left( {\varphi_{2}^{{\left( {i + 1} \right)}} \left( {0,t} \right) + \frac{{\partial u_{3}^{{\left( {i + 1} \right)}} \left( {0,t} \right)}}{{\partial x_{1}^{{\left( {i + 1} \right)}} }}} \right)\sin \theta ; \hfill \\ G^{\left( i \right)} J^{\left( i \right)} \frac{{\partial^{2} \beta^{\left( i \right)} }}{{\partial x_{1}^{\left( i \right)2} }} = G^{{\left( {i + 1} \right)}} J^{{\left( {i + 1} \right)}} \frac{{\partial^{2} \beta^{{\left( {i + 1} \right)}} }}{{\partial x_{1}^{{\left( {i + 1} \right)2}} }}. \hfill \\ \end{gathered} $$

(B.2)

For the folded SS beam, these continuity conditions are substituted in Eq. (5) to determine the equations of motion. Similarly, for folded MDF beams, these conditions are substituted in Eq. (6) to determine the equations of motion.

Appendix 3

3.1 Modal experiments

The six handles of the hexagon specimens (see Figs. 3 and

Fig. 23

Experimental test setup for testing hexagon specimens. a Scanning laser vibrometer (MSA-100-3D, Polytec, Irvine, CA) b HD SS specimen clamped in the fixture c HD MDF specimen clamped in the fixture

23) are clamped in customized-built fixtures. For the SS specimen, the six handles of the specimen are clamped in the aluminum fixture with grooves to restrict the in-plane vibration and the cap is bolted from top to inhibit out-of-plane motion during actuation as shown in Fig. 23. Similarly, the MDF specimen is clamped in the 3-D printed fixture made from polylactic acid (PLA) plastic (Gizmodorks, Temple City, CA). Also, the handles of the MDF specimen are epoxied in the grooves designed in the fixture using a 50,133 plastic bonder (J-B Weld, Atlanta, GA) to avoid any slippage at the handles.

To experimentally determine the mode shapes and frequencies on these complex specimens, scanning laser vibrometry is chosen as it is a non-contact measurement technique [17, 22, 31]. The fixture assembly with the specimen is bolted to the x/y stage of the scanning laser vibrometer (MSA-100-3D, Polytec, Irvine, CA) as shown in Fig. 23. To actuate the specimen, piezo actuator (P-885.91, Physik Instrumente GmbH & Co.KG, Germany) is used which is glued to the fixture instead of the specimen to avoid adding mass to the specimen which will alter the dynamics of the kerf structure. The scanning laser vibrometer is used to perform a modal analysis with the input of 8 V chirp excitation from the piezo actuator.

As the surface of the SS specimen is shiny so the specimen is sprayed with an occlusion spray to avoid the mirror effect, which will lead to good quality measurement. In the case of the SS specimen, the velocity output range for scanning laser vibrometer is kept 10 mm/s with a sampling rate of 15.65 kHz. A Fast Fourier Transform (FFT) is performed within a selected bandwidth between 1 and 6250 Hz. For the SS specimen, 744 points on the surface of the SS are used as measurement locations, each scanning point and FFT averaged 12 times. For the MDF specimen, the velocity output range for the vibrometer is 20 mm/s with a sampling rate of 12.5 kHz. The bandwidth is 1–5000 Hz and the number of points on the MDF specimen is kept similar to the SS specimen. As compared to the test on the SS specimen, each point is averaged 8 times during the test. The frequency response function (FRF) for each data point, average FRF is obtained and stored in a file that is post-processed in the PSV software (Polytec, Irvine, CA) to extract mode shapes and resonance frequencies.

3.2 Creep experiments

Uniaxial creep tests are performed on MDF dog-bone specimens to characterize the viscoelastic properties. The creep tests are performed at constant room temperature (25 °C) and 50% of the ultimate tensile strength of MDF. A constant uniaxial load is applied to the dog-bone specimens for 2 h at room temperature. A linear viscoelastic model is used to capture the creep behavior (Fig. 

