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In this paper, an approximate analytical approach is developed for the determination of natural longitudinal frequencies of a cantilever-cracked beam based on the Lagrange inversion theorem.

The crack is modeled by an equivalent axial spring with stiffness according to Castigliano's theorem. Thus, an implicit frequency equation corresponding to cantilever-cracked bar is obtained. The resulting equation is solved using the Lagrange inversion theorem.

Effect of different crack depths and crack positions on natural frequencies of the cracked beam is analyzed. It is shown that an increase in the crack depth ratio produces a decrease in the fundamental longitudinal natural frequency of a cracked bar. Furthermore, approximate analytical results are compared with those obtained numerically as well as from experimental tests.

A new approximate analytical expression of a fundamental longitudinal frequency, as a function of crack depth and crack location, is obtained.

The purpose of this paper is to introduce a new numerical approach for studying a cantilever bar having a transverse crack. The crack is modeled by an elastic longitudinal spring with a stiffness K according to Castiglione’s theorem.

The bar is excited by different longitudinal impulse forces. The considered problem based on the differential equation of motion is solved by the method of characteristics (MOC) after splitting the second-order motion equation into two first-order equivalent equations.

In this study, effects of the crack size and crack’s position on the reflected waves from the crack are investigated. The results indicate that the presence of the crack in the cantilever bar generates additional waves caused by the reflection of the incident wave by the crack.

A numerical approach developed in this paper is used for detecting the extent of the damage in cracked bars by the measurement of the difference between the dynamic response of an uncracked bar and a cracked bar.

The purpose of this paper is to investigate the influence that different components of relational capital (marketing capability, open innovation with business and scientific partners, technological reputation, brand) have on customer performance (CP). Moreover, the moderating effect of absorptive capacity on such relationships is tested.

First, the direct relationship between the different components of relational capital and CP is analyzed through a linear regression model. Then, to test the moderating effect, two distinct regression analyses are conducted into two sub-samples, defined according to the level of absorptive capacity. The authors carried out these analyses on a sample of 150 small- and medium-sized enterprises (SMEs) in the medium- and high-tech B2B context.

Results of this study prove that CP is enhanced through firm marketing capability, open innovation with business partners and technological reputation, while brand and open innovation with scientific partner do not have an association with CP. In particular, the impact of marketing capability and open innovation with business actors on CP is greater for firms with higher absorptive capacity.

This paper, highlighting the relevance of relational capital and absorptive capacity in improving CP, enhances our knowledge about the factors that help to strengthen the relationships with customers, which is an under-investigated issue especially for SMEs competing in B2B industries, and extends our knowledge on open innovation practices.

Findings of this paper suggest that, to achieve better CP, managers should pay special attention to nurturing their marketing capability and high-quality relationships with external actors and invest in absorptive capacity to enhance the positive effect of such linkages.

This work, combining the external perspective of relational capital and the internal organizational dimension of absorptive capacity, provides valuable insights about the knowledge and resource mix that firms might rely on to achieve better customer satisfaction and loyalty.