If the block-and-spring model is subjected to the periodic force F=F0 cosωt, show that the diffe

If the block-and-spring model is subjected to the periodic...

Key Concepts

Method of Undetermined Coefficients

This technique is a systematic method used to find a particular solution for linear differential equations with constant coefficients when the non-homogeneous term is a function such as a polynomial, exponential, sine, or cosine. By assuming a form for the particular solution based on the forcing term and determining the unknown coefficients, one can efficiently solve for the response induced by the external driving force.

Homogeneous and Particular Solutions

The general solution of a linear non-homogeneous differential equation is composed of two parts: the homogeneous solution, which solves the equation without the forcing term, and the particular solution, which directly accounts for the forcing function. The homogeneous part represents the natural or transient response of the system, while the particular part represents the steady-state behavior under the influence of an external force.

Resonance

Resonance occurs when the frequency of the external forcing function matches the natural frequency of the system, leading to a significant amplification of the oscillatory response. Understanding resonance is critical, as it can result in large amplitude oscillations that may lead to system instability or failure if not properly managed, and it plays an important role in the design and analysis of physical systems.

Second-Order Linear Differential Equations

This concept refers to differential equations that involve the second derivative of an unknown function with respect to a variable, typically time. Such equations often describe dynamic systems like oscillations, where the acceleration (or second derivative) is linked to position and external forces. They are fundamental in physics for modeling systems such as mechanical oscillators and circuits.

Forced Oscillations

Forced oscillation describes a system that is driven by an external time-dependent force, leading to a response that is the sum of the system's natural behavior and a forced component. This is essential for understanding real-world systems where external factors continuously impact the system's dynamics, leading to behavior such as beats or resonance.

Transcript

00:01 And this problem, we have a block here that's attached to ground through a spring.

00:09 If we draw a freebar diagram of the block, we have its weight and a normal force from the surface that's counteracts the weight.

00:17 It's not moving vertically, and it's sliding without friction on this surface here.

00:23 There's an external low at f acting on it.

00:26 And then we have the force in the spring acting on the block also.

00:31 And so i'm calling this positive x.

00:35 And so this is the displacement of the spring here, or the change in length of the spring is also going to be that.

00:42 So if we buy a diagram here, it says f minus fs must be mx double dot.

00:49 Fs is x times, or k times x.

00:53 So our equation governing the motion becomes mx double dot plus kx equals f not cosine capital omega t.

01:01 Now in the book, they use a little omega here, and i used to do that, but then after teaching a class that's focused purely on vibrations for many years, i realized students would oftentimes forget to write omega -n, or they'd forget the n or not see the n here, and then they'd get omega, and then they'd get confused between omega -n and n, so i decided just to use a capital -o -omega here.

01:27 So this is the forcing frequency.

01:28 I'm going to always use capital -omega for it.

01:31 So that it's very distinct from the natural frequency, which is a little omega -n.

01:38 We can then divide through this by m, and so we get x double dot plus omega -n squared x equals f -not over m cosine omega -t.

01:49 Now they asked for the general solution of this equation.

01:54 Well, the general solution consists of a homogeneous in a particular solution.

02:00 The homogeneous solution, as you can kind of see, derived in the book is harmonic, so waves oscillating in time with two unknown amplitude, two unknown coefficients here which you use initial conditions to solve for.

02:17 So you need an initial condition on the displacement and the velocity and then you could solve for these.

02:24 Then you also have a particular solution.

02:26 So this solution satisfies the equation when this is zero...

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