Fig. 24

Uniaxial creep responses of MDF samples

24) using the Prony parameters on the time-dependent compliance $D(t) = D(0) + \sum\nolimits_{i = 1}^{N} {D_{i} \left( {1 - e^{{ - t/\tau_{ci} }} } \right)}$. The instantaneous compliance $D(0) = 1/E_{0}$, where E₀ is the elastic modulus of the MDF given in Table 1. The time-dependent parameters are then calibrated by fitting the data in Fig. 24. The time parameters $\tau_{ci}$ in the Prony function with three terms are determined as 100, 1000, and 5000 s, respectively, and the calibrated values for D_i are 10^–3, 2 × 10^–3, 7 × 10^–3 ksi⁻¹, respectively. The beam element model discussed above is expressed in terms of a relaxation modulus, it is then necessary to obtain the relaxation modulus of the MDF material from the creep responses. The time-dependent relaxation modulus of the following form $E(t) = E(\infty ) + \sum\nolimits_{i = 1}^{N} {E_{i} e^{{ - t/\tau_{i} }} }$ is considered and the material parameters are determined by using a Laplace transform method, $\hat{E}(s)\hat{D}(s) = 1/s^{2}$, where s is the transform variable, $\hat{E}(s)$ and $\hat{D}(s)$ are the Laplace transforms of E(t) and D(t), respectively. The time-dependent relaxation moduli are given in Table 2.

Appendix 4

Responses of hexagon kerf domain with periodic boundary conditions

By using Floquet–Bloch theorem for wave propagation [22, 32, 33], the complex displacements on the hexagonal domain unit cell are as follows:

$$ q_{rt} = q_{lb} e^{{i\left( {k_{1} \left( {1 + \sin \frac{\theta }{2}} \right) + k_{3} \sin \theta } \right)a}}, \quad q_{t} = q_{b} e^{{i\left( {2k_{3} \sin \theta } \right)a}}, $$

(D.1)

where the subscripts $r,l,b,$ and $t$ represent displacements corresponding to right, left, bottom, and top, respectively. The double subscripts represent displacements of the handles: for example, $rt$ denotes the right top handle as shown in Fig. 

Fig. 25

Unit cell showing nomenclature used in Floquet conditions (top); mode shapes showing normalized displacement and natural frequencies of MDF hexagonal domain with periodic boundary conditions (bottom)

25. The side length of the unit cell is denoted as $a$ and $\theta$ is the angle subtended between the beams in the unit cell as mentioned earlier. ${k}_{1}$ and ${k}_{3}$ are components of the wave vector of the plane wave. The above-mentioned Floquet conditions are prescribed on the MDF hexagonal domain unit cell. The nonzero modes at (${k}_{1}=0, {k}_{3}=0)$ and corresponding frequencies are determined (see Fig. 25). From the results, it can be noticed that the unit cell with periodic boundary conditions shows both in-plane and out-of-plane mode shapes. However, as expected, the mode shapes and modal frequencies change as compared to the unit cell with clamped boundary conditions. The resonance frequencies decrease compared to the unit cell with clamped boundary conditions which shows us that the structure is becoming more compliant. Also, more out-of-plane mode shapes are observed in the initial modes. These responses are expected since adding more cells to form larger kerf panels leads to more compliant panels and out-of-plane deformations are easier to achieve in the larger panels.

Appendix 5

Influences of pre-deformed stresses

We performed an additional analysis to examine the extension of pre-deformations on the dynamic response of the kerf unit cell. One triangular unit cell of SS hexagon domain is actuated by prescribing 1 mm and 0.5 mm out-of-plane displacements, and a modal analysis was performed. The modal behaviors (mode shapes and modal frequencies) remain the same when the unit cell is deformed by 0.5 mm and 1 mm, as shown in Fig. 

Fig. 26

Comparison of modal frequencies with different actuation levels

26. The SS hexagon domain actuated by 1 mm undergoes higher maximum principal stress compared to the SS hexagon domain actuated by 0.5 mm as shown in Fig. 

Fig. 27

Principal stresses in deformed unit cells

27. The stresses are kept below the yield stress of the stainless steel material (Table 1). It can be also be noticed that due to marginal pre-deformation, most of the hexagon domain does not undergo any stress except a certain region of the actuated triangle unit cell. Therefore, the stresses due to pre-deformation do not have a significant effect on the modal analysis.