Analyze of Mechanical Systems

       In mechanical systems, motion can be of different types i.e. Translational, Rotational or combination of both. The equations governing such motion in mechanical systems are often directly or indirectly governed by Newton's laws of motion.

1.Translational Motion

       Consider a mechanical system in which motion is taking place along a straight line. Such systems are of translational type. These systems are characterized by displacement, linear velocity and linear acceleration.

Note : According to Newton's law of motion, sum of forces applied on rigid body or system must be equal to sum of forces consumed to produce displacement, velocity and acceleration in various elements of the system.

       The following elements are dominantly involved in the analyze of translational motion systems.

              i)  Mass           ii)  Spring               iii)  Friction

2. Mass (M)

      This is the property of the system itself which stores the kinetic energy of the translational motion. Mass has no power to store the potential energy. It is measured in kilograms (kg). the displacement of mass always takes place in the direction of the applied force results in inertial force. This force is always proportional to the acceleration produced in mass (M) by the applied force.

      Consider a mass 'M' as shown in the Fig.1 having zero friction with surface, shown by rollers.

      The applied force f(t) produces displacement x(t) in the direction of the applied force f(t). force required for the same is proportional to acceleration.

      Taking Laplace and neglecting initial conditions we can write,

      Also mass cannot store potential energy so there cannot be consumption of force in the mass e.g. if two masses are directly connected to each other as shown in the Fig.2 and if force f(t) is applied to mass M₁ then mass M₂ will also display by same amount as M₁.

      Due to mass, there cannot be any change in force from one mass to other hence no change in displacement.

Note: The displacement of rigidly connected masses is always same.

3. Linear Spring

      In actual mechanical system there may be an actual spring or indication of spring action because of elastic cable or a belt. Now spring has a property to store the potential energy. The force required to cause the displacement is proportional to the net displacement in the spring. All springs are basically nonlinear in nature but for small deformations their behaviours can be approximated as linear one. Hence assuming linear spring constant 'K' for the spring, we can write equation for the spring in the system.

      Consider a spring having negligible mass and connected to a rigid support. Its spring constant be 'K' as shown in the Fig.3.

      Force required to cause displacement x(t) in the spring is proportional to displacement.

      Now consider the spring connected between the two moving elements having masses M₁ and M₂ as shown in the Fig.4 where force is applied to mass M₁.

      Now mass M₁ will get displaced by x₁(t) but mass M₂ will get displaced by x₂(t) as spring of constant K will store some potential energy and will be the cause for change in displacement. Consider free body diagram of spring as shown in the Fig.5.

       Net displacement in the spring is x₁(t) – x₂(t) and opposing force by the spring is proportional to the net displacement i.e. x₁(t) – x₂(t).

Note : The spring between the moving points causes a change in displacement from one point to another.

       Spring behaves exactly same in rotational systems, only the linear spring constant becomes torsional spring constant but denoted as 'K' only.

4. Friction

      Whenever there is a motion, there exists a friction. Friction may be between moving element and fixed support or between two moving surfaces. Friction is also nonlinear in nature. It can be divided into three types,

              i) Viscous friction         ii) Static friction      iii) Coulomb friction

      Viscous friction as dominant out of the three is generally considered, neglecting other two types. Viscous friction is assumed to be linear, with frictional constant 'B'. This has linear relationship with relative velocity between two moving surfaces.

      The friction is generally shown by a dash-pot or a damper as shown in the Fig.6.

      This is the symbolic representation of a friction.

       Consider a mass M as shown in the Fig.7 having friction with a support with a constant 'B' represented by a dash-pot.

       Friction will oppose the motion of mass M and opposing force is proportional to velocity of mass M.

       Taking Laplace and neglecting initial conditions,

      Similarly if friction is between two moving surfaces, it is shown in the Fig. 8.

      In such a case, opposing force is given by,

      Taking Laplace,

       Thus if the force applied to mass M₂ is f(t) then due to friction between the masses M₁ and M₂, the force getting transmitted to M₁ is always less than f(t). hence the displacement of mass M₁ is different than the displacement of mass M₂.

Note : The friction between two moving points, causes a change in displacement from one point to other.

      Frictional force also behaves exactly in same manner, in rotational systems, only linear frictional constant becomes torsional frictional constant but denoted by same symbol 'B' only.