Mass Spring Damper System Mechanical Engineering Notes | EduRev

Mechanical Vibrations

Mechanical Engineering : Mass Spring Damper System Mechanical Engineering Notes | EduRev

The document Mass Spring Damper System Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Mechanical Vibrations.
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Mass-Spring-Damper System 

The most basic system that is used as a model for vibrational analysis is a block of mass m connected to a linear spring (with spring constant K and unstretched length ℓ0) and a viscous damper (with damping coefficient c). In addition, an external force P(t) is applied to the block and the displacement of the block is measured from the inertially fixed point O, where O is the point where the spring is unstretched. Finally, the spring and damper are both attached at the inertially fixed point Q. This system is shown in Fig. 1–1 Denoting unit vector in the direction from O to Q as Eand the inertial reference frame of the ground by F, the inertial acceleration of the block is given as

 F= x¨Ex                                                                                                 (1–1)

Next, the forces exerted by the spring and damper are given, respectively, as

F = −K(ℓ − ℓ0)u                                                                                     (1–2)
Ff = −cvrel                                                                                                 (1–3)

First, because the spring is attached at point Q, we have

ℓ = ║  r − r║                                                                                          (1–4)

 

Mass Spring Damper System Mechanical Engineering Notes | EduRev

Figure 1–1 Block of mass m sliding without friction along a horizontal surface connected to a linear spring and a linear viscous damper.

where r and rQ are the positions of the block and the attachment points of the spring, respectively. Using a coordinate system with its origin at point O at Ex as the first principal direction, we have

r = xE                                                                                                                       (1–5)
rQ = ℓ0Ex                                                                                                                     (1–6)

Therefore,

Mass Spring Damper System Mechanical Engineering Notes | EduRev

Then, because x < ℓ0 we have

|x − ℓ0| = ℓ0 − x                                                                                                          (1–8)

Finally, the unit vector in the direction from the attachment point of the spring to the position of the block is

Mass Spring Damper System Mechanical Engineering Notes | EduRev

The force in the linear spring is then given as

Mass Spring Damper System Mechanical Engineering Notes | EduRev 

Next, because the ground is already assumed to be inertial, the relative velocity between the block and the ground is simply the velocity of the block, i.e.,

Mass Spring Damper System Mechanical Engineering Notes | EduRev                                                                                                      (1–11)

Therefore, the force exerted by the viscous damper is obtained as

Mass Spring Damper System Mechanical Engineering Notes | EduRev                                                                                                        (1–12)

The resultant external force acting on the particle is then obtained as

Mass Spring Damper System Mechanical Engineering Notes | EduRev                      (1–13)

Applying Newton’s second law to the particle, we obtain

Mass Spring Damper System Mechanical Engineering Notes | EduRev                                                                     (1–14)

Dropping Ex from Eq. (1–14) and rearranging, we obtain the differential equation of motion as

Mass Spring Damper System Mechanical Engineering Notes | EduRev                                                                                  (1–15)

Now historically it has been the case that the differential equation has been written in a form that is normalized by the mass, i.e., we divide Eq. (1–15) by m to obtain

Mass Spring Damper System Mechanical Engineering Notes | EduRev

where p(t) = P (t)/m. Furthermore, it is common practice to define the quantities K/m and c/m as follows:

Mass Spring Damper System Mechanical Engineering Notes | EduRev
The quantities ωn and ζ are called the natural frequency and damping ratio of the system, respectively. In terms of the natural frequency and damping ratio, the differential equation of motion for the mass-spring-damper system can be written in the so called standard form as

Mass Spring Damper System Mechanical Engineering Notes | EduRev

 

It is seen that Eq. (1–17) is a second-order linear constant coefficient ordinary differential equation. Often, the term “constant coefficient” is replaced with the term time-invariant, i.e., we say that Eq. (1–17) is a called a second-order linear time-invariant (LTI) ordinary differential equation. The terminology “time invariant” stems from the fact that, for a given input p(t) and a given set of initial conditions Mass Spring Damper System Mechanical Engineering Notes | EduRev at the initial time t = t0 is the same as the solution to the input p(t + τ) for the initial conditions Mass Spring Damper System Mechanical Engineering Notes | EduRev at the (shifted) initial time t = t0 + τ. Because of this fact associated with an LTI system, without loss of generality we can assume that the initial time is zero, i.e., t0 = 0. Thus, when studying the zero input response of an LTI system we can restrict our attention to initial conditions Mass Spring Damper System Mechanical Engineering Notes | EduRev.

 

General Solution of a Second-Order LTI Differential Equation 

Eq. (1–17) can be written as

Mass Spring Damper System Mechanical Engineering Notes | EduRev

which can be further written as

Mass Spring Damper System Mechanical Engineering Notes | EduRev

Now let

Mass Spring Damper System Mechanical Engineering Notes | EduRev

Then we can view the system of Eq. (1–17) as a system of the form

Lx = f                                                                                                                   (1–21)

It is seen that the operator L defined in Eq. (1–20) is linear because

L(αx1 + βx2) = αL(x1) + βL(x2)                                                                            (1–22)

for all constants α and β. Then it is seen that Eq. (1–21) is a linear system whose general solution is of then form Eq. (1–17) is given as

x(t) = xh(t) + xp(t)                                                                                               (1–23)

here xh(t) is the homogeneous solution (i.e., the solution for a particular set of initial conditions Mass Spring Damper System Mechanical Engineering Notes | EduRev with a zero input function p(t) ≡ 0) while xp(t) is the particular solution (i.e., the solution for zero initial conditions (x(t0), x(t0) = (0, 0) and an arbitrary input function p(t) ≠ 0). The homogeneous solution and particular solutions are also called the zero input response and zero initial condition response, respectively. The general solution x(t) to a second-order LTI system is then given as the sum of the zero input response and the zero initial condition response. Because the zero input response satisfies Eq. (1–17) when p(t) ≡ 0, we have

 

Mass Spring Damper System Mechanical Engineering Notes | EduRev

 

Contrariwise, because the zero initial condition response satisfies Eq. (1–17) when p(t) ≠ 0 and the initial conditions are zero, we have

Mass Spring Damper System Mechanical Engineering Notes | EduRev

 

From the preceding discussion, it is seen that studying the general response of a secondorder LTI system amounts to studying independently the zero input response and the zero initial condition response. Consequently, the study of single degree-of-freedom vibrations amounts to quantifying the zero input response and the zero initial condition response. In this remainder of this chapter we study in detail the zero input response of a second-order LTI system that arises in the study of mechanical vibrations.

 

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