Dynamics of the Disk-Pendulum Coupled System With Vertical Excitation

This paper investigates the static and dynamic characteristics of the semi-elliptical rocking disk on which a pendulum pinned. This coupled system’s response is also analyzed analytically and numerically when a vertical harmonic excitation is applied to the bottom of the rocking disk. Lagrange’s Equation is used to derive the motion equations of the disk-pendulum coupled system. The second derivative test for the system’s potential energy shows how the location of the pendulum’s pivotal point affects the number and stability of equilibria, and the change of location presents different bifurcation diagrams for different geometries of the rocking disk. For both vertically excited and unforced cases, the coupled system shows chaos easily, but the proper chosen parameters can still help the system reach and keep the steady state. For the steady state of the vertically excited rocking disk without a pendulum, the variation of the excitation’s amplitude and frequency result in the hysteresis for the amplitude of the response. When a pendulum is pinned on the rocking disk, three major categories of steady states are presently in the numerical way.

The distribution of sales revenues from pharmaceutical innovation.

OBJECTIVE: This report updates our earlier work on the returns to pharmaceutical research and development (R&D) in the US (1980 to 1984), which showed that the returns distributions are highly skewed. It evaluates a more recent cohort of new drug introductions in the US (1988 to 1992) and examines how the returns distribution is emerging for drugs with life cycles concentrated in the 1990s versus the 1980s. DESIGN AND SETTING: Methods were described in detail in our earlier reports. The current sample included 110 new drug entities (including 28 orphan drugs), and sales data were obtained for the period 1988 to 1998, which represented between 7 and 11 years of sales for the drugs included. 20 years was chosen as the expected market life for this cohort, and a 2-step procedure was used to project future sales for the drugs--during the period until patent expiry and then beyond patent expiry until the 20-year time-horizon was completed. Thus, the values in the first half of the life cycle are essentially based on realised sales, while those in the second half are projected using information on patent expiry and other inputs. MAIN OUTCOME MEASURES AND RESULTS: Peak annual sales for the top decile of drugs introduced between 1988 and 1992 in the US amounted to almost $US1.1 billion compared with peak sales of less than $US175 million (1992 values) for the mean compound. In particular, the top decile accounted for 56% of overall sales revenue. Although the sales distributions were skewed in both our earlier and current analysis, the top decile in the later time-period exhibited more rapid rates of growth after launch, a peak that was more than 50% greater in real terms than for the 1980 to 1984 cohort, and a faster rate of expected decline in sales after patent expiry. One factor contributing to the distribution of sales revenues becoming more skewed over time is the orphan drug phenomenon (i.e. most of the orphan drugs are concentrated at the bottom of the distribution). CONCLUSION: The distribution of sales revenues for new drug compounds is highly skewed in nature. In this regard, the top decile of new drugs accounts for more than half of the total sales generated by the 1988 to 1992 cohort analysed. Furthermore, the distribution of sales revenues for this cohort is more skewed than that of the 1980 to 1984 cohort we analysed in previous research.