Building frequency fluctuations from continuous monitoring of ambient vibrations and their relationship to temperature variations

Abstract

Building frequencies (fundamental and higher modes) are a critical parameter especially in the field of structural health monitoring mainly based on the stability of the structural dynamic parameters of individual building (frequencies, damping and modes shape). One of the most used methods to find out these parameters is based on the use on ambient vibration analysis. In this work, we study the fluctuations over a month period of the fundamental frequencies (transverse and longitudinal) of a 3.5-story RC-building made of 2 identical units connected by a structural joint. Time independent building frequencies is a strong assumption; as illustrated by our experiment showing that over an observation period of a month, building frequencies fluctuate of about 3.5 %. A clear correlation is found between the building frequency fluctuations and temperature variations, with a phase-shift interpreted as the characteristic time of heat diffusion within the walls. This allows: (1) determining the thermal diffusivity of the structure, (2) inferring its relative stiffness variations, and (3) showing that its Young modulus varies linearly with temperature.

Appendix 1: Determination of the thermal diffusivity of the medium

In 1D, the Fourier law of thermal conduction reads

$$\frac{{\partial {\text{T}}}}{{\partial {\text{t}}}} =\upalpha\frac{{\partial {\text{T}}^{2} }}{{\partial {\text{x}}^{2} }} ,$$

(2)

where $\upalpha$ is the thermal diffusivity given by

$$\upalpha = \frac{\uplambda}{{\uprho{\text{C}}_{\text{p}} }},$$

(3)

where $\lambda$ is the thermal conductivity, $\rho$ the density and $C_{p}$ the specific heat capacity.

A simple dimension analysis on Eq. (2) shows that the thermal diffusion time $\tau_{D}$ is roughly given by

$$\tau_{D} = \frac{{L^{2} }}{\alpha },$$

(4)

where $L$ is a characteristic distance of the 1D structure. Equation (4) represents the diffusion time for heat to cross a structure of width $L$. It is comparable to the phase lag $\phi$ defined in the main text.

A more rigorous expression can be obtained using the Fourier transform of the temperature field $T$

$$T\left( {x,t} \right) = \frac{1}{{\left( {2\pi } \right)^{2} }}\mathop \int \limits_{ - \infty }^{ + \infty } dk\mathop \int \limits_{ - \infty }^{ + \infty } dw\tilde{T} \left( {k,w} \right)e^{{i\left( {kx - \omega t} \right)}} ,$$

(5)

Putting Eq. (5) into (2) yields the following dispersion equation

$$i\omega = \alpha k^{2} ,$$

(6)

that can be injected back in Eq. (5) to give

$$T\left( {x,t} \right) = \frac{1}{{\left( {2\pi } \right)}}\mathop \int \limits_{ - \infty }^{ + \infty } dk\tilde{T}\left( k \right)e^{ikx} e^{ - t/\tau \left( k \right)} ,$$

(7)

where

$$\tau \left( k \right) = \frac{1}{{\alpha k^{2} }},$$

(8)

Equation (7) shows that the temperature field is a superposition of spatially oscillating modes decaying with time with a characteristic time given by $\tau \left( k \right)$ for mode $k$. Considering that the dominating mode corresponds to the wavelength $L$ where $L$ is the width of the wall, the characteristic diffusion time $\tau \left( L \right)$ is given by

$$\tau \left( L \right) = \frac{1}{\alpha } \left( {\frac{L}{2\pi }} \right)^{2} ,$$

(9)

When the width $L$ of the wall and the phase lag $\phi$ are known, we can use Eq. (9) using $\tau \left( L \right)$ = $\phi$ to estimate the thermal diffusivity $\alpha$ of the building walls

$$\alpha = \frac{1}{\tau \left( L \right)} \left( {\frac{L}{2\pi }} \right)^{2} ,$$

(10)

1.1 Implications for the variations of the elastic properties

To a first order, the resonance frequency of the building is given by

$$f_{0} = \frac{1}{2\pi }\sqrt {\frac{K}{M}} ,$$

(11)

where $K$ is the stiffness of the building and $M$ its mass. The relative variations of frequency are given by

$$\frac{{\Delta f_{0} }}{{f_{0} }} = \frac{1}{2}\left( {\frac{{\Delta K}}{K} - \frac{{\Delta M}}{M}} \right),$$

(12)

We assume that the variations of the building mass are negligible so that Eq. (12) reduces to

$$\frac{{\Delta f_{0} }}{{f_{0} }} = \frac{1}{2}\frac{{\Delta K}}{K},$$

(13)

The relative variations in the elastic parameter $K$ are hence given by

$$\frac{{\Delta K}}{K} = 2\frac{{\Delta f_{0} }}{{f_{0} }},$$

(14)

1.2 Variations of the resonance frequency with temperature

Figure 5 show the longitudinal and transverse resonance frequency $f$ as a function of temperature. A linear trend is observed suggesting that

$$f_{0} \left( T \right) = a + bT,$$

(15)

where $a$ and $b$ are two parameters independent of temperature, at least in the temperature range considered in our experiment.

Combining Eq. (15) with (11) yields

$$K\left( T \right) = \left( {2\pi } \right)^{2} M\left( {a + bT} \right)^{2} ,$$

(16)

where, again, the variations of mass with temperature are neglected ($\frac{\partial M}{\partial T} = 0$). Note that Eq. (15) predicts a quadratic dependence of the stiffness on temperature.

If the stiffness of the building K and the Young modulus $E$ are proportional, then

$$E\left( T \right) = E\left( {T_{0} } \right)\frac{K\left( T \right)}{{K\left( {T_{0} } \right)}},$$

(17)

where $T_{0}$ is a reference temperature in the range considered in our experiment. We can combine Eq. (17) with (16) to get the quadratic expression

$$E\left( T \right) = E\left( {T_{0} } \right)\left[ {\frac{a + bT}{{a + bT_{0} }}} \right]^{2} ,$$

(18)

Considering the values of $a$ and $b$ given earlier and given that $T$ varies from −0.5 to 26.4 °C, it is straightforward to show that Eq. (18) simplifies into

$$E\left( T \right) \approx E\left( {T_{0} } \right)\left[ {1 + \frac{2b}{a}\left( {T - T_{0} } \right)} \right],$$

(19)

Equation (${\text{A}}18$) predicts a linear dependence of $E$ with $T$.