Refinement of Housner’s model on rocking blocks

Abstract

Housner published a simple model for the rocking block more than five decades ago (Housner in Bull Seismol Soc Am 53:403–417, 1963), which is widely used also for modeling stone and masonry columns and arches. In this paper we investigate the reasons of the well-known fact that experiments show lower energy loss during impact than it is predicted by Housner’s model. It was found that a reasonable explanation for the difference is that in the original model the best case scenario was assumed: that impact occurs at the edges, which results in the maximum energy loss. In reality, due to the unevenness of the surfaces, or due to the presence of aggregates between the interfaces, rocking may occur with consecutive impacts, which reduces the energy loss. This hypothesis was also verified by experiments.

Appendix: Housner’s model when the two axes of rotation are at arbitrary locations

Here, we give the simple extension of Housner’s model, when the location of the axis of rotation before impact (P ₁) and after the impact (P ₂) are not at the edges of the block but at arbitrary positions (Fig. 14). Immediately before impact (rotation around axis P ₁) the angular momentum about axis P ₂ is

$$ L_{\text{b}} = m\omega_{b} \left( {\frac{{\left( {2b} \right)^{2} }}{12} + \frac{{\left( {2h} \right)^{2} }}{12} + h^{2} + x_{1} x_{2} } \right), $$

(3)

while after impact (rotation around axis P ₂) the moment of momentum about axis P ₂ is:

$$ L_{\text{a}} = m\omega_{a} \left( {\frac{{\left( {2b} \right)^{2} }}{12} + \frac{{\left( {2h} \right)^{2} }}{12} + h^{2} + x_{2}^{2} } \right), $$

(4)

where m is the mass of the block, and x ₁ and x ₂ are the locations of the axes measured from the middle of the edge. From the condition that L _a = L _b, we obtain the following expression for the angular velocity:

$$ \omega_{\text{a}} = \mu \omega_{\text{b}} , \quad \mu = \frac{{2h^{2} + 0.5b^{2} + 1.5x_{1} x_{2} }}{{2h^{2} + 0.5b^{2} + 1.5x_{2}^{2} }}. $$

(5)

Fig. 14

Housner’s model for a rocking block if rotation occurs around two axes of arbitrary position

For x ₁ = –b and x ₂ = b Eqs. (1) and (5) are identical.

If the corners are cut (Fig. 14), and we set x ₁ = –b ₂ and x ₂ = b ₂, Eq. (5) results in

$$ \omega_{\text{a}} = \mu_{\text{Hous}} \omega_{\text{b}} ,\quad \mu_{\text{HousC}} = \frac{{2h^{2} + 0.5b^{2} - 1.5b_{2}^{2} }}{{2h^{2} + 0.5b^{2} + 1.5b_{2}^{2} }}. $$

(6)

Now we apply Eq. (5) in two steps. First, x ₁ = –b ₂ and x ₂ = 0, i.e. the block rotates at the left corner and then impact occurs at the middle. Equation (5) gives:

$$ \omega_{{{\text{a}}1}} = \omega_{{{\text{b}}1}} ; $$

(7)

and second: x ₁ = 0 and x ₂ = b ₂, i.e. the block rotates at the middle and then impact occurs at the right corner:

$$ \omega_{{{\text{a}}2}} = \omega_{{{\text{b}}2}} \frac{{2h^{2} + 0.5b^{2} }}{{2h^{2} + 0.5b^{2} + 1.5b_{2}^{2} }}. $$

(8)

By setting ω _a = ω _a2, ω _b = ω _b1, ω _a1 = ω _b2, from Eqs. (7) and (8) we obtain an expression for the change in the angular velocity, if rocking occurs in two steps, according to the geometry shown in Fig. 7b:

$$ \omega_{\text{a}} = \mu_{\text{HousC}}^{{2{\text{imp}}}} \omega_{\text{b}} , \quad \mu_{\text{HousC}}^{{2{\text{imp}}}} = \frac{{2h^{2} + 0.5b^{2} }}{{2h^{2} + 0.5b^{2} + 1.5b_{2}^{2} }}. $$

(9)

If the width of the block is identical to the width of the base (b = b ₂) Eq. (9) simplifies to

$$ \omega_{\text{a}} = \mu_{\text{HousC}}^{{2{\text{imp}}}} \omega_{\text{b}} ,\quad \mu_{\text{HousC}}^{{2{\text{imp}}}} = \frac{{2h^{2} + 0.5b^{2} }}{{2h^{2} + 2b^{2} }}. $$

(10)

Since the kinetic energy is proportional to the square of the angular velocity, the relative loss in kinetic energy during rocking can be calculated as:

$$ \eta = \frac{{\omega_{b}^{2} - \omega_{a}^{2} }}{{\omega_{b}^{2} }} = 1 - \mu^{2} , $$

(11)

where ω _b and ω _a are the angular velocities before and after rocking, and μ is the angular velocity ratio defined by Eqs. (1), (6), (9) and (10).