Pendulum physical

Until now we have considered the theory of the mathematical model of the real pendulum. Next, suppose that a solid body of finite dimensions swings in the plane XZ around a horizontal axis, and that the motion takes place with the angular velocity o(t), Fig. 3.4. 

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Thus, have we derived an equation of a motion of the physical pendulum and found parameters, which describe the swinging around the fixed axis. Introducing the ratio ... 

Consider a physical pendulum, as represented in Fig. 1. A mass m is attached by a spring to a rigid support. The spring is characterized by a force... 

In case of a real pendulum the density and viscosity of air should also be introduced into the relevance list. Both contain mass in their dimensions. However, this would unnecessarily complicate the problem at this step. Therefore we will consider a physical pendulum with a point mass in a vacuum. 

The freely swinging leg during walking can be visualized as a physical pendulum pivoted at the top end. The period of oscillation of this physical pendulum is (Davidovits, 1975)... 

The period of oscillation of a physical pendulum (the leg) moving in response to a torque is given by the moment of inertia of the leg and the maximum torque ... 

There is an optimum speed of walking. Faster than this speed, additional muscular energy is required to propel the body forward. Moving slower than the optimal speed requires additional muscular energy to retard leg movement. Thus, the optimal speed is related to the rate at which the leg can swing forward. Simple analysis of the leg as a physical pendulum shows that the optimal walking speed is related to leg length ... 

Chapter 9 gives a review on how multibody systems can be modelled by means of multibond graphs. A major contribution of the chapter is a procedure that provides a minimum number of break variables in multibond graphs with ZCPs. For the state variables and these break variables (also called semi-state variables) a DAE system can be formulated that can be solved by means of the backward differentiation formula (BDF) method implemented in the widely used DASSL code. The approach is illustrated by means of a multibond graph with ZCPs of the planar physical pendulum example. 

If K is extracted from the equation for the calculation of the period (7) of oscillations of physical pendulum,... 

Any body having the possibihty to oscillate freely under a gravitational force around a horizontal axis, not passing through the body s CM, is referred to as a physical pendulum. In this case, all points of a rigid body move along an arc of concentric circles. Consequently, for the description of a physical pendulum s oscillations, the rotational laws of dynamics should be applied. 

Let an axis of rotation z pass horizontally through point O (Figure 2.10) perpendicular to the plane of drawing. Also, let the physical pendulum be deflected from the position of equilibrium by angle a, which, as previously, is considered to be small. Then, the main law of dynamics of rotational motion can be written as... 

Sign corresponds to the accepted sign rule for the returning force moment of the Oz axis. Thereby, the differential equation for small physical pendulum oscillations according to eqs. (2.4.10) and (2.4.11) can be written as... 

Comparing this expression with eq. (2.4.1) we can conclude that the physical pendulum makes harmonic oscillations with cyclic frequency... 

The length of such a mathematical pendulum, which is equal to the physical pendulum s oscillation period, is called the reduced length of a physical pendulum. An expression for the reduced length of a physical pendulum can be found by comparing eqs. (2.4.9) and (2.4.14) ... 

Point Oj on the line OC (Figure 2.10) at a distance L from the axis of rotation z is called the center of swing of the physical pendulum. It is noteworthy that if a pendulum is turned over and hung up on the horizontal axis passing through the point Oj the period of its oscillation does not change, point 0 being the new center of oscillation. We will leave the proof of this property as an exercise for the reader. 

Solution The frequency of the physical pendulum s oscillation and the period of its oscillations can be found according to eqs. (2.4.14) and (2.4.13). There are three values to define total weight m, the Ml of the pendulum relative to the axis of oscillation and the distance from the axis of oscillations up to the center of weights /. Oscillation axis z passes through the center of the rod perpendicular to the plane of drawing let us direct an axis x vertically downward (parallel to the rod) and super-... 

Remember that the oscillation axis coincides with the position of the oscillation axis, i.e., the coordinate of the CM numerically coincides with 4. The weight of a physical pendulum is equal to m = m + m2 + m3 (0.9 kg). 

We shall find the MI of a physical pendulum relative to oscillation axis as the sum of the moments of inertia of three bodies ... 

A physical pendulum consists of a rod of mass m and length / = 1 m and of two small balls of masses m and 2m fixed to the rod at lengths HI and I, respectively. The pendulum makes small oscillations relative to a horizontal axis passing perpendicularly to the rod through the middle of the rod. Determine the frequency vof the harmonic pendulum oscillations. 

A mathematical pendulum of / = 40 cm in length and a physical pendulum in the form of a thin straight rod of length /2 = 60 cm oscillate around a common horizontal axis. Find the distance a between the rod CM and the oscillation axis. 

